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The Seventh Marcel Grossmann Meeting [Hardcover ed.]
 9810229941, 9789810229948

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THE SEVENTH

MARCEL GROSSMANN MEETING On recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories

Also published by World Scientific:

PROCEEDINGS OF THE SIXTH

MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY PART A & PART B Eds. Humitaka Sata and Takashi Nakamura Series Ed. Rema Ruffini

THE SEVENTH

MARCEL GROSSMANN MEETING On recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories

Proceedings of the Meeting held at Stanford University 24-30 July 1994

Editors

Robert T. Jantzen Department of Mathematical Sciences Villanova University Villanova, PA 19085 USA

G. Mac Keiser Gravity Probe B Relativity Mission W. W Hansen Experimental Physics Laboratory Stanford University Stanford, CA 94305 USA

Series Editor

Remo Ruffini

h

World Scientific

I r Singapore· New Jersey· London· Hong Kong

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Published by

World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THE SEVENTH MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any in/ormation storage and retrieval system now known or to be invented, without written permission/rom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 9789814533485

Printed in the United States of America.

v THE MARCEL GROSSMANN MEETINGS The Marcel Grossmann Meetings have been conceived with the aim of reviewing recent developments in gravitation and general relativity, with major emphasis on mathematical foundations and physical predictions. Their main objective is to bring together scientists from diverse backgrounds in order to deepen our understanding of spacetime structure and review the status of experiments testing Einstein's theory of gravitation. Publications in the Series of Proceedings Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity these volumes (Stanford, USA, 1994) Edited by R.T. Jantzen and G.M. Keiser World Scientific, 1996 Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity (Kyoto, Japan, 1991) Edited by H. Sato and T. Nakamura World Scientific, 1992 Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity (Perth, Australia, 1988) Edited by D.G. Blair and M.J. Buckingham World Scientific, 1989 Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity (Rome, Italy, 1985) Edited by R. Ruffini World Scientific, 1986 Proceedings of the Third Marcel Grossmann Meeting on General Relativity (Shanghai, People's Republic of China, 1982) Edited by Hu Ning Science Press - Beijing and North-Holland Publishing Company, 1983 Proceedings of the Second Marcel Grossmann Meeting on General Relativity (Trieste, Italy, 1979) Edited by R. Ruffini North-Holland Publishing Company, 1982 Proceedings of the First Marcel Grossmann Meeting on General Relativity (Trieste, Italy, 1976) Edited by R. Ruffini North-Holland Publishing Company, 1977 Series Editor: REMO RUFFINI

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ORGANIZING BODIES OF THE SEVENTH MARCEL GROSSMANN MEETING:

INTERNATIONAL ORGANIZING COMMITTEE David Blair, Yvonne Choquet-Bruhat, Jurgen Ehlers, Francis Everitt, Fang Li Zhi, Stephen Hawking, Yuval Ne'eman, Remo Ruffini (Chair), Abdus Salam, Humitaka Sato, Rashid Sunayev, Steven Weinberg

LOCAL ORGANIZING. COMMITTEE Mac Keiser (Chair), BIas Cabrera, Paolo Carini, Daniel DeBra, Lance Dixon, Francis Everitt, Monica Jarnot, Renata Kallosh, Tom Langenstein, Roger Romani, Robert Wagoner, Yueming Xiao

SPONSORS International Center for Relativistic Astrophysics (ICRA) International Centre for Theoretical Physics (ICTP) International Science Foundation (ISF) International Union of Pure and Applied Physics (IUPAP) United Nations Office for Outer Space Science (UN) National Science Foundation (NSF) Office of the Dean of Research, Stanford University Institute for International Studies, Stanford University Lockheed Missiles and Space Company, Inc. Hewlett-Packard Company Domaine Chandon

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The recipients of the Marcel Grossman Awards (from left): Peter Stockman, Jim Wilson, and Subrahmanyam Chandrasekhar.

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MARCEL GROSSMANN AWARDS

SEVENTH MARCEL GROSSMANN MEETING

Institutional Award SPACE TELESCOPE SCIENCE INSTITUTE "for its critical role in the direction and operation of the Hubble Space Telescope, a truly unique international laboratory for the investigation and testing of general relativity in the context of modern astrophysics and cosmology"

Individ ual Awards SUBRAHMANYAN CHANDRASEKHAR "for his contributions to the analysis of gravitational phenomena from Newton to Einstein and especially for leading the way to relativistic astrophysics with the concept of critical mass for gravitational collapse"

JIM WILSON ''for having built on his experience in nuclear physics, thermonuclear reactions, and extensive numerical simulation to create a new testing ground for the novel concepts of relativistic astrophysics"

Each recipient is presented with a silver casting of the TEST sculpture by the artist A. Pierelli. The original casting was presented to His Holiness Pope John Paul II on the first occasion of the Marcel Grossmann Awards.

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FOURTH MARCEL GROSSMANN MEETING

Institutional Award THE VATICAN OBSERVATORY

Individ ual A wards WILLIAM FAIRBANK ABDUSSALAM

FIFTH MARCEL GROSSMANN MEETING

Institutional Award THE UNIVERSITY OF WESTERN AUSTRALIA

Individ ual A wards SATIO HAYAKAWA JOHN ARCHIBALD WHEELER

SIXTH MARCEL GROSSMANN MEETING

Institutional Award RESEARCH INSTITUTE FOR THEORETICAL PHYSICS (Hiroshima)

Individ ual A wards MINORD ODA STEPHEN HAWKING

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Photograph of the TEST sculpture of A. Pierelli by S. Takahashi.

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Subrahmanyan Chandrasekhar delivering the opening public lecture "The Series Paintings of Claude Monet and the Landscape of General Relativity."

Previous Marcel Grossmann Award recipient John Wheeler introducing the evening public lecture series.

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PREFACE The Seventh Marcel Grossmann Meeting (MG7), dedicated to C.W. Francis Everitt on the occasion of his 60th birthday, took place for the first time on the American continent at Stanford University, Stanford, California, July 24-29, 1994. The opening ceremonies began the morning of July 25 with welcoming addresses by Remo Ruffini, Francis Everitt, and John Howe (on behalf of the NASA Administrator Daniel Goldin). The three Marcel Grossmann Awards were presented by Remo Ruffini and Humitaka Sato. The Space Telescope Science Institute was the recipient of the institutional award, which was accepted on behalf of the Institute by its director Peter Stockman. The individual award recipients were Subrahmanyan Chandrasekhar and Jim Wilson. Each received a silver replica of the TEST (Traction of Events in Space-Time) sculpture by Attilio Pierelli. The audience received an elegantly printed award pamphlet including pictures of the individual recipients and some spectacular glossy photographs of the TEST sculpture taken by the artist Shu Takahashi, who helped design the two meeting posters and the meeting tote bag. Also included in the pamphlet was an essay by Anna Imponente (reproduced in these proceedings) about the interaction between science and art which is concretely embodied in the TEST sculpture, the three-dimensional extension of the Marcel Grossmann Meeting logo used in the meeting posters and other promotional materials since the fourth meeting. After the opening ceremonies, the meeting began with the first of four public talks of the series "Science and Society - Thoughts for the New Millennium." John Wheeler, acting in his capacity as the series speaker presenter, introduced Professor Chandrasekhar's talk "The Series Paintings of Claude Monet and the Landscape of General Relativity."* The scientific program included five days of 26 morning plenary talks and four afternoons of a total of 40 parallel sessions including approximately 400 presented papers. The idea of a volunteer rapporteur report on the area of research associated with each parallel session was introduced for the first time at this meeting. Presented either by the chairperson or someone invited by the chairperson, these talks had the goal of summarizing the state of affairs in the specific area of the session. No poster sessions were held. Instead, everyone not able to be allotted a longer time to speak was guaranteed their "at least five minutes of fame" to announce their work for further discussion on an individual level. Close to 559 registered participants and 120 registered accompanying persons were present for the meeting during a week of truly fine weather at Stanford. Though certain parts of the long-term planning process for this meeting got off to a late start, things seemed to come together at the last minute to make another successful gathering of a diverse group of scientists from all over the world. Crucial phases "This talk was published in pamphlet form as the dedication address for the Inter-University Centre for Astronomy and Astrophysics in Pune, India, delivered on December 28, 1992.

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of the advanced preparations were managed efficiently by Monica Jarnot, who also hosted two special events for the accompanying persons. Organizational matters before, during, and after the meeting were handled effectively by Jill Higgins, Gary Bertolucci, and their organization Meeting Planning Associates. The work of Nadine Brown on the conference registration, participant database, and manuscript archival process was invaluable. The combined efforts of the local organizing committee chaired by Mac Keiser and many members of the Gravity Probe B Relativity Mission led by Francis Everitt went into making the meeting become a reality. An enthusiastic group of graduate student volunteers both from Stanford and around the world, coordinated by Paolo Carini, contributed to the success of the day-to-day operation of the conference. The evening program included the opening night organ concert by Robert Bates at the Stanford Memorial Church, the banquet, and the remaining three lectures of the Science and Society series. These were given by Fang Li Zhi ("Decline of Central Totem and New Horizons"), Wolfgang Panofsky ("Copernican Humility, Change and the Creation of Purpose") and Yuval Ne'eman ("The Physical Heritage of the Cold War"). The last two of these are reproduced at the end of these proceedings. The midweek banquet was held in an elegant setting amidst spectacular natural beauty at the Olympic Club in Lakeside overlooking the Pacific Ocean and San Francisco. Those present were treated to a wonderful after-dinner talk by Ken Nordtvedt recounting anecdotes about his early days at Stanford when the idea of the gyroscope experiment was first taking form. The meeting was closed on July 29 with a reminiscence by Remo Ruffini of the days of the renaissance of relativity and of the special role played by various individuals at Stanford University in the early seventies in generating the birth of this series of meetings. This talk has been expanded upon in "An Excursus on Experimental Gravitation from Space and Relativistic Astrophysics in Honour of Francis Everitt's 60'th Birthday" reproduced at the beginning of these proceedings. A warm expression of thanks was given to the participants, speakers, and organizers, and to all those whose financial help made possible the realization of this seventh meeting.

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Professor Yuval Ne'eman delivering his public lecture "Copernican Humility, Change and the Creation of Purpose."

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Francis Everitt

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An Excursus on Experimental Gravitation from Space and Relativistic Astrophysics in Honour of Francis Everitt's 60'th Birthday These two volumes of the Proceedings of the Seventh Marcel Grossmann Meeting are dedicated to Francis Everitt in honour of his sixtieth birthday. In many ways Francis's life is intertwined with the maturing of experimental general relativity, both from the ground and from space, and has close ties with major theoretical developments in gravitation physics and astrophysics. Since the beginning his activities have interacted with the scientific achievements of a large and distinguished set of scientists. I take this occasion to recount some of his story and weave it into the narrative of some of my own reminiscences of the times during which the Stanford projects were initiated and developed, and to discuss how the Stanford group interacted with the scientific communities nationally and internationally. A part of this story is the birth of this series of Marcel Grossmann Meetings aimed at bringing the scientists working in gravitational physics together with mathematicians, theoretical physicists, and astrophysicists to stimulate each other's progress. The first section sketches Francis's studies in England and his arrival in the United States as a research associate and the second section covers his first activities in the new Stanford ambiance and the origin of the gyroscope experiment, as well as his first encounter with Irwin Shapiro. In the third section I recall significant moments of the Princeton-Stanford collaboration, including my first visit to Stanford, the ESRO Interlaken meeting, the development of the physics and astrophysics of Black Holes, the birth and flourishing of X-ray astronomy and its impact on relativistic astrophysics, the identification of Cygnus Xl as a Black Hole, my spending a quarter at Stanford and finally the XVI Solvay Conference and the LXV Varenna School. In the fourth section I recall the foundation of the Marcel Grossmann Meetings and my return to Italy and in the fifth section my first contacts with China and the Third Marcel Grossmann Meeting (MG3) in China. The last section describes the birth of the International Center for Theoretical Physics (ICRA) and MG4, MG5, and MG6-from China to Stanford via Perth and Kyoto. 1. Francis Everitt: A Life in Science Francis Everitt was born on March 8,1934 in Sevenoaks, England, the youngest in a family of four boys and one girl. He was educated at the ancient local school founded in 1418 and then at Imperial College, London, where he obtained his bachelor's degree in 1955 and his doctorate for research in paleomagnetism under P.M.S. Blackett in 1959. Francis entered the field of space and gravitational physics in 1962 when he moved to Stanford University and he has revealed that his interest in this and in the history and philosophy of science seems to have been prepared for through his family and physics background. His father was a patent attorney trained in engineering under Silvanus Thompson, the electrical engineer and biographer of

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Kelvin. As a growing boy Francis recalls his-father's reminiscences of Thompson, including Thompson's anecdotes about Kelvin, and Thompson's encounter with Tesla. Francis's father also read many scientific books including Einstein's The Meaning of Relativity, and he took the side of Einstein in the Bohr-Einstein debate. An uncle from the other side of Francis's family had been on the mathematics faculty of Imperial College. Francis's eldest brother Robin is also a mathematician and a former schoolmaster, while another brother is a distinguished historian. In fact, Francis had the unusual experience of having Robin as his mathematics master for two years in school while the two of them were living in the same house, an experience which Francis says was probably more difficult for his brother than for him. It was Robin who in 1949 picked up from a secondhand bookshop the book The Life of James Clerk Maxwell by Lewis Campbell and William Garnett [1] which provided the background for Francis's biographical interest in Maxwell and played a major role in his becoming a physicist. When Francis entered Imperial College in 1952, the Physics Department was at a low point following the retirement of G.P. Thomson and had still not recovered from the absence of so many of its faculty during World War II. A year later Blackett arrived "like a whirlwind" bringing many people and programs from Manchester, including the very new rock-magnetism group that he had formed with the former radio astronomer J .A. Clegg. This was the group that Francis decided to join at the end of his undergraduate period, although his first publication was neither in rock-magnetism nor in English. In 1955, Francis spent a summer at the PhysikalischTechnische Bundesanstalt in Braunschweig doing research on electron optics with K.J. Hanssen, an experience which led to a short paper in German on a new method for measuring the principal planes and focal lengths of electrostatic electron lenses. Blackett had entered rock magnetism in an unexpected way. During the course of his earlier cosmic ray research he had became interested in the effects of the Earth's and other planetary and stellar magnetic fields, and was led to revive and quantify an earlier conjecture of Schuster's proposing a fundamental connection between magnetism and the rotation of a gravitating body. To test this hypothesis Blackett found it necessary to invent a new magnetometer much more sensitive than any previously existing ones. He did this by effectively reinventing Kelvin's astatic magnetometer. His paper describing this work remains in Francis's eyes one of the great classic works in optimizing instrument design. Blackett's enormous influence on Francis during this period later emerged as an important part of the foundation underlying the design of the Gravity Probe B experiment, although Francis admits that he had to unlearn one strong prejudice he picked up from Blackett, who resisted the world of modern electronics. Having invented a magnetometer a factor of 1000 more sensitive than any other in existence, and only a factor of 100 less sensitive than modern SQUIDS, Blackett naturally wanted to put it to some use. He was led to recognize that it could be applied to measuring the very weak magnetic moments of sedimentary rocks. The story is told, certainly a believable one, that he asked a geologist's advice on what book to read to learn some geology and on being referred to Arthur Holmes' classic

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Principles of Physical Geology (1944) he read it in a weekend while making fifteen or twenty pages of detailed notes. This led to the establishment of two research groups, one with Blackett under Clegg and one at Cambridge and later Newcastle under Blackett's former student S.K. Runcorn, who died so tragically in 1995. These two groups produced many of the early key papers that opened up the new field of plate tectonics. As is not infrequently the case, there was intense rivalry between them: the only joint publication they ever produced was the one that Francis wrote with John Belshe of Cambridge on the paleomagnetism of the British Carboniferous system. It concluded that during the Carboniferous period (rv 300 million years ago) Britain had been roughly 10 degrees south of the equator. Francis wrote nine more papers on rock-magnetism, including one on an application to geology which resulted in a pleasant correspondence with Arthur Holmes, and three which developed experimentally and theoretically a new understanding of thermoremanent magnetization, namely the magnetism acquired when a body such as an igneous rock cools in a weak magnetic field. In reflecting on Blackett's influence, Francis praises him not just for his brilliance in instrument design but for his extraordinary directness. To someone like Francis who had always been disposed to assume that science is subtle and speculative, this quality was very unexpected. Blackett had been trained as a naval officer and had fought in the Battle of Jutland, and one of his favorite sayings had been that you should "treat your research like a military campaign." Since Francis's maternal grandfather had been a torpedo engineer with the Royal Navy, this message did gradually sink in. The question Francis had to face after completing his Ph.D. and spending about 18 months as a post-doctoral associate at Imperial College was whether or not to "turn himself into a geologist." Despite much advice not to "throwaway all your experience," he concluded that he preferred to return to physics, and after many discussions with many people of whom the most influential was Philip Morrison, he decided to try low temperature physics. After a seemingly chance encounter that Francis views as miraculous, an opportunity arose to work with Kenneth Atkins at the University of Pennsylvania ("Penn"), and Francis moved to the United States for "two to three" years that have now stretched into thirty-five. At Penn Francis designed his own elegant apparatus to investigate the possible existence of "third sound", a surface wave in thin films of superfiuid helium, as predicted by Atkins. Within twelve months he, Atkins, and Denenstein had determined all the major properties of third sound through ellipsometry measurement of the thickness of the film. This technique had been invented by L.E. Jackson, and Jackson's student Llewellyn Grimes had sought earlier to apply it to the same problem. While Francis was at Penn, the physics department there arranged a three-day visit by William Fairbank to deliver their Mary Amanda Wood lectures, and it was through this occasion that Francis met Fairbank and first heard about Leonard Schiff's suggestion of the orbiting gyroscope experiment [2]. Francis likes to tell the story of Atkins' introduction of Fairbank in the lecture series. He said that

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whenever a low temperature physicist heard of an experiment that was completely impossible, the next news six months later was that Bill Fairbank had just finished it and had done so with even higher precision than the one that had seemed impossible. Privately Atkins also told Francis that finally Bill Fairbank had gone too far. His last suggestion really was impossible. It was to develop a new kind of gyroscope for the Schiff experiment. This encounter with Bill Fairbank signaled the entrance of Francis into the arena of gravitational physics experiments. II. Development of the Scientific Basis for Gravitational Physics Experiments ILL Early Impressions Like so many other important scientific movements, the new life in gravitational physics that came towards the end of the 1950's seemed to rise almost spontaneously in several places. The group at Princeton led by Robert Dicke was one influential source. Another one was R.V. Pound whose 1959 redshift experiment depended on the Mossbauer effect. Also related were Joe Weber's early reflections on the detection of gravitational waves, which began while he spent a sabbatical at Princeton in 1957. Schiff seems to have entered the picture almost independently from the delayed influence of Robert Oppenheimer, with whom he worked on Mach's principle in 1939. Bill Fairbank's interest was aroused by a remark of Bryce DeWitt's at the Chapel Hill gravitation meeting in 1957 at which DeWitt tossed a piece of chalk in the air and caught it saying (with some hyperbole as well as a parabola) that there was only one experiment in gravitational physics and that we do it over and over again as he had just done. Another factor was the early Gravity Foundation essay by Philip Morrison and Thomas Gold, who sought to explain the absence of significant quantities of antimatter in our universe on the basis of the hypothesis that matter and antimatter repel each other gravitationally. Francis says that in addition to his meeting with Bill Fairbank, three things drew him to gravity. First there was a proposal made by a young faculty member at Penn to measure gravitational waves using the Mossbauer effect. Second there were Kenneth Atkins's discussions with him about the distinction between active and passive gravitational mass during the time Atkins was writing a general physics textbook. Third there was a remark attributed to T.D. Lee to the effect that the next important field to work in would be gravitation. But of all influences, the decisive one was Fairbank's. The first thing that struck Francis upon arriving at Stanford, after he had become accustomed to Fairbank's relaxed approach to administrative questions, was the extraordinary range of Bill's interests. The diversity of experiments and students within the group was amazing. Particularly memorable were Bascom Deaver's completed measurement on quantized flux, Morris Bol's search for the London moment, which reached fruition in 1963, Fred Witteborn's experiment to measure the free fall of the electron and (though this was never done) on the positron, and George Hess's experiment to detect quantized vortices in superfluid helium. There were many more. Francis has offered his own appraisal of Bill in the article "The

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Creative Imagination of an Experimental Physicist" [3) in the volume Near Zero in honour of Bill's 65th birthday. Three other traits of Bill Fairbank's besides his imagination made a deep impression on Francis. One was the value of memory. Bill had a wonderful ability for recalling numbers for physical quantities during conversations at the blackboard and using them to make rapid and relevant order of magnitude calculations. Another was his ease at revealing his ignorance. In early exchanges Francis recalls feeling almost embarassed on Bill's behalf as he asked absurdly naive questions of some visiting expert. But as discussions advanced it became clear that there was method in this naivete, for often after an hour or so there would emerge out of the chaos a sudden profound question from Bill that no expert had ever thought of. Closely related was Bill's third trait, his willingness to collaborate with people of entirely different backgrounds. Here Stanford offered him an almost unique opportunity. 11.2. The Stanford Ambiance: Physics and Engineering For historical reasons that are interesting and complex, as discussed by Francis in his article on Gravity Probe B in the book Big Science [4), Stanford University has had a long record of collaboration between physicists and engineers. W.W. Hansen, perhaps Stanford's greatest physicist despite his tragically early death from beryllium poisoning at the age of 43, had begun as an electrical engineer. He is now mainly remembered as the inventor and builder of the first electron linear accelerator, completed in 1949 just after his death, the machine on which Robert Hofstadter performed his classic experiment determining nuclear form factors. Earlier he had been responsible for rescuing the Varian brothers from obscurity and providing them with the space and intellectual guidance they needed for inventing and developing the klystron. No less important was his close work with Felix Bloch in inventing nuclear magnetic resonance. It is not often remembered that the patent for nuclear magnetic resonance was in the names of both Hansen and Bloch (in that order). Whether consciously or unconsciously Fairbank continued this tradition. From the beginning he had talked about gyroscopes with Robert Cannon, the newly arrived head of the guidance and control group in the Stanford Department of Aeronautics and Astronautics. This began the long and fruitful collaboration between physicists and engineers without which it is inconceivable that Gravity Probe B could have been developed. None of this had been clear to Francis before he arrived, but with the strong engineering background in his family and at Imperial College he found the connection easy to embrace, and was able to establish a close intellectual rapport with Daniel DeBra, Richard Van Patten and other members of the Stanford Department of Aeronautics and Astronautics. Later the facilities of HEPL (the Hansen Experimental Physics Laboratory) growing out of Hansen's earlier initiative were again to prove essential, and have been part of the Stanford ambiance from which Francis and his colleagues have benefited.

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II.3. Schiff and the Origin of the Gyroscope Experiment In addition to Fairbank and this fertile collaborative atmosphere at Stanford, there was Leonard Schiff. Schiff's interest in questions bearing on relativity began in 1939 when, under stimulus from Robert Oppenheimer with whom he was then working, he investigated a question in electromagnetism related to Mach's principle. He then moved into other fields, particularly nuclear physics, but in 1958 quite independently of Bill Fairbank's interest in the question, he examined the MorrisonGold conjecture about a repulsive gravitational interaction between matter and antimatter. He provided an ingenious counterargument to the effect that in nuclear interactions there are virtual antiparticles present, and that if gravitation acted negatively on these there would be a violation of the equivalence principle already detectable in the Eotvos and Dicke experiments. This line of reasoning led Schiff to consider the more general question of the experimental foundations of gravitational theories. In October 1959 he wrote a paper for the American Journal of Physics that was simultaneously short, important, pessimistic, partly wrong, and profoundly influential on both Schiff himself and others. Its essential point was that the experimental evidence for general relativity was weaker than was usually assumed. Not only was the gravitational redshift deducible from equivalence principle arguments, but so also (according to Schiff) was the gravitational deflection of light. The famous Einstein factor of two, which had long been thought to be a strict consequence of the full general theory of relativity, seemed to arise simply from Einstein's having forgotten the special relativistic space-contraction in his early calculation of 1911. Schiff's pessimistic conclusion was that it would not be possible to make any interesting test of general relativity with clocks or photons until one attained second order accuracy in the measurement. Such a second order test would have required a clock with a stability of 1 part in 1018 , which is still beyond our reach nearly four decades later. Remarkably, within two months of submitting this gloomy paper for publication, Schiff had conceived of two entirely new tests of general relativity involving massive bodies, the geodetic and frame-dragging measurements with an orbiting gyroscope. Even more remarkably, at the same time independently of Schiff, George Pugh had conceived of the same two tests in a document not widely known, and had invented the concept of a drag-free satellite to help perform them. Francis has analyzed this striking example of "simultaneous discovery" in his article on the history of the Gravity Probe B experiment in Near Zero [5]. Pugh had become interested in the subject after hearing a short talk by H. Yilmaz at the 1959 American Physical Society meeting in New York describing his theory of gravitation. This is a clear lesson that sometimes very important experimental ideas can emerge from dubious theories. Schiff's argument that special relativity combined with the hypothesis that gravitation is to be explained through a metric theory would automatically yield the Einstein deflection of light was incorrect. Scalar-tensor theories are metric theories and give a different answer. The argument was interesting, however, and not so simplistically false as is sometimes thought. In combination with work by

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H.P. Robertson, it led to a re-examination of-the framework for understanding experimental relativity that Eddington had invented in 1922. This framework for gravitational theories, now called the parametrized post-Newtonian (PPN) theory, emerged in stages. The next step after Schiff was taken by Kenneth Nordtvedt who had been Schiff's student. This was then followed by the work of Clifford Will, Wei Tou Ni, and others at Caltech and elsewhere. Schiff's American Journal of Physics article was also interesting for the disagreement it caused between Leonard Schiff and Robert Dicke. Dicke's long referee's report on it was transformed into a companion article published alongside Schiff's original article, after which Dicke introduced his own framework for gravitational theories in 1962. In a wonderfully terse footnote to his original text, Schiff introduced what has since become known as "Schiff's conjecture." The dispute with Dicke was related to a proposal by Ramsey and Vessot to perform a clock test of the Einstein gravitational redshift effect in a satellite. Dicke had feared that Schiff's argument would discourage NASA from doing such an experiment. Fortunately it did not and in 1976 Vessot and Levine performed the "Gravity Probe A" clock test in a suborbital rocket, reaching a precision of 1.4 parts in 104 • Gravity Probe A was one of the most beautiful experiments NASA has ever performed.

11.4. Encounters with Irwin Shapiro Another very important step forward came in 1964 when Irwin Shapiro conceived of his "fourth test of general relativity," the radar time-delay experiment. The title of Shapiro's paper gave rise to an amusing controversy. Schiff objected to calling the measurement a fourth test, partly because he considered his own twin gyroscope experiments to be the real fourth and fifth tests of Einstein's theory, and partly because he argued on the basis of the PPN formalism that the time delay and light deflection experiments measure the same parameter. Shapiro answered these objections by stating that his test would be the fourth to be performed, not the fourth to be suggested, and that the time delay and light deflecton tests should be treated as distinct, as most relativists now agree. Francis recalls that during a 1966 visit to the Boston area connected with the design of the first experimental dewar vessel for Gravity Probe B, he used the occasion to visit Shapiro. The meeting persuaded him to "abandon any Stanford provincialism." It was also on that occasion that Shapiro made his first (but not his last or most important) suggestion for reducing the error in the Gravity Probe B measurement due to uncertainty in the proper motion of the guide star. He pointed out that one can remove the proper motion error using observations of two gyroscopes at different altitudes. Twenty years later Shapiro suggested to Francis yet another method of subtracting the proper motion error based on finding a guide star that is both a visible and radio source. Radio astronomical observations by Very Long Baseline interferometers (VLBI) then enable one to measure proper motion of the guide star with respect to remote quasars, sharply reducing the error. Actually the idea of using VLBI techniques in order to pin down the proper motion of the guide star had occurred to Shapiro already in the late seventies at the time he was chairing the National

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Academy Committeee of gravitational physics. There are few better illustrations than this one of the range of technological developments needed to perform Gravity Probe B. Even the method of fixing the position of the star depends on a technique that had not yet been invented when Schiff first proposed the experiment. The Shapiro time delay measurement is still the most precise test of any positive effect of general relativity. It reached its greatest precision of 1 part in a thousand in the observations on the Viking Lander in 1976, the same year in which Gravity Probe A was flown and a year that may be rightly seen as bringing to a close the first era in testing general relativity from space. III. On a Princeton-Stanford Collaboration: 1968-1915 IILL My Arrival in Princeton and the First Stanford Visit In 1967 after spending a few months in Hamburg with Pasqual Jordan, I was invited to Princeton by John Wheeler and joined his group as a European Space Research Organization (ESRO) postdoctoral fellow. Ulrich Gerlach, Hans Ohanian, and Frank Zerilli were just finishing their Ph.D.'s with him at the time. Among the memorable visitors at Princeton that year was Brandon Carter, who presented his elegant analysis of the geodesics in Kerr-Newman geometries which soon became a basic mathematical tool for the exploration of gravitationally collapsed objects. The year 1968 was one of the single most important years to date in the history of experimental relativity. This was the year that Jocelyn Bell discovered pulsars while working with a new array conceived by and build under the guidance of Tony Hewis. The following year the Apollo astronauts planted the U.S. retroreflectors on the Moon, which along with the Soviet-landed French reflectors made possible the lunar laser ranging experiment. This experiment, thanks notably to ideas of Kenneth Nordtvedt, would lead to yet another important test of general relativity in the following years. As a consequence of the pulsar discovery many research centers worldwide jumped into the study of neutron stars. Wheeler himself, who had just published a comprehensive book [6] on this subject, boldly led us ahead of the crowd. On the one hand he recommended reaching a profound understanding of relativity through a deeper geometrical approach. I still remember receiving from him an extended essay on Teichmiiller spaces which left me speechless for several days. On the other hand he also directed us toward the vastly unexplored field of "continued gravitational contraction" first studied by Oppenheimer and Snyder. The fate of a star with mass larger then the critical mass against gravitational collapse became in those days the topic of research of a small number of scientists, no more than fifty worldwide (compared to the many hundreds working today on related topics in some nations alone). In addition to our relativity group in Princeton, there was one in Cambridge, England, one in Moscow, and a newly formed one, also originating from Wheeler, at Caltech. In June 1968 after an unforgettable car trip across the States, I visited Stanford and began a friendship with Francis that proved more than enduring. During that first visit to Stanford, I wrote a letter to Edoardo Amaldi in Rome describing

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my encounter with Bill Fairbank, Bill Hamilton, and Francis, and it was in part through later meetings with Fairbank and in part through a fuller appreciation of the astrophysical implications that Amaldi decided to enter the field of gravitational wave research. Hamilton and Fairbank had already begun thinking about a cryogenic bar detector in 1967. The year 1968 saw Joe Weber's first announcement of his apparent detection of gravitational waves through coincidence measurements between room temperature bars at Maryland and Chicago. Later that same year I attended the international conference GR5 on general relativity in Tbilisi, then in Soviet Georgia, where for the first time I met Andrey Sakharov, Yacov Borisovich Zel'dovich, and many members of their research groups, including Igor Novikov and Rashid Sunyaev. In the years that followed, those acquaintances and the one that was soon to come with Evgenij Lifshitz, became very important for me and they often stimulated my own research and the common interests with the Stanford group. Schiff was also present at the Tiblisi meeting and gave an elegant talk on the gyroscope experiment, still remembered by Russian physicists. III.2. The First European Physical Society Meeting and the ESRO Interlaken Meeting April 1969 saw the very impressive first meeting of the European Physical Society in Florence. In those days universities allover the world had been affected by an "intellectual fever" and that "fever" was present as well in Florence. For us this turned out not to be a bad thing after all. During a forced interruption of the meeting while sitting on the steps of "Palazzo della Signoria" watching a demonstration in which thousands of workers and students marched past carrying red flags, Roger Penrose told me about some interesting work he and his student Roger Floyd were doing on energy extraction in the field of a Kerr solution of the field equations of general relativity. Soon after I started a lively correspondence with Floyd on this subject. When I returned to Princeton, Johnny Wheeler and I decided to carefully examine this problem. Wheeler's first suggestion was to systematically analyze all orbits in the Kerr metric using an "effective potential technique" which turned out to be very powerful in this application. Years later Evgenij Lifshitz graciously dedicated a problem to this work of ours in his last edition of the Landau-Lifshitz treatise The Classical Theory of Fields. An important occasion to let some of these new ideas and contacts flourish came at the Interlaken Meeting of September 1969 called by Hermann Bondi, then ESRO's Director General. Francis, Leonard Schiff, John Wheeler and I all attended this meeting. This was Francis's first meeting with Wheeler, and the first occasion he had to give a detailed account of the progress on Gravity Probe B in the presence of Leonard Schiff. A large number of challenging questions were asked, especially by Tommy Gold, and Francis seemed to have a detailed quantitative answer to everyone of them. Schiff told us after that he was "very impressed." Meanwhile Joe Weber had just claimed to have some evidence of a sidereal correlation in the coincidence between his detectors in Maryland and Chicago. Wheeler, Francis, and

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I had already had long discussions about this·on the flight from New York to Paris en route to Interlaken, in those days still a very pleasant daylight trip. In the joint report that Wheeler and I wrote for the meeting [7], we introduced new concepts which in due course contributed to making the Black Hole not just a mathematical solution of the field equations of general relativity but an object of great relevance to physics and astrophysics. In particular we obtained expressions for the binding energies of particles co-rotating and counter-rotating in the field of a Kerr Black Hole, we introduced the concept of the "ergosphere," giving explicit examples of energy extraction processes, and we estimated the spectrum and the gravitational energy radiated by a particle falling in the background field of a Black Hole. We also presented a theoretical framework for estimating the cross-section and directionality of gravitational radiation detectors and consequently pointed out how difficult it was to interpret Weber's results straightforwardly as gravitational wave signals. The meeting was tremendously helpful in developing a strategy in this new field of relativistic astrophysics we all sensed was on the verge of beginning, both theoretically and experimentally, both from the ground and from space. There was a great deal of enthusiasm among the participant scientists, so well selected by Hermann Bondi, himself a distinguished relativist. Francis and I pondered the many discussions that had been held at Interlaken during long walks in the Maritime Alps near my home town of La Brigue before returning to the United States. This was the first of many happy visits that Francis paid to La Brigue. III.3. The Mass-Energy Formula for Black Holes In April 1968 a very special 16 year old high school student, Demetrios Christodoulou, had arrived from Greece and was admitted to Princeton as an undergraduate student. By September 1968 he had already been admitted to graduate school and he passed his general exam in October 1969. In addition Jacob Bekenstein, Bahram Mashoon, Clifford Rhoades, Bill Unruh, and Robert Wald were graduate students at the time. I became the thesis advisor of Demetrios and Cliff. Also working with us at Princeton was Daniel Wilkins, Francis's graduate student at Stanford who had developed an elegant correction for the effects of the Earth's oblateness on the gyroscope experiment and had joined us in the study of Black Holes. The "process of extraction of gravitational and electromagnetic energy from Black Holes," the "mass-energy formulae of Black Holes" found with Demetrios [8, 9], the "upper limit to the maximum mass of a neutron star" found with Cliff [10], and the "Wilkins effect" found with Daniel [11] all emerged from this very fortunate resonance of ideas, intense work, and personalities. It was during the writing of my paper with Demetrios on the electrodynamics of Black Holes that we emphasized with Wheeler over and over again the analogy between thermodynamics and Black Hole physics. This topic became the specific thesis topic of Wheeler's student Jacob Bekenstein. Through a profound set of gedanken experiments, Jacob pushed further the analogy between thermodynamics and Black Hole physics. Demetrios and I had formally established the existence of reversible and irreversible transformations in Black Hole physics as well as the mono-

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tonic increase, as occurs for entropy in thermodynamics, of the irreducible mass mirr of a Black Hole (from which the word irreducible arises) also formally established independently by Hawking in his area theorem [12]. The complete equivalence between the two results immediately follows from the identity relating the surface area 8 = 161T"mirr2 of the Black Hole to min, as suggested by Bryce DeWitt and confirmed in a quick calculation by Demetrios and myself. Jacob went one step further proposing that the area of the Black Hole S measured in Planck-Wheeler units should indeed be identified with entropy. He did this by formulating a statistical interpretation of Black Hole entropy and introducing the first generalized form of the first law of thermodynamics in physical processes involving Black Holes. These topics even today, more than twenty years later, still inspire lively debate! Jacob's proposal was extremely interesting and very intriguing at the time and remains so for me in some ways even today. The proposal certainly was not contradictory but I could not find a necessity for transforming it into an identity. This entire matter became the subject of even more lively discussion after Stephen Hawking proposed a physical process which if true would transform all these theoretical conjectures into physical reality: the Black Hole quantum evaporation process. This topic also inspires lively debate some two decades later. It is likely that these issues will be clarified once there is a theory encompassing both General Relativity and Relativistic Quantum Field Theories. Like the photoelectric effect at the beginning of the century was of paramount importance for the subsequent development of quantum field theories, so too the existence of this problematic relating quantum fields and thermodynamics to Black Hole physics promises to be a most important motivation and formidable testing ground for looking at the unification of these fundamental physical theories. Throughout this period I stayed in constant contact with Stanford and followed the developments of Gravity Probe B under Francis and of the gravitational wave program led by Fairbank and Hamilton both at Stanford and at Baton Rouge. Especially important was Leonard Schiff's rapidly growing interest in strong gravity and relativistic astrophysics. At Stanford a new group of graduate students very committed to this research field was forming. In addition to Daniel, there were Larry Smarr, Mark Peterson, Wick Haxton, and others. In the words of Michael May, director of Lawrence Livermore Laboratories on sabbatical at Princeton in 1970, Princeton had become a "Mecca" of relativistic astrophysics. It was partly through Michael that we established a very important additional collaboration between our group and the San Francisco Bay Area. Jim Leblanc and Jim Wilson had started some impressive work in numerical astrophysics using the Livermore supercomputing facilities. Wilson's work extending it to the relativistic regime soon became of paramount importance: a realistic testing ground for new theoretical ideas and models in relativistic astrophysics. With great enthusiasm Jim was bringing to bear on this new field of research his broad knowledge not only of numerical techniques but particularly of nuclear physics and hydrodynamics, with splendid results for all of us. From his daily work at Livermore, Jim knew the cross-sections and order of magnitude estimates of physics regimes far

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from my own scientific expertise. The writing of the Interlaken proceedings took Johnny and me more than one year of very intense work. On January 1971 Physics Today presented an excerpt of our report in our joint article "Introducing the Black Hole." The cover of that issue was also dedicated to our article with a painting by the German born artist Helmut Wimmer working at the New York Hayden Planetarium. I twice discussed with him the "Black Hole" concept. He later told me he could not understand the concept at all. He was almost desperate, and then one morning he woke up at four o'clock and the idea was suddenly clear. He limned in the painting in ten minutes. When I saw the painting I found it to be perfect and beautiful; only the sequence of colours in the iris had to be reversed. He offered me the painting, but I gladly accepted the first proof of the cover and asked him to donate the original to Princeton University where it still hangs in the Physics and Math library. Our complete Interlaken report finally appeared in the book Black Holes, Gravitational Waves and Cosmology [13J which Wheeler and I published with Martin Rees. Suddenly January 19, 1971 Leonard Schiff died. In the years that followed we increasingly missed him and the uniquely subtle role he played in the development of both experimental and theoretical research in general relativity at Stanford and in the establishment of that very special entente between Princeton and Stanford. IlIA. The Uhuru Satellite and the Flourishing of Relativistic Astrophysics The successful launch, operation, and superb data collection from the UHURU satellite, developed at American Science and Engineering by Riccardo Giacconi and his team and launched by NASA still in 1971, marked an epoch in the development of the new field of relativistic astrophysics. All the theoretical work that preceded it, started by Oppenheimer, Landau, Gamow, and others, could finally be confronted with a wealth of high quality data from continuous observations of galactic Xray sources with good sensitivity and time and angular resolution. Hundreds of neutron stars were observed accreting matter from a normal star, permitting the first measurements of neutron star masses and yielding the first information about their magnetospheres. Meanwhile, the absolute upper limit on the maximum mass of a neutron star that Rhoades and I had just found offered a decisive step for the identification of a Black Hole in our galaxy. I still remember the excitement of Gloria Lubkin of Physics Today at a Washington meeting of the American Physical Society telling me "Remo, go ... go and listen to the talk of your countryman Giacconi-it seems he has observed the objects Wheeler and you just introduced in Physics Today." I was very impressed by Riccardo but surprised that he had almost completely forgotten his native Italian language! All of this work came to full visibility in July, 1971 at the "Les Houches" conference beautifully organized by Cecile and Bryce de Witt. The lecturers were Jim Bardeen, Brandon Carter, Steven Hawking, Igor Novikov, Kip Thorne, and myself. Herbert Gursky, a close collaborator of Riccardo at American Science and Engineering, came and delivered some beautiful lectures on the observations from the Uhuru satellite. In my lectures I ventured to develop

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the concept of Black Holes as "alive", not -just as energy sinks but as sources of cosmic energy through their magnetosphere and their rotational energy. This concept originated in my work with Demetrios pointing out that up to 50 percent of the energy of a Black Hole could be stored as rotational and electromagnetic energy and could be extracted in principle [9). More complicated were the difficulties encountered in translating the term "Black Hole" into other languages. The literal Italian translation was considered obscene. A well known journalist had insisted instead on naming them "Ie buche nere" but I supported the literal masculine version "i buchi neri," which was finally accepted. In the French translation major opposition came from Cecile, who refused the literal translation and proposed the less impudent alternative "les astres occlus." The French title of the proceedings of that impressive school actually used this tamer version. But in due course the French also adopted the literal translation of the name we introduced in the article with Wheeler and they have happily used it ever since: "trou noir"! III.5. The Identification of Cygnus Xl as a Black Hole On my return to Princeton from Les Rouches I found Wheeler more and more involved in superspace, spacetime topologies, and Teichmiiller spaces. I chose instead to concentrate on relativistic astrophysics. My strategy was to identify theoretical arguments to firmly distinguish Black Holes from neutron stars, and then to zero in on the observational data that might lead to the first identification of a Black Hole. We needed to sharpen our understanding of the range of masses permitting the existence of neutron stars, and to determine what would make Black Holes, up till then just a mathematical concept, into physically credible and meaningful astrophysical systems. Is a Black Hole stable? Can its horizon be associated with the surface of a meaningful astrophysical object? How does a Black Hole react to an implosion of particles and how does it behave during the emission of gravitational radiation? What characteristic signature can carry signals from processes occurring in the ergosphere and magnetosphere of a Black Hole? With answers to these questions one might hope to discriminate between neutron stars and Black Holes. In this context, the work with Clifford Rhoades on the critical mass of a neutron star seemed central to us, and I worked out some refinements of the initial proof in order to establish 3.2 solar masses as an absolute upper limit solely on the basis of causality and a fiducial density, independent of the largely unknown interactions of matter at supranuclear densities. Meanwhile Marc Davis, formerly an MIT undergraduate then a graduate student at Princeton, had found a clever method of numerically integrating the Zerilli equations governing perturbations around a Black Hole. That technical breakthrough allowed us to explore a large variety of physical processes around a Black Hole with unprecedented rigour and accuracy, using the Regge-Wheeler and Zerilli approaches. Almost daily encounters with Tullio Regge, then professor at the Institute for Advanced Study, and the deep friendship with Frank Zerilli made this work even more pleasant. One of the most important results was the identification by Marc, myself, and the Brazilian theoretician Jaime Tiomno of the precise form

xxx of the burst of gravitational waves emitted· by a particle falling into a Black Hole. The existence of a precursor, a main burst, and a ringing tail in the overall burst structure of a gravitational wave signal emitted in a process of gravitational collapse that we established then in our specific idealized example has been consistently confirmed. In all subsequent articles, in hundreds of numerical computations, and even in more complex processes like the collision of two Black Holes, results have fit this pattern. These investigations have become part of the wider analysis of the intensities and structures of possible gravitational signals to be seen in detectors like the network of cooled bars at Stanford, Louisiana State University, and Rome and in optical interferometers like the Laser Interferometric Gravitational Observatory (LIGO), the French-Italian project VIRGO, and the German-British observatory GEO 600. In addition Frank Zerilli and I collaborated in developing the complete nonlinear perturbations of the Einstein-Maxwell fields around a Black Hole. We found two new phenomena: the "gravitationally induced electromagnetic radiation" originating from gravitational radiation impinging on the electromagnetic structure of a Black Hole, and "electromagnetic induced gravitational radiation" originating from electromagnetic radiation impinging on the gravitational structure of a Black Hole. These investigations on radiation processes and perturbations of Black Holes had in my mind a dual significance. They were significant for the study of gravitational radiation, but even more they were a direct testing ground for the stability of the Black Hole horizons and the so called "uniqueness conjecture" of axially symmetric Black Hole field configurations. In the following years this conjecture was proved mathematically by Brandon Carter. It offered an additional critical test for distinguishing Black Holes from neutron stars-unlike neutron stars, no regular pulsations should be expected from an accreting Black Hole in a binary system. Finally, in parallel with these activities two refreshing things happened which gave me great serenity. The first was a series of weekly encounters with Kurt Godel at the Institute for Advanced Studies in Princeton. It was really marvelous to have the chance to discuss with Godel himself the motivations for his classic work in cosmology, to be confronted with his profound technical knowledge of mathematics and philosophy and the boldness of some of his views on the universe, and to enjoy his evaluation of the recent progress in general relativity that he had most surprisingly been following in great detail. The second was that a brilliant group of undergraduate students started to work with me on various topics in general relativity for their junior theses: Robert Jantzen, Mark Johnston, Richard Hanni, and Robert Leach, having been recruited earlier by Jim Isenberg to fill a student initiated seminar on differential geometrical techniques in relativity that I taught. After my experience with Demetrios, I was convinced that the teaching of general relativity should be introduced as soon as possible in the educational career of a physicist in order to exploit the full novelty of the alternative geometrical interpretation of nature. The work with Mark Johnston [14] led to the article in which the diagram of the twisted family of orbits around a Kerr Black Hole was first published, later becoming the logo for the Marcel Gross-

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mann Meetings, reproduced in the article TEST: Traction of Events in Space-Time by Anna Imponente in these proceedings. Rick Hanni initiated with me the study of the electric and magnetic lines of force of a test particle being captured by a Black Hole[15]. This work, inspired in discussions with Wheeler, offered insight in the electrodynamical properties of the Black Hole horizon preceding later studies by many years. Bob Jantzen and I studied and translated from Italian into English Bianchi's original papers on spatial symmetry groups and began a critical re-examination of Bianchi cosmology, inspired by Godel's original papers on rotating cosmologies and discussion with the man himself at the Institute for Advanced Study. These activities and conceptual considerations with the graduate and undergraduate students supplied a secure theoretical basis for the identification of the Black Hole signatures. Complementary to these theoretical activities were the multiwavelength observational data on binary X-ray sources which in the meantime had became available in X-rays from space observatories and, equally important, in radio and optical wavelengths from the Earth's surface. On these grounds I felt confident in proposing the identification of Cygnus Xl with a Black Hole using the paradigm I proposed in [16]. I also made an oral presentation on this matter in New York as an invited talk at the 1972 Texas Relativistic Astrophysics Symposiun in a memorable session chaired by John Wheeler. The New York Academy of Sciences which hosted the Texas symposium had just given me their most prestigious Cressy Morrison Award for my work on neutron stars and Black Holes. Much to their dismay I never wrote the paper for the proceedings since it was contained in [16].

Ill. 5. Encounters During a Quarter at Stanford This work in relativistic astrophysics might seem far removed from the confluence of physics and space engineering occupying Francis's attention. In fact our contacts grew ever closer. In the spring of 1973, I spent a quarter at Stanford, following an earlier delayed invitation from Leonard Schiff, then no longer with us. I enjoyed every minute of my three months at Stanford. The lectures I gave introduced me to a very exciting group of students, including Blas Cabrera, Kyle Baker, Wick Haxton, and Mark Peterson, all of whom have gone on to fine careers, and I had the opportunity to interact more closely than ever in various ways with Bill Fairbank and Francis. With Bill and his brilliant circle of graduate students and research associates, which included John Madey, Mike McAshan, Steve Boughn, Ho Jung Paik, Blas Cabrera, and then Rick Hanni who had decided to do graduate work at Stanford, the discussions often went far into the night. At the time, Bill was concentrating much of his attention on the cryogenic gravity wave detector in a collaborative effort with Bill Hamilton at Louisiana State University. The first small bar had been cooled down, and Ho Jung was working on the magnetic transducer for it. This gave us the opportunity to have many discussions about sources of gravitational radiation, and also strengthened the ties with Amaldi's group in Rome, leading to the three-way Stanford-LSU-Rome collaboration. The goal we set, on the basis of a better understanding of the cross-sections of the detectors and of the astrophysical

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setting, was to create an international network of detectors capable of revealing a gravitational wave signal corresponding to an energy of one percent of a solar mass emitted anywhere in our galaxy. Robin Giffard, who contributed so much to the practice and theory of gravitational wave antennas, joined Stanford a year later. Another consuming interest of Bill's at this time was his brilliantly conceived but ultimately ill-fated experimental search for free quarks. Meanwhile Francis and his colleagues in the gyroscope program, including the three Johns-John Lipa, John Anderson, and John Nikirk-were engaged in the arduous task of developing their first operational cryogenic gyroscopes. Slowly the many new technologies were coming together. What most impressed me was the range of thinking that went into every part of the program involving Francis's physics group, the industrial studies, and the work of Dan DeBra's guidance and control group in the Stanford Department of Aeronautics and Astronautics, which had recently launched the first drag-free satellite. The close cooperation between Bill's group and Francis's group continued through BIas Cabrera's brilliant work on ultralow magnetic field technology and John Anderson's work on SQUID's. It was during this time that I also learned from Francis and his student Paul Worden about their remarkable concept of the orbiting equivalence principle experiment that has now matured into the proposed NASA-ESA (European Space Agency) MiniSTEP program. Francis and I were to have many further discussions about this mission leading to the joint proposal that he, Paul, I and others submitted to the ESA M2 competition in November, 1989. Two unexpected events also occurred. The first happened during the silence of a night in the Stanford Library. I was quietly approached by a man known to generations of Stanford people, Mr. Chang, who had come to the Physics Department in 1937 as a graduate student from China and had remained there ever since in a strange combination of roles which included acting as a resident guardian of the Department. Then in his seventies Mr. Chang had a very impressive knowledge of the history of physics and he asked me: "Professor, why don't you ever mention Marcel Grossmann in your lectures?" He then showed me the extraordinary two part article published jointly by Einstein and Grossmann in 1913: Part I, a physical section by Einstein in which he attempted the first geometrization of a physical interaction, and Part lI,a mathematical section by Grossmann in which he showed that Einstein's ideas could be expressed mathematically using the absolute differential calculus originally developed by the Italian geometers Ricci-Curbastro and Levi-Civita [17]. Together these two parts laid the foundation for the general theory of relativity. This unique interaction between mathematics, geometry, and physics remained fixed in my mind as a point of reference for any future progress in the fundamental understanding of the laws of nature. The second event was the visit of Werner Heisenberg and his wife. I had met Heisenberg in Munich the previous year, but the occasion to spend three days with him and his wife visiting the Stanford group, discussing with Bill and Francis their experiments and leisurely walking through the campus and through the hills of Portola Valley, gave me the opportunity to learn much more about him, of his

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relations with Einstein, Fermi, and Chandrasekhar, and of his own role in some of the intricacies of recent European history. I was very pleased to hear Heisenberg recall his encounters with Fermi both in Leipzig and then in Rome after Fermi's appointment to the Chair of Theoretical Physics especially created for him by Orso Mario Corbino. Heisenberg's very lengthy and detailed recollection of the work of Chandrasekhar and of his own encounter with Chandrasekhar and Fermi in Chicago was summarized by Heisenberg in the pregnant phrase "you are right, the work of Chandra has not yet been recognized enough." The meaning of that "yet" became clear to me when Chandra later received the Nobel Prize in 1983. Also very impressive were the hours of Heisenberg's intense discussions with Bill on the many aspects of the applications of magnetic quantized fluxes to fundamental research ranging from medical physics to space science to the study of the basic constituents of matter. On the other hand I found Heisenberg's considerations on the birth of quantum mechanics versus the birth of general relativity to be very surprising. Even more startling were some of Heisenberg's recollections about his detention at Farm Hill in England at the end of the war, and about the efforts in Germany and the United States to build the first atomic bomb. They were extreme, revealing, and very credible--and at variance with all published statements on the subject. I cross checked them with Johnny Wheeler and Edoardo Amaldi and this was also interesting. I was intrigued by these statements of Heisenberg at the time as I still am now after more than twenty years have passed. Since I consider them somewhat direct and confidential, I will record them in due course at some future occasion when I will feel more at ease about doing so. On the margins of this time of great excitement, however, I sensed that the Stanford ambience had changed and was almost tense. The very beautiful equilibrium established by Leonard Schiff was gone. Fairbank's heroic and at times ill-fated motto was: "the stronger the wind, the stronger we saiH" Every single achievement seemed to require a sequence of ad hoc actions and sometimes even a confrontation. III. 6. The XVI Solvay Conference and the LXV Varenna School Two extremely gifted students from the Paris based Ecole Normale Superieure came to Princeton as ESA fellows to work with me: Nathalie Deruelle in the academic year 1973-1974 and Thibault Damour in 1974-1976. Using their proverbially excellent French mathematical craftmanship, and their exceptional physical insight, we studied a variety of physical processes occurring near the horizon of a Black Hole. Nathalie and I studied some aspects of quantum particle creation near a Black Hole, while Thibault and I developed some elegant general analytical treatments of vacuum polarizations and electro dynamical processes around Black Holes, generalizing the Sauter-Heisenberg and Schwinger formalisms to Black Hole physics. Thibault introduced a new surface-based (or "membrane") viewpoint in Black Hole physics. He came along with me on many of my trips to Stanford and we collaborated with Jim Wilson at Livermore producing a treatment of magnetohydrodynamics around a Black Hole which still serves as a standard today [18]. This work followed another important result Jim and I found together in showing for the first time how to ex-

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tract energy from a rotating Black Hole with ·the torque created by an appropriate magnetosphere [19]. The year 1974 saw a great event in relativistic astrophysics, the discovery of the binary pulsar by Joe Taylor and Richard Hulse. Nothing could have more beautifully connected our research on relativistic astrophysics with the kind of work Francis and his colleagues were doing with sophisticated new technologies to test Einstein's theory in the weak gravitational field of the Earth. Thibault and I still remember the great excitement on learning of the discovery during an astrophysics lunch at the Institute for Advanced Study in Princeton. It gave us the opportunity to write a paper representing the point of ideal contact between the Stanford work and direct observations in an astrophysical setting. It is well known that Thibault subsequently became a leader in the precise interpretation of relativistic effects in binary pulsars. We argued specifically that one could apply these accurate measurements of gravitomagnetism obtained in Earth orbiting experiments to provide interpretations and estimates of strong gravitational field effects and we illustrated this theme with exact estimates. We considered this discussion so urgent that we asked Andre Lichnerowicz to publish it quickly in the Annals of the French Academy of Sciences [20]. The occasion to present these results systematically came on two successive occasions. The first was the Sixteenth Solvay Conference on Physics [21]. The second, also connected with Stanford, was the annual meeting of the American Association for the Advancement of Science held in San Francisco in February 1974. This meeting led to the book Neutron Stars, Black Holes and Binary X-Ray Sources [22]. I then decided to travel and lecture in the Pacific area, first in Japan at Kyoto where I was warmly received by a newly created relativistic astrophysics group led by Humitaka Sato [23]. From Kyoto I reached Rome via the Trans-Siberian railroad, stopping in Moscow to see Novikov and Zel'dovich. With Riccardo Giacconi I then co directed the LXC International "Enrico Fermi" School on the Physics and Astrophysics of Neutron Stars and Black Holes held in Varenna in July 1975 [24]. I considered this school and its proceedings the occasion to conclude an intense activity on these subjects and look for new problematics. I departed again for the Pacific traveling to Tahiti, Morea, and the Fiji Islands, and then again all the way down to Perth, Australia. I had been invited by Michael Buckingham, a close collaborator of Bill Fairbank, to lecture at the beautiful Perth campus, so reminiscent of Stanford. At that time Michael had promoted an interest in relativistic theories in his department, and a research group was formed led by a young Ph.D.-David Blair. It was during my stop in Rome on my way to Perth that Edoardo Amaldi told me that "the time has come for you to return to Rome" and I answered "I will think about it," and Amaldi replied "No, I think the time has come."

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IV. The Foundation of the Marcel Grossmann Meetings Possibly the most important opportunity provided by my return to Italy was the occasion to meet often and constructively with Abdus Salam. I was very eager to bring the field of relativistic astrophysics to a larger number of scientists as a tool in reaching a deeper understanding of the fundamental laws of nature. In this Salam was an enthusiastic ally. But there was also another aspect of this effort about which both Salam and I felt very strongly: the broadening of fundamental thinking to much wider groups of people internationally. We came up with the motto: "In understanding the laws of nature, no country can afford the luxury of having another country think for it." It is not important if the actual contribution of a country is small or very specialized. What is essential is the exercise of the right to discovery and original thinking which so greatly ennobles the human existence. We decided to act on two different levels. Knowing my love for the mountains and for adventurous trips, Salam asked me to help create a very special summer school in Pakistan. He selected a beautiful small village called N athiagali in the Indukush mountain range which used to be the summer capital of the North West Frontier province under British rule. The coats of arms of the English battalions carved in the mountains and the millenary forest still populated with jaguars and mountain lions were reminiscent of Kipling's dreamlike descriptions. We were especially concerned about the fate of so many Ph.D. 's coming from third world countries who upon receiving their degrees from leading universities worldwide, after working at the forefront of scientific research, were returning to their home institutions and having difficulty even finding electricity. They were completely isolated scientifically and intellectually. With a grant from the Swedish Academy, we started a school on "Physics and Contemporary Needs" bringing together scientists and university professors from institutions in a broad belt from Indonesia to Morocco. As lecturers we invited scientists who had made the most current scientific discoveries and were especially brilliant communicators. Our second step was in a totally different direction: in nourishing that most promising discovery process created by a fortunate resonance at the interface between mathematics and physics that we had seen occasionally developing in the field of relativistic astrophysics. Essential to that process was not only the theoretical and geometrical work but also the development of new technologies leading to crucial precise measurements, from the ground and from space, and new observations both at subnuclear and astrophysical scales. As I had discussed with Heisenberg, I considered a most remarkable example of such resonant interaction between mathematics, geometry, and physics the very work of Einstein and Grossmann [17) which paved the way to the birth of general relativity. In celebration of that interaction and as an omen to success Abdus and I decided to initiate the "Marcel Grossmann Meetings", eliciting contributions from general relativity to relativistic field theories, from pure mathematics to astrophysical observations, from numerical and algebraic techniques to new theoretical tools with the only common focus being the progress in determining the basic physical laws of nature. The First Marcel Grossmann Meeting was held in Trieste in 1976, for which

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Francis wrote his remarkable paper on "Gravitation, Relativity and Precise Experimentation" [25] which was put in the interesting perspective of the history of science and rooted in the many principles that he and others had been thinking about for Gravity Probe B, the Satellite Test of the Equivalence Principle (STEP), and other tests of general relativity. This article revealed yet another aspect of Francis's activities: as a historian of science. In 1960 he wrote two papers on Maxwell's scientific work, one of which "Maxwell, Osborne, Reynolds and the Radiometer" [26] is remarkable for being the first occasion in which unpublished references and reports about papers found by Francis in the Archives of the Royal Society were used to disentangle a complicated historical question. As a result of this work Francis was asked by the Dictionary of Scientific Biography (DSB) to write an article on Maxwell. Thomas Kuhn called that article the single most important one among the approximately 5000 included in the Dictionary. Francis was asked to turn the article in a short book [27]. Francis also contributed articles for the DSB on Fritz and Heinz London and later on Schiff. In 1976 he was awarded a Guggenheim fellowship which resulted in articles on Maxwell's scientific creativity [28] and on Maxwell's work on Saturn's rings and on molecules and gases [29]. Many people have wondered how Francis managed to combine work at such a high level in the history of science with the development of Gravity Probe B. His reply is that it that having this outlet probably enabled him to pass through some of the difficult early days of the experiment. Maxwell himself set the example by being one of the greatest 19th century physicists who also managed to write his own edition on the work of Cavendish [30]. Combining two careers can benefit both. By 1976 fourteen years had passed since Francis had come to Stanford, and twelve years since NASA had begun funding the research effort on the program. The first gyro operations had began in 1974 and the first precise measurements of the London moment readout in a spinning superconducting gyroscope were made early in 1975. Blas Cabrera had developed and demonstrated the technology for producing ultralow magnetic fields below 10- 7 gauss. Paul Worden had conceived of the basic idea of the STEP satellite in his doctoral dissertation and had demonstrated some of the technologies in the laboratory. The prototype Gravity Probe B telescope had been built and an artificial star for testing it was under construction. Working with Stanford and the NASA Marshall Center, Ball Aerospace had performed their first Mission Definition Study of a flight program in 1971. Marshall Center developed the ball manufacturing technology and successfully produced gyros which were spherical to better than 1 microinch (7 parts in 107 in diameter). It might have seemed that there was little more to do. In fact all the most difficult technical, management, and political issues of developing a flight program remained. Some of these subtle questions are discussed in Francis's article "Background to History: The Transition from Little Physics to Big Physics in the Gravity Probe B Relativity Gyroscope Program" [4].

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V. The First Contact With China In 1978 I received an award from the Australian government to lecture in universities allover Australia and it was on my return to Rome that I was invited to visit China by Chou Pei Yuan, President of the Chinese Association of Science and Technology and a close friend of Abdus. It was during this long visit to many University campuses which were still recovering from the destruction of the so-called cultural revolution that I met a young astrophysics professor named Fang Li Zhi. It was Fang who accompanied me during the entire trip and translated my lectures. Fang had in common with Demetrios Christodoulou, Evgenij Lifshitz, T.D. Lee and a few others the fact that he obtained his Ph.D. at 19. The many similarities in our thinking and identical scientific interests convinced both of us that we were "twins," in spite of our quite different origin and upbringing! We decided to publish our common scientific interests in a small book which was well received in China and abroad. A "first" collaboration between a Chinese scientist and a Westerner [31]. The realities I saw in that first visit to China had a strong impact on the rest of my life. In 1979 the second Marcel Grossmann meeting took place in Trieste. Its first International Organizing Committee included Y. Choquet Bruhat, Chou Pei Yuan, J. Ehlers, S. Hawking, A.R. Kadoura, Y. Ne'emann, myself, A. Salam, H. Sato, S. Weinberg, and Ya.B. Zeldovich. We had memorable lectures from P.A.M. Dirac, E.M. Lifshitz, C.N. Yang, J.H. Taylor, and Amaldi and for the first time the enthusiastic participation of a Chinese delegation. We decided to hold the next Marcel Grossmann Meeting in China. Meanwhile during the same year the Space Sciences Board of the U.S. National Research Council appointed an Ad Hoc Committee on Gravitational Physics Experiments in Space chaired by Irwin Shapiro. In a memorable turn of phrase Shapiro stated that he chose the committee members as a collection of "cancelling biases." In 1981 the Committee issued its report with Space Science Board approval. Its highest priority recommendation was to perform a precise measurement of the frame-dragging effect through the Gravity Probe B experiment. This recommendation was re-endorsed when the Committee was convened again in 1982 to review the outcome of the NASA-Stanford Phase B study of the program, which had just been completed. Meanwhile still in 1981, NASA had conducted a five-day technology review at Stanford which gave the program the highest technical reading. The year 1981 was also memorable to both Francis and Bill Fairbank for being the first time Gravity Probe B was cancelled and then reinstated with Congressional support. This began Francis's initiation into the ways of the U.S. Congress. VI. MG meetings From China to Stanford and the Founding of lCRA In 1982 the Marcel Grossmann Meeting left Italy for China. MG3 was the first international scientific meeting held in China in the early days of its opening to the West. In spite of many revolutions over the ages there has been a profound continuity in the actions of Chinese officials and a marked sense of history. Traditionally China has always been open to international exchanges with beneficial results. The

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case of Marco Polo is the one most quoted,at'least in the West, although the most important for his cultural and scientific impact is probably the case in the early seventeen century of Ri Ma To (the Jesuit Matteo Ricci) who died as a minister of the Celestial Empire. One can find a discussion of his activities in the beautiful book by P. M. D'Elia provocatively titled Galileo in China [32]. It was not surprising therefore that in the halls of the hotels and in the streets we repeatedly read the phrase written in red letters "Welcome friends from allover the world." But getting them to recognize the fact that an ensemble of friends does not make an international meeting, which by its own nature transcends the concept of friendship and must guarantee the right of any interested qualified scientist to participate, independent of race, religion, or political opinion, was one of the most difficult tasks I had to accomplish in my life. After all, this right is a consequence of the Renaissance, the Enlightenment, and Liberalism in Western culture, concepts which are foreign to Chinese society. Officials had at first refused to allow participants from some countries with no diplomatic relations with China, namely Israel and South Korea. I finally realized that the denial was not due to bureaucratic difficulty but to a different conceptual way of interpreting international relations. After an entire week of fierce bargaining, during which I often made use of wise advice given to me by Mr. Chang and read chapters of the Ri Ma To memoirs [33] every evening, also this difficulty was overcome-all the "problematic" scientists were admitted by the invention of a special passport agreed to by the Chinese ambassador in Rome and myself and signed by Abdus Salam! Many Chinese officials are still grateful today for those efforts, since after the MG3 experience the organization of international meetings in China became almost routine. Needless to say the meeting was a great success due to the great interest of the Chinese people in science and education and to the incomparable beauty of that country and its traditions. At Stanford in 1984, after many discussions with NASA at the highest level, the decision was made to initiate development in the following fiscal year of flight hardware for Gravity Probe B through the Shuttle Test of the Relativity Experiment (STORE). Critical to Francis and the program development were Brad Parkinson's decision to return to Stanford as a Professor and Program Manager and John Turneaure's decision to accept an appointment as leader of the Gravity Probe B hardware development group. Without this troika of leaders working together, the program would have been simply impossible. An additional occasion to review the work going on at Stanford in fundamental physics from space, including the beautiful lambda point experiment conducted by John Lipa as well as Gravity Probe B and STEP, came from my being asked to participate in a task group formed by the Space Science Board of the US National Academy of Science from 1985 to 1987. Rene' Pellat from France, Minoru Oda from Japan, and I had the good fortune to be invited from abroad and collaborate in this work with Bill Fairbank, Rainer Weiss, and Bob Schrieffer. The goal was to formulate guidelines for the NASA long term scientific program from 1995 to 2015 in fundamental physics and chemistry [34].

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1985 was also the year of MG4, returning' to Italy in Rome, and the meeting in which the Marcel Grossmann Awards were initiated. The institutional award was given to the Vatican Observatory for its continous work on stellar evolution and for its heritage from the Osservatorio del Collegio Romano directed by the Jesuit Angelo Secchi, the founder of the classification of stars according to their spectra. The two individual awards went to William Fairbank and to Abdus Salam for their contributions to the understanding of the fundamental laws of physics. So many scientists coming to Rome gave us the opportunity to formalize the scientific relations I had developed since my return to the University of Rome "la Sapienza." With the cooperation of George Coyne, the brilliant director of the Vatican Observatory, Fang Li-Zhi, Francis Everitt, Riccardo Giacconi, and Abdus Salam, we founded the International Center for Relativistic Astrophysics (ICRA) in Rome. In addition to "la Sapienza" the member institutions of ICRA are the University of Hofei in China, Stanford University, the University of Washington at Seattle, and the Space Telescope Science Institute in the USA, and the Specola Vaticana (Vatican Observatory), the International Center for Theoretical Physics (ICTP), and the Third World Academy of Science (TWAS) in Europe. In 1988 MG5 was held in Perth, Australia, at the splendid campus of the University of Western Australia in Nedlands on the banks of the Swan River. The hosting group there led by David Blair had pioneered new approaches and obtained new results in the field of gravitational wave detectors. The institutional award went to the University of Western Australia for its contribution to relativistic astrophysics beginning with the Wallal observations during the solar eclipse of 1919 confirming Einstein's prediction of the deflection of starlight by the sun and continuing with the development of the southern hemisphere link in the worldwide chain of laboratories seeking to observe gravitational radiation, also predicted by Einstein. The individual awards went to Satio Hayakawa for advancing relativistic astrophysics both as an observer of gamma, X- and infrared radiations and of cosmic rays and as a pioneer in the field of binary X-ray sources and to John Archibald Wheeler for leading several generations of scientists to a deeper understanding of spacetime structure and, with his geometrodynamics, expanding Einstein's vision. During these same years a work of great importance for gravitational physics was Joe Taylor's progressive refinement of his experimental work on the binary pulsar leading to the beautiful proof of gravitational wave damping. It is interesting that Taylor had already established this result at the fifteen percent level at the time of the Shapiro report. Thibault Damour, first in collaboration with Nathalie Deruelle, and then in direct association with Joe Taylor has been particularly involved in developing theoretical aspects of binary pulsar experiments. One result not obvious at first sight is that the large self-energies of the two bodies allow not only tests of weak field effects but also of some strong field effects. These take the form of null tests that can set limits on alternative theories of gravitation which could not be obtained from solar-system experiments. Meanwhile, there has been progess in several other directions, including the development of gravitational wave antennas of both the cryogenic bar and laser

xl

types. Progress has also continued on lunar laser ranging with observations being taken up in both Europe and the U.S. The principal center for laser ranging is now the CERGA center in France. Francis has recalled receiving a memorable phone call from me about STEP on November 2, 1989. At that time STEP was still a very small laboratory research effort at Stanford by Paul Worden and his colleague Matthew Bye. For several years, Francis and I had been having discussions about whether a flight program might somehow be developed as a U.S.-Italy collaboration. My phone call was to explain that there was an opportunity for a wider European collaboration through ESA's M2 announcement. I had first learned about this from Dr. Roger Bonnet, the Director of Science for ESA, who was present in my office at the time of the call. Casually Francis had asked when proposals were due. The answer was November 30, less than a month away. Fortunately, Francis was already committed to a trip to Europe on November 16, so with great speed he and Paul Worden prepared a draft proposal that he was able to take to colleagues in Rome, Pisa, Paris, and London, and send to others including those in Norway and Germany. The theory team most happily included Thibault. The numerous European improvements and suggestions were safely incorporated and the proposal reached ESA Headquarters just in time. To our delight STEP was selected for an assessment and then a Phase A study. In 1991 MG6 was held in Kyoto, Japan. The institutional award went to the Research Institute of Theoretical Physics (RITP) for keeping alive research in relativity, cosmology, and relativistic field theory and the development of a school of international acclaim. The individual awards went to Minoru Oda for participating in the pioneering work of the early sixties in X-ray astronomy and for his subsequent molding of an agile and diversified Japanese scientific space program investigating the deepest aspects of relativistic astrophysics and to Stephen Hawking for his contributions to the understanding of spacetime singularities and of the large scale structure of the Universe and of its quantum origins. This brief sketch of past Marcel Grossmann Meetings interwoven with Francis's story and that of GP-B finally brings us back to Stanford with MG7, held there in 1994 and organized by the GP-B group. The Marcel Grossmann Awards went to the Space Telescope Science Institute and to Subrahmanyan Chandrasekhar and Jim Wilson, the citations for which appear in these proceedings. Francis could have hardly imagined when he entered the field of gravitational experiments in space in 1962 how much there was to do or how wide the interest would become. Much work still remains to be done both in the theoretical and experimental fields. While writing these remarks, I read the article "Earthly Politics Boosts Space Probe" in the magazine Science [35], and looking at the diagram in it captioned Time Travel: A History of Gravity Probe B, I could not refrain from asking Francis: how did you manage to go through all this successfully? He answered by quoting from a speech Winston Churchill had made in 1942 at his old school Harrow. Churchill said that in looking back on his own career he could see that he had done some things successfully and others not so well, and in pondering its lessons he

xli

concluded that they could be summarized in nine simple words. They were: "Never give up, never give up, never, never, never." On behalf of the organizing committees of the Marcel Grossmann Meetings I am happy to dedicate these proceedings to Francis Everitt: ad majora! Remo Ruffini

References 1. Lewis Campbell and William Garnett, The Life of James Clerk Maxwell, 1882. 2. L.I. Schiff, Proc. Nat. Acad. Sci. 45, 69 (1960). 3. C.W.F. Everitt, "The Creative Imagination of an Experimental Physicist," in Near Zero: New Frontiers of Physics, J.D. Fairbank, B.S. Deaver, Jr., C.W.F. Everitt, P.F. Michelson, eds., W.H. Freeman, New York, 1988. 4. C.W.F. Everitt, "Background and History: The Transition from Little Physics to Big Physics in the Gravity Probe B Relativity Gyroscope Program," in Big Science, The Growth of Large Scale Research, Peter Galison and Bruce Hevly, eds., Stanford, 1992. 5. C.W.F. Everitt, "The Stanford Relativity Gyroscope Experiment (A): History and Overview," in Near Zero: New Frontiers of Physics, J.D. Fairbank, B.S. Deaver, Jr., C.W.F. Everitt, P.F. Michelson, eds., W.H. Freeman, New York, 1988. 6. B. Harrison, K.S. Thorne, M. Wakano, and J.A. Wheeler, Gravitation theory and Gravitational Collapse, University of Chicago Press, Chicago, 1965. 7. R. Ruffini and J. A. Wheeler, "Relativistic Cosmology From Space Platforms," in Proceedings of the Conference on Space Physics, V. Hardy and H. Moore, eds., E.S.R.O., Paris, France, 1971. 8. D. Christodoulou and R. Ruffini, "On the Electrodynamics of Collapsed Objects," Bull. Amer. Phys. Soc. Ser. II 16, 612 (1971). 9. D. Christodoulou and R. Ruffini, "Reversible Transformations of a Charged Black Hole," Phys. Rev. 4, 3552 (1971). 10. C. Rhoades, Jr. and R. Ruffini, "Maximum Mass of a Neutron Star," Phys. Rev. Lett. 32, 3 (1974). 11. D. Wilkins, "Bound Geodesics in the Kerr Metric" Phys. Rev. 5, 814 (1972). 12. S. Hawking, "Black Holes in General Relativity," Commun. Math. Phys. 25, 152 (1972). 13. M. Rees, R. Ruffini, and J.A. Wheeler, Black Holes, Gravitational Waves and Cosmology, Gordon and Breach, New York, 1974. 14. M. Johnston and R. Ruffini, "Generalized Wilkins Effect and Selected Orbits in a Kerr-Nev.'lD.an Geometry," Phys. Rev. DI0, 2324 (1974). 15. R.S. Hanni and R. Ruffini, "Lines of Force of a Point Charge Near a Schwarzschild Black Hole," Phys. Rev. 8,3259 (1973).

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16. R. Leach and R. Ruffini, "On the Masses of X-Ray Sources," Astrophys. J. Lett. 180, L15 (1973). 17. A. Einstein and M. Grossmann, "Entwurf einer verallgemeinerten Relativitatstheorie und einer Theorie der Gravitation," "Physikalischer Teil," Zeit. Math. Phys. 62, 1 (1913); "Mathematischer Teil," Zeit. Math. Phys. 62, 23 (1913). 18. T. Damour, R.S. Hanni, R. Ruffini, and J.R. Wilson, Phys. Rev. D17, 1518 (1978). 19. R. Ruffini and J. Wilson, Phys. Rev. D12, 2959, (1975). 20. T. Damour and R. Ruffini, "Sur certaines verifications nouvelles de la Relativite renerale rendues possibles par la decouverte d'un pulsar membre d'un systeme binaire," Comptes Rend. Acad. Sc. Paris A279, 971 (1974). 21. Proceedings of the Sixteenth Solvay Conference on Physics at the University of Bruxelles, September, 1979, Editions de l'Universite' de Bruxelles, 1974. 22. Neutron Stars, Black Holes and Binary X-Ray Sources, H. Gursky and R. Ruffini, eds. and coauthors, H. Reidel, Amsterdam, 1975. 23. H. Sato and R. Ruffini, Black Holes. Ultimate State of Stars and General Relativity, Chuo Koron Sha-Tokyo, 1976 (in Japanese). 24. "Physics and Astrophysics of Neutron Stars and Black Holes: Proceedings of the LXV INternational "Enrico Fermi" Varenna Summer School of 1975, R. Giacconi and R. Ruffini, eds. and coauthors, North Holland, Amsterdam,1978. 25. C.W.F. Everitt, "Gravitation, Relativity and Precise Experimentation," in Proceedings of the First Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini, North Holland, Amsterdam, 1977. 26. C.W.F. Everitt, "Maxwell, Osborne Reynolds, and the Radiometer," in Historical Studies of the Physical Sciences, Vol 1, 105, University of Pennsylvania Press, Philadelphia, 1969. 27. C.W.F. Everitt, James Clerk Maxwell, Physicist and Natural Philosopher, Scribners, New York, 1975. 28. S.G. Brush, C.W.F. Everitt, and E.W. Garber, Maxwell on Saturn Rings, MIT Press, Cambridge, MA, 1983. 29. E.W. Garber, S.G. Brush, and C.W.F. Everitt, Maxwell on Molecules and Gases, MIT Press, Cambridge, MA, 1986. 30. J.C. Maxwell, Unpublished Electrical Research by the Honourable Henry Cavendish, Cambridge, 1879. 31. L.Z. Fang and R. Ruffini, Basic Concepts in Relativistic Astrophysics, Shanghai Science and Technology Press, Shanghai, 1981 (in Chinese); English translation by World Scientific, 1983. 32. P.M. D'Elia, Galileo in China, Harvard University Press, Cambridge, Mass., 1960. 33. P.M. D'Elia, Fonti Ricciane, Storia dell'introduzione del Cristianesimo in Cina, Vol. I, Poligrafico dello Stato, Rome, 1942 XX, Vol. II, 1949. 34. Space Science in the Twenty-First Century: Imperatives for the Decades 1995 to 2015, National Academy of Sciences, Washington, D.C., 1988. 35. A. Lawler, "Earthly Politics Boosts Space Probe," Science 267, 1756 (1995).

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Humitaka Sato presenting the Grossmann Award to Jim Wilson.

Remo Ruffini introducing the lecture by Demetrios Christodoulou.

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CONTENTS Publications in this Series

v

. . . . .

Organizing Committees and Sponsors

VI!

Marcel Grossmann Awards

IX

Preface

. Xlll

An Excursus on Experimental Gravitation From Space and Relativistic Astrophysics in Honour of Francis Everitt's 60th Birthday . . . . . . . . . . . . . .

xvii

PART A

PLENARY SESSIONS Chairperson: Humitaka Sato

Was Einstein 100 Percent Right? THIBAULT DAMOUR.

3

. . . . .

Relativistic Fluids and Gravitational Collapse DEMETRIOS CHRISTODOULOU

. . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chairperson: David Finkelstein

Do We Understand Black Hole Entropy? JACOB BEKENSTEIN.

. . . . . . ..

.......................

39

Unitary Rules for Black Hole Evaporation ANDY STROMINGER.

. . . . . . . . 59

. . . . . . . . .

Recent Mathematical Developments in Quantum General Relativity ABHAY AsHTEKAR

. . . . . . . ..

...........

. . . . . . . . 75

Black Hole and Particle-Like Solutions of the Einstein-Yang-Mills Equations JOEL SMOLLER, ARTHUR G. WASSERMAN . . . . . . . . . . . . . . . . . . . . . . . 88

Gravitational Topological Charge and the Gravibreather VLADIMIR BELINSKY

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . 96

Quantum Gravity, the Planck Lattice and the Standard Model GIULIANO PREPARATA

............ .

102

Chairperson: Abraham Taub

The Lunar Orbit as Probe of Relativistic Gravity KEN NORDTVEDT

............ .

119

xlvi Gravitoelectromagnetism: Just a Big Word? ROBERT T. JANTZEN, PAOLO CARINI, DONATO BINI . . . . . . . . . . .

133

Approach to Gravitational Wave Physics Via the Regge-Wheeler-Teukolsky Equations . . . . . . . . . . .

153

Chaos, Regularity, Noise in Self-Gravitating Systems HENRY E. KANDRUP . . . . . . . . . . . . . .

167

MISAO SASAKI

. . . . . .

Fully Relativistic 3D Numerical Simulations of Coalescing Binary Neutron Stars TAKASHI NAKAMURA

...................... .

183

Chairperson: Ray Weiss The Advantages and Difficulties of Space Experiments for Testing General Relativity G. MAC KEISER

. . .

. .

. . . . . . . . . . . .

.

. . . . . .

207

Real Performance and Real Promise-Resonant Gravitational Antennas BILL HAMILTON

... . .

. . . . . . . . . . . .

. .

. . . . .

222

Chairperson: Jim Bardeen COBE Observations of Cosmic Background Anisotropies GEORGE SMOOT

. . . . .

241

. . . . . . . . . . . .

Primordial Nucleosynthesis as a Probe of Early Universe KATSUHIKO SATO, N. TERASAWA

253

........ .

Cosmic Rays from 1017 eV to Beyond 101g eV: Evidence From the Fly's Eye Experiment and Ground Arrays PIERRE SOKOLSKY

. .

. .

. . . . . . . . . . . .

. .

. . . . . .

264

PARALLEL SESSIONS Exact Solutions Chairperson: V. Belinsky

Exact (3+1)-Dimensional Gravitational Soliton Solutions of the Einstein Equation CHI Au, LI-ZHI FANG, F.T. To . . . . . . . . . . . . . . . . . . . . . .

289

Radiating Dyon In Einstein-Maxwell Theory

291

A. CHAMORRO, K.S. VIRBHADRA

Geodesic Incompleteness and the Kinked De Sitter Spacetime K.A. DUNN, T.A. HARRIOTT, J.G. WILLIAMS

..... .

294

Bondi-Sachs Metrics and Exact Solutions S.J. FLETCHER, A.W.-C. LUN . . . . .

296

xlvii A Radiating Generalization of the Kerr Solution

299

DIETRICH KRAMER . . .'. . . . . . . . . .

Cylindrical Black Hole in General Relativity and Black Holes in Lower Dimensional Gravity Theories JosE P.S. LEMOS

........................... .

302

Quantum Cosmology for the Bianchi Type IX Models

304

ALFREDO MAciAS, MICHAEL P. RYAN, JR.

Metric of an Axisymmetric Neutron Star

306

VLADIMIR S. MANKO, J. MARTIN, E. RulZ

Dilaton Solutions with Arbitrary Electromagnetic Field TONATIUH MATOS

308

. . . . . . . . . . . . . . . .

Extended N-Soliton Electrovac Solution 310

E. RUIZ, V.S. MANKO, J. MARTIN . .

A New Exact Solution of Einstein Field Equations YONGJIU WANG, ZHIMING TANG, HONGJUN Guo, HONGWEI Yu, JILIANG JING .

313

Multi-Dimensional Gravitational Theories Chairperson: M. Demianski Kaluza-Klein Solitons: New Singularities and Origin Definitions 317

ANDREW P. BILLYARD, PAUL S. WESSON . . . . . . . . . .

Nonstandard-Dimensional Gravitational Theories

319

MAREK DEMIANSKI . . . . . . . . . . . . .

A Systematic Approach to the Study of Dimensional Reductions of Self-Dual Yang-Mills Equations MARIO C.

DiAZ,

GEORGE A.J. SPARLING . . . . . . . . . . . . . . . . .

325

The Role of the Internal Metric in Generalized Kaluza-Klein Theories W. DRECHSLER, D. HARTLEY . . . . . . . . . . . . . . . . .

328

The Classical Tests in (4+1)-Gravity

330

D. KALLIGAS, P. WESSON, C.W.F EVERITT

Space-Time-Matter

333

BAHRAM MASHHOON, HONGYA LIU, PAUL S. WESSON

Fermi-Walker Coordinates in 2+1 Dimensional Gravity

336

PIETRO MENOTTI, DOMENICO SEMINARA . . . . . .

Variation of Fundamental Constants and Red-Shift in 6-D Cosmological Models AMERICO PERAZA ALVAREZ

. . . . . . . . . . . . . . . .

. . . . . . .

339

xlviii Locally Supersymmetric Action in aD-Dimensional Superspace

341

P. SALGADO . . . . . . . . . . . . . .

Toward Chiral Supergravity and Unification

343

KAZUNARI SHIMA. . . . . . . . . . . .

Extreme Regimes of Gravitational Theory Chairperson: L. Halpern

Deformations, Large and Small, of Relativistic Membranes RICCARDO CAPOVILLA, JEMAL GUVEN

347

....... .

Highly Relativistic, Self-Gravitating Perfect Fluids with Differential Rotation F.J. CHINEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

349

Is a Gravitational Geon Compatible With General Relativity? F.r. COOPERSTOCK, V. FARAONI, G.P. PERRY . . . . . .

351

Extragalactic Background Radiation and the Time Symmetry of the Universe DAVID A. CRAIG . . . . . . . . . . . . . . . . . . . . . . .

. . . .

355

Stability Test for Cauchy Horizons

358

T.M. HELLIWELL, D.A. KONKOWSKI

Stability Tests for Mild Singularities

360

D.A. KONKOWSKI, T.M. HELLIWELL

Atomic Transitions Induced by Time-Dependent Spacetime Geometries FABRlZIO PINTO

. . . . . . . . . . . . . . . . . . . . . . .

362

.

Approximation of Static Skyrmions from Connections in Compactified Minkowski Space M. ROSENBAUM, A.A. MINZONI

..................... .

366

Effects of Topology on the Gravitational Wave

369

JIRO SODA . . . . . . . . . . . . . . . .

Mathematical Foundations Chairperson: R. Schoen

Geometry Gravitation and Matter in Extreme Limits

373

LEOPOLD HALPERN . . . . . . . . . . . . . . .

A Blunder in Quantum Field Theory MASAYOSHI MIZOUCHI

377

.. . . . .

A Tensorial Lax Pair Equation and Integrable Systems in Relativity and Classical Mechanics KJELL ROSQUIST . . . . . . . . . . . . . . . . . . . . . . .

. .

. . . .

379

xlix Stability of Non-Abelian Black Holes and Catastrophe Theory

386

TAKASHI TACHIZAWA, KEI-IcHI MAEDA, TAKASHI TORI! . .

The Role of Weyl Geometry in the Causal Interpretation of Quantum Mechanics W.R. WOOD, G. PAPINI

..... . . . . . . . . . . . . . .

. .

. . .

389

Quantum Probabilities Related to Wave Function's Behavior with Respect to Spacetime Domain NORIFUMI YAMADA . . . . . . . . . . . . . . . . . . . . . .

. . . . .

392

Macroscopic Gravity

394

ROUSTAM M. ZALALETDINOV

Mathematical Relativity Chairperson: T. Dray

Algebraic Computing in General Relativity

T.

DRAY

401

............ .

A Class of Hyperbolic Gauge Conditions C. BONA, J. STELA, J. MASSO, E. SEIDEL

410

Mass and Angular Momentum from Taub Numbers

413

EDWARD N. GLASS . . . . . . . . . . . . . .

Ricci Fall-Off in Static, Globally Hyperbolic, Geodesically Complete, Ricci-Positive Spacetimes STEVEN G. HARRIS, DAVID GARFINKLE

............... .

415

MapleTensor: A New System for Performing Indicial and Component Tensor Calculations by Computer M. KAVIAN, R.G. McLENAGHAN, K.O. GEDDES

........... .

418

Similarity Reductions of the Field Equations for Stationary Axisymmetric Rigidly Rotating Perfect Fluids BEN LANGTON, EDWARD FACKERELL . . . . . . . . . . . . . .

. .

420

MathTensor: Doing Tensor Analysis and Differential Forms by Computer LEONARD PARKER

. . . . . . . . . . . . . . . . . . . . . .

. .

422

Static Axisymmetric Approach to the Two Black Hole Head-on Collision SAMUEL ROCHA DE OLIVEIRA

. . . . . . . . . . . . . . . . . . .

Spinors, Jets, and the Einstein Equations C.G. TORRE . . . . . . . . . . . . .

426 431

Positive Energy Proof for Asymptotically Anti-de Sitter Spacetimes

E.

WOOLGAR

...................... .

434

Phase Space and Gravitation Chairperson: G. Bisnovatyi-Kogan

Vlasov Equation in Spacetime Tangent Bundle

439

HOWARD E. BRANDT . . . . . . . . .

Chaos in Static Axisymmetric Spacetimes

441

KEI-IcHI MAEDA, YASUHIDE SOTA, SHINGO SUZUKI

Global Properties of Energy Truncated Spheroidal Stellar Systems MARIA TERESA MENNA, GIUSEPPE PUCACCO, REMO RUFFINI

444

Stability of Dense Stellar Clusters Against Relativistic Collapse: Maxwellian Distribution Functions with Different Cutoff Parameters MARCO MERAFINA, GENNADY S. BISNOVATYI-KOGAN, REMO RUFFINI, ENRICO VESPERINI . . . . . . . . . . . . . . . . ...... .

447

Landau Damping in Semi-Degenerate Gravitating Systems R. RUFFINI, M. CAPALBI, S. FILIPPI, J.G. GAO, L.A. SANCHEZ

451

Motion of Spin-l/2 Particles in External Gravitational and Electromagnetic Fields STAMATIS VOKOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

454

Scale-Invariant Phase Space and the Conformal Group JAMES T. WHEELER

............. .

457

Spacetime Coarse Grainings in Relativistic Particle Motion JOHN T. WHELAN

................. .

460

Alternate Gravitational Theories Chairperson: Y. Fujii

Summary of the Session: Alternative Gravitational Theories YASUNORI FUJII

. . . . . . . . . . . . . . .

. . . .

465

A Bianchi-Type IX Cosmological Solution in Brans-Dicke Theory P.AG. CHAUVET, J.L. CERVANTES-COTA . . . . . . . . . .

467

General Relativity and Empirical Success R. DISALLE, W.L. HARPER, S.R. VALLURI

470

The Fokker de Sitter Gyroscopic Precession in (0, A"" )-Field Theory WANG TAK FUNG, ALFRED XIN Yu

............. .

Gravitation and Spin, Minimal Coupling and Beyond RICHARD T. HAMMOND . . . . . . . . . . . . .

472 474

How Does Generalization of Einstein Theory Enhance Growth of Density Perturbations TOSHINARI HIRAI, KEI-IcHI MAEDA

.................. .

477

Ii Observational Limits on Scalar-Tensor Theories D. KALLIGAS, K. NORDTVEDT, R.V. WAGONER

480

The Gravitational Radiation of !l-Field Theory Yu YIU LAM . . . . . . . . . . . . . . .

484

Higgs Mechanism and the Structure of the Energy-Momentum Tensor in Einstein Gravity and Conformal Gravity PHILIP D. MANNEHEIM, DEMOSTHENES KAZANAS

............ .

486

The Equivalence Principle, Mach's Principle and a Possible Link Between LorentzInvariant Scalar-Field Theory of Gravity and General Relativity P. MAZILU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . .

489

Gravity Coupled to Vector Coulomb Fields SHIN-I CHI NAKARIKI, KAZUMI FUKUMA, TETSUO FUKUI, MASAYOSHI MIZOUCHI, TERUYA OHTANI . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491

Crossing Symmetry is Incompatible with General Relativity H. PIERRE NOYES

. . . . . . . . . . . . . . . . . .

Quantum Behavior in Asymmetric, Weyl-Like Cartan Geometries J .E. RANKIN . . . . . . . . . . . . . . . . . . . . . . .

493 496

Isotopic Quantization of Gravity and its Universal Isopoincare Symmetry RUGGERO MARIA SANTILLI

................... .

500

Motion of a Spinning Test Particle in the !l-Field Metric JUDY TAM HONG . . . . . . . . . . . . . . . . .

506

Inflation in !l-Field Theory LEE WA TSAN, ALFRED XIN Yu

509

Advances on a Unified Physics-Astrophysics Theory Based on a Standing-Wave Particle Model 511

RAFAEL A. VERA

Can Gravity Explain Galactic Mysteries? VADIM V. ZHYTNIKOV

....... .

514

Gravitomagnetism Chairperson: D. Theiss Gravitoelectromagnetism: Further Applications DONATO BINI, PAOLO CARINI, ROBERT T. JANTZEN

519

Gravitoelectromagnetism and Inertial Forces in Relativity PAOLO CARINI, DONATO BINI, ROBERT T. JANTZEN . .

522

The Question of Linearized Gravity and Maxwell-Newtonian Approximation C.Y. Lo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

525

Iii

On The Electromagnetism-Gravitation Analogy NIKOLAI V. MITSKIEVICH . . . . . . . . . .

528

The Gravitational and Electrostatic Fields Far from an Isolated Einstein-Maxwell Source J.H. OCARIZ, HECTOR RAGO

530

Thomas Precession in Spacetime Geometries ABRAHAM A. UNGAR . . . . . . . . . .

532

Perturbations of Stars and Black Holes Chairperson: V. Ferrari Nonradial Pulsations of Compact Objects in General Relativity: An Overview VALERIA FERRARI . . . . . . . . . . . . . . . . . . . . . . . . . .

537

Aspects of Black-Hole Scattering NILS ANDERSON . . . . . . .

560

Transformations Between the Regge-Wheeler and Bardeen-Press Equations J.F.Q. FERNANDES, A.W.-C LUN . . . . . . . . . . . . . . . . . .

564

Discharge of the Electromagnetic Field of a Reissner-Nordstrom Black Hole RHETT HERMAN, WILLIAM A. HISCOCK . . . . . . . . . . . . . . .

567

Classically Unstable Modes of an Isolated Black Hole OSAMU KABURAKI . . . . . . . . . . . . . . .

570

Non-Radial Oscillations of a Slowly Rotating Relativistic Star YASUFUMI KOJIMA . . . . . . . . . . . . . . . . . . .

572

Quasinormal Frequencies of Step Potentials HANS-PETER NOLLERT . . . . . . . . .

574

Quasinormal Modes of Neutron Stars and Black Holes: Kin or Strangers? HANS-PETER NOLLERT, MARKUS LEINS, MICHAEL SOFFEL . . . . . .

577

Quasi-Normal Modes and Late Time Behaviour of Waves in Gravitational Systems W.-M. SUEN, E.S.C. CHING, P.T. LEUNG, K. YOUNG . . . . . . . . . . . .

580

Small Disturbances in Black Hole Magnetosphere TOSHIO UCHIDA . . . . . . . . . . . . . .

584

Groups in General Relativity Chairperson: M.P. Ryan, Jr. Spacetimes Admitting a 3-Parameter Group of Homotheties JAUME CAROT, ALICIA M. SINTES . . . . . . . . . . .

589

SO(2,4) Gauge Symmetry and the Einstein-Cartan-Maxwell Equations D.M. KERRICK. . . . . . . . . . . . . . . . . . . . . . . . .

592

liii Conformal Motions in Bianchi I Spacetimes D. B. LORTAN, S.D. MAHARAJ

594

A Generalization of the Noether Theorem NIKOLAI V. MITSKIEVICH, SAMIR A. SIDAWI

597

A Generalization of Ashtekar and the MacDowell-Mansouri Proposals OCTAVIO OBREGON, J.A. NIETO, J. SOCORRO . . . . . . . .

599

Bubbles with an 0(3)-Symmetric Scalar Field in Curved Spacetime NOBUYUKI SAKAI, YOONBAI KIM, KEI-IcHi MAEDA . . . . . .

601

Connections on Principal Bundles Over Space-Time with Structure Groups SL(2,C) and O(3,C) A.H. TAUB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

604

Tensor Multinomials and Einstein-Maxwell Theory CHRIS VUILLE . . . . . . . . . . . . . . . .

606

Numerical Relativity Chairperson: M. Choptuik Numerical Instabilities Associated with Gauge Modes MIGUEL ALCUBIERRE, BERNARD F. SCHUTZ

611

Modelling Moving Black Holes GABRIELLE ALLEN, MIGUEL ALCUBIERRE, SIMON FARRAR, BERNARD F. SCHUTZ, LEE ASTON WILD . . . . . . . .

615

Rotating Black Hole Spacetimes STEVEN R. BRANDT, EDWARD SEIDEL

619

Masses of Spindle and Cylindrical Naked Singularities TAKESHI CHIBA, KEN-IcHI NAKAO, TAKASHI NAKAMURA

622

Initial Data for Quasi-Circular Orbits of Black-Hole Binaries GREGORY B. COOK. . . . . . . . . . . . . . . . . .

624

Finite Differencing on the Sphere ROBERTO GOMEZ, PHILIPP OS PAPADOPOULOS, JEFFREY WINICOUR . . . . . . . . . .

626

Critical Trapping of a Spherical Scalar Field SEAN A. HAYWARD . . . . . . . . . . .

629

A 3-D Apparent Horizon Finder JOAN MASSO, JOSEPH LIBSON, EDWARD SEIDEL, WAI-Mo SUEN

631

Adaptative Mesh Refinement in Numerical Relativity JOAN MASSO, EDWARD SEIDEL, PAUL WALKER . .

634

liv Singularity Avoidance in Numerical Black Hole Spacetimes JOAN MASSO, PETER ANNINOS, GREG DAUES, EDWARD SEIDEL, WAI-Mo SUEN

637

Numerical Integration of the Robinson-Trautman Equation D.A. PRAGER, A.W.-C. LUN

640

.......... .

3-D Numerical Relativity at NCSA EDWARD SEIDEL, PETER ANNINOS, KAREN CAMARDA, JOAN MASSO, WAI-Mo SUEN, MALCOLM TOBIAS, JOHN TOWNS . . . . . . . . .

644

Event Horizons of Numerical Black Holes WAI-Mo SUEN, PETER ANNINOS, DAVID BERNSTEIN, STEVE BRANDT, JOSEPH LIBSON, JOAN MASSO, EDWARD SEIDEL, LARRY SMARR, PAUL WALKER

648

Interface Behaviour in an Adaptive Mesh for Hyperbolic Equations LEE ASHTON WILD, MIGUEL ALCUBIERRE, GABRIELLE ALLEN, BERNARD

F.

SCHUTZ

651

Do One Dimensional Numerical Models of Supernovae Have Anything to Do With the Real World? JAMES R. WILSON, DOUGLAS S. MILLER, RoNALD W. MAYLE

654

Relativistic Binary Neutron Star Coalescence JAMES R. WILSON, GRANT J. MATHEWS

658

Large Scale Structure Theory Chairperson: W. Stoeger

Robertson-Walker Models with Bulk Viscosity in Brans-Dicke Theory AROONKUMAR BEES HAM

. . . . . . . . . . . . . . . . . . .

665

Decoherence of Homogeneous and Isotropic Metrics in the Presence of Massive Vector Fields ORFEU BERTOLAMI, P.V. MONIZ

.................. .

668

The Missing Mass Problem in a Multiconnected Hyperspace-Time CLAUDE GAUTHIER . . . . . . . . . . . . . . . . . . .

672

Observational Constraints on Primordial Perturbation Spectra STEFAN GOTTLOBER

. . . . . . . . . . . . . . . . . .

674

Numerical and Classical Cosmology Chairperson: B. Berger

Developments in Numerical Cosmology BEVERLY K. BERGER . . . . . . . .

679

Numerical Simulation of Unpolarized Gowdy Spacetimes DAVID GARFINKLE, BEVERLY BERGER, CARRIE SWIFT, VIJAYA SWAMY, VINCENT MONCRIEF, BORO GRUBISIC . . . . . . . . . . . . . . .

699

Iv Colliding Gravitational Waves in Friedmann-Robertson-Walker Backgrounds J .B. GRIFFITHS . . . . . . . . . . . . . . . . . . . . . . . . . .

702

A Numerical Kinetic Theory Approach for Two-Fluid Cosmological Models H. HARLESTON, H. QUEVEDO, R.A. SUSSMAN ........... .

705

Continuous Time Dynamics and Iterative Maps of Expansion Normalized Variables for Mixmaster Cosmologies D.W. HOBILL, T.D. CREIGHTON . . . . . . . . . . . . . . . .

708

On Properties of Large-Scale Inhomogeneities of Gravitational and Scalar Fields in the Vicinity of Cosmological Singularity A.A. KIRILLOV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

711

Rotational Perturbations of a Friedmann Universe CHRISTIAN KLEIN . . . . . . . . . . . . . .

713

Tolman Bondi Spacetimes from a Different Viewpoint D.R. MATRAVERS, R. MAARTENS, N.P. HUMPHREYS

716

A Quantum-Driven-Time (QDT) Quantization of the Taub Cosmology WARNER A. MILLER, ARKADY KHEYFETS . . . . . . . . . . . .

718

On The General Behaviour of the Universe Near the Cosmological Singularity GIOVANNI MONTANI . . . . . . . . . . . . . . . . . . . . . .

722

The Contour of Integration in Quantum Cosmology and Two Simplicial Minisuperspace Models PETER A. MORSE . . . . . . . . . . . . . . . . . . . . . . .

726

Chaotic Inflation and Baryogenesis in Supergravity H. MURAYAMA, HIROSHI SUZUKI, T. YANAGIDA, JUN'ICHI YOKOYAMA

729

Evolution of a Perturbed Anisotropic Universe HYERIM NOH . . . . . . . . . . . . . .

733

Bianchi V Models in N=2, D=5 Supergravity LUIS O. PIMENTEL, J. SOCORRO . . . . .

735

Cylindrical Solutions in N=2, D=5 Supergravity LUIS O. PIMENTEL . . . . . . . . .

738

Weyl Equation in Godel Type Universes LUIS O. PIMENTEL, ABEL CAMACHO, ALFREDO MACiAS

741

Inflation in a Planar Universe HISA-AKI SHINKAI, KEI-IcHI MAEDA

744

Heat Transport in an Inhomogeneous Spherically Symmetric Universe JOSEPH TRIGINER, DIEGO PAVON . . . . . . . . . . . . . . .

747

Ivi Quantum Gravity Chairperson: R. Kallosh Exact Duality in String Effective Action .

751

Canonical Formulation and Finiteness of N=l Supergravity with Supermatter P.D. D'EATH . . . . . . . . . . . . . . . . . . . . . . . . . . . .

753

ERIC BERGSHOEFF

. .

.

.

. . . .

A Noncritical String Approach to Black Holes, Time and Quantum Dynamics JOHN ELLIS

755

............................ .

Average Effective Potential in Quantum Gravity ROBERTO FLOREANINI

.........

758

.

Pair Creating Black Holes

761

JEROME P. GAUNTLETT .

IWP Solutions in Dilation-Axion Gravity DAVID KASTOR.

.

. .

. .

. . . . .

764

.

Bosonic Physical States in N=l Supergravity?

767

MIGUEL E. ORTIZ, S.M. CARROLL, D.Z. FREEDMAN, D.N. PAGE

Interpretation of Exact Quantum Taub Model Solutions in Terms of Classical Instanton Solutions MICHAEL P. RYAN, JR.

.

.

. . . . .

.

.

.

.

.

. .

. .

. .

. .

. .

. .

.

770

Quantum Effects in 2+ 1 Dimensional Black Holes ALAN R. STEIF

.............

773

.

Knots and Quantum Gravity Chairperson: J. Baez Knots and Quantum Gravity: Progress and Prospects JOHN C. BAEZ . .

. .

.

.

. .

. .

. .

.

.

.

.

779

.

Moduli Space Cohomology and Wavefunctionals in 3D Quantum Gravity ROGER BROOKS

.

. .

.

.

. . . .

. .

.

.

.

.

. .

.

.

. .

.

.

.

.

798

Skein Relations From Homotopy Transformations in Chern-Simons Theory BERND BRUGMANN . .

. .

. . . .

. .

.

.

.

.

. .

.

. .

.

. .

. .

801

Geometries Induced by Surfaces: An Algorithm on the 3-Dimensional Simplex Lattice JUNICHI IWASAKI . .

.

.

.

. .

Vassiliev Invariants and Quantum Gravity LOUIS H. KAUFFMAN . . . . . . . . .

803 805

Ivii

Wilson Loop Coordinates for 2+1 Gravity

807

R. LOLL . . . . . . . . . . . . .

Topological Feynman Rules for QGD 811

HUGO A. MORALES-TECOTL, CARLO ROVELLI

Gravity and Quantum Theory Chairperson: S. Carlip

Gravity and Quantum Theory

817

STEVE CARLIP . . . . . . .

New Diagrammatic Method for Quantum Field Theory in the Heisenberg Picture MITSUO ABE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

823

Charge and Central Charge in Chern-Simons Field Theory MAXIMO BANADOS

. . . . . . . . . . . . . . . . .

825

On the Hamiltonian Formulation of Higher Dimensional Chern-Simons Gravity MAXIMO BANADOS, LUIS J. GARAY

......

. . . . . . . . . . . .

828

Anomaly in 2D-Gravity Coupled to Matter Fields DANIEL CANGEMI

............ .

831

Fundamental Constants and the Problem of Time ALBERTO CARLINI, JEFF GREENSITE . . . . .

834

The Statistical Mechanics of the (2+1)-Dimensional Black Hole STEVE CARLIP . . . . . . . . . . . . . . . . . . . . .

837

Solution Generating Methods in Stringy Gravity DMITRI GAL'TSOV, ALBERTO GARciA

840

On Black Hole Solutions in Multidimensional Gravitational Models V.D. IVASHCHUK, V.N. MELNIKOV. . . . . . . . . . . . . .

842

Finite Temperature Effects on Hyperbolic Space-Time K. KIRSTEN, G. COGNOLA, L. VANZO, S. ZERBINI .

844

Mass Generation in Self-Interacting 1/>4 Scalar Theories K. KIRSTEN, E. ELIZALDE, G. COGNOLA, L. VANZO, S. ZERBINI

846

Black Holes as Quantum Membranes MICHELE MAGGIORE

. . . . . . .

848

The Spectral Analysis Inner Product for Quantum Gravity DONALD MAROLF

................ .

851

Quantum Fluctuations of Black Hole Geometry KOUJI NAKAMURA, SHIGELU KONNO, YOSHIMI OSHIRO, AKIRA TOMIMATSU

854

Iviii Effective Potential and Dynamical Symmetry Breaking in 20 Induced Gravity Yu.1. SHIL'NOV, E. ELIZALDE, S.D. ODINTZOV, A. R;OMEO

.............

The Einstein-Ostrogradsky-Dirac Hamiltonian JONATHAN Z. SIMON . . . . . . . . . . .

857 859

A General Supergravity Formalism for a Naturally Flat Inflation Potential EWAN D. STEWART . . . . . . . . . . . . . . . . . . . . . . . .

862

A Quadratic Spinor Lagrangian for General Relativity ROH-SUAN TUNG, JAMES M. NESTER

.....

865

String Theory in General Relativity Chairperson: L. Dixon On a Possible Solution for the Polonyi Problem in String Cosmology M.C. BENTO, O. BERTOLAMI

................ .

869

ALE Instantons in String Theory MASSIMO BIANCHI, F. FUCITO, G.C. ROSSI, M. MARTELLINI

873

Cherenkov Radiation From Superconducting Cosmic Strings DMITRI GAL'TSOV, YURI GRATS . . . . . . . . . . . .

876

Stringy Black Holes from Sigma-Models DMITRI GAL'TSOV, ALBERTO GARciA, OLEG KECHKIN

878

Black Holes and Sphalerons in Low Energy Effective String Theory GEORGE LAVRELASHVILI

................. .

880

Exact Four-Dimensional Dyonic Black Holes and Bertotti-Robinson Spacetimes in String Theory DAVID A. LOWE, ANDREW STROMINGER

. . . . . . . . . . . . . . . . . .

883

Duality and Supersymmetry TOMAS ORTiN

889

. . . . . .

Black Hole Entropy in Superstring Theory JOHN UGLUM

891

........... .

PARTB Quantum Fields in General Relativity Chairperson: L. Parker Effects of Quantum Fields on the Spacetime Geometries, Temperatures and Entropies of Static Black Holes PAUL R. ANDERSON, HOOMAN BAHRANI, WILLIAM A. HISCOCK, JANET WHITESELL, JAMES W. YORK, JR . . . . . . . . . . . . . . . . . . . . . . . . .

897

lix On the Existence of Black Hole Evaporation V. BELINSKY. . . . . . . . . . • . . .

900

Renormalization Group and Coarse-Graining in the Einstein Universe ALFIO BONANNO . . .

.

.

. . .

.

.

.

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. . . . . .

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.

903

.

Radiation from a 2-D Evaporating Black Hole

908

SUKANTA BOSE, LEONARD PARKER

Black Holes Can Change Mass When Nucleating Vacuum Phase Transitions THOMAS J. BRUECKNER, WILLIAM A. HISCOCK . . . . . . .

. .

.

.

915

.

Spinors Without Spinors

918

TEVIAN DRAY, CORINNE A. MANOGUE, JORG SCHRAY

Averaged Energy Conditions and Quantum Inequality Restrictions on Negative Energy

921

L.H. FORD, T.A. ROMAN . . . . . . . . . . . . . . . . . . . . . . . .

Relic Gravitons, Lightcone Fluctuations, and Ultraviolet Divergences L.H. FORD

924

.................... .

Extreme Black Holes: A Third Venue for Quantum Gravity?

928

WILLIAM A. HISCOCK, DANIEL J. LORANZ, PAUL R. ANDERSON

Graphical Representation of Invariants and Application to Quantum Gravity SHOICHI ICHINOSE

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. . . . . .

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931

. .

Two Regularization Methods of Energy-Momentum Tensor in Curved Space-Time SANG PYo KIM ..... . . . . . . . . . . . . . . . . . . . . . . .

.

934

Entropy Increase for Black Holes in Higher Curvature Gravity ROBERT MYERS, TED JACOBSON, GUNGWON KANG

937

Stringy Twists of the Taub-Nut Metric ROBERT MYERS, CLIFFORD V. JOHNSON

940

Black Holes, the Wheeler-De Witt Equation and the Semi-Classical Approximation MIGUEL E. ORTIZ . . . . . . . . . . . . . . . . . . . . . . . .

943

Chronology Protection and Quantized Field in Generalized Misner Space TSUNEFUMI TANAKA, WILLIAM A. HISCOCK ...........

947

.

Topologically Induced Chaos in the Universe ROMAN TOMASCHITZ

.........

.

950

Semiclassical Black Hole in Thermal Equilibrium with a Nonconformal Scalar Field JANET WHITESELL, WILLIAM A. HISCOCK, PAUL R. ANDERSON, JAMES W. YORK, JR..

.

.

. .

.

.

. .................. .

952

Ix

Quantum Radiation and Accelerated Frames Chairperson: U. Gerlach Paired Accelerated Frames ULRICH H. GERLACH . .

957

The Equivalence Principle and The Boulware State in 1+1 Dimensions WARREN G. ANDERSON . . . . . . . . . . . . . . . . . . . . .

977

Quantum Cosmology Chairperson: D. Page Probabilities Don't Matter DON N. PAGE . . . . . Inflation and Initial Singularities ARVIND BORDE . . . . . . .

983

. . . . . . . . . . . . . . . . . . . . 1003

Physical States for Super Quantum Cosmology RICCARDO CAPOVILLA, JEMAL GUVEN, OCTAVIO OBREGON

. . . . . . 1005

Quantization of Bianchi Models in N = 1 Supergravity with a Cosmological Constant A.D.Y. CHENG, P.D. D'EATH, P.R.L.V. MONIZ . . . . . . . . . . . . . . . . . . 1007

=

Finite N 1 Supergravity P.D. D'EATH . . . . .

. . . . . 1010

Quantum Fluctuations of Planck Mass As Mutation Mechanism in a Theory of Evolution of the Universe JUAN GARCiA-BELLIDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 Thermodynamic Stability of a Multi-Bubble Cosmological Model GERALD HORWITZ, OLEG FONAREV . . . . . . . . . . . .

. . . . . . 1014

Exact Solutions in Classical and Quantum Multidimensional Cosmology and Chaotic Behaviour Near the Singularity V.D. IVASHCHUK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 Quantization of Bianchi Class A Models in Supergravity and the Probability Density Function of the Universe HUGH LUCKOCK, CHRIS OLIWA

. . . . 1020

Creation of a Child Universe by a False-Vacuum Bubble with Out-Going Negative Energy Radiation TAKASHI MISHIMA, HIROMI SUZUKI, NORIAKI YOSHINO . . . . . . . . . . . . . . . 1023 Decoherence in Quantum Cosmology MASA-AKI SAKAGAMI, TAKASHI OKAMURA, HIROTO KUBOTANI . . . . . . . . . . . . 1026

Ixi

Quantum State During and After the False Vacuum Decay MISAO SASAKI, TAKAHIRO TANAKA, KAZUHIRO YAMAMOTO, JUN'ICHI YOKOYAMA

...

1028

Conformally Invariant Cosmology YURI V. SHTANOV

. . . . . . 1030

...•.

Quantum State After O(4)-Symmetric Bubble Nucleation with Gravitational Effects

. . . . . . 1032

TAKAHIRO TANAKA, MISAO SASAKI

Neutron Stars Chairperson: S. Tstlruta

Neutron Star Session Report: Some Selected Problems on Neutron Stars SACHIKO TSURUTA

. . . . . . . . . . . . . . . . . . . ..

.•.......

Low-Frequency Modes of Pulsation of Relativistic Accretion Disks JAMES R. IpSER • . . . . . . . . . . . . . . . . . . . . .

1037

. . . . . 1048

The Effects of Superfiuid Hydrodynamics on the Stability of Rotating Neutrons Stars LEE LINDBLOM, GREGORY MENDELL . . . . . . . . . . . . . .

. . . . . 1051

Self-Trapping of Superfiuid Vortices and the Origin of Pulsar Glitches YUKO MOCHIZUKI, TAKEO IZUYAMA

. . . . . . . . . . . . . . . . • . . . . . . .

1053

Dynamical Coherence in Neutron Stars E. DEL GIUDICE, R. MELE, C. GUALDI, G. MANGANO, G. MIELE . . . 1055

G. PREPARATA,

Neutron Stars with Strong Magnetic Fields LETAO QIN, SACHIKO TSURUTA

. . . . . . . . • . . . . . . . . . . . .

1064

Formation of Bosonic Compact Objects

. . . . 1067

EDWARD SEIDEL, WAI-Mo SUEN . . .

An Estimate of the Effect ofIntermediate-Range Forces on the Structure of Neutron Stars CHRIS VUILLE

. . . . 1070

. . . . . . . . . . . . . . . . . . .

Generation and Detection of Gravitational Waves Chairperson: J. Jpser

Modulation of Gravitational Waveforms from Merging Binaries Caused by SpinInduced Orbital Precession THEOCHARIS A. ApOSTOLATOS

..........................

1075

Stochastic Gravitational Wave Background KATHERINE COMPTON, DAVID NICHOLSON, BERNARD

F.

SCHUTZ . . . . . . . . . . .

1078

Ixii Efficient Gravitational Wave Chirp Detection and Estimation via Time-Frequency Analysis and Edge Detection M. FEO, V. PIERRO, I.M. PINTO, M. RICCIARDI . . . . . . . . . . . . . . . . . . 1086 Prospects for Deducing the Equation of State of Nuclear Matter from GravitationalWave Measurements of Neutron-Star Binary Coalescence DANIEL KENNEFICK, DUSTIN LAURENCE, KIP S. THORNE . . . . . . . . . . . . . . 1090 Preparation for the Signal Search in the VIRGO Data B. MOURS, B. CARON, A. DOMINJON, F. MARION, L. MASSONNET, R. MORAND, M. YVERT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 A Possible Method for Estimating the Distances to Sources of Gravitational Waves PETER R. SAULSON, WILLIAM J. STARTIN . . . . . . . . . . . . . . . . . . . . . 1096

Cosmic Background Radiation Chairperson: S. Torres Hot and Cold Spots in the Cobe-DMR Maps LAURA CAYON . . . . . . . . . . . . .

. . . . . . . . 1101

Infrared Cutoff in Initial Density Perturbations and Cosmic Temperature Fluctuations LI-ZHI FANG, YI-PENG JING . . . . . . . . . . . . . . . . . . . Using Cosmic Variance and the COBE DMR Results to Constrain the Inflation Effective Potential HANNU KURKI-SUONIO, GRANT J. MATHEWS . . . . . . . . . . . . . . .

. . . . . 1104

. . . . 1106

A New Calculation of the Sachs-Wolfe Effect and a Proof That Almost Isotropy of the Background Radiation Implies the Universe Is Almost FLRW WILLIAM R. STOEGER, S.J., GEORGE F.R. ELLIS . . . . . . . . . . . . . . . . . . 1108 The Imprint of n on the Cosmic Microwave Background NAOSHI SUGIYAMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111

Gravitational Lenses Chairperson: A. Petters Mathematical Aspects of Gravitational Lensing ARLIE O. PETTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117 Gravitational Lenses and Plastic Simulators RONALD J. ADLER, WILLIAM C. BARBER, MARK E. REDAR The Odd Image Theorem DANIEL HENRY GOTTLIEB

. . . . . . . . . . . . . 1138

. . . . . . . . . . . . . . . . . . . 1141

Effects of the Metric of Realistic Model Universes MARK W. JACOBS, ROBERT V. WAGONER . . . . . . . . . . . . . . . . . . . . . 1144

Ixiii Gravitational Lensing Effect of Wormhole SUNG-WON KIM, Y.M. CHO

. . . . . 1147

.....

Huygens' Principle for Weyl's Neutrino Equation and Maxwell's Equations in Petrov Type III Space-Times R.G. McLENAGHAN, F.D. SASSE

. . . . . 1149

Lensing in (4+1) Gravity H. LIM, PAUL S. WESSON

...............

1151

................................

1153

JAMES M. OVERDUIN, PAUL

Criteria for Multiple Imaging in Lorentzian Manifolds VOLKER PERLICK

Geometry and Matter in (4+1) Gravity PAUL S. WESSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1156

Astroparticle Physics Chairperson: J. Frieman

Creation of a Scalar Potential in 2D Dilaton Gravity KLAUS BEHRNDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1161

Lasing Axion Clusters . . . . . . . . . . . . . . . 1164

THOMAS W. KEPHART, THOMAS J. WEILER

Evolution Equations for Scale-Dependent Effective Actions M. REUTER

................... .

. . . . . . . 1168

A Solvable Model for Anharmonic Evolution of Cosmic Axion Overdensities KARL STROBL, THOMAS J. WEILER

. . . . . . . . . . . . . . . . . . . . . . . .

1171

Fluid Cosmology with Decay and Production of Particles WINFRIED ZIMDAHL, DIEGO PAVON

........................

1175

Hubble Space Telescope Results Chairperson: D. Macchetto

Supernovae as Distance Indicators for Cosmology DAVID BRANCH

..................

1181

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1184

..............

High Energy Phenomena at the Nuclei of Galaxies WILLIAM B. SPARKS

. . .

Millisecond Pulsars Chairperson: M. Tavani

Millisecond Pulsars: Open Issues and Future Perspectives MARCO TAVANI

................................

1189

Timing of an Array of Millisecond Pulsars DONALD C. BACKER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199

Ixiv

High Precision Metrology from Pulsar PSR J1713+0747 R.S. FOSTER, F. CAMILO, A. WOLSZCZAN . . . . . . . . . . . . . . . . 1209 Searches for Millisecond Pulsars A.G. LYNE

. . . . . . . . . . 1213

........ .

Companion Evaporation in VLMXBS and Binary Millisecond Pulsars JACOB SHAHAM

.................................

1223

Binary Pulsars and Neutron Star Masses . . . . . . . . . . . . . . . . . . . 1231

S.E. THORSETT, Z. ARZOUMANIAN . .

Compact Objects, Theory and Formation Chairperson: J. Wilson

Inner Structure of a Spherical Black Hole . . . . 1239

S. DROZ, A. BONNANO, W. ISRAEL, S.M. MORSINK

Ernst-Like Formalism and New Differentially Rotating Solutions for Perfect Fluids in General Relativity L.M. GONZALEZ-RoMERO . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

1242

Quasilocal Energy and Rotating Black Holes ERIK A. MARTINEZ . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 1245

A Critique of the Expanding Photosphere Method MARCOS J. MONTES, ROBERT V. WAGONER

. . . . . 1247

Neutrino Flux Through the Conversion From Two-Flavor to Three-Flavor Quark Matter in a Supernova Core QUIHE PENG, ZIGAO DAI, TAN Lu . . . . . . . . . . . . . . . . . . . . . . . . .

1250

The Gravitational Entropy of Collapsed Objects . . . . . . 1253

QUIHE PENG . . . . . . . . . . . . . . . .

Axisymmetric Neutrino Radiation and the Mechanism of Supernova Explosions ...............

1256

..............

1261

. . . . . . . . . .

1282

......................

1285

TETSUYA SHIMIZU, SHOICHI YAMADA, KATSUHIKO SATO

Accreting Compact Objects Chairperson: L. Stella / P. Michelson

The Search for Black Holes in X-Ray Binaries: An Update LUIGI STELLA, G.L. ISRAEL, S. MEREGHETTl, D. RICCI Accretion of Matter with Magnetic Field onto a Black Hole G.S. BISNOVATYI-KoGAN

. . . . . . . . . . . . . . .

Highly Variable Magnetohydrodynamic Accretion onto a Black Hole KOUICHI HIROTANI, AKIRA TOMIMATSU

Ixv Relativistic Hadrons and the Origin of Relativistic Outflows in Active Galactic Nuclei DEMOSTHENES KAZANAS, JOHN CONTOPOULOS

...................

1287

Post-Newtonian Oscillations and Gravitational Radiation of a Rotating Disk of Dusk WILHELM KLEY, GERHARD SCHAFER . . . . . . . . . . . . . . . . . . . . . . . .

1290

Quasar Redshifts Revisited From the Schwarzschild-Like Model .............

1294

........... . . . . . . . . . .

1296

W. PITTER, Z. MORANTES, A. AGUILLON, R. URDANETA

Processes of Matter Flow Nearby the Massive Compact Object MARINA RoMANOVA, RICHARD LOVELACE

Trans-Fast MHD Accretion onto Black Hole MASAAKI TAKAHASHI . . .

. . . . . . .

. . . . . . . . . . . . . 1298

Relativistic Diskoseismology: A New Signature of Black Holes . . . . . 1301

ROBERT WAGONER, MICHAEL NOWAK, CHRISTOPHER PEREZ

Problems in the Theory of Old Neutron Stars Accreting the Interstellar Medium S. ZANE, A. TREVES, M. COLPI, R. TUROLLA, L. ZAMPIERI . . . . . . . . . . . . . 1304 Galactic X and 'Y Ray Sources Chairperson: P. Laurent / R. Ramaty Very Short 'Y Bursts and Primordial Black Hole Evaporation DAVID B. CLINE

. . . . .

. . . . . . . . . .

1309

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1314

. . . . . . . . . . . . . . . . . .

Hard X-Ray Observations of Black Holes and Neutron Stars PHILIPPE LAURENT . . . .

MACHOs and Dark Matter Chairperson: C. Stubbs Production and Detection of Dark Matter Candidates: The PVLAS Experiment D. BAKALOV, G. CANTATORE, G. CARUGNO, S. CARUSOTTO, P. FAVARON, F. DELLA VALLE, U. GASTALDI, E. IACOPINI, E. MILOTTI, R. ONOFRIO, R. PENGO, F. PERRONE, G. PETRUCCI, E. POLACCO, C. Rizzo, G. Ruoso, E. ZAVATTINI, G. ZAVATTINI . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 A Possible Scenario for a Baryonic Dark Halo FRANCESCO DE PAOLIS, GABRIELE INGROSSO, PHILIPPE JETZER, MARCO RONCADELLI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1331

Dark Matter in NGC 4472 FRANCESCO DE PAOLIS, GABRIELLE INGROSSO

. . . . . . . 1333

Gravitational Microlensing and the Mass of the Dark Compact Halo Objects PHILIPPE JETZER.

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1336

Ixvi Results from EROS (Experience de Recherche d'Objets Sombres) MARC MONIEZ . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . .

1338

Gravitational Wave Interferometers Chairperson: N. Kawashima Functions of Proto-Type Laser Interferometers when KM Class Antenna Construction Started . ....

1345

AKITO ARAYA, K. TSUBONO, N. MIO, S. TELADA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1347

NOBUKI KAWASHIMA

. . . . . . . . . . . . . . . . . . . . . .

Frequency Stabilization of a Nd:YAG Laser Using an Independently Suspended Fabry-Perot Cavity

Sensitivity of the LIGO Interferometer to Mirror Misalignment and Method for Automatic Alignment . ....

1349

JIM HOUGH, G.P. NEWTON, N.A. ROBERTSON, H. WARD, A.M. CAMPBELL, J.E. LOGAN, D.1. ROBERTSON, K.A. STRAIN, K. DANZMANN, H. LUCK, A. RUDIGER, R. SCHILLING, M. SCHREMPEL, W. WINKLER, J.R.J. BENNETT, V. KOSE, M. KUHNE, BERNARD F. SCHUTZ, D. NICHOLSON, J. SHUTTLEWORTH, H. WELLING, P. AUFMUTH, R. RINKLEFF, A. TUNNERMANN, B. WILLKE . . . . . . .

1352

YARON HEFETZ, NERGIS MAVALVALA . . . . . . . . . . . . . . .

GEO 600: A 600 m Laser Interferometric Gravitational Wave Antenna

On a Possibility of Detecting Gravitational Waves Using Berry's Phase for Photons NIKOLAI V. MITSKIEVICH, ALEXANDER I. NESTEROV . . . . . . . . . . . . . . . . 1364 Development of 100m DL Type Laser Interferometer Gravitational Wave Antenna (TENKO-lOO) EI-ICHI MIZUNO, N. KAWASHIMA, R. TAKAHASHI, S. MIYOKI, K. SUEHIRO, S. KAWAMURA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1366

On Topology Change in (2+1)-Dimensional Gravity ALEXANDER I. NESTEROV, VLADIMIR N. EFREMOV

. . . . . . . . . . . . . . . . .

1368

M. OHASHI, M.-K. FUJIMOTO, T. YAMAZAKI, M. FUKUSHIMA, A. ARAYA, S. TELADA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1370

A Recombined Fabry-Perot Prototype

Dilaton Black Holes with Electric Charge MALIK RAKHMANOV

.......

. .

. . . . . . 1372

Frequency Doubling of Nd:YAG Laser Light for Use in Gravitational Wave Detectors .....

1375

GIUSEPPE Ruoso, GUIDO ZAVATTINI . . . . . . . . . . . . . . . . . . . . . . . .

1379

S. ROWAN, A.M. CAMPBELL, J. HOUGH, G.P. NEWTON, M. GRAY, J. HONG

Frequency Locking of a ND:YAG Laser Using the Laser Itself as Optical Phase Modulator

Ixvii Status of the LIGO 40-Meter Interferometer R.L. SAVAGE, JR. . . . . . . . . . . .

. . . . . . . . . . . 1381

Scattering of Optical Components for Gravitational Wave Detectors ROLAND SCHILLING . . . . . . . . . . . . . . . . . . . . .

. . . . 1383

Laser Beam Geometry Fluctuations and Techniques for Their Suppression in Laser Interferometric Gravitational Wave Detectors KENNETH SKELDON, K.A. STRAIN, H. WARD, J. HOUGH . . . . . . . . . . . . . . . 1385 Shot Noise Limited Sensitivity of the LIGO 40 Meter Interferometer ROBERT E. SPERO . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . 1388

Imperfect Spherical Mirrors in the 100 m Delay-Line Interferometer R. TAKAHASHI, S. MIYOKI, E. MIZUNO, E. SUEHIRO, S. KAWAMURA, N. KAWASHIMA. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 1390

The Glasgow Prototype Laser Interferometric Gravitational Wave Detector - A Progress Report H. WARD. E MORRISON, D.1. ROBERTSON, J. HOUGH, S. KILLBOURN, G.P. NEWTON, N.A. ROBERTSON, K.A. STRAIN . . . . . . . . . . . . . . . . . . 1392 Gravitational Wave Interferometers Chairperson: A. Giazotto A New Technique for .Low-Frequency Horizontal Vibration Isolation in the Suspension of a Laser-Interferometric Gravity Wave Detector MARK A. BARTON, KAZUAKI KURODA . . . . . . . . . . . . . . .

. . . . . 1399

Concepts for Extending the Ultimate Sensitivity of Interferometric Gravitational Wave Detectors Using Non-transmissive Optics with Diffractive or Holographic Coupling R.W.P. DREVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 Brownian Motion of a Torsion Pendulum GABRIELA I. GONZALES, PETER R. SAULSON

. . . . . . 1407

Towards the Development of Monolithic Fused Silica Suspensions for Laser Interferometric Gravitational Wave Detectors J.E. LOGAN, J. HOUGH, N.A. ROBERTSON, K. DANZMANN, R. HUTCHINS. Reduction of Outgassing Rate of Vacuum Ducts for Large-Scale Interferometric Gravitational-Wave Antennas: Application of Pre-Baking and TiN Coating N. MATUDA, Y. SAITO, Y. OGAWA, K. AKAISHI, G. HORIKOSHI . . . . . .

. . . . . 1410

. . . . 1413

AIGO: A Southern Hemisphere Second Generation Gravitational Wave Observatory D.E. MCCLELLAND, D.G. BLAIR, R.J. SANDEMAN . . . . . . . . . . . . . . . . . 1415 Toward the Development of a Second Generation Long Baseline Laser Interferometer for Gravitational Wave Detection D.E. MCCLELLAND, H.-A. BACHOR, M.B. GRAY, C.C. HARB, A.J. STEVENSON . . . . 1418

Ixviii Active/Passive Vibration Isolation at 1Hz and Above Part I: Single Stage, Six Degrees of Freedom D.B. NEWELL, S.J. RICHMAN, R.T. STEBBINS, P.G. NELSON, J.E. MASON, P.L. BENDER, J.E. FALLER . . . . . . . . . Operation of the Explorer Detector of the Rome Group GIOVANNI VITTORIO PALLOTTINO . . . . . . . . .

· . . . . 1421

. . . . . . . 1424

Active/Passive Vibration Isolation at 1 Hz and Above Part II: Main Stages S.J. RIcHMAN, D.B. NEWELL, R.T. STEBBINS, P.L. BENDER, J.E. FALLER . . . . . . 1426 Seismic Isolation in the LIGO 40 Meter Interferometer ROBERT E. SPERO, L.A. SIEVERS . . . . . . . .

. . . . . . . . . . . 1429

300-m Laser Interferometer Gravitational Wave Detector in Japan KIMIO TSUBONO . . . . . . . . . . . . . . . . . . . . . .

· . . . . 1432

Measurement of Optical Path Fluctuations due to Residual Gas in the LIGO 40 Meter Interferometer M.E. ZUCKER, STANLEY E. WHITCOMB '" . . . . . . . . . . . . . . . . . . . 1434

Gravitational Wave Bar Detectors Chairperson: D. Blair Resonant Bar Gravitational Wave Detectors DAVID G. BLAIR . . . . . . . . . . . .

· . . . . 1439

A Capacitive B.A.E. Transducer Coupled to the ALTAIR Gravitational Wave Antenna: Results and Perspectives PAOLO BONIFAZI, MASSIMO VISCO . . . . . . . . . . . . . . . . . .

. . . . . 1441

Resonant Antenna for Monitoring Gravitational Wave Bursts in Our Galaxy KAZUAKI KURODA, NOBUYUKI KANDA, MARK A. BARTON, JIRO ARAFUNE, NAOTO KONDO, NORIKATSU MIO, KIMIO TSUBONO . . . . . . . . . . . .

. . . . 1444

Low Noise Temperature Operation of the Niobium Gravitational Wave Antenna at the University of Western Australia NICK P. LINTHORNE, D.G. BLAIR, 1.S. HENG, E.N. IVANOV, F. VAN KANN, M.E. TOBAR, P.J. TURNER . . . . . . . . . . . . . . . . . . . . . , . . . . . 1446 Optimal Reconstruction of the Input Signal in Resonant Gravitational Wave Detectors A. ORTOLAN, M. BIASOTTO, G. MARON, L. TAFFARELLO, G. VEDOVATO, M. CERDONIO, G.A. PRODI, S. VITALE, J.P. ZENDRI . . . . . . . . . . . . . . . 1459 An Interferometric Sapphire Transducer for Characterising Vibration Isolators H. PENG, L. Ju, D.G. BLAIR . . . . . . . . . . . . . . . . . . . . . . . . . . . 1461

Ixix Operation and Developments Concerning ALLEGRO-The Resonant Mass Antenna at Louisiana State University N. SOLOMONSON, Z.K. GENG, W.O. HAMILTON, W.W. JOHNSON, E. MAUCELI, S. MERKOWITZ, A. MORSE, B. PRICE . . . . . . . . . . . . . . . . . . . . . . . 1463 Signal-To-Noise Analysis for a Spherical Gravitational Wave Antenna Instrumented with Multiple Transducers THOMAS R. STEVENSON . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 1466

Massive and Three-Mode Niobium-Sapphire Resonant-Bar Gravitational Wave Antennas MICHAEL E. TOBAR, D.G. BLAIR . . . . . . . . . . . . . . . . . . . . . . . . . 1469 Supernova 1987A Rome Maryland Gravitational Radiation Antenna Observations J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472

Gravitational Wave Bar Detectors Chairperson: M. Cerdonio A 100 Ton lOmK Spherical Gravitational Wave Detector GIORGIO FROSSATI . . . . . . . . . . . . . . . .

. . . . . . . 1477

Solution of the Inverse Problem for a Spherical Gravitational Wave Antenna NADJA S. MAGALHAES, WARREN JOHNSON, CARLOS FRAJUCA, ODYLIO D. AGUIAR. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 1481

Wideband Resonant-Lever Transducer for Massive Spherical Gravitational Wave Detectors . . . . . 1483 Ho JUNG PAIK, GREGORY M. HARRY, THOMAS STEVENSON The Ultracryogenic Gravitational Wave Antenna Nautilus G. PIZZELLA, P. ASTONE, M. BASSAN, P. BONIFAZI, P. CARELLI, G. CASTELLANO, E. COCCIA, C. COS MELLI , V. FAFONE, S. FRASCA, A. MARINI, G. MAZZITELLI, 1. MODENA, P. MODESTINO, G.V. PALLOTTINO, P. RAPAGNANI, F. RICCI, F. RoNGA, M. VISCO, L. VOTANO . . . . . . . . . . . . . 1486 Status of the Auriga G.W. Antenna G.A. PRODI, M. CERDONIO, L. FRANCESCHINI, G. FONTANA, R. MEZZENA, S. PAOLI, S. VITALE, J.P. ZENDRI, M. BIASOTTO, M. LOLLO, R. MACCHIETTO, G. MARON, A. ORTOLAN, M. STROLLO, G. VEDOVATO, M. BONALDI, P. FALFERI, E. CAVALLINI, P.L. FORTINI, E. MONTANARI, L. TAFFARELLO, A. COLOMBO, D. PASCOLI, B. TIVERON . . . . . . . . . . . . . .'. . . . . . . . . . . . . . . 1488 Multimode Optical Transducer for Massive Resonant Gravitational Wave Detectors JEAN-PAUL RICHARD . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 1490

Progress Towards a Two Mode Superconducting Transducer for Resonant Mass Gravitational Wave Detectors N. SOLOMONSON, Z.K. GENG, W.O. HAMILTON, W.W. JOHNSON . . . . . . . . . . . 1492

Ixx

Behaviour of a DC Squid Tightly Coupled to a High-Q Resonant Transducer . . . . . . . . . . . . . . . . 1495

THOMAS R. STEVENSON, HANS J. HAUCKE .

Lunar Laser Ranging and Space Geodesy Chairperson: R. Rheinhard / P.J. She/us

Testing General Relativity with Lunar Laser Ranging JOHN F. CHANDLER, ROBERT D. REASENBERG, IRWIN

I. SHAPIRO

. . . . . . 1501

The Modified Hughes-Drever Experiment and the Strong Equivalence Principle BIPING GONG

. . . . . . . . . .

. . . . . . . . . . . .

. . . ..

. . . . . . 1505

A VLBI Measurement of the Solar Gravitational Deflection of Radio Waves D.E. LEBACH, 1.1. SHAPIRO, M.1. RATNER, J.1. DAVIS, B.E. COREY, T.A. HERRING. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 1515

Determination of Relativistic Quantities by Analyzing Lunar Laser Ranging Data J. MULLER, M. SCHNEIDER, M. SOFFEL, H. RUDER . . . . . . . . . . . . . . . . . 1517

A Mission Concept to Measure Second-Order Relativistic ,Effects, Solar Angular Momentum and Low-Frequency Gravitational Waves WEI-Tou NI, AN-MING Wu, JOW-TSONG SHY

. . . . . . . . . . . . . . . . . . . 1519

Laser Astrodynamics and Geodesy WEI-Tou NI, JOW-TSONG SHY, SU-AN Wu, Fu-JENG LIN, SHIE-EI WANG, SUN-KuN KING, AN-MING Wu, SHIAO-MIN TSENG, CHING-YUN REN, CHANG-HUEI Wu, CHWEN-SHELL Ho, CHING-TING LEE, TZE-JANG CHEN

. . . . . 1522

General Relativity Parameters: The Relevance of Lunar Laser Data From the McDonald Laser Ranging Station PETER J. SHELUS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525

Frames of Reference in Relativistic Celestial Mechanics SLAVA G. TURYSHEV

. . . . .

. .

. . . . . . . .

. . . . . . . . . . . . . 1527

Relativity Parameters Determined from Lunar Laser Ranging JAMES G. WILLIAMS, X.X. NEWHALL, JEAN O. DICKEY

. . . . . . . . . . . . . 1529

Equivalence Principle and Inertial Frames Chairperson: J. Turneaure

The Relativity Mission Gyroscopes SAPS BUCHMAN, C.W. FRANCIS EVERITT, BRADFORD PARKINSON, J.P. TURNEAURE, G.M. KEISER, Y. XIAO, D. BARDAS, B. MUHLFELDER, DALE GILL, PING ZHOU, P. BAYER, C. GRAY, R. BRUMLEY . . . . . . . . . . . . . 1533

Induced Composition-Dependence in BD Theory YASUNORI FUJII

. . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . 1536

Ixxi

The Scientific Significance of Long-Lifetime High-Accuracy Version of the Relativity-Gyro Experiment with Altitude-Chance Capability BENJAMIN LANGE

. . . . 1538

......................

Status of the Cryogenic Inertial Reference System for the Gravity Probe B Mission . . . . . . . . . . . . . . . . 1542

J.A. LIPA, D.-H. Gwo, S. WANG, R.K. KIRSCHMAN

A Status Report on the Relativity Mission B. MUHLFELDER, C.W.F. EVERITT, B.W. PARKINSON, J.P. TURNEAURE, D. BARDAS, S. BUCHMAN, D.B. DEBRA, H. DOUGHERTY, D. GILL, G.B. GREEN, G.M. GUTT, J. Gwo, N.J. KASDIN, G.M. KEISER, T. LANGENSTEIN, J.A. LIPA, J.M. LOCKHART, R.T. PARMELY, M.A. TABER, R.A. VAN PATTEN, Y.M. XIAO, S. WANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545

Geodesy with Step Comparison of Various Mission Options J. MULLER, H. FROHLICH, R. RUMMEL, N. SNEEUW .

. . . . . . . . . . . . . . 1551

The Guide Star for the Gyroscope Relativity Mission MICHAEL 1. RATNER, D.E. LEBACH, 1.1. SHAPIRO, N. BARTEL

. . . . . . . 1553

Gravitomagnetic Curvature Effects and the Motion of Local Inertial Frames DIETMAR S. THEISS

........................

-......

1555

Measurement of the Earth's Gravitomagnetic Field Using Space Gradiometry DIETMAR S. THEISS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558

Scaling Laws for Rotating Fluid Shapes RoDNEY H. TORll, J.E. MCCuAN, P.J. NESCHLEBA, P.1. LIN, R. FINN . . . . . . . . 1561 Accelerometric Reference Sensor for Interferometer Space Antenna P. TOUBOUL, A. RUDIGER

.. . . . . . . . . . . . . . ..

. . . . . . . . . 1563

Servo-Controlled Accelerometers for Gravitational Experiments in Space . . . . . . . . . . . . . . . . . . . . 1565

P. TOUBOUL, B. FOULON, A. BERNARD

Error Analysis for the QuickSTEP Experiment PAUL W. WORDEN, JR., MICHAEL A. SWARTWOUT

. . . . . . . . . . . . . . . . . 1567

The QuickSTEP Instrument and Science PAUL W. WORDEN, JR. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . 1569

Readout Schemes for the GP-B Experiment Y.M. XIAO, G.M. KEISER

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574

Precision Gravitational Measurements Chairperson: P. Boynton Progress Report on a Rotatable Torsion Balance for a New Test of the Equivalence Principle M.A. BEILBY, N. KRISHNAN, R.D. NEWMAN

. . . . . . . . . . . . . . . . . . . . 1579

Ixxii Equivalence Principle and Feeble Yukawa-like Forces RAMANATH COWSIK, N. KRISHNAN, S.N. TANDON, C.S. UNNIKRISHNAN . . . . . . . . 1583 The Velocity of Particles of Galactic Dark Matter RAMANATH COWSIK, CHARU RATNAM, P. BHATTACHARJEE

. . . . . . . . 1597

Precise Experimental Free Fall Tests of the Weak Principle of Equivalence H. DITTUS, R. GREGER, S.T. LOCHMANN, P. MAZILU . . . . . . . . . . . . 1599 Search for Fundamentally New Interactions with Ranges> lcm J.H. GUNDLACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601 Cryogenic Torsion Balances - A Pilot Study N. KRISHNAN, M.K. BANTEL, M.A. BEILBY, A. HUNT, R.D. NEWMAN . . . . . . . . 1604 Testing Gravity with Atom Beam Interferometry CLAUS LAMMERZAHL . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . 1607

Solar-Neutrino Experiments with Null Results J. Luo, X. CHEN, J.G. LI, S.H. FAN, F.Y. LI

. . . . . . . . . . . . . . . . . . . 1610

Detection of Non-Newtonian Forces M.W. MOORE, P.E. BOYNTON

. . . . . . 1612

Future Torsion Balances for Gravitation Experiments: What Are Their Limits? RILEY D. NEWMAN, M. BANTEL, M. BEILBY, N. KRISHNAN, E. SIRAGUSA, P.E. BOYNTON, A. GOODSON . . . . . . . . . . . . . . . . . . . . . . . . 1619 Search For Anomalous Spin-Dependent Forces Using a dc Squid With a Paramagnetic Salt WEI-Tou NI, SHEAU-SHI PAN, HSIEN-CHI YEH, LI-SHING Hou, JULING WAN

. . . . . 1625

Test of Quantum Electrodynamics and Search For Light Scalar/Pseudoscalar Particles Using Ultra-High Sensitive Interferometers WEI-Tou NI, SUN-KuN KING, HUNG-WEN CHEN, JOW-TSONG SHY, NORIKATSU MIO, KIMIO TSUBONO, T.C.P. CHUI . . . . . . . . . . . . . . . . . . 1628 Photon Fall in a Gravitational Field at LIGO JAMES T. WHEELER . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 1631

Space Gravitational Wave ExperiInents Chairperson: R. H ellings Pioneer 10 1993 Gravitational Wave Experiment J.D. ANDERSON, E.L. LAU, J.W. ARMSTRONG

. . . . . 1637

Search for Gravitational Waves By Doppler Tracking of Interplanetary Spacecraft BRUNO BERTOTTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638 LISA: Laser Interferometer Space Antenna for Gravitational Wave Measurements KARSTEN DANZMANN . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . 1641

Ixxiii Thermal Stability Analysis for a Heliocentric Gravitational Radiation Detection Mission W. M. FOLKNER, P. McELROY, R. MIYAKE, P.L. BENDER, R.T. STEBBINS, W. SUPPER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655 Cassini Relativity Experiments GIACOMO GIAMPIERI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658 Optics for Laser Transponders D.l. ROBERTSON, J. HOUGH, A. McLAREN, H. WARD, K. DANZMANN Algorithms for Unequal-Arm Michelson Interferometers MASSIMO TINTO, GIACOMO GIAMPIERI, RONALD W. HELLINGS, PETER BENDER, JAMES E. FALLER . . . . . . . . . One-Arm Interferometric Detectors of Gravitational Waves MASSIMO TINTO, FRANK B. ESTABROOK . . . . . . .

. . . . . . . . 1665

. . . . . . . . . . . 1668

. . . . . . . . . . . . . . 1671

PUBLIC TALKS Copernican Humility, Change and the Creation of Purpose YEVAL NE'EMAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677 The Physical Heritage of the Cold War WOLFGANG K.H. PANOFSKY

. . . . . . . . . . . . . . . . . . . . . . 1717

TEST: Traction of Events in Space-Time ANNA IMPONENTE . . . . .

1738

List of Participants

1744

Author Index. . .

1760

PLENARY SESSIONS

Chairperson: Humitaka Sato

3

WAS EINSTEIN 100% RIGHT?

Thibault DAMOUR Institut des Hautes Etudes Scientifiques 91440 Bures-sur- Yvette, France and DARC, CNRS - Observatoire de Paris 92195 Meudon, France

ABSTRACT The confrontation between General Relativity and experimental results, notably binary pulsar data, is summarized and its significance discussed. The agreement between experiment and theory is numerically very impressive. However, some recent theoretical

findings (existence of non-perturbative strong-field effects, natural cosmological attraction toward zero scalar couplings) suggest that the present agreement between Einstein's theory and experiment might be a red herring and provide new motivations for improving the experimental tests of gravity.

1. Introduction

General Relativity can be thought of as defined by two postulates. One postulate states that the action functional describing the propagation and self-interaction of the gravitational field is 4

Sgravitation

[gl'lI]

=

c G 167r

J-c-..;g 4

d x

R(g)_

(1)

A second postulate states that the action functional describing the coupling of all the (fermionic and bosonic) fields describing matter and its electro-weak and strong interactions is a (minimal) deformation of the special relativistic action functional used by particle physicists (the so called "Standard Model"), obtained by replacing

4

everywhere the flat Minkowski metric fl''' = diag( -1, +1, +1, +1) by gl',,(x A ) and the partial derivatives al' == ajaxl' by g-covariant derivatives 'V I'" [With the usual subtlety that one must also introduce a field oforthonormal frames, a "vierbein", for writing down the fermionic terms]. Schematically, one has Smatter

[Tj!, A, H, g]

-21

=

d4x -c- Jg

J

C matte.,

" -Tj! H Tj!, IDI' HI 2 - V(H) - 'L..Y

(2a)

(2b)

where FI''' denotes the curvature of a U(l), SU(2) or SU(3) Yang-Mills connection AI" FI''' = gl'Ot g"P Fop, g* being a (bare) gauge coupling constant; DI' == 'V I' + AI'; Tj! denotes a fermion field (lepton or quark, coming in various flavours and three generations); 71' denotes four Dirac matrices such that 71' 7" + 7" 71' = 2g l''' 114 , and H denotes the Higgs doublet of scalar fields, with y some (bare Yukawa) coupling constants. Einstein's theory of gravitation is then defined by extremizing the total action functional, Stat

[g, Tj!, A, H] =

Sgravitation

[g]

+ Smatter

[Tj!, A, H, g].

(3)

Although, seen from a wider perspective, the two postulates (1) and (2) follow from the unique requirement that the gravitational interaction be mediated only by massless spin-2 excitations [1], the decomposition in two postulates is convenient for discussing the theoretical significance of various tests of General Relativity. Let us discuss in turn the experimental tests of the coupling of matter to gravity (postulate (2)), and the experimental tests of the dynamics of the gravitational field (postulate (1)). For more details and references we refer the reader to [2] or [3]. 2. Experimental tests of the coupling between matter and gravity The fact that the matter Lagrangian (2b) depends only on a symmetric tensor 9 1',,( x) and its first derivatives (i.e. the postulate of a "metric coupling" between matter and gravity) is a strong assumption (often referred to as the "equivalence principle") which has many observable consequences for the behaviour of localized test systems embedded in given, external gravitational fields. Indeed, using a theorem of

5 Fermi and Cartan [4] (stating the existence of coordinate systems such that, along any given time-like curve, the metric components can be set to their Minkowski values, and their first derivatives made to vanish), one· derives from the postulate (2) the following observable consequences: C 1 : Constancy of the "constants" : the outcome of local non-gravitational experiments, referred to local standards, depends only on the values of the coupling constants and mass scales entering the Standard Model. [In particular, the cosmological evolution of the universe at large has no influence on local experiments]. C 2 : Local Lorentz invariance : local non-gravitational experiments exhibit no preferred directions in spacetime [i.e. neither spacelike ones (isotropy), nor timelike ones (boost invariance)]. C 3 : "Principle of geodesics" and universality of free fall: small, electrically neutral, non self-gravitating bodies follow geodesics of the external spacetime (V, g). In particular, two test bodies dropped at the same location and with the same velocity in an external gravitational field fall in the same way, independently of their masses and compositions. C 4 : Universality of gravitational redshift : when intercompared by means of electromagnetic signals, two identically constructed clocks located at two different positions in a static external Newtonian potential U(x) exhibit, independently of their nature and constitution, the difference in clock rate:

(4) Many experiments or observations have tested the observable consequences C1 C4 and found them to hold within the experimental errors. Many sorts of data (from spectral lines in distant galaxies to a natural fission reactor phenomenon which took place in Gabon two billion years ago) have been used to set limits on a possible time variation of the basic coupling constants of the Standard Model. The best results concern the fine-structure constant Q for the variation of which a conservative upper bound is [5]

I~I < 10- 15 yr- 1 ,

(5)

which is much smaller than the cosmological time scale'" 10- 10 yr- 1 • Any "isotropy of space" having a direct effect on the energy levels of atomic nuclei has been constrained to the impressive 10- 27 level [6). The universality of free fall has been verified at the 3 x 10- 12 level for laboratory bodies [7) and at the 10- 12

6

level for the gravitational accelerations of the Moon and the Earth toward the Sun [8]. The "gravitational redshift" of clock rates given by eq. (4) has been verified at the 10-4 level by comparing a hydrogen-maser clocldlying on a rocket up to an altitude '" 10 000 km to a similar clock on the ground. In conclusion, the main observable consequences of the Einsteinian postulate (2) concerning the coupling between matter and gravity ("equivalence principle") have been verified with high precision by all experiments to date. The traditional view (first put forward by Fierz [10]) is that the extremely high precision of free fall experiments (10- 12 level) strongly suggests that the coupling between matter and gravity is exactly of the "metric" form (2), but leaves open possibilities more general than eq. (1) for the spin-content and dynamics of the fields mediating the gravitational interaction. We shall provisionally adopt this conclusion to discuss the tests of the other Einsteinian postulate, eq. (1). However, we shall emphasize at the end that recent theoretical findings suggest a rather different view. 3. Tests of the dynamics of the gravitational field in the weak field regime Let us now consider the experimental tests of the dynamics of the gravitational field, defined in General Relativity by the action functional (1). Following first the traditional view, it is convenient to enlarge our framework by embedding General Relativity within the class of the most natural relativistic theories of gravitation which satisfy exactly the matter-coupling tests discussed above while differing in the description of the degrees of freedom of the gravitational field. This class of theories are the metrically-coupled tensor-scalar theories, first introduced by Fierz [10] in a work where he noticed that the class of non-metrically-coupled tensor-scalar theories previously introduced by Jordan [11] would generically entail unacceptably large violations of the consequence C 1 . [The fact that it would, by the same token, entail even larger violations of the consequence C 3 was, probably, first noticed by Dicke in subsequent work]. The metrically-coupled (or equivalence-principle respecting) tensor-scalar theories are defined by keeping the postulate (2), but replacing the postulate (1) by demanding that the "physical" metric gp." be a composite object of the form

(6)

g;"

where the dynamics of the "Einstein" metric is defined by the action functional (1) (written with the replacement gp" -+ g;,,) and where cp is a massless scalar field. [More

7

generally, one can consider several massless scalar fields, with an action functional of the form of a general nonlinear (j model [12]]. In other words, the action functional describing the dynamics of the spin 2 and spinO degrees of freedom contained in this generalized theory of gravitation reads

(7) Here, G .. denotes some bare gravitational coupling constant. This class of theories contains an arbitrary function, the "coupling function" A( cp). When A( cp) = const., the scalar field is not coupled to matter and one falls back (with suitable boundary conditions) on Einstein's theory. The simple, one-parameter subclass A( cp) = exp( 00 cp) with 00 E IR is the Jordan-Fierz-Brans-Dicke theory. In the general case, one can define the (field-dependent) coupling strength of cp to matter by o ( cp )

_ 0 In A( cp )

=

ocp

.

(8)

It is possible to work out in detail the observable consequences of tensor-scalar theories and to contrast them with the general relativistic case (see ref. [12] fora recent treatment ). Let us now consider the experimental tests of the dynamics of the gravitational field that can be performed in the solar system. Because the planets move with slow velocities (vjc '" 10-4 ) in a very weak gravitational potential (Ujc 2 '" (vjC)2 '" 10-8 ), solar system tests allow us only to probe the quasi-static, weak-field regime of relativistic gravity (technically called the "post-Newtonian" limit). In this limit all solar-system gravitational experiments, interpreted within tensor-scalar theories, differ from Einstein's predictions only through the appearance of two "post-Einstein" parameters "f and 73 (related to the usually considered post-Newtonian parameters through"f == ,-I, 73 == 13 - 1). The parameters "f and 73 vanish in General Relativity, and are given in tensor-scalar theories by

,= -

2

02 0

1 + o~'

(9a) (9b)

where 00 == o(CPo), 130 == oO(CPo)jocpo; CPo denoting the cosmologically-determined value of the scalar field far away from the solar system. Essentially, the parameter "f

8

depends only on the linearized structure of the gravitational theory (and is a direct measure of its field content, i.e. whether it is pure spin 2 or contains an admixture of spin 0), while the parameter 73 parametrizes some of the quadratic nonlinearities in the field equations (cubic vertex of the gravitational field). All currently perfonned gravitational experiments in the solar system, including perihelion advances of planetary orbits, the bending and delay of electromagnetic signals passing near the Sun, and very accurate range data to the Moon obtained by laser echoes, are compatible with the general relativistic predictions '1 = 0 = 73 and give upper bounds on both I'll and (i.e. on possible fractional deviations from General Relativity) of order 10-3 [8], [13]. Recently, the parametrization of the weak-field deviations between generic tensor-multi-scalar theories and Einstein's theory has been extended to the postpost-Newtonian order [14]. Only two post-post-Einstein parameters, representing a deeper layer of structure of the gravitational interaction, show up. See [14] for a detailed discussion, including the consequences for the interpretation of future, higher-precision solar-system tests.

Ipl

4. Tests of the dynamics of the gravitational field in the strong field regime In spite of the diversity, number and often high precision of solar system tests, they have an important qualitative weakness: they probe neither the radiation properties nor the strong-field aspects of relativistic gravity. Fortunately, the discovery [15] and continuous observational study of pulsars in gravitationally bound binary orbits has opened up an entirely new testing ground for relativistic gravity, giving us an experimental handle on the regime of strong and/or radiative gravitational fields. The fact that binary pulsar data allow one to probe the propagation properties of the gravitational field is well known. This comes directly from the fact that the finite velocity of propagation of the gravitational interaction between the pulsar and its companion generates damping-like terms in the equations of motion, i.e. tenns which are directed against the velocities. [This can be understood heuristically by considering that the finite velocity of propagation must cause the gravitational force on the pulsar to make an angle with the instantaneous position of the companion [16], and was verified by a careful derivation of the general relativistic equations of motion of binary systems of compact objects [17]]. These damping forces cause the binary orbit to shrink and its orbital period Pb to decrease. The remarkable stability of the pulsar clock, together with the cleanliness of the binary pulsar system,

9

has allowed Taylor and collaborators to measure the secular orbital period decay Fb == dPb/ dt [IS], thereby giving us a direct experimental probe of the damping terms present in the equations of motion. Note that, contrary to what is commonly stated, the link between the observed quantity Fb and the propagation properties of the gravitational interaction is quite direct. [It appears indirect only when one goes through the common but unnecessary detour of a heuristic reasoning based on the consideration of the energy lost in the gravitational waves emitted at infinity). The fact that binary pulsar data allow one to probe strong-field aspects of relativistic gravity is less well known. The a priori reason for saying that they should is that the surface gravitational potential of a neutron star Gm/c 2 R ~ 0.2 is a mere factor 2.5 below the black hole limit (and a factor'" 108 above the surface potential of the Earth). It has been recently shown [19] that a self-gravity as strong as that of a neutron star can naturally (i.e. without fine tuning of parameters) induce orderunity deviations from general relativistic predictions in the orbital dynamics of a binary pulsar thanks to the existence of nonperturbative strong-field effects in tensorscalar theories. [The adjective "nonperturbative" refers here to the fact that this phenomenon is nonanalytic in the coupling strength of scalar field, eq. (S), which can be as small as wished in the weak-field limit). As far as we know, this is the first example where large deviations from General Relativity, induced by strong selfgravity effects, occur in a theory which contains only positive energy excitations and whose post-Newtonian limit can be arbitrarily close to that of General Relativity. A comprehensive account of the use of binary pulsars as laboratories for testing strong-field gravity has been recently given [20). Two complementary approaches can be pursued: a phenomenological one ("Parametrized Post-Keplerian" formalism), or a theory-dependent one [12], [20). The phenomenological analysis of binary pulsar timing data consists in fitting the observed sequence of pulse arrival times to the generic 00 timing formula [21) whose functional form has been shown to be common to the whole class of tensor-multi-scalar theories. The least-squares fit between the timing data and the parameter-dependent DD timing formula allows one to measure, besides some "Keplerian" parameters ("orbital period" Pb , "eccentricity" e, ... ), a maximum of eight "post-Keplerian" parameters; k, -y, Fb , r, s, he, e and x. Here, k == wPb/27r is the fractional periastron advance per orbit, -y a time dilation parameter (not to be confused with its postNewtonian namesake), Fb the orbital period derivative mentioned above, and r and s the "range" and "shape" parameters of the gravitational time delay caused by the

10

companion. The important point is that the post-Keplerian parameters can be measured without assuming any specific theory of gravity. Now, each specific relativistic theory of gravity predicts that, for instance, k,:Y, Pb, r and s (to quote parameters that have been successfully measured from some binary pulsar data) are some theorydependent functions of the (unknown) masses ml, m2 of the pulsar and its companion. Therefore, in our example, the five simultaneous phenomenological measurements of k, /, Pb, rand s determine, for each given theory, five corresponding theory-dependent curves in the ml - m2 plane (through the 5 equations kmeasured = ktheorY(mt,m2), etc ... ). This yields three (3 = 5 - 2) tests of the specified theory, according to whether the five curves meet at one point in the mass plane, as they should. In the most general (and optimistic) case, discussed in [20], one can phenomenologically analyze both timing data and pulse-structure data (pulse shape and polarization) to extract up to nineteen post-Keplerian parameters. Simultaneous measurement of these 19 parameters in one binary pulsar system would yield 15 tests of relativistic gravity (where one must subtract 4 because, besides the two unknown masses mt, m2, generic post-Keplerian parameters can depend upon the two unknown Euler angles determining the direction of the spin of the pulsar). The theoretical significance of these tests depends upon the physics lying behind the post-Keplerian parameters involved in the tests. For instance, as we said above, a test involving Pb probes the propagation (and helicity) properties of the gravitational interaction. But a test involving, say, k, I, r or s probes (as shown by combining the results of [12] and [19]) strong self-gravity effects independently of radiative effects. Besides the phenomenological analysis of binary pulsar data, one can also adopt a theory-dependent methodology [12], [20]. The idea here is to work from the start within a certain finite-dimensional "space of theories", i.e. within a specific class of gravitational theories labelled by some theory parameters. Then by fitting the raw pulsar data to the predictions of the considered class of theories, one can determine which regions of theory-space are compatible (at say the 90% confidence level) with the available experimental data. This method can be viewed as a strong-field generalization of the parametrized post-Newtonian formalism [2] used to analyze solarsystem experiments. In fact, under the assumption that strong-gravity effects in neutron stars can be expanded in powers of the "compactness" C A == -2 In m A/ In G G mA/c2 R A , Ref. [12] has shown that the observable predictions of generic tensormulti-scalar theories could be parametrized by a sequence of "theory parameters",

a

7,

7J , /32 , /3' , /3/1 , /33 , (/3/3') ...

a

(10)

11

representing deeper and deeper layers of structure of the relativistic gravitational interaction beyond the first-order post-Newtonian level parametrized by '7 and 71 (the second layer /32,/3' parametrizing the second-order post-Newtonian level [14], etc... ). A specific two-parameter subclass of tensor-hi-scalar theories T(/3', /3") has been given special consideration [12], [20]. After having reviewed the theory of pulsar tests, let us briefly summarize the current experimental situation. Concerning the first discovered binary pulsar PSR1913+ 16 [15], it has been possible to measure with accuracy the three post-Keplerian parameters k" and Pb. From what was said above, these three simultaneous measurements yield one test of gravitation theories. After subtracting a small ('V 10- 14 level in Pb !), but significant, perturbing effect caused by the Galaxy [22], one finds that General Relativity passes this (k - , - Pbh913+16 test with complete success at the 3.5 x 10- 3 level [23], [18). This beautiful confirmation of General Relativity is an embarrassment of riches in that it probes, at the same time, the propagation and strong-field properties of relativistic gravity! If the timing accuracy of PSR1913 + 16 could improve by a significant factor two more post-Keplerian parameters (r and s) would become measurable and would allow one to probe separately the propagation and strong-field aspects [23). Fortunately, the recent discovery of the binary pulsar PSR1534 + 12 [24] (which is significantly stronger than PSR1913 + 16 and has a more favourably oriented orbit) has opened a new testing ground, in which it has been possible, already after one year of data taking, to probe strong-field gravity independently of radiative effects. A phenomenological analysis of the timing data of PSR1534 + 12 has allowed one to measure the four post-Keplerian parameters k", r and s [23]. From what was said above, these four simultaneous measurements yield two tests of strong-field gravity, without mixing of radiative effects. General Relativity is found to pass these tests with complete success within the measurement accuracy [23], [18]. More recently, it has been possible to extract also the "radiative" parameter Pb from the timing data of PSR1534 + 12. Again, General Relativity is found to be fully consistent (at the current 'V 20% level) with the additional test provided by the Pb measurement [25]. Note that this gives our second direct experimental confirmation that the gravitational interaction propagates as predicted by Einstein's theory. Moreover, an analysis of the pulse shape of PSR1534+ 12 has shown that the misalignment between the spin vector of the pulsar and the orbital angular momentum was greater than 8° [20]. This opens the possibility that this system will soon allow one to test the spin precession induced by gravitational spin-orbit coupling.

12

To end this brief summary, let us mention that a comprehensive theory-dependent analysis of all available pulsar data has been performed, and has led to significant bounds on the strong-field parameters /3', /3" [23]. In spite of the impressive agreement between the predictions of General Relativity in the strong-field regime and all current binary pulsar data, the number and precision of present strong-field tests is still rather small, and it is important to continue obtaining and/or improving such tests, especially in view of the results of [19} which prove that such tests are logically independent from solar-system tests. For a general review of the use of pulsars as physics laboratories the reader can consult Ref. [26].

5. Was Einstein 100% right ? Summarizing the experimental evidence discussed above, we can say that Einstein's postulate of a pure metric coupling between matter and gravity ("equivalence principle") appears to be, at least, 99.9999999999% right (because of universality-offree-fall experiments), while Einstein's postulate (1) for the field content and dynamics of the gravitational field appears to be, at least, 99.9% correct both in the quasi-staticweak-field limit appropriate to solar-system experiments, and in the radiative-strongfield regime explored by binary pulsar experiments. Should one apply Ockham's razor and decide that Einstein must have been 100% right, and then stop testing General Relativity? My answer is definitely, no ! First, one should continue testing a basic physical theory such as General Relativity to the utmost precision available simply because it is one of the essential pillars of the framework of physics. Second, some very crucial qualitative features of General Relativity have not yet been verified : in particular the existence of black holes, and the direct detection on Earth of gravitational waves. [Hopefully, the LIGO/VIRGO network of interferometric detectors will observe gravitational waves early in the next century]. Last, some recent theoretical findings suggest that the current level of precision of the experimental tests of gravity might be naturally (i.e. without fine tuning of parameters) compatible with Einstein being actually only 50%, or even 33% right! By this we mean that the correct theory of gravity could involve, on the same fundamental level as the Einsteinian tensor field g;,,, a massless scalar field [5up -

Eup(local)ITol - [5down - Edown(1ocal)ITol

(17)

where To is TBH blueshifted to the local observer's frame: TolTBH = E(local)1 E( 00). Frolov and Page regard the "up" and "down" systems as strictly equivalent by time reversal invariance. The "up" states coming out of the past horizon are supposed to be in equilibrium at global temperature TBH and thus at To in the local observer's frame. The "down" modes form the same system, but in some other state. Frolov and Page recall that when 5 and E are properties of a thermodynamic system in any state, and To is some fixed temperature, 5 - E ITo attains its maximum when the system is in equilibrium at temperature To. Thus, conclude Frolov and Page, the r.h.s. of inequality (17) must be positive, and the GSL (2) follows. How general is the Frolov-Page proof of the GSL ? It is, of course, limited by its reliance on the semiclassical approximation (classical geometry driven by averages of quantum stress tensor). This weakness is remediable. Fiola, Preskill, Strominger and Trivedi54 have recently devised a proof of the GSL in 1 +1 dimension dilaton gravity which goes beyond semiclassical considerations. However, that proof is restricted to very special situations, and works only if a new type of entropy is ascribed to coherent radiation states. Frolov and Page's proof certainly has a wider applicability. But it does have a loophole: the assumed equivalence of "up" and "down" systems by time reversal invariance. The eternal black hole (Kruskal spacetime) is time-reversal invariant as assumed; the realistic radiating black hole is not (a time reverted black hole is not a black hole). Can one project this equivalence of systems from the former to the later? What is involved in the statement that 5 - E ITo is maximum at equilibrium at temperature To ? The state of the matter and radiation is encoded in some a density operator {J. In terms of the hamiltonian iI, E = Tr(piI) while 5 = -Tr(pIn p). Thus the variation 8(5 - ToE) under a small variation 8p is

(18) so that 5 - ToE has an extremum under variations that preserve Tr p = 1 where p satisfies if + To In p+ To - ). = 0 with), a Lagrange multiplier. Obviously there is a unique solution p ex: exp( - iI ITo), i. e., there is one extremum of 5 - To E, a

53

thermal (equilibrium) state with temperature To. This extremum is a maximum since for fixed E, S attains a maximum in equilibrium. Thus the r.h.s. of Eq. (17) is indeed nonnegative provided the "up" and "down" states are described by the selfsame hamiltonian. For the eternal black hole time reversal invariance does indeed guarantee equivalence of "up" and "down" hamiltonians. Compare now an evaporating black hole made by collapse with an eternal black hole of like parameters. Assuming a complete set of states, each of the relevant hamiltonians can be expanded in the usual form iI = L: Ij)UIEj" IT the time variation of the evaporating black hole's parameters may be ignored, the "down" states and eigenenergies for the two black holes are in detailed correspondance, so that the "down" hamiltonians are equivalent. Thus the "down" hamiltonian for the evaporating black hole is equivalent to the "up" hamiltonian of the eternal black hole. But the equivalence cannot be carried one step further. The Hawking "up" states from an evaporating black hole emerge through the time dependent geometry of the collapsing object. Thus they cannot be put into exact correspondance with "up" states for the eternal black hole which emerge right into the stationary geometry. This is particularly true of early emerging "up" states. Thus the exact equivalence of "up" and "down" hamiltonians for the realistic evaporating black hole is in question since the comparison must be over a complete set of states. The above mathematical nicety may well prove irrelevant for the Frolov-Page proof when it is the scattering of microscopic systems off the black hole which is under consideration. However, for events involving an evaporating black hole and macroscopic objects, the sets of "up" and "down" modes are distinctly different. Macroscopic objects are bound states of many quanta of elementary fields. As discussed below, over the black hole evaporation lifetime such an object occurs in the Hawking radiance only with exponentially small probability. Thus even if emitted, the object is emitted by a black hole whose parameters cannot be regarded as stationary even in rough approximation. The comparison of the realistic and eternal black holes is thus murky since the former evolves drastically over the relevant time span. The equivalence of the "up" and "down" hamiltonians is thus unclear, and inequality (17) cannot be exploited. 9.9. The Universal Entropy Bound from the GSL As just mentioned, the Frolov-Page proof is unconvincing for a situation where macroscopic matter falls into a black hole. Such a situation occurs frequently, e.g., astrophysical accretion onto a black hole. The Sorkin proof does seem to apply. Thus I assume that the GSL is also valid in such a situation. There are then interesting consequences. First consider dropping a spherical macroscopic system of mass E, radius Rand entropy s into a Schwarzschild black hole of mass M ~ E from a large distance D ~ M away. The black hole gains mass E, which it then proceeds to radiate over time T. At the end of this process the black hole is back at mass M. Were

54

the emission reversible, the radiated entropy would be E /TBH • As mentioned, the emission is actually irreversible, and the entropy emitted is a factor 11 > 1 larger. Thus the overall change in generalized entropy is (19) From numerical work Page56 estimates 11 = 1.35 - 1.64 depending on the species radiated. One can certainly choose M larger than R by an order of magnitude, say, so that the system will fall into the hole without being torn up: M = 'YR with "I = a few. Thus, if the GSL is obeyed, the restriction s

< 811'Y7r RE /n

(20)

must be valid. It is clear from the argument that there is no need for 11"1 to be arbitrarily large. Thus from the GSL one infers a bound on the entropy of a rather arbitrary - but not strongly gravitating - system in terms of its total gravitating energy and size. Note that this bound is compatible with bound (13) which comes from statistical mechanics in flat spacetime. One objection that could be raised to the above line of argument is that Hawking radiation pressure might keep the system from being absorbed by the hole, thus obviating the conclusion. This is not so. Approximate the Hawking radiance as black body radiance of temperature n/(87rM) from a sphere of radius 2M, the energy flux at distance r from the hole is

n

F( r) = -61-,-44-0""-(7r-M r )"""'"2 ---C

(21)

resulting in a radiation force frad(r) = 7r R2 F(r) on the infalling sphere. Writing the Newtonian gravitational force as fgrav(r) = ME/r 2 one sees that frad(r) nR2 fgrav(r) = 61,4407r 2 M3E

(22)

The size of a macroscopic system always exceeds its Compton length. Thus for any macroscopic sphere able to fall whole into the hole n/ E < R < M. Therefore, frad(r)/grad(r) ~ 1 throughout the fall until very close to the hole where the Newtonian approximations used must fail. By then the game is up, and the system must surely be swallowed up. It is also clear that the system falls essentially geodesically (more on this below). The objection might be refurbished by relying on the radiation pressure of a large number of massless species to overpower gravity and drive the system away. So let me pretend the number of species in nature is large. However, the relevant number, n, is the number of species actually represented in the radiation flowing out during the time that the sphere is falling in. I shall take D to be such that the infall time equals the time T to radiate energy E. Then the number of radiation

55 species into which E is converted is also n. Thus from Eq. (21) one sees that the hole radiates the energy E in time 7 ~ 5x 104 EM 2h- I n- l . Since D ~ (37/.../2)2/3 M I /3, one checks that D ~ 2.2 x 103(ME/nh)2/3M. Now, the typical Hawking quantum bears an energy of order TBH , so the number of quanta radiated is ~ 87rME/h. Since a species will be effective at braking the fall only if represented by at least one quantum, one has n < 87rME/h. As a result, D ~ M as required by the discussion. Multiplying the ratio (22) by n and recalling that h/E < R < M, one sees that (23) Radiation pressure thus fails to modify appreciably the geodesic fall of the sphere, and bound (20) follows. I conclude that the GSL requires for its functioning a property of ordinary macroscopic matter encapsulated in bound (20). This is consistent with the tighter and more definite bound (13) established from statistical arguments in flat spacetime. This last granted, the GSL is seen to be safe from the invasion of a black hole's airspace by macroscopic entropy-bearing objects. It is interesting that this profoundly gravitational law "knows" about prosaic physics. This last statement has been at the heart of a protracted controversy60,61 in which Unruh and Wald have argued that the GSL can take care of itself with no help from the entropy bound by means of the buoyancy of objects in the Unruh acceleration radiation. Yet in the gedanken experiment above buoyancy is irrelevant: the sphere falls freely, radiation pressure makes a small perturbation to its unaccelerated worldline, and so there is no Unruh-Wald buoyancy. Evidently, the GSL's functioning does depend on properties of ordinary matter. (For a recent demonstration that the entropy bound (13) follows from the GSL even in circumstances where buoyancy is present see Ref. 62 and references cited therein.)

:/.4.

Do Black Holes Emit TV Sets ?

Nothing illustrated so well to my generation the force of the "no hair" principle than Wheeler's proverbial TV set falling into a black hole. 63 But if a black hole can radiate, are TVs emitted in the Hawking radiance? The thermodynamic notion that anything can be found in a thermal radiation bath would suggest the answer is yes. This principle, however, must be applied cautiously. First, a system of energy E appears spontaneously in a thermal bath only when the temperature is at least of order E. A black hole cannot be hotter than the Planck-Wheeler temperature. Thus the only TVs that could be expected to appear are those lighter than the Planck-Wheeler mass'" 10-5 gm. Further, the TV should be recalcitrant to dissociation. In the primordial plasma at redshift z = 105 there were no hydrogen atoms, not because it was not in equilibrium, but because the corresponding temperature of 3 x 105 OK is way above the ioniztion temperature of hydrogen. There were 4He nuclei then because their dissociation temperature is way above 3 x 105 OK. According to all this logic, Wheeler

56 TVs weighing much less than the Planck-Wheeler mass, and having a very high dissociation temperature, should show up in Hawking radiance whose temperature is of order of the TV's rest energy. Yet, as I show now, TV sets or other macroscopic systems do not occur measurably in any Hawking radiance. Suppose a macroscopic object (a TV for short) of size R has rest energy E and a degeneracy factor g. The last reflects the complexity of the composite system, so that 9 could be very large. The object will get emitted in an available Hawking mode with probability 9 exp( - E ITBH ). Actually, if the TV is measurably excited at temperature TBH one should replace 9 by an appropriate partition function; I ignore such complications. Over the Hawking evaporation lifetime'" M3 In there emerge of order M2 In "up" modes of each species. Thus the probability that the hole emits a TV over its lifetime amounts to p '" (M2 In)g exp( -811" M E In). Obviously In 9 plays the role of internal entropy of the object. From the bound (13) one may infer that lng < 211"REln. Thus p < (M2In)exp[211"(R - 4M)Eln]. However, in order for the TV to be emitted whole it must be smaller than the hole: R < 2M. Hence p < (M2 In) exp( -411" M E In). But obviously the particles composing the TV (masses ~ E) must have Compton lengths smaller than R < 2M so that EM In ~ 1. It follows that the argument of the exponent is very large, so that p is exponentially small. Thus in practice an evaporating black hole does not emit TVs or any macroscopic objects. This "selection rule" depends on the bound on entropy. 4. Acknowledgements

I thank F. Englert, G. Horwitz, D. Page, R. Parent ani , C. Rosenzweig, M. Schiffer and B. Whiting for critiques and enlightment, U. Bekenstein for help with the graphics, and the Israel Science Foundation, administered by the Israel Academy of Sciences and Humanities, for support. 5. References

J. D. Bekenstein, Lett. Nuov. Cimento 4 (1972) 737. J. D. Bekenstein, Phys. Rev. D7 (1973) 2333. J. D. Bekenstein, Phys. Rev. D9 (1974) 3292. S. W. Hawking, Nature (Physical Science) 248 (1974) 30. S. Hawking, Commun. Math. Phys. 43 (1975) 199. S. W. Hawking, Phys. Rev. D14 (1976) 2460. D. Christodoulou, Phys. Rev. Lett. 25 (1970) 1596; D. Christodoulou and R. Ruffini, Phys. Rev. D4 (1971) 3552. 8. J. D. Bekenstein, Physics Today 33 (1980) January issue, p. 24. 9. R. Penrose and R. M. Floyd, Nature (Physical Science) 229 (1971) 177. 10~ S. W. Hawking, Phys. Rev. Lett. 26 (1971) 1344. 1. 2. 3. 4. 5. 6. 7.

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11. B. Carter, Nature 238 (1972) 71; B. Carter, in Black Holes, eds. B. DeWitt and C. M. DeWitt (Gordon and Breach, NY, 1973). 12. J. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161. 13. W. Israel, in 300 Years of Gravitation, eds. S. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987). 14. I. Moss, Phys. Rev. Lett. 69 (1992) 1852; M. Visser, Phys. Rev. D48 (1993) 583; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D47 (1993) 5400 15. J. D. Bekenstein, Phys. Rev. D12 (1975) 3077. 16. S. W. Hawking, Phys. Rev. D13 (1976) 191. 17. R. Sorkin, Phys. Rev. Lett. 56 (1986) 1885. 18. L. Parker, Phys. Rev. D12 (1975) 1519; R. M. Wald, Commun. Math. Phys. 45 (1975) 9. 19. C.G. Callan, S.B. Giddings, J.A. Harvey, and A. Strominger, Phys. Rev. D45 (1992) R1005; J. Russo, 1. Susskind and L. Thorlacius, Phys. Rev. D46 (1992) 3444 and 47 (1993) 533. 20. C. Shannon and W. Weaver,The Mathematical Theory of Communication (University of illinois Press, Urbana, 1949). 21. J. A. Wheeler, A Journey into Gravity and Spacetime (Freeman, NY, 1990). 22. G. 't Hooft, Nucl. Phys. B335 (1990) 138. 23. 1. Susskind, L. Thorlacius and R. Uglum, Phys. Rev. D48 (1993) 3743. 24. S. Giddings, Phys. Rev. D49 (1994) 4078. 25. M. Visser, Phys. Rev. D46 (1992) 2445. 26. R. M. Wald, Phys. Rev. D48 (1993) 3427. 27. T. Jacobson, G. Kang and R. Myers, Phys. Rev. D49 (1994) 6587; M. Visser, Phys. Rev. D48 (1993) 5697. 28. C. Baiiados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 72 (1994) 957; L. Susskind, "Some Speculations About Black Hole Entropy in String Theory", hep-thj9309145 (1993). 29. G. W. Gibbons and S. W. Hawking, Phys. Rev. D15 (1977) 2752. 30. J. D. Bekenstein, in To Fulfill a Vision, ed. Y. Ne'eman (Addison-Wesley, Reading, Mass., 1981). 31. J. S. Dowker, "Remarks on Geometric Entropy", hep-thj9401159 (1994). 32. W. Israel, Phys. Lett. AA57 (1976) 107. 33. L. Bombelli, R. Koul, J. Lee and R. Sorkin, Phys. Rev. D34 (1986) 373. 34. M. Srednicki, Phys. Rev. D71 (1993) 66. 35. C. Holzhey, Princeton University thesis (unpublished, 1993). 36. C. Callan and F. Wilczek, "On Geometric Entropy", hep-thj9401072 (1994). 37. D. Kabat and D. J. Strassler, "Comment on Entropy and Area", hep-thj9401125 (1994).

58 38. C. Holzhey, F. Larsen and F. Wilczek, "Geometric and Renormalized Entropy in Conformal Field Theory", hep-thj9403108 (1994). 39. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, N. J., 1955). 40. U. H. Danielsson and M. Schiffer, Phys. Rev. 048 (1993) 4779. 41. J. D. Bekenstein, Phys. Rev. D30 (1984) 1669. 42. J. D. Bekenstein and M. Schiffer, Phys. Rev. 039 (1989) 1109. 43. J. D. Bekenstein and M. Schiffer, Int. J. Mod. Phys. Cl (1990) 355. 44. J. D. Bekenstein, Phys. Rev. D23 (1981) 287. 45. L. Susskind and J. Uglum, "Black Hole Entropy in Canonical Quantum Gravity and String Theory", hep-thj9401070 (1994). 46. R. D. Sorkin, in General Relativity and Gravitation, eds. B. Bertotti, F. de Felice and A. Pascolini (Consiglio Nazionale della Ricerche, Rome, 1983), Vol. 2. 47. G. 't Hooft, talk at Santa Barbara Conference on Quantum Aspects of Black Holes (unpublished, 1993). 48. T. Jacobson, "Black Hole Entropy and Induced Gravity", gr-qcj9404039 (1994). 49. V. Frolov, "Why the Entropy of a Black Hole is A/4 ?", gr-qc/9406037 (1994). 50. V. Frolov and I. Novikov, Phys. Rev. 048 (1993) 4545. 51. J. M. York, Phys. Rev. D33 (1986) 2092. 52. D. Page, Phys. Rev. Lett. 44 (1980) 301. 53. G.'t Hooft, Nucl. Phys. B256 (1985) 727. 54. T. M. Fiola, J. Preskill, A. Strominger and S. P. Trivedi, "Black Hole Thermodynamics and Information Loss in Two Dimensions", hep-thj9403137 (1994). 55. G. L. Sewell, Phys. Lett. A122 (1987) 309. 56. D. N. Page, Phys. Rev. D14 (1976) 3260. 57. R. M. Wald, in Black Hole Physics, eds. V. de Sabbata and Z. Zhang (NATO ASI series, Vol. 364, 1992). 58. V. P. Frolov and D. N. Page, Phys. Rev. Lett. 71 (1993) 3902. 59. W. H. Zurek and K. S. Thorne, Phys. Rev. Lett. 54 (1985) 2171. 60. W. G. Unruh and R. M. Wald, Phys. Rev. D25 (1982) 942 and D27 (1983) 2271. 61. J. D. Bekenstein, Phys. Rev. D26 (1982) 950 and D27 (1983) 2262. 62. J. D. Bekenstein, Phys. Rev. D49 (1994) 1912. 63. R. Ruffini and J. A. Wheeler, Physics Today 24 (1971) January issue, p. 30.

59 UNITARY RULES FOR BLACK HOLE EVAPORATION ANDREW STROMINGERt Department of Physics, University of California, Santa Barbara, CA 93106-9530 ABSTRACT Hawking has proposed non-unitary rules for computing the probabilistic outcome of black hole formation. It is shown that the usual interpretation of these rules violates the superposition principle and energy conservation. Refinements of Hawking's rules are found which restore both the superposition principle and energy conservation, but leave completely unaltered Hawking's prediction of a thermal emission spectrum prior to the endpoint of black hole evaporation. These new rules violate clustering. They further imply the existence of superselection sectors, within each of which clustering is restored and a unitary S-matrix is shown to exist.

1. Introduction: Low-Energy Approach to Black Hole

Formation jEvaporation Consider the formation of a black hole of mass M, where M is much greater than the Planck mass M p , from the collapse of low-energy (i. e. sub-planckian) matter. Everyone agrees that the black hole will evaporate due to Hawking radiation 1 , and that virtually all (up to corrections of order of the outgoing quanta will have energies well below Mp. Since both the in- and out-states can be adequately described by a low-energy, sub-planckian, effective field theory, it is natural to seek a low-energy effective description of the scattering interaction. However, there is no guarantee that this effective description can be derived from the laws of lowenergy physics alone, since the dynamics of gravitational collapse inexorably lead to regions of high curvature where Planck-scale physics is important. Nevertheless, it should be possible to summarize our ignorance about Planck scale physics in a phenomenological boundary condition (or generalization thereof) which governs how low energy quanta enter or exit the planckian region. In principle this effective description should be derived by a coarse-graining procedure from a complete theory of quantum gravity such as string theory. But this is not feasible in practice. Instead we shall find that the possible descriptions can be highly constrained by low-energy considerations alone. Using this latter approach, we shall be led to a new and satisfying effective description of black hole formation/ evaporation2 . A classic example of this type of approach is the analysis of the Callan-Rubakov effect,3,4 in which charged S-wave fermions are scattered off of" a GUT magnetic monopole. Even at energies well below the GUT scale, the scattering cannot be directly computed from a low energy effective field theory, because the fermions are inexorably compressed into a small region in the monopole core in which GUT interactions become important. Initially the GUT scale physics was analyzed in some detail. The results were then coarse-grained and summarized in an effective

¥f)

t

Email address: [email protected]

60

boundary condition for fermion scattering at the origin. It was subsequently realized that the detailed GUT scale analysis was l~rgely unnecessary for understanding the low-energy scattering: up to a few free parameters (a matrix in flavor space) the effective description is determined by low-energy symmetries. We now turn to the black hole problem with this philosophy in mind. Classically, black holes are stable, but quantum mechanically they slowly evaporate and shrink.! Hawking has calculated the outgoing radiation state using lowenergy effective field theory together with the adiabatic approximation. Although they have certainly been questioned a , both of these approximations would seem to be valid as long as the black hole mass M is well above the Planck mass Mp. The calculation requires (by causality) only the exterior black hole geometry. It follows 5 that the outgoing Hawking radiation carries little information about what has fallen into the black hole, at least prior to the evaporation "endpoint" at which the black hole shrinks to the Planck mass and the approximations break down. For example, in a theory with an exact chocolate-vanilla flavor symmetry, the outgoing radiation prior to the endpoint is identical for black holes formed from vanilla or chocolate matter, and so information about the flavor of the initial state cannot be obtained from this radiation. Once the black hole reaches the Planck mass, quantum gravity must be solved to continue the evolution. As quantum gravity is poorly understood, it might seem that one should simply give up on the problem at this point. However, as discussed above, it still makes sense to ask what a low-energy experimentalist who makes black holes and measures the outgoing radiation could observe, and to try to describe this by an effective field theory. In the following sections we discuss several possibilities. 2. Remnants? One logically possible outcome of gravitational collapse is that planckian physics shuts off the Hawking radiation when the black hole reaches the Planck mass, and the information about the initial state is eternally stored in a planckian remnant. As there are infinite numbers of ways of forming black holes and letting them evaporate, this remnant must have an infinite number of quantum states in order to encode the information in the initial state. In an effective field theory these remnants would resemble an infinite number of species of stable particles. This raises the so-called "pair-production problem". Since the remnants carry mass b , it must be possible to pair-produce them in a gravitational field. Naively the total pair-production rate is proportional to the number of remnant species, and therefore infinite. It is easy to hide a Planck-mass particle, but it is hard to hide an infinite number of them. Thus it would seem that remnants can be experimentally ruled out by the observed absence of copious pair-production. a

b

The author's view on this controversy can be found in [5J. Massless remnants would create even worse difficulties.

61

However, an error in this logic 6 may be briefly summarized as follows c . It was shown7 that the quantum versions of the charged Reissner-Nordstrom solution have an infinite degeneracy of stable "remnant states" which for large charge can (unlike their neutral planckian cousins discussed above) be described with weakly-coupled, semiclassical perturbation theory. This infinite degeneracy potentially leads to unacceptable pair-production, so the Reissner-Nordstrom remnants provide a good laboratory for analyzing the pair-production problemd . It can be seen that the different remnant states differ solely by the action of a local operator in a region which is near or inside of a horizon and causally disparate from the external observer. Causality (i. e. the fact that operators commute at spacelike separations) implies that the causally disparate states can not be distinguished in a finite-time scattering experiment. This is certainly at odds with the naive low-energy effective description in which each remnant state is represented as a distinct particle species, and could therefore all be distinguished e.g. in finite-time interference experiments. The naive description must therefore fail e . This failure can be traced to ultra nonlocal (in time) interactions along the remnant worldline. One may also expect that the infinite degeneracy of states lying in a causally distant region could not have a divergent effect on any finite-time pair production process. This expectation was borne out in the euclidean instanton calculation of the pair-production rate 9 , which yielded a finite result. While certainly more remains to be understood on this topic, it is clear that the standard argument that infinite pair-production is inevitable for all types of remnants is too naive, and it is plausible that in some theories the pair production rate is finite. Further discussion can be found in references [10,11] and the recent reviews [12,13]. A more inescapable objection to eternal remnants is the lack of any plausible mechanism to stabilize them. In quantum mechanics what is not forbidden is compulsory. In the absence of a conservation law it is hard to understand why matrix elements connecting a massive remnant to the vacuum plus outgoing radiation should be exactly zero. Nature contains no example of such unexplained zeroes. Moreover, a formal representation of quantum gravity as a sum-over-geometriesand-topologies certainly includes such processes. Eternal remnants are therefore highly unnatural. An alternative possibility is that the "Planck soup" which forms when the black hole reaches the Planck mass continues to radiate in a manner governed by planckian dynamics until all the mass is dissipated. In principle, as we do not understand the dynamics, the radiation emitted by the Planck soup could be correlated with the earlier Hawking emissions and return all the information back out to infinity. Energy C In [6]dilaton rather than Reissner-Nordstrom black holes were considered. As stressed in [7], the argument is cleaner in the Reissner-Nordstrom context because regions of strongly coupled or planckian dynamics can be avoided. d This was also stressed in [9]. e A more subtle type of low-energy effective description where the remnant interior is described by an entire two-dimensional field theory, may still workS.

62

conservation implies that the total energy of the radiation emitted by the Planck soup is itself of the order of the Planck m~ss, and thus small relative to the initial mass of the black hole. It is very hard to encode all the information in the initial state with this small available energy. The only way to accomplish this is to access very low-energy, long-wavelength states, which requires a long decay time. This leads to a lower bound of 7 '" M4 (in Planck units) for the decay time of the Planck soup.lO For a macroscopic black hole this far exceeds the lifetime of the universe. Hence, it is not possible for the information to be emitted in a planckian burst at the end of the evaporation process. In this scenario one necessarily has a longlived, but not eternal, remnant. Note that our discussion required no knowledge of planckian dynamics. This is a prime example of how low-energy considerations highly constrain the possible outcome of gravitational collapse. Of course, long-lived remnants are implausible without an explanation for their long lifetime, or a mechanism for the Planck soup to reradiate the information. We shall encounter both below.

3. Information Destruction Faced with the apparent unpalatability of remnants, Hawking 5 argued in favor of a different possibility, depicted in fig. 1. The black hole disappears in a time of order the Planck time after shrinking to the Planck mass, and the infalling information disappears with it. After all, in practice, information often escapes to inaccessible regions of spacetime, even in the absence of gravity. The inclusion of gravity, Hawking argues, implies information is lost in principle as well as in practice.

Fig. 2: Hawking's rule for density matrix superscattering for single black hole formation. The left (right) side of the diagram represents the evolution of the ket(bra) of the density matrix. The trace over the part of the Hilbert space which falls into the hlack hole is schematically represented by sewing together the left and right black hole interiors.

63

r

j-

Fig. 1: Collapsing radiation forms an apparent black hole (shaded region) which evaporates, shrinks down to T = 0 at XE, and subsequently disappears. The dashed wavy line is the region at which Planck-scale physics becomes important, and is just prior to the classical singularity. According to Hawking, information which crosses the event horizon is irretrievably lost.

Since information is lost in this proposal, there can be no unitary S-matrix mapping in-states to out-states. Rather, Hawking suggests that a "superscattering" matrix, denoted "$", which maps in-density matrices (of the general form p = L Pij j7fi)(7fjj ) to out-density matrices can be constructed as

$

= trBHS

st .

(1)

$ will not in general preserve the entropy -trplnp. In components, $ acts on an in-density matrix as ($[p]) kl = ($hl ij Pij. S here is a unitary operator which maps the in-Hilbert space to the product of the out-Hilbert space with the Hilbert space of states which falls into the black hole (defined, for example, as quantum states on the event horizon in fig. 1). trBH is the instruction to trace over these latter unobservable states. Expressions of the form (1) are familiar in physics, and arise, for example, in the computation of e+ e- scattering in which the spins of the final state are not measured. A diagrammatic representation of Hawking'S prescription for the case of one black hole appears in fig. 2.

64

Fig. 3: Hawking's rule for superscattering of two black holes involves two traces, one for each black hole.

It is implicit in Hawking's proposal that the probabilistic outcome of the formation/evaporation of an isolated black hole near the spacetime location Xl can in this manner be computed from the portion of the quantum state which collapses to form the black hole. In this case the outcome of forming a second black hole at a greatly spatially or temporally separated location X2 is uncorrelated and the two-black hole $-matrix can be decomposed into a product of single black hole $matrices (In other words, probabilities cluster.) The corresponding diagrammatic representation of $ for the case of two black holes is given in fig. 3.

4. The Superposition Principle

In fact as it stands Hawking's proposal is in conflict with the superposition principle f . To see this note that there are inevitably non-zero but possibly small quantum fluctuations in the location Xl where the black hole is formed. trBH instructs one to trace by equating the black hole interior states of the bra and the ket in the density matrix, independently of the precise location where the black hole is formed. Now Xl cannot be an observable of the black hole interior Hilbert space, since by translation invariance the interior state of the black hole does not depend on where it was formed. Hence the trace will include contributions from black holes interiors which are in the same quantum state, but which were formed at slightly different spacetime locations. f The arguments of this and the following section may be related to those employed in a somewhat different context in references [15,10J

65

r+

r -

1

+ -

2

1

+ -

2

Fig. 4: Superscattering of an initial coherent superposition of semiclassical states which form black holes near widely separated locations x, and X2. The superposition principle and translation invariance imply that all four diagrams contribute.

This phenomenon is more pronounced in initial states for which the fluctuations in the location of the black hole are not small. Such states can certainly be constructed. For example, let the in-state be the coherent superposition

(2) where IXi) is a semiclassical initial state which collapses to form a black hole near Xi, and Xl and X2 are very widely separated spacetime locations. By continuity the construction of $ must include terms which equate the interior black hole bra-state formed at Xl with the ket-state formed at X2. There are then four terms in fJ as illustrated in fig. 4. It may already seem rather strange that $ should contain such correlations between widely separated events, but matters become even worse when one considers a semiclassical initial state lXI, X2) which collapses to form two black holes at the widely separated locations Xl and X2. The superposition principle then requires that the cross diagram of fig. 5 be added to the diagram of fig. 3. g To see this, g This extra cross diagram will be small if the parts of the incoming states which form the two black holes are very different and the black hole interiors have a correspondingly small probability of being in the same state. On the other hand if they differ only by a translation, fig. 5 will be similar in size to fig. 3.

66

Fig. 5: The superposition principle implies that for two black holes this cross diagram must be added to that of fig. 3, correlating widely separated experiments.

consider a smooth one-parameter family of initial states IXI (s), X2(S)) in which the locations Xl and X2 are interchanged as the parameter s runs from zero to one. Let the in-state be

(3) Then the diagrams of fig. 3 and fig. 5 are interchanged as s goes from 0 to 1 in the ket-state, so neither can be invariantly excluded. Thus the superposition principle implies that one cannot, in the manner Hawking suggests, compute the probabilistic outcome of a single experiment in which a black hole is formed. Knowledge of all past and future black hole formation events is apparently required to compute the superscattering matrix (although we shall see below that this is not as unphysical as it seems). Again, it is striking that low-energy reasoning highly constrains possible outcomes of black hole formation without requiring knowledge of planckian dynamics. Note that our conclusions about difficulties with the usual interpretation of Hawking's proposal have derived from consideration of superpositions of semiclassical states which form black holes. These difficulties have not been so evident in previous discussions simply because such superpositions are not usually considered. 5. Energy Conservation

Although the superposition principle is restored with the extra cross diagram of fig. 5, correlations are introduced between arbitrarily widely separated experiments, and clustering is violated. ll Thus we seem to be faced with a choice: abandon the superposition principle or abandon clustering. In fact we shall see below that the breakdown of clustering is a blessing in disguise, but first we need to introduce a second refinement of Hawking's prescription required by energy conservationh . h

I am grateful to S. Giddings for emphasizing to me the importance of under-

67 j+

Fig. 6: When the evolution of spacelike slices (denoted by the dashed lines) reaches the endpoint x E, the incoming slice, and the quantum state on the slice, is split into exterior and interior portions. This splitting process is described using the operator qi J (qi K ) which annihilates (creates) an incoming (outgoing) asymptotically flat slice in the J'th (I'th) quantum state and qii which creates an interior slice in the i'th quantum state.

In computing the $-matrix, complete spacelike slices are split into interior and exterior portions when they encounter the evaporation endpoint at XE, as illustrated in fig. 6. One imagines that the Hilbert space on these slices is also split into the product of two corresponding interior and exterior Hilbert spaces. This requires some new boundary conditions originating at XE: an incoming light ray just prior to XE falls into the black hole, while an incoming light ray just after XE reflects through the origin and back out to null infinity. (Explicit examples of such boundary conditions exist in 1 + 1 dimensions,12 but for our present purposes an explicit form will not be needed.) Implementing this in practice immediately runs afoul of the Anderson-DeWitt 13 problem. These authors considered the propagation of a free conformal field in 1 + 1 dimensions on the trousers spacetime of fig. 7 in which (as in the black hole case) spacelike slices are split into two portions at some fixed time ts, when reflecting boundary conditions are turned on at x = O. They find that the vacuum state for t < ts evolves to a state with infinite energy for t > ts. This is not surprising since the Hamiltonian changes at an infinite rate at t = ts. This phenomenon is not peculiar to two dimensions. A change in the Hamiltonian in the form of new boundary conditions at a fixed spacetime location violates standing energy conservation in this context.

68

v~

~~

%

,~

,,

,,

---------~------------

____ t=t

5

vacuum

Fig. 7: Anderson and DeWitt studied a free field propagating on a geometry which is split into two at time t = t, by reflecting boundary conditions at x = O. The sudden change in the Hamiltonian produces infinite energy pulses which propagate along the dashed lines.

general covariance and therefore energy conservation. This problem should be expected to affect the separation of Hilbert space into interior and exterior portions at the evaporation endpoint x E for the black hole case. Indeed the most concrete description given of this splitting process - that in the 1 + 1 dimensional RST model 14_ suffers from exactly this problem. Energy is not conserved in this model because the quantum state of the matter field acquires infinite energy as it is propagated past x E 15. To remedy this, a smooth energy-conserving method of splitting the incoming Hilbert space into two portions is needed. A physical example of a system which exhibits such a smooth splitting is given by cosmic string decay. Consider, e.g. a magnetic flux tube described by a Nielsen-Olesen vortex. At low energies it is described by a 1 + 1 dimensional quantum field theory whose massless fields are the transverse excitations X(a) of the string. Next suppose that the string can decay by the formation of a heavy monopole-anti-monopole pair which divides the string into two parts. Clearly such a process can occur and will conserve energy. It cannot, however, be simply described by propagating the 1 + 1 dimensional fields on the fixed geometry of fig. 7 (or superpositions thereof), as analyzed by Anderson and DeWitt. Rather, the decay rate depends on the final state after the split through initial and final wave function overlaps appearing in decay matrix elements, and the decay time is thus correlated with the quantum states on the two final strings. This decay process may be conveniently and approximately (at low energies) described by the interaction Hamiltonian (see fig. 8) Hint

= L I,J,K

g PIJK(PI.) = -00, and A(r>.) = O. That is, the A-orbit crashes at r>., (A(r>.) = 0). 3. The equations (2), (3) are fiercely non-linear. Writing ~(r) r,and u(r)

=1-

w 2 (r), (1), (2) become

(2)'

r2 Aw"

(3)'

rA'

+ ~w' + uw = 0

+ 2W,2 A = ~/r.

= r(I

- A) - (1 - w 2 )/

90

If we define the region f by f={(w,w',A,r)€R 4 :w 2 < I,A > O,r > O,(w,w')

=I (O,O)},

the equations define a "flow" on f, and solutions we seek must lie in f. In this section we consider the following problem: Find "particle-like" solutions, that is, non-singular solutions defined for all r > 0, and, as r -> 00, they should have the following asymptotic form: 2M 2M [A(r), w2 (r), T- 2 (r)] '" [1 - -,1, (1 - -)], as r r

r

-> 00,

where M = const. In this case solutions have finite ADM mass 2M, and the metric is asymptotically Mirkowskian. Now if ~(r) > (friction), then no crash occurs. Thus, even though ~(r) is really, ~(r, w, A), we can say general things about ~; in fact, we have the

°

~-Proposition

([6, 7]): If r ~ 1 + € (€ > 0), or w 2

::::;

1, then there exists au>

such that ~(r) ~ u. In these cases, there exist." > 0, m > with A{r, A) ~ ." and (w'(r, A)) ~ m.

°

°

(both independent of A)

The existence particle-like solutions follows from the No Crash Lemma ([7]). If

is a sequence of orbit segments in f, with uniformly bounded rotation, and if A,,(b,,)

->

f, then the (backwards) orbit through P is in f for a ~ r ~ b, and has (the same) bound on its rotation. One application of the above lemma is Theorem 1 there exists an infinite number of particle-like solutions, A,,(r), and the nth-orbit has rotation number -n7l". Sketch of Proof: We first show that if A is near zero, then the A-orbit doesn't crash in the region w 2 < 1, and exits f through w = -1; i.e. there is a number p>. such that w(p>., A) = -1, w'(r, A) < if < r ~ p>., and A(r, A) > if ~ r ~ p>.; c.f. Figure p = (tv, w', A, b)



°°

°°

1.

(w(r, A), w'(r, A) Figure 1.

91

Let .AI, be the sup of those .A for which the .A-orbit has this behavior. Since orbits crash in the region w > 0 for.A > 2, We see.A l ::; 2. ApR lying the No-Crash Lemma to the .AI-orbit shows that the.A l orbit is a I-connector (rotation -71"). Let :AI be the sup of the.A for which the .A-orbit is a I-connector; by our No Crash Lemma, the :AI-orbit is a I-connector, the ~largest" I-connector; see Figure 2. w'

Figure 2. Now let.A > :A near Xl' We prove that the .A-orbit exits 2. [The idea of this proof is to consider the function

and observe that 0 ::; P(w) ::; ~ if w 2

::;

r

l. Thus if H

via w = +1, as depicted in Figure

>

~ =

maxP, then the ~kinetic

r 2 w ,2

energy", -2- cannot be zero. Thus for .A near XI,.A > Xl, we show and when w

= 0, H

>

~.

HI

> 0 in

Since it is easy to show that the .A-orbit must stay in

Wi

> 0

r

(as it

doesn't crash if r > 1), it must enter the region w' > O. It then follows that the .A-orbit exits r via w = 1, Wi > 0, as depicted.] Now let .A2 be the sup of those .A for which the .A-orbit exits r as depicted above; again.A2 ::; 2, and by the No Crash Lemma, one shows that the .A2-orbit is a 2-connector, (rotation -271"). If X2 is the largest .A such that the .A-orbit is a 2-connector, then we use our No Crash Lemma to prove that the X2-orbit is a 2-connector; the "largest" 2-connector. Taking.A > X2,.A near X2 , we prove that the .A-orbit exits depicted in Figure 3. Thus "" I

----~--~----~------_*----~~v

Figure 3.

r

as

92 let '\3 be the sup of those ,\'s for which the '\-orbit has this behavior, and continue this argument. (The proof is completed by induction). Note: Bartnik and McKinnon ([1]) obtained the first particle-like solutions numerically. Kiinzle and Masood, ([3]), added both rigorous and numerical information about these solutions, but didn't provide an existence proof. The first rigorous existence proof was given in [4]. We next discuss the "physical" significance of our above theorem: (a) The Yang/Mills (SV(2) "weak") repulsive force balances the gravitational attractive force and prevents the formation of singularities in 4 - d spacetime. (This result is not true for the pure Einstein equations G ij = 0, (the Schwarzschild Solution has a non-removable singularity at r = 0) nor for the pure Yang/Mills equations d* Fij = 0 (see [3]), nor for the coupled Einstein/Maxwell equations, where G = U(I) (since the Reissner-Nordstrom Solution is singular at r = 0), nor for the coupled EYM equations with any compact gauge group, in 3 - d spacetime; see ([4]). (b) All or our solutions An( r) have finite (ADM) mass j i.e. J.ln = lim r(1 - An( r)) < 00. r-oo

These masses are derived from our solutions, and are not arbitrary constants. They satisfy

o < !!. < J.ln < Ji < 00, for some constants!!. and Ji. (c) The Yang/Mills field strength, . fy An (r ) , satls

IF:nl--

0(1) and r2

T.oon _-

IFni and the energy density T~, for each of our solutions 0(1) as r r2

-+ 00,

. I.e., t h ey decay f aster t h an 1 at r2

infinity. 4. Another application of these ideas is to the existence of Black-Hole solutions of the EYM equations. Recall that black-hole solutions are solutions of the field equations which describe collapsed masses. They contain a closed "event horizon" , and inside this event horizon, there is a singularity of the gravitational field which forms during collapse. The model for black-hole solutions is the celebrated Schwarzschild solution of the pure Einstein equations, R;j = 0; namely

At r = 2M, a singularity occurs: 900 = 0,911 = 00. To generalize this to our situation, we say that we have a black-hole at some p > 0 (the event horizon), for the metric (1), provided that the following hold:

93 (i) A(p)

= 0,

(ii) T_OO lim fL(r)

and A(r) > 0 for r > p,

= lim

T-OO

(iii) (A(r),T- 2 (r))

r(1 - A(r))

= 71


0 in the region w 2 r/'r

:::;

l.

The idea is to first show that if the N-orbit crashes, then N :2: X, and then to show that the X -orbit crashes. Thus, by the "No-Crash" lemma, the set of A-orbits and A < A' cannot have bounded rotation. The A = 0 orbit has 0 rotation, so, by a result in [7],

94 for every integer k, there is a Thus Ak :S

A~,O


1, for all r > p.

n-oo

The convergence is uniform on compact r-intervals. (Here r(l - A,,(r))

= p.n(r).)

Note that if p :S 1 (small black-holes, or particle-like solutions), the (ADM, limiting) mass outside the black hole is 2 - p -+ 2 as p -+ 0, while if p > 1, this mass is p-l --+ o as p --+ 00. We shall sketch the proof. Thus fix p ~ 0, and pick R > 1 + p. Then as discussed in §3, there are numbers Ti > and m > 0, independent of Ti, such that

°

95 1 2': An(R) 2': TJ, Iw~(R)1 ~ m and w;(R) ~ 1. Consider the sequence of points Pn = (wn(R),w~(R),An(R),R),n = 1,2, .... OU!; bounds imply that Pn has a convergent subsequence

It is not difficult to prove that in fact, P = (0,0, A, R). Now the unique solution of (I), (2) through P is a Reissner-Nordstrom (RN) solution: w(r) == O,ARN(r) = 1 - ; + -!r. Also, c2 2: 4; otherwise ARN(r) > 0 in r > 0 and ARN(r) -> 00 as r -+ O. Since the An(R) are near ARN( R) if n is large, we have An (r) > 1 if r is near 0 and n is large, and this is not possible. Similarly, c > o. Now consider the case p > 1. Here ARN cannot be the extreme RN solution, for if so,

ARN(r)

=

(T~2l)2, and ARN(p) > 0, but An(P)

= o.

Thus ARN(r) has two roots, and one

shows that these roots are p and p-l. Thus ARN(r) = 1 - (p+~-l)

ARN(r), which implies 1 -

I'n;T) -+

ARN(r), so I-Ln(r)

-+

+ -!r, so if r

(p + p-l) -~, as n

> p, An(r) -+ 00.

Next, if p ~ I, one shows that ARN(r) is the extreme RN solution; ARN(r)

r2, so if r > 1, An(r)

-+

ARN(r), and thus I-Ln(r)

->

2 - ~.

-+

= (r -

1)2 /

96

GRAVITATIONAL TOPOLOGICAL CHARGE AND THE GRAVIBREATHER V. BELINSKY National Institute for Nuclear Physics (INFN) and International Center for Relativistic Astrophysics, University of Rome, 1-00185 Roma, ITALY ABSTRACT It is shown that the notions of gravisolitons and antigravisolitons with respect to

some topological charge can be introduced into General Relativity. The existence of attractive forces between two gravitational solitons with charges of opposite sign and repulsive forces between solitons of the same charge is established. The exact Gravibreather solution representing a bound state of a gravisoliton and an antigravisoliton is constructed.

1.

Introduction

In 1978 the procedure of application of the Inverse Scattering Method to General Relativity and of calculation of the exact soli tonic solutions to the Einstein equations was described. 1 From that time not many significant developments have occurred in this field. The reason is that despite the fact that from the mathematical point of view gravisolitons have the full status of solitons, their physical properties differ significantly from their nongravitational relatives: 1. A gravisoliton's amplitude and shape are not preserved in time. 2. The velocity of a gravisoliton also changes in time, and moreover there is no clear definition of velocity. 3. In general the gravisoliton field has discontinuities of the first derivatives on certain isotropic surfaces (although these singularities leave the invariants regular). 4. There is no evolution of gravisolitons from a free state at t = - 0 0 to an another free state at t = +00 because of the unavoidable existence of a cosmological singularity in between. Therefore the description of the collision process needs some special care. 5. There do not exist definitions of the mass and energy for gravisolitons. It is worth recalling that any description of nongravitational solitons starts with the declaration of its finite energy status. 6. For a long time it was not clear whether the gravisoliton represents a topological object or whether some topological charge can be associated with it. In what follows I suggest the answer to the last problem and as a result, along with the clarification of the gravisoliton's topological properties, one can get a qualitative understanding of a gravisoliton head-on collision process and how to define the gravisoliton's velocity. However, the solution of many of the above problems, especially the exact formulation of the gravisoliton energetics, must await future developments. It can be shown that in spite of essential pecularities in the physical properties of

97

gravitational solitons there remains some analogy between them and Sine-Gordon (SG) kinks which may be exploited to clarify a number of physical aspects of a gravisoliton's behaviour. In particular the topological structure of SG kinks can be generalized for gravisolitons in a precise mathematical way. This allows one to show that a gravisoliton is a topological object and that it carries a topological charge. There exist gravisolitons and antigravisolitons with respect to this charge. There is attraction between gravisolitons and antigravisolitons and repulsion between gravisolitons of the same charge. There also exists a bound state of a gravisoliton and an antigravisoliton which oscillates in time and which is a direct analog of the Sine-Gordon breather. I call such a solution the Gravibreather. This talk is based on a previous article 2 and for those who are interested in the details I suggest looking at it together with the original article on gravitational solitons. l

2.

The integrable ansatz in General Relativity

It was shown 1 that the Einstein equations in vacuum can be integrated exactly for the metric

(1) where a, b = 1,2. The equation for the matrix g (with components gab) has the form

(2) where 17

1

(3)

= 2(z - t),

If we know the matrix g, the integration of the equations for the coefficient f does not present any serious difficulties so we can simply ignore it here. The trace of eq.(2) gives a,e'l/=O . Along with the solution a of this wave equation we also need a second independent solution f3 of the same equation. These two solutions can be represented in the form a = a( 0

+ b( 17)

,

f3

=

a(O - b(17) ,

(4)

where a and b are arbitrary functions. We restrict ourselves to the case where a is time-like (i.e., a,ea,'I/ < 0 ) everywhere in space-time (the variable f3 is then automatically space-like everywhere). Then a(> 0) can be used as a cosmological time and a = 0 corresponds to the cosmological singularity. We also postulate that a, f3 form a single patch of the natural coordinates which cover the whole related two-dimensional section of the maximally extended physical space-time and each pair of real numbers a, f3 from the ranges a > 0, -00 < f3 < +00 represent one and only one point of this space-time and vice-versa. The variables t, z we consider as a second equivalent coordinate system in the same space-time, which means that the map between a, j3 and t, z is everywhere smooth and one-to-one in both directions.

98 The solutions for the matrix g( ~, rt) can· be found from the solutions of the linear spectral problem related to eq.(2). We construct a "Schroedinger" eigenvalueeigenfunction equation for the matrix W(A,~, 'rJ) with some complex spectral parameter A. In this equation the unknown matrix g( ~, rt) plays the role of a "potential". For each type of analytical structure of the matrix W in the complex A-plane (each such type corresponds to some definite set of scattering data for this spectral problem) we can reconstruct the "potential" g. To do this technically we need to have some (usually as simple as possible) particular solution go( ~, 'rJ), Woe A, ~, 77) of our problem which we call the background or seed solution. Given that solution we represent the matrix W in the form W = XWo introducing in this way a new unknown matrix X in place of W. It turns out that the analytic structure in the A-plane of the solutions of our linear spectral problem can be formulated in most clear and simple way in terms of the matrix X. For example, the purely solitonic solutions for the metric components 9 correspond to the following simple pole structure of the matrix X:

x=

t

k=l

Rk(~''rJ)

A - I1k(~' rt)

.

(5)

The number of poles n in X corresponds to the number of solitons we introduce into the background solution go. Inserting this form for X into our linear spectral problem equations and taking into account the reality and symmetry conditions for the metric matrix g, we find by purely algebraic manipulations the exact solutions for the matrices Rk(~, 77) and pole trajectories I1k(~' 77). The solution for 9 can then be expressed in terms of the value of the matrix X( A, ~, rt) at the point A = 0 by the following simple formula:

(6) This procedure gives the functions

I1k(~'

rt) as the roots of the quadratic equation: (7)

where Wk are arbitrary (in general complex) constants. The requirement that 9 be real and symmetric gives some restrictions on 11k: all 11k should be either real or (if some of them are complex) should appear only in complex-conjugate pairs. For each complex pole 11k, there should also exist another pole I1k+l of the matrix X which must be complex-conjugate to 11k: l1k+l = jik. Accordingly, for each real pole 11k the constant Wk in (7) should be real and for each conjugate pair of poles 11k, l1k+l, the constants Wk and Wk+l should complex-conjugates of each other. It is important to emphasize that each 11k comes from a quadratic equation, so each time we introduce a pole (or soliton) we have two possibilities for the choice of 11k. If 11k is the root to the equation (7) then a 2 11k l is also a solution. If the absolute value Il1kl of the first solution belongs to the interval [0, a], the modulus la 211k 1 of the second solution is out of this interval. We can introduce the notation l1~n and l1%ut for these two possibilities. All poles in the complex A-plane are 1

I1r

99

located inside the circle 1>,1 = a and all f-C~ut are outside of this circle. It is evident that at each fixed space-time point each pole f-Ck can be classified either as f.L~n or as f.L~ut. The crucial statement now is that the same is true also for the entire pole trajectory f-Ck( ~, 'Tf). It can be seen that the property of f.Lk to be f-C~n or f-C~ut is global. If at some space-time point one has f-Ck = f.L~n then f-Ck will be f-Ctn everywhere in space-time and the same for f.L~ut. In other words the pole trajectory cannot cross the circle 1).1 = Q.

3.

Gravisoliton's topological charge

So, there exist two one-soli tonic solutions for the metric g: the first one for and the second for f.L1 = l1~ut. It can be shown now that there is enough evidence to consider these two solutions as belonging to different topological sectors and, consequently, having different topological indices. In another words it can be assumed that there is no homotopy between these two solutions. If such a homotopy were to exist it should also exist in the limit where the function Q tends to be constant. However, in this limit the theory based on eq. (2) tends to agree with Sine-Gordon theory, for which solutions with f.Lin and f.L~ut emerge as well-known topologically different solutions associated with topological charges plus one and mInus one. To verify these assertions we need to pass from the metric matrix 9 to more appropriate field variables which don't depend on arbitrary linear transformations of the ignorable coordinates xl and x 2 . It is possible to construct such invariants only from the matrices K = g,eg- 1 and J = g,."g-l. The first three nontrivial quantities of this kind are Tr(K2), Tr(J 2) and Tr(KJ). The simplest invariants TrK = 2a,ea-1 and TrJ = 2a,."a- 1 are trivial, because they do not carry any information about the soliton's behaviour. It is convenient, however, to deal with some combinations RI, R2 and cosw of all of these invariants which we introduce by the relations: 111

= l1~n

4Ri = 2Tr(K2) - (TrK)2 ,

(8)

4~ = 2Tr(J 2) - (TrJ)2 ,

(9)

4R1R2COSW = 2Tr(KJ) - (TrK)(TrJ) .

(10)

It can be seen that from the Einstein equation (2) follows a self-consistent and closed system of three differential equations containing only the invariants Rb R2 andw. In the limit Q ...... const (from (4) and (7) it follows that in this limit also (3 ...... const and 11k ...... const) this system results in constant values for R1 and R2 and an exact Sine-Gordon equation for the field w. Accordingly, a direct calculation shows that the substitution of the one-soli tonic solution for the matrix 9 (constructed in the way explained in section 2) into the right hand side of the formulas (8)-(10) in the limit of constant a results in certain constant values for R1 and R2 and in the

100

following Sine-Gordon kink for the field w: a:2 + J-l2 2a:J-ll w=4arctanexp[1 RIR21 2 ~(z+ 2 2t+const)J. a: - J-ll a: + J-l

~

(11)

It follows from this expression that the topological charge T for this kink is:

2 T = sign(a: - J-li)

(12)

and it is associated with some real topological property: its value depends only on the position of the pole point>. = J-ll with respect to the circle 1>'1 = a:. Now we see how important the phenomenon of global confinement of each pole trajectory J-lk( ~, 1]) (either inside or outside of this circle) is also in the general case for an arbitrary function a:( ~, 1]), a phenomenon we already mentioned in the previous section. Due to this confinement property and the limiting correspondence (12) we can generalize the notion of topological charge to gravitational solitons directly in terms of solitonic solutions for the matrix g. The result is that even if a: i= const the relation (12) still makes sense. In other words the existence of pole trajectory confinement permits us to set up a regular map between the set of gravisolitons and Sine-Gordon kinks, a map which determines the same homotopy relations between gravisolitons as between the corresponding Sine-Gordon kinks. There also exists a more direct way to see the origin of gravisolitonic topological charge. One can consider the field w( mod 271") as the main gravisolitonic characteristic and then find appropriate time-like and space-like variables 7 and p for which the map w( 70, p) for each fixed value of 70 has for each gravisolitonic solution the appropriate nontrivial Brouwer degree which does not depend on 70. This is the natural generalization of what occurs in Sine-Gordon theory. In the previously cited article 2 it was shown that this can indeed be done for the one-solitonic solution. In this case the function w( 70, p) acts as a regular map between the one-dimensional p-space and the w-circle: the map is one-to-one in both directions and the angle w covers exactly once the segment [0,271"] when p runs between its natural boundaries. It turns out that for this map the integer quantity sign (a: 2 - J-li) indeed corresponds to the Brouwer degree. 4.

Dynamical manifestation of the topological charge

Continuing the previous development we may call the one-soliton solution g(J-ld for the case J-ll = J-lin a gravisoliton (S) and for the case J-ll = J-l~ut an antigravisoliton (A). However, the real physical manifestation of topological charge can be seen only in the collision process of two such objects. If the notion of topological charge has been introduced in an appropriate way, attractive forces in the system SA and repulsion for the combinations SS or AA should be expected to exist. It is not simple to see this in a direct way, but we can use the same approach as in SineGordon theory. First of all we need to show the existence of three types of double gravisoliton solutions: the first one, describing the SS-scattering state, a second for

101

the SA-scattering process and a third describing the oscillatory (in time) bound state of two gravisolitons. The latter, if it exists, can be called the Gravibreather (a term which captures the analogy between this solution and the well-known SineGordon breather). If it turned out that the Gravibreather can only represent the SA-bound state, i.e., if for the combinations SS and AA it would be impossible to have a solution of this kind, this would be proof of the existence of an attraction between gravisoliton and antigravisoliton and a repulsion between gravisolitons of the same charge. If so, the real metric of correct signature for the Gravibreather should follow from the SA-scattering solution by its analytic continuation to purely imaginary values of the relative collision velocity of the colliding gravisolitons. At the same time the analogous analytic continuation of the SS "and AA type solutions should lead to an unphysical (complex) metric tensor. It can be shown that all this is really the case. 2 However, here I will not go into the detailed proofs. The reason why all this is so is clear is again due to the existence of a regular map between the space of solutions to the Sine-Gordon equation and the space of solutions for the metric matrix g. Consequently that fact that in SineGordon theory there exist all three aforementioned solutions and that the breather represents the SA bound state only is exactly mirrored in our gravitational anzsatz (although the picture in the latter case is quantitatively much more cumbersome). One example of an exact gravibreather solution has been given 2 and some numerical results describing its behaviour are available. 3 References 1. V. Belinsky and V. Zakharov, Sov. Phys. JETP 48 (1978) 985. 2. V. Belinsky, Phys. Rev. D44 (1991) 3109. 3. P. Kordas, Phys. Rev. D48 (1993) 5013.

102

QUANTUM GRAVITY, THE PLANCK LATTICE AND THE STANDARD MODEL G. Preparata Dipartimento di Fisica dell'Universita di Milano and INFN, Sezione di Milano, via G. Celoria 16, 20133 Milano, Italy (E-mail: preparata@mi.injn.it) ABSTRACT A possible ground state of Quantum Gravity is Wheeler's "space-time foam", which can be modeled as a "Planck-lattice", a space-time cubic lattice of lattice constant ap ~ 1033cm , the Planck length. I analyse the structure of the Standard Model defined on the Planck Lattice, in the light of the "no-go" theorem of Nielsen and Ninomiya, which requires an extension of the continuum model through Nambu-Jona Lasinio terms, quadrilinear in the Fermi-fields. As a result, a theory of masses (of both fermions and gauge bosons) is seen to emerge that, without Higgs excitations, agrees well with observations.

1.

Introduction

The Standard Model (SM) of fundamental particle interactions represents the achievement of a long scientific path that through three decades of theory and experiment finally unifies the three different types of interactions (strong, electromagnetic and weak) in a single and well defined theoretical structure, that of Gauge Theories (GT). It is remarkable that throughout the intellectual adventure that leads to the firm establishment of the SM, Quantum Gravity (QG) is seen to play essentially no role: the gravitational interactions, after all, at the typical space-time distances of particle interactions ("-' 10 13 cm) are so incredibly tiny that the phenomenology of the SM can well do without them. Even though, and this is a tantalizing thought, the structure of QG too is that of a GT, whose gauge group, as Einstein and his friend Marcel Grossmann showed at the beginning of our Century, is just the Poincare group. It is only recently, after the completion of the SM synthesis, achieved at the beginning of the Eighties with the "announced" discovery of the weak gauge bosons (W±, ZO), that the widespread theoretical urge to go "beyond" the SM has put the limelight on QG as one of the "new" interactions that would be unified with those of the SM within the plethora of supersymmetric models, which have been proposed since the mid-Seventies. Curiously enough, the various supersymmetric extensions of the SM were so prodigal in new particles and interactions, that even the now discredited "Fifth Force" cquld easily and "naturally" be accommodated in their wide and flexible

103

framework. However, such renewed interest in QG as a limited, particular sector of an all-embracing, unified Theory of Everything (TOE), the main aim of Superstring Theories, is quite different in character from that line of thought which, at the periphery of particle physics, has been pursued by J .A. Wheeler and his school, that has been called Geometrodynamics. 1 The main focus of this latter research .pr.ogram, in fact, is to obtain a deeper understanding of the structure of space-time, the arena of fundamental particle interactions, that according to Einstein's General Relativity (GR) is fully determined by the dynamics of the gravitational field. But, as Wheeler correctly remarks, a quantized gravitational field does acquire an independent dynamical role, i.e. it is no more uniquely determined by the distribution of matter (the energy-momentum tensor). Thus it may well happen that space-time itself, i.e. the web of relationships among different physical events, owing to the fundamental quantum-field fluctuations acquires some very peculiar structure, for instance that of a "foam", whose discontinuities have the size of the Planck length ap ~ 10- 33 cm. In Wheeler's view, differently from what is being contemplated in Supersymmetric and String Theories, QG is not a sector of an extension of the SM, out rather the dynamical theory of space-time upon which the matter and fields of the SM stand and evolve. It is just this far reaching view that I will follow in this talk with the goal to understand if and how the SM can be formulated upon a hypothesized peculiar "foam" structure, that can be modeled by a simple discrete space-time Planck Lattice (PL), whose lattice constant is ap. This line of research has been pursued in the last three years in collaboration with the bright chinese theorist She-Sheng Xue.

2.

SM: what if space-time were a Planck lattice?

Let me first remind you very briefly what is the SM on a continuous, Minkowskian space-time (CSM). Its Lagrangian is (1)

where the gauge lagrangian is

CG(X) =

-~

(t c~vcaJJ.v + t A~vAiJJ.v + BJJ.vBJJ.V) + LWgiJpiJ!q + LW1iJpiJ!I, (2) a=l

.=1

I

q

and describes the fundamental matter fields iJ! 9 (quarks) and iJ! I (leptons) coupled to the gauge fields of the colour SU(3)-group (C~v)' the SU(2)L-group (A~/,) and the U(1)y-group (BJJ.v): TIl

¥

. n { 1 ) YL 1 YR}. Ifi Ti = '1:::t + 191'f "2 (1 -1'5 2 +"2 (1 + 1'5) 2"" + 192"1' 2 (1 -

1'5)

.

+ 193

qa Aa2'

(3)

with (4)

104

ol'A" - o"AI' + ig2 [AI" A,,] , OI'C" - O"CI' + ig3 [CI" C,,].

(5)

(6)

All this is of course very beautiful and elegant, giving blood and flesh to the simple and very general idea that all fundamental symmetries must be defined locally, thus tying intimately symmetries to space-time. As for the Higgs Lagrangian, its simplest form is (7)

with

a weak isospin doublet, (8)

and with the "ad hoc" negative mass squared term to ensure "spontaneous electroweak symmetry breaking" . And this is of course very ugly for: (a) the elegance of the gauge principle, as applied to the fundamental building blocks of matter leads to a massless world and in CSM no dynamical mechanism has been found that, respecting the Ockam's razor, would produce the observed massive world by using the two fundamental building blocks only: (b) to generate mass no better way has been found than graft upon the beautifully simple gauge lagrangian LC (2) the ugly Higgs-mechanism, induced by LHiggs (7), that (i) is motivated by an unpretentious pedagogical modeL developed with the aim to impressionistically describe the main features of the Meissner effect of superconductivity; (ii) extends in a totally arbitrary fashion the fundamental building blocks of nature, spin-l/2 fermions and vector bosons, to include a very odd scalar field ¢>; (iii) introduces the instability that leads to spontaneous chiral symmetry breaking by means of a totally arbitrary "negative mass squared" in liif (9);

105

(iv) introduces scalar local fields that in QFT are very peculiar objects, there being no way, besides "tuning", to keep its mass from diverging like the ultraviolet cut-off Auv. This is one of the main reasons why Supersymmetry, that could cure such disease, survived that remarkable "physics fasting" that goes on since more than twenty years. (c) the regularizationjrenormalization program of the QFT defined by bad shape, for

.c CSM

is in

(i) the lattice regularization is blocked by the Nielsen-Ninomiya "no-go" theorem (see below); (ii) dimensional regularization is still incapable of metabolizing no less than IS!

In spite of all these difficulties the low energy phenomenology of electroweak interactions is fundamentally decoupled from all such flaws, that are all of a conceptuaL theoretical nature, and stands as a beautiful confirmation of the simple and powerful 8U(2)L @ U(l)y gauge principle. And this is the strong message of 20 years of electroweak interactions. In view of all this, the question of the title of this Section appears much less philosophical and may well lead us to a SM without all the defects that have been exposed above, as I shall try to argue in the rest of this Lecture. As mentioned in the Introduction, I side completely with John Wheeler who sees the violent quantum fluctuations at the Planck scale to tear continuous space-time apart and create a foam like structure, full of voids and discontinuities*. Thus our question can be reformulated as: what if Wheeler's idea were right and space-time at distances smaller than ap would dissolve into the nothingness of wormholes? A positive answer would lead us to make the following hypothesis: spacetime is not a 4-dimensional continuum but rather a (random) Planck lattice. As a consequence the SM and all QFT's, that so far have been defined on a continuous manifold, must be reformulated as Lattice QFT's. It is amusing to note that our hypothesis, if right, would turn the various "theories of everything" into theories of nothing, for at the scales where superstrings become relevant space-time would dissolve. But if we are to formulate the SM as a chiral Lattice Gauge Theory (LGT), our research program clashes immediately with the "no-go" theorem of Nielsen and Ninomiya,3 which informs us that we cannot simply transcribe the usual SM on a lattice, for when a LGT is chiral the low energy spectrum (the massless fermions) gets doubled in such a way that the fundamental chiral symmetry is violated. The physical origin of such unpleasant result is the peculiarity of the dispersion relations in a discrete *The analogy with the non-trivial vacuum structure that has been demonstl'ated 2 to emerge in another non-abelian GT, QeD, is to my mind particularly relevant. I hope to be able to make it more precise in the near future

106

= !(1 -

space-time: for a Weyl fermion [\IlL

w(i)

15)\II] the free-field dispersion relation is

= .!. L a

(10)

sin(pia),

i

yielding low-energy (o) X(c..>o)

+ QX(2D-c..>0)

0 (20)

K(2D-c..>0) X(2D-c..>0)

+ QX(c..>o)

0

with the natural "anomalistic" frequency (frequency of perigee occurence) gi ven by the vanishing of the determinant of coeffients of these coupled homogeneous equations: =

(21)

0

matrix inverse being indicated where appropriate. Since the anomalistic frequency obtained from Eq. (21) is very close to the sidereal frequency (2n/(~-~o) = 8.9 years), a sidereal frequency perturbation is substantially amplified (being close to resonance) and fulfills:

og(c..» K(2D-c..» X(2D-c..»

+ QX(c..»

(22)

0

with solution: 1

K(c..»

-Q K(2~-c..» Q

0 g(c..»

(23)

which gives a sidereal radial amplitude of about:

A (c..»

• -l. log(c..» I

(24)

2 c..> (c..> - c..> 0)

Measurement of this sidereal amplitude to about a centimeter precision with lunar laser ranging data constrains the magni tude of any differential, cosmically originated (fixed in inertial space) acceleration between the Earth and Moon to be less than 3 10- 14 cm/sec 2 15,16,21. If "dark matter" produces the bulk of our solar system's acceleration of about 10- 8 cm/sec 2 toward the galactic center, then the free-fall rate of ordinary matter toward "dark matter" is universal to significant precision --- implying that any non-metric long range interaction between the two kinds of matter is very weak, if not zero. The synodic month period perturbations are given by:

xeD) + QX(D) with solution: K(D)

=

g(D)

(25)

(26)

128

To adequate approximation for our present needs, this inverse matrix is given by: 1 - 2o - + •• 1 K(D) +(}



1 [ 2Ca>Cl

01 1-76) + ...

1

o

-2 +2- + ..

(,)

(,)

C

-2+2-+ ..

(27)

o

4 - 2 - + •. (,)

(,)

Solution of Eq. (21) yields the leading Newtonian and post-Newtonian order contributions to the anomalistic frequency:

G'm _ 20 2

II!

p3

_

0

225

16

O~D -

(')0

2 G 2m (5y+4) _ _ + •..

(28)

C 2 p4

Combining this with Eq. (16) gives the more observationally useful perigee precession rate: ..

3 02 225 0 3 - - + - - - - + ••• 4 (,)2 32 (,)3

(29)

+ (2+2y-P)

GIn

c 2p

+ ...

The de Sitter precession term is the dominant relativistic correction, but an additional relativistic correction from the solar tidal acceleration is about ten percent as large, while the Earth's own relativistic contribution is a further order of magnitude smaller. In General Relativity: + •••

(30)

M/R and m/p are about 1/13.4,10- 8 and 10- 11 , respectively. The quarter century of lunar laser ranging has achieved measurements of the sidereal and synodic frequencies to accuracies: Q/6),

'"

2 . 10-12

and

(31)

thereby resulting in a one percent confirmation of the de Sitter precession as corrected by the additional relativistic terms l7 ,18.

129

The radial amplitude of the "evection" perturbation in Eq. (20) is given as a ratio to the "eccentric" radial amplitude, and only this ratio has useful theoretical significance. The overall amplitude of the unforced natural motion is, like its phase, a free initial condition of the motion. Solving Eq. (20) gives:

(32)

s-l:il!(1 _ 21 +4{i -3 8

6

6)

GM)

c 2R

+ ...

In General Relativity theory this represents about a 1.5 centimeter modification in the "evection" amplitude for given "eccentric" amplitude. The "variation" amplitude is also rescaled by general relativistic corrections. The orbital perturbation of frequency 2D is the dominant part of the total tidal distortion of the lunar orbit first studied in its entirety by Hill":

X2D (0)

..

P

~ (0)

cos2Do

To leading order amplitudes are:

P ..

02 6)2:

o(

P -1 - 1:

the

~

+

T

tt (0) sin2.Da

radial

and

tangential

"variation"

+ ... )

(33)

02

T .. -2£p (..!! 6)2 0 8

+

~l! 12 6)

+ •.• )

The radial amplitude of "variation" in General Relativity is about 4 centimeters less than in Newtonian theory. However, the relationship between this radial amplitude and mean orbital radius given by Eq. (33) is actually employed in reverse of what one might expect: because the mean orbital radius is so weakly determined (due to strong correlations wi th the positions of lunar reflector stations which are poorly determined because of almost total absence of change of lunar "face") the well-measured radial "variation" amplitude is used to estimate the Moon's mean orbital radius. Perhaps the most significant consequence of the "variation" perturbation of the lunar orbit is its interaction with the synodic month period perturbation of the lunar orbit. 2D frequency perturbations mix with D frequency perturbations to produce not only 3D frequency perturbations (of little interest to us here), but they also feedback to enhance the D

130

frequency perturbations. This is quantified by solving Eq. (26) with the perturbing acceleration given by the appropriate terms from Eq. (14). The resulting synodic month period amplitude is: A(D)

'"

Po

(34)

The

determinant

IK'(D) +

QI

is

surprisingly

small

when

evaluated for the lunar orbit: this manifests itself through a common amplification denominator which almost doubles the size of all lunar synodic month period perturbations. For an orbit outside the Moon (p=l. 8pe with period of about two months) the indicated determinant goes through zero, and a resonant amplification singularity occurs. For such an orbit the synodic month (driving) frequency becomes equal to the anomalistic (natural) frequency of the orbit. If a very high performance drag-free system can be developed for a satellite, laser ranging to a satellite in an orbit close to this resonant orbit could perhaps provide particularly precise tests of post-Newtonian gravity. Being of some possible interest for the future, the above quantitative features of the amplification of synodic month amplitudes by interaction with the tidal distortion of the orbi t have been confirmed by computer integration of the complete equations of motion. The finite general relativistic correction to the synodic month amplitude in Eq. (34) (proportional to y+2) is, including amplification factor, about 5 centimeters in magnitude. The overwhelmingly dominant sensi tivi ty of the lunar orbi t to possible post-Newtonian modifications of General Relati vi ty is through the term in Eq. (34) proportional to the Earth's gravitational self energy as given in Eq. (5). This term, though it vanishes in General Relati vi ty, presently provides the strongest experimental test of the non-linear structure of gravitational theory. After amplification it

131

becomes: () A (D)

..

(4

P- 3 - Y )

(12.8

meters)

(35)

The realistic experimental uncertainty in the measurement of this ampl i tude is about 1.3 cm'6 - '9 , so: (36) Using the independent experimental bound on y obtained from solar system observations of light propagation20 :

Iy -1 1

~

2 ' 10-3

(37)

then permits use of Eq. (36) as an excellent bound on B:

(38) If the hypothesis being tested is the possibility of a BransDicke type of scalar-tensor theory of gravity, however, the PPN

p

coeffient is necessarily one, so the constraint from

lunar laser ranging on the PPN y coeffient is twice as strong as the constraint from light propagation in the solar system. With: 1-y

=

1

2 + (o)S-D

the equivalent constraint on the Brans-Dicke dimensionless parameter is: CJ)S-D

>

1000

(39)

This work was supported in part by N.A.S.A. Contract NASW-4840 and also by the Gravity Probe B program at Stanford University.

132

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

W. De Sitter, Mon. Not. Roy. Ast. Soc. 77 (1916) 155. R. Baierlein, Phys. Rev. 162 (1967) 1275. K. Nordtvedt, Phys. Rev. 169 (1968) 1017. K. Nordtvedt, Phys. Rev. 170 (1968) 1186. K. Nordtvedt, Astrophys. J. 161 (1970) 1059. K. Nordtvedt, Phys. Rev. 043 (1991) 3131. K. Nordtvedt, Phvs. Rev. 047(8) (1993) 3633. B. Bertotti, I. Ciufolini, and P. Bender, Phys. Rev. Lett. 58 (1987) 1062. I. Shapiro, R. Reasenberg, J. Chandler, and R. Babcock, Phys. Rev. Lett. 61 (1988) 2643. B. Shahid-Saless, Phys. Rev. 046 (1992) 5404. C. Will and K. Nordtvedt, Astrophys. J. 177 (1972) 757. G. Hill, Amer. J. Math. 1 (1878) 18. K. Nordtvedt, Astrophys. J. 297 (1985) 390. K. Nordtvedt, Astrophys. J. 407 (1993) 758. K. Nordtvedt, Astrophys. J. 437(1) (1994) 529. J. Muller, M. Schneider, M. Soffel, and H. Ruder, these proceedings (1994). J. Dickey, P. Bender, J. Faller, X Newhall, R. Ricklefs, J. Ries, J. Shelus, C. Veillet, A. Whipple, J. Williams, and C. Yoder, Science 265 (1994) 482. X X Newhall, J. G. Williams, and J. O. Dickey, these proceedings (1994). J. Chandler, these proceedings (1994). I. Shapiro, in General Relativity and Gravitation, 1989, ed. N. Ashby, D. F. Bartlett, and W. Wyss (Cambridge University Press 1990) p. 313. K. Nordtvedt, J. Muller and M. Soffel. Astron.Astrophys. Lett. (1995) in press.

133

GRAVITOELECTROMAGNETISM: JUST A BIG WORD? ROBERT T. JANTZEN Department of Mathematical Sciences, Villanova University, Villanova, PA 19085, USA, and ICRA, Dipartimento di Fisica, University of Rome, 1-00185 Roma, ITALY PAOLO CARlNI GP-B, Hansen Labs, Stanford University, Stanford, CA 94305, USA and International Center for Relativistic Astrophysics, University of Rome, 1-00185 Roma, Italy and DONATO BINI Physics Department and International Center for Relativistic Astrophysics, University of Rome, 1-00185 Roma, Italy ABSTRACT Arguments are made in favor of broadening the scope of the various approaches to splitting spacetime into a single common framework in which measured quantities, derivative operations, and adapted coordinate systems are clearly understood in terms of associated test observer families. This "relativity of splitting formalisms" for fully nonlinear gravitational theory has been tagged with the name "gravitoelectromagnetism" because of the well known analogy between its linearization and electromagnetism, and it allows relationships between the various approaches to be better understood and makes it easier to extrapolate familiarity with one approach to the others. This is important since particular problems or particular features of those problems in gravitational theory are better suited to different approaches, and the present barriers between the proponents of each individual approach sometimes prevent the best match from occurring.

1.

Introduction

Before I explain what I mean by this big word "gravitoelectromagnetism" which is still not a part of our relativity jargon, I must admit my reluctance to give a plenary talk on mathematical formalism, which is what I am about to do. I am not specialized in the various difficult and challenging physical problems that my remarks may touch upon, but I believe that the mathematical tools of GEM do help us better understand the way in which spacetime structure enters some of those difficult calculations. To justify my talk I can look to the short objective statement found at the beginning of the proceedings of each Marcel Grossmann meeting, where one finds the two phrases " ... emphasis on mathematical foundations ... " and " ... deepen our understanding of spacetime structure ... ". The conference itself derives its name from the standard lore about Einstein's mathematician friend who supposedly pointed out some mathematical tools crucial for the development of general relativity. In a similar way I wish to point out some mathematical tools for splitting spacetime which together are important in many different applications in general relativity and which separately are continuously being used, but which have failed

134

GR

spatial Geometry

} observer-space geometry

gravito- Electricity } 0 b server ki nemat'lCS gravitoagnetism

M

Figure 1: GEM: Spacetime Splitting as a Nonlinear Analogy/Generalization of Electromagnetism, or the Relativity of Splitting Formalisms

to find a common home as a standard part of the foundations of general relativity and are not even always recognized when they are being used. There are two difficulties which limit the accessibility of this whole set of tools. • The first is the existence of a "literature horizon" beyond which many of the books and articles which do discuss these tools have fallen. The quantity of literature in relativity and gravitation, as in other fields, has wildly grown over the decades and many results are simply buried in the past, lost in the sheer volume of publications. References familiar to one generation are often less so or unknown to the successive one, an effect which increases with the passage of time . • The second difficulty is, even if one is successful in breaking through the liter· ature horizon and in mining its hidden wealth for the relevant gems, one ofte:a finds either antiquated notation or lack of a common mathematical framework in which to formulate a coherent approach to all of the various pieces one finds, some familiar and some not. What we have attempted to do is to describe such a common framework and notation, making more precise certain notions and their relationships to each other, and it seems natural to use the word "gravitoelectromagnetism" (GEM) to refer to this way of looking at general relativity. With the increasing widespread use of the terms "gravitoelectric (GE) field", "gravitomagnetic (GM) field", and "gravitomagnetism", I think most of us are aware of the analogy between the linear theory of electromagnetism and the linearized theory of general relativityl and related approximate theories, an analogy which seems to lend itself immediately the longer word "gravitoelectromagnetism." The analogy is in fact with the space-plustime splitting of electromagnetism in flat spacetime. The GE field is associated with local accelerations and in the post-Newtonian limit, related to the Newtonian gravitational field, while the GM field is associated with local rotations and the so called effect of the "dragging of inertial frames," and is a new feature of relativistic gravitational theories compared to Newtonian gravity. The splitting of fully nonlinear general relativity gives rise to a nonlinear analogy with flat space electromagnetism whose linearization leads to the more familiar linear analogy, but this is not well known at all. The rich structure of general relativity and curved spacetime allows many variations of the simple act of splitting

135

spacetime into space plus time. Unfortunately these variations have developed in isolation from each other in styles which make it difficult to relate to each other or analyze in terms of their geometric relationships. Each of the various approaches to "splitting spacetime" is equivalent to describing what a family of test observers in spacetime measure along a certain family of trajectories in spacetime, with respect to a certain class of adapted spacetime frames evolving in a specified way along those trajectories. These latter assumptions are equivalent to the choice of derivative used to measure evolution along that family of trajectories. Of course to treat these questions properly, one needs to introduce a precise mathematical description which necessarily involves some investment of time to become familiar with, but the end result is a concise framework within which one can unambiguously study otherwise confusing issues. The details are completely straightforward. One need only reformulate the mathematics of special relativity, usually treated in terms of the affine structure of Minkowski spacetime, in such a way that it respects the manifold structure of this spacetime, leading automatically to the appropriate (often multiple) generalizations to curved spacetimes. This natural marriage of special relativity, curved manifolds, and modern mathematical methods, and the interpretation of adapted coordinate systems in this context, though not conceptually difficult, has not found its way into the standard toolbox of relativists, even though parts of it find widespread application in gravitational physics. Before discussing the possibilities in more detail, it is useful to mention three collections of names which are associated with the most visible splitting approaches, each of which is anchored in the literature by high profile books or articles. 1. ADM: Arnowit-Deser-Misner,2 MTW Gravitation 3 (Wheeler: 4 lapse and shift). [two of the most well known anagrams in relativity] [ADM approach motivated by quantum gravity] "slicing approach" .

2. Landau-Lifshitz, Classical Theory of Fields. 5 [roots in 40's edition, reports stationary case of 50's work] "threading approach"

3. Ehlers-Hawking-ELLIS, cosmology review articles.6-9 [kinematical quantities of a unit timelike vector field] "congruence approach" Of course these are only the tip of the iceberg so to speak, with foundations whose ''first generation" of authors might also include among many others: Einstein, Bergmann, Lichnerowicz, Moller, Zel'manov, Cattaneo, Ferrarese, Choquet-Bruhat, and Dirac (see Ref. [10] for references). Each of these three approaches ignores the existence of the others. Each has its own peculiar established notation that makes comparisons more difficult. Each has certain applications which seem more natural arenas for their use.

136

classical GR foundations quantum gravity minisuperspace cosmology

exact solution techniques black holes perturbation problems: FRW gauge invariant cosmology isolated systems

inertial forces parametric manifolds

initial value problem, degrees of freedom, dynamics canonical approach, Ashtekar variables exact solutions, qualitative analysis, multidimensional theories, classical and quantum 2 or 1 Killing vector cases membrane paradyme, numerical work Bardeen Ellis, Bruni, et al Ehlers et al, Newtonian limit, PN initial value problem Damour et al PN Celestial Mechanics Thorne, Forward, PN Theory Abramowicz et al Perjes, Boersman and Dray

1 1 l

2, 1 1 1 3 2,3 2 2 2 2

Table 1: Some applications of spacetime splitting. The number in the last column refers to the three splitting schools listed in the text.

Listing some ongoing applications helps justify giving some attention here to the whole idea of spacetime splittings and their relationships. In these applications and many others one or more of these splittings naturally occurs or is an important tool. Though Einstein made a great leap forward by unifying space and time into a single object, we can only experience it through our space-pIus-time perspective as observers within it, and such splittings help us interpret spacetime geometry in terms of that perspective. This is probably the most important reason why spacetime splittings occur so frequently in general relativity. Table 1 is a short list of some topics in which spacetime splittings play an important role. Okay, so perhaps we can agree that spacetime splitting is a widespread activity, although those of you who actually do it probably think whatever you are presently doing is just fine and we don't really need to talk about the bigger picture. It is exactly this attitude which has maintained the fragmentation that has characterized the topic for decades. It would be fun to try to trace the history of these ideas in detail, but I think it is a better investment of time to try and communicate some sense of what a common framework for them is and how it can help us better understand certain aspects of general relativity.

2.

Splitting: The Basics

First, given the unified concept of spacetime which is the arena of general relativity, what does it mean to speak of space and time separately? These are in fact

137

complementary notions related to two distinct ways of characterizing time itself. The first of these is embodied in our wristwatches that most of us are probably wearing. It is our own local time reference that we carry with us whereever we go and use to mark off events along our worldline in spacetime. The second notion might be exemplified by a VCR setting to record a TV program (an analogy not meant to be taken too seriously!). At a certain moment of time within a given geographical area, the program begins, for everyone in that area that cares to tune in. This is a synchronization of their local times, which is also a way of identifying the concept of space within spacetime. In order to get started, let's use these two notions of time to give a broad characterization of the different splitting approaches, as sketched in Table 2. One can either do a partial splitting or a full splitting. A partial splitting (choice of time gauge) is based on a choice of independent local time or globally synchronized time (space) respectively made by specifying a distribution of local time directions through a unit timelike vector field u, or by specifying an integrable distribution of spacelike local rest spaces LRSu, where u is a vorticity-free unit timelike vector field orthogonal to each such space. Both partial splittings are equivalent to specifying only a timelike congruence (the worldlines of u) or only a spacelike slicing (the integrable hypersurfaces of the distribution of 3-spaces orthogonal to u). The additional complementary choice of space in a full splitting determines the spatial gauge freedom for the given choice of time in each category, so that one has a pair consisting of a slicing and a threading with the causality condition imposed on the one associated with the choice of time. A full splitting is most easily described locally by using an adapted coordinate system {t, x a } which incorporates the additional structure of a choice of parametrization for the family of slices (time function), and a choice of parametrization for the threading congruence curves (spatial coordinate system). It seems reasonable to call this structure modulo spatial coordinate transformations a parametrized nonlinear reference frame, where the word "parametrization" refers to the specific choice of time function. Some of the variations of the main splitting schemes found in the older literature depend on this additional time function rather than just the family of slices. Given a partial or full splitting of spacetime, two key ideas characterize the splitting process: measurement and evolution. One interprets u as the 4-velocity of a family of test observers which "measure" both spacetime tensor fields and spacetime differential operators and tensor equations involving these quantities. This is done as in special relativity but independently on each tangent space rather than globally on all of spacetime at once as on Minkowski spacetime referred to a family of inertial observers in special relativity theory. It is accomplished simply by orthogonal projection of everything in sight, based on the underlying orthogonal decomposition of each tangent space into a local time direction (along u) and an orthogonal local rest space LRSuo The temporal projection, being associated with a I-dimensional subspace, may be simplified to a scalar projection, thus leading to a family of "spatial" tensor fields or "spatial" tensor operators of different ranks when decomposing a single spacetime tensor or tensor operator. The contraction of

138

time: 1

space: 3

("single-observer" time)

("moment of time")

time: 1

space: 3

1((

PARTIAL SPLITTING u or LRSu GE, GM fields

(3) congruence p.o.v.

FULL SPLITTING parametrized nonlinear reference frame: {t,x a } GE, GM fields and potentials

time

+ space: 1 + 3

z$

L __ _

,_~

@ (4) hypersurface p.o.v. space

+ time:

3

+1

~ '

I

(2) threading p.o.v.

(1) slicing p.o.v.

TIME gauge

timelike observers +-+ threading

spacelike local rest spaces (timelike normal observers) +-+ slicing

SPACE gauge

arbitrary synchronization of observer times +-+ slicing

arbitrary identification of "points of space" +-+ threading

Table 2: A characterization ofthe different points of view (p.o.v.) that may be adopted in splitting spacetime. Solid lines in diagrams imply the use of the appropriate causality condition while dashed lines indicate that no causality condition is assumed. The hypersurface p.o.v. is essentially equivalent to the vorticity-free congruence p.o.v.

139

any index of a spatial tensor field with u a Qr U a is zero. Evolution is the description of how fields "evolve in time" and is equivalent differentially to specifying a direction of differentiation along which the evolution will take place, easily given as the tangent to a congruence of evolution curves, as well as a way of evolving a reference spatial frame along those curves to measure the evolution against. This information can be packaged in a single temporal derivative operator along the evolution congruence which acts on the spaces of spatial fields. The congruence of measurement worldlines and the congruence of evolution curves may coincide (congruence, hypersurface, and threading points of view) or be independent (slicing point of view). The threading point of view is just a more explicit representation of the congruence point of view which takes advantage of the additional (arbitrary) synchronization information represented by the slicing. The slicing point of view is instead a 2-congruence approach, one for measurement and one for evolution, so that the test observers are in relative motion with respect to the curves describing the evolution. One may introduce the quotient space of the spacetime by the threading curves, or "computational 3-space," and the quotient space by the observer worldlines, or "observer 3-space." These coincide except for the slicing point of view. In the full splitting one has a I-parameter family of embeddings of the computational 3-space into the original spacetime generating the nonlinear reference frame and natural isomorphisms between the computational 3-space tangent spaces and the corresponding local rest spaces associated with the test observers on the spacetime, enabling one to consider the spatial measured tensor fields as time-dependent tensor fields on the computational 3-space.

3.

Splitting: A Few Details

The measurement process is just an orthogonal decomposition based on the following representation of the identity tensor in terms of the temporal and spatial projections associated with the test observers with 4-velocity u a

(1) Applied to a vector field it leads to a scalar and a spatial vector, once one discards factors of u a ( or in general of U a as well) with free indices

(2) This decomposition may be extended to any rank tensor field in an obvious way, yielding a family of spatial tensor fields of all ranks up to the original rank. For a single nonzero mass test particle world line with timelike unit 4-velocity U a and corresponding 4-momentum pa = mU a , this process yields the usual special relativistic quantities, namely the gamma factor, relative velocity, energy, and spatial momentum a _ (-y(U, u), ,(U, u )v(U, u )a) , (3) pa _ (E(U,u),p(U,u)a) ,

u

140

where ,(U,u) = [1-llv(U,U)Wt 1 / 2 and IIXlj = IX!3X!311/2. For the spacetime covariant and contravariant metrics, the only nontrivial fields this yields are the spatial such metrics (let X· and XU be the index-free notation kernel symbols for tensors whose indices have all been lowered and raised respectively) (4) P(u)"'!3 = [P(u)g]"'!3 , while the only nonzero field resulting from measuring the oriented unit volume 4-form 17"'!3-yslgll/2€"'!3-YS is the unit spatial volume 3-form

(5) which may be used to introduce the spatial cross product of two vectors

(6) and to introduce the spatial duality operation *u on antisymmetric spatial tensor fields. The spatial dot product is defined analogously

(7) One may also measure differential operators: the covariant derivative \7, the exterior derivative d, and the Lie derivative £. In this process certain spatial differential operators arise, where a spatial operator is one which maps the space of spatial tensor fields into itself. These may be distinguished as spatial or temporal derivative operators, according to the direction along which they differentiate. It is convenient to introduce the spatial covariant derivative \7(u) = P(u)\7, which is a spatial derivative operator, and the spatial Lie derivative £(u)X = P(u)£ X, from which both Lie derivatives along spatial and temporal directions may be obtained. In each case all free indices are spatially projected after the spacetime derivative acts on a tensor field. The spatial Fermi-Walker derivative along u, \7(fw)(U) = P(u)\7 u (so named since it coincides with the spacetime Fermi-Walker derivative along u when acting on spatial tensor fields), and the temporal Lie derivative \7(1ie)(U) = P(u)£u are both temporal derivative operators. With the spatial covariant derivative, spatial dot product and spatial cross product, and obvious definitions of gradw curl u , and divu, one can mirror all the usual operations of 3-dimensional vector analysis, or with the introduction of the spatial exterior derivative d( u) = P( u)d and the spatial Lie bracket [X, Y](u) = P(u)[X, Y], all of the corresponding exterior derivative algebra. The various covariant derivatives of the spacetime and spatial metric are all zero

so index raising/lowering co=utes with these derivatives. The covariant derivative \7!3u'" may be measured to yield the so called kinematical quantities of u"'.5-8 The

141

measured fields are a scalar (zero), a vector,: the acceleration a(u)a = [\7(fw)(U)U]C", a I-form (zero), and a spatial tensor \7 (U),8u a == - k( u yx,8, the vanishing of the two fields due to the unit condition on u(Jt: u(Jtu(Jt = -1The spatial covariant derivative of u(Jt may in turn be decomposed into its symmetric and antisymmetric parts, yielding the vorticity (rotation) tensor w( u )(Jt,8 and the expansion tensor B(u)(Jt,8 (recall [uP](Jt = u(Jt) 1

P

w(u)(Jt,8 = -\7(u)[(Jtu,8) = 2[d(u)u ](Jt,8 ,

(9)

B(u)a,8 = \7(u)«(Jtu,8) = \7(Jie)(u)g(Jt,8 = \7(Jie)(u)P(u)(Jt,8 . The spatial dual of the vorticity tensor yields the spatial vorticity (rotation) vector [w(u)](Jt = w(u)(Jt (the "overarrow" on the kernel symbol avoids ambiguity)

(10) The expansion tensor may itself decomposed into its pure trace, the expansion scalar 0( u) = B( u)(Jt,8 = \7 (Jtu(Jt, and the tracefree part, the shear tensor a( u)(Jt,8 =

B(u)(Jtp -

~e(u)c5(Jt,8

.

The expansion, shear, and rotation describe the relative motion of neighboring test observers with respect to a set of Fermi-propagated spatial triad vectors along each test observer world line, which is encoded in the relationship between the two types of temporal derivatives

(ll) A vector field Y(Jt is called a "connecting vector" if £uY(Jt = 0, i.e., if it is invariant under dragging along by the flow of u (Jt. If y(Jt is small compared to the characteristic distances over which u(Jt itself varies, it may be interpreted as connecting a point on a given observer worldline to a point on a neighboring one whose position is identified with the tip of y(Jt in the tangent space. The spatial projection X(Jt = [PC u )y](Jt may be interpreted as the spatial position vector of this neighboring observer in the local rest space LRSu , i.e., the position of the neighboring test observer as seen by the first one. It satisfies \7 (Jie) ( u )X(Jt = 0, which means that compared to a set of orthonormal spatial frame vectors {e::'} which are Fermi-Walker transported along the u congruence: \7 (fwi u )e:: = 0, the "relative position vector" xa of neighboring observers undergoes a combined scaling, (volume-preserving) deformation, and rotation of the local rest space LRSu whose rates are determined respectively by the expansion scalar, the shear tensor, and the vorticity tensor.&-9 The index-free formula dUP(X, Y) = XuP(Y) - YuP(X) - uP([X, Y]) applied to spatial vector fields X and Y (orthogonal to u a ) immediately reduces to

2W(u)· (X

XU

Y) = 2w(u)P(X, Y) = duP(X, Y) = -uP([X, Y]) .

(12)

The measurement of the Lie bracket of two spatial vector fields then becomes

[X, y](Jt = [X, Y](u)(Jt + 2wP(u)(X, Y)u(Jt .

(13)

142

-xr;{y, X]( u ).6.a.6.b - Y . 6 . b L J Y.6.b X.6.a Figure 2: The commutator as a closer of quadilaterals. Measuring the "closer" commutator expression yields the spatial closer of the spatial projection of the quadrilateral and the synchronization defect of the "closed loop."

The spatial Lie bracket of two spatial vector fields describes the "closure of their quadrilateral" (see box 9.2 of Misner, Thome, and Wheeler 3 and Fig. 2) projected into the observer local rest space, while (twice) the vorticity tensor evaluated on them describes the failure of the two paths from the origin to the open vertex to remain synchronized with respect to the observer, equaling the change in observer proper time between the two events. Looked instead as a closed loop around the quadrilateral plus the commutator closer, the latter factor describes the synchronization defect (change in observer proper time from the beginning to the end) of the spatially closed loop in the local rest space of the observer.

4.

Measuring the Intrinsic Derivative Along a Curve

Suppose one has an arbitrary parametrized curve c(A) in spacetime with tangent vector e'(A)'" = [de(A)/dA]'" and one wishes to measure the intrinsic or absolute derivative D / dA along this curve. This is important for measuring the equations of motion of a test particle following a geodesic or under the influence of some applied force or force field, or for studying more general curves in spacetime. For example, one may easily introduce a Serret-Frenet frame along any nonnull curve to study the differential properties of the curve itsel£ll and then repeat the process from the point of view of the family of test observers. The intrinsic derivative along the curve is defined so that if X'" is an arbitrary vector field on spacetime, then (notationally suppressing the dependence of both sides on c( A))

(14) It is then restricted to vector fields defined only along the curve so that when extended smoothly to a vector field defined on the spacetime the previous result is obtained, i.e., by using the second of the two formulas. Measuring the tangent vector

e'(A)'"

+-+

(-e'(A)I3u(3, [P(u)e'(A)l"') ,

(15)

one can introduce two different new parametrizations of the curve valid respectively when this tangent vector is not orthogonal to or proportional to u itself (not purely

143

de( A)j df(c'(A),U)

[-u . e'(A)]u

de(A)jdA

u

v(e'(A), U) P(U)e'(A)

I

j

= e'(A)

de(A)jdr(cl(A),U)

i/(e'(A), U)

Figure 3: Measuring the tangent vector to a curve and re-parametrizing the curve in terms of an observer-proper-time or observer-spatial-arclength parameter.

spatial or temporal)

dr(c'(A),U)j dA = -e'( A)fJ UfJ ,

df(c'(A),U)jdA = IIP(u)e'(A)11

(16)

as illustrated in Fig. 3. The first re-parametrization corresponds to the (continuous) sequence of observer proper-time differentials of the I-parameter family of observers which cross paths with this curve, while the second corresponds to the sequence of spatial arclength differentials of the relative motion seen by this family. One may introduce the relative velocity

(17) and the unit vector i/( e'(A), u Y>: = Ilv( e'(A), u )11-1v( e'(A), u y>: specifying the direction of relative motion as long as the tangent vector is not spatial. Then the two new parameters are related to each other by

dC(c'(A),U)jdr(c'(A),U) = Ilv(c'(A),u)11

(18)

as long as the tangent is not purely temporal or purely spatial. Using the chain rule one can re-parametrize the intrinsic derivative to correspond to these two new parametrizations to obtain two new derivatives DXO. = r above) is the proper-time-parametrized world line of a test particle of nonzero mass m with 4-velocity UO: = [dc(r)/dr]O: and acceleration a(U)O: = DUO:/dr. (Only if UO: is itself a vector field on spacetime does DUO:/dr = 'VUUO:.) If it is moving in spacetime under the influence of a force f(U)O:, which measures to ("((U, u )P(U, u), "((U, u )F(U, U)0:), where PCU, u) is the observed power and F(U, u)O: the observed spatial force as in special relativity, one can measure the equation of motion ma(U)O: = feU, u )0:, leading to a scalar and spatial vector equation. For example, if the test particle has electric charge q and the applied force is the Lorentz force f(U)O: = qfO:pUP due to an electromagnetic field fo:p, then the spatial force is F(U, u)O: = q{ E( u t + B( u)O: pv(U, u )p}

= q{E(u)O: + [v(U,u) xuB(u)]O:,

(26)

in terms of the observed electric and magnetic vector fields. The measured equation of motion is then

D(tem)(U, u)p(U, Ut/dT(U,u) = F~~m)(U, ut + F(U, u)O: , D(tem)(U, u)E(U, U)/dT(U,u) = [F~~m)(U, u)p + FCU, u)p] v(U, u)f3 ,

(27)

where, for example, the Fermi-Walker apparent spatial gravitational force is

F~~)(U, u)O: = m"((U, u){ -a( u)O: + k( u)O: pv(U, u)f3} = m,,(CU, u ){g( u)O: + H(fw) (u t pv(U, u )p} = m"((U, u){g( u)O: + ~[v(U, u)

H(u)t + H~)(u)O:pv(U, u)i3} . (28) The clear analogy with electromagnetism leads g( u) = -a( u) to be called the gravitoelectric (GE) field, H( u)O: = 2w( u)O: the gravitomagnetic (GM) vector field (its coefficient 1/2 above has the value 1 for the other choices), and H(tem)(u)O:p the Xu

gravitomagnetic tensor field, whose symmetric part is proportional to the expansion tensor of u which in turn is proportional to the spatial Lie derivative of either the spacetime or spatial metric. (The gravitoelectromagnetic terminology is due to Thorne. 13, H) Thus the additional feature of spatial geometry in gravitation contributes its effect on the left hand side in the spatial intrinsic derivative through its spatial derivatives as well as on the right hand side through its Lie (temporal) derivative in the symmetric part of the GM tensor field (and of course through spatial inner products). The electromagnetic analogy extends to the measurement of the Einstein equations (together with the identity d2 = 0), which leads to four equations (two pair of scalar and spatial vector equations) which are a nonlinear generalization for

u'

147

the gravito-vector fields of equations analog9us to the corresponding four measured Maxwell equations for the electric and magnetic vector fields and an spatial symmetric tensor equation due to the additional feature of spatial geometry.l0 The linearized version of these equations leads to equations rather similar to Maxwell's equations 1, 10, 15, 16

6.

Measured Potentials for the Gravito-Fields

The test observer covariant 4-velocity U a and the spatial metric P( U )a/3 act as potentials for the spatial gravitational force terms and spatial connection which appear in the equation of motion of a test particle. Although the GE and GM vector fields result from the measurement of the exterior derivative of the 4-potential [du b]a/3, in order to have a scalar and spatial vector potential for the gravito-vector fields analogous to the electromagnetic case, one must introduce a full splitting of spacetime, namely a parametrized nonlinear reference frame consisting of a parametrized slicing and a threading, with the appropriate causality properties for each point of view. A parametrization for the threading (a spatial coordinate system) in addition provides explicit potentials for the spatial connection coefficients themselves. Suppose {t, x a } are local coordinates adapted to the parametrized nonlinear reference frame. These provide an explicit representation of the associated threading and/or slicing points of view. One can then parametrize the spacetime metric components in this local coordinate system in terms of the observer orthogonal decomposition of the tangent space, introducing the lapse functions, shift vector field and I-form, and spatial metrics threading: slicing:

ds 2 = _M2(dt - M adx a)2 + 'Yabdxadxb , ds 2 = _N 2 dt 2 + gab(dx a + Nadt)(dx b + Nbdt) .

(29)

Fig. 4 compares the geometrical interpretation of the lapse and shift in the threading and slicing points of view, using some index-free notation (no indices but Xl> for Xa or X for xa when ambiguity exists). In analogy with electromagnetism, these quantities act as the scalar and vector potentials respectively for the gravitovector fields threading:

slicing:

g(m)a = -V'(m)alnM - V'(lie)(m)8/Ot Ma , H(m)a = M[curlmMr , g(n)a = -V'(n)alnN , H(n,8/Ot)a = N- 1 [curl n N]a .

(30)

Note the absense of a vector potential term in the slicing GE field and keep in mind from Fig. 4 that the relative velocities between the test observer and the orthogonalslicing/threading directions are respectively MM a and N- 1 N a when comparing the two GM vector relationships. One can also express the gravitomagnetic tensor field and the spatial connection coefficients in terms of the measured metric quantities.

~

The Seventh Marcel Grossmann Meeting Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 09/18/17. For personal use only.

T( threading)

n

N- 1 8/8t

Nn

8/8t = Nn

LRSn

Xi +6x i

Xi t

+ 6t

LRSm

Xi - 6x i t

Xi

+ 6t

6t8/8t

LRSn

t

t

THREADING 6t = Mi6xi, 67m = M6t Figure 3. See adjacent page for caption.

SLICING

6x i

= N i 6t,

67n

= N6t

+N

149 Figure 4: 'Thp: Tangent space relationship of observer splitting to the parametrized nonlinear reference frame

splitting. In each case the tangent space horizontal axis is tangent to the slicing and the vertical axis is orthogonal to it (nonnull slice!). Each of the three pairs of parallel lines in the threading diagram is a cross-section of a pair of parallel 3-planes in the tangent space representing the corresponding I-form as described by Burke,17 with the three pairs related to each other by I-form addition describing the relationship of the differential dt to the threading observer. This is analogous to the vector relationship between {)/ &t and the slicing observer 4-velocity. Each of these defines the lapse and shift in the corresponding point of view. Bottom: Infinitesimal displacement relationship of observer splitting to the parametrized nonlinear reference frame. In the threading p.o.v. the shift I-form describes the change in t towards the observer local rest space as one moves along the slicing, while in the slicing p.o.v. the shift vector field describes the change in xi away from the observer local time direction as one advances along the threading. The lapse in each case converts the coordinate time changes to observer proper time changes.

The slicing GM vector field still must be defined, and differs from the vanishing GM vector field of the corresponding hypersurface point of view (zero vorticity) due to the new choice of evolution direction along the distinct threading rather than along the normal congruence to the slicing. This brings us to the question of evolution in the slicing point of view, which is a hybrid or bi-congruence approach. In this point of view the measurement of tensor equations involving covariant derivatives naturally leads to measured expressions involving the Fermi-Walker temporal derivative which may be re-expressed in terms of the Lie temporal derivative V (fw)( n )XlX = V' (Jie)( n )XlX + O( n)lX pXP which in turn may be re-expressed in terms of the threading temporal derivative V' (Jie) (n) = V' (Jie) ( n, 8/ Ot) - N- 1£( n) N' where V(Jie)(n,8/Ot) = N- 1P(n)£(n)8/Ot. This spatially projected Lie derivative along the threading is the evolution operator in the slicing point of view, and the slicing spatial intrinsic derivative is defined by using this temporal operator in its equivalent decomposition when acting on spacetime tensor fields. Given a test particle word line with 4-velocity

UlX = ,(U, n){ nlX + v(U, n )"'} = ,(U, n){N- 18/Ot + [v(U, n)lX - N- 1 N"'n ,

(31)

the latter decomposition into threading and slicing parts carries over to the slicing spatial intrinsic derivative

D(Jie)(U, n, 8/Ot)X'" /dr = ,(U, n) [VCJie) (n, 8/Ot)X'" + V'(n)[v(U, n) _ N-1 N]X~ = DCJie)(U, n)XlX/dr - ,(U,n)~H(Jie)(n, 8/Ot)"'pXP] ,

(32) where the first equality only holds for spacetime vector fields and the difference term b.HCJie)(n,8/Ot)"'pXP = N-1[V(n)N - £(n)N]XP ,

[b.H(Jie)(n, 8/Ot)]"'p = N-1[V(n)NJt

-

nl

JZ dzz 2e- iz jt

[:3 (eiZz2~(n-l)(z»)']' [:3 (eiZz2~(n-l)(z»)']'

-jt JZ dzz 2e- iz nl

(23)

If the above indefinite integrals can be explicitly performed, it is easy to implement the correct boundary condition on it to obtain ~}n). It was shown that this is indeed

159

the case for n = 1 and 2,11 and the boundary condition is that ~}n) be also regular at z = 0 at least for n :5 3. We then find for n = 1, c(I)

_

0 for all infinitesimal perturbations, linear stability is guaranteed. Alternatively, if 8(2) H is indeterminate, one cannot necessarily infer a linear instability, but one does expect at least nonlinear instability and/or instability

173

in the presence of dissipation. 23 Indeed, one can actually prove that generic rotating axisymmetric equilibria are always unstable towards dissipation, as provided, e.g., by the emission of gravitational radiation. 24 This is the collisionless analogue of the theorem 25 that all rotating perfect fluid stars are unstable. Neither for collisionless nor collisional systems is there any guarantee that the timescale associated with this instability is sufficiently short to be of interest astronomically. However, it is significant that general relativity triggers a generic instability which, apparently, is insensitive to the form of the self-gravitating matter. The astronomical implications of this instability are currently under investigation. This general approach to stability can also be adapted to the consideration of steady-state equilibria, such as an homogeneous and isotropic Friedman cosmology, where it provides an interesting derivation of the Jeans instability.26 Viewed Newtonianly, such an expanding Universe corresponds in the "inertial" frame to a system characterised by a time-independent Hamiltonian which finds itself in a time-dependent steady state. This explicit time-dependence can be removed by effecting a time-dependent canonical transformation into the average "comoving" frame. This transformation leads to a new time-dependent Hamiltonian H( t). However, the first variation 6(1) H vanishes identically, and the second variation 6(2) H can be shown to satisfy d6(2) H / dt ::; O. A simple Liapounov argument therefore guarantees that the existence of negative energy perturbations 6(2) H < 0 for sufficiently long wavelengths must imply an instability: If the system be perturbed in such a fashion that 6(2) H < 0, the energy can only become more negative, so that the "distance" from equilibrium, as probed by the magnitude of 6(2) H, can only Increase. 4.

Transient ensemble dynamics

In studying the properties of orbits in a fixed potential, it is natural to apply the technology of nonlinear dynamics, as has been developed over the past several decades. However, in many settings involving gravitational systems, the utilisation of this technology may require a new twist. The standard formulation of nonlinear dynamics typically involves a theory of asymptotic orbital dynamics, which focuses primarily, if not exclusively, on the long time behaviour of individual orbits. However, for many astronomical systems this may not be appropriate. For example, in terms of their natural timescale, galaxies are relatively young objects, only'" 100 - 200 crossing times tcr in age, so that it is not at all obvious that an asymptotic t --+ 00 limit is well motivated physically. Thus, e.g., standard estimates of Liapounov exponents typically require integrations for times t 2:: 104t cr , a period that is orders of magnitude longer than the age of the Universe, tHo Moreover, it is arguably true that, in many astronomical systems, individual orbits are not the fundamental objects of interest. It is, for example, obvious that one cannot track individual orbits of stars within a galaxy. All that one can detect are properties like the overall brightness distribution which reflect the contributions of many

174

stars. Similarly, it is evident that one must "focus on collections of orbits if he or she wishes ultimately to address the problem of self-consistency. For these reasons, it would seem more natural to consider instead a theory of transient ensemble dynamics, which focuses on the statistical properties of ensembles of orbits, restricting attention exclusively to short timescales, t < tH, and recognising that much of the observed behaviour may be intrinsically transient in character. This distinction may not be all that important for integrable, or near-integrable potentials, which contain only regular orbits. However, there is no reason to assume that the bulk potential associated with a self-gravitating system is integrable, or even near-integrable, and there are indications from numerical experiments that objects like rotating elliptical galaxies and barred spiral galaxies may admit large numbers of stochastic, or chaotic, orbits. 27,28 Roughly, regular orbits correspond to orbits that have regular shapes and are characterised by simple topologies, e.g., box orbits in two dimensions with trajectories that resemble a Lissajous figure or tube orbits in three dimensions that are restricted to a region with the topology of a torus. By contrast, stochastic orbits are manifestly irregular in shape and appear to "run all over" phase space. Unlike regular orbits, stochastic orbits exhibit an exponentially sensitive dependence on initial conditions, as manifest by the fact that such orbits are characterised by a positive Liapounov exponent. 29 The crucial point to be illustrated below is that, at least when considering stochastic orbits, the transient ensemble perspective can prove extremely important. Consider, for example, the scattering of photons incident on a multi-black hole system; or similarly, consider a star moving in a nonspherical potential, supposing that the star is unbound but that, owing to the shape of the potential, it can only escape in certain directions. For each of these problems, one discovers that, in certain phase space regions, the direction and time of escape to infinity exhibits a very sensitive dependence on initial conditions, in fact a fractal dependence, this being an example of what the nonlinear dynamicist would call chaotic scattering. 30 Naively, one might anticipate that this complex microscopic behaviour would lead to an equally complex description when considering the evolution of ensembles of orbits. However, this is not always the case. At least in certain cases, one observes instead striking regularities, which lead to a simple scaling behaviour,31 that may actually be universal. 32 The very fact that the microscopic evolution is complex appears to be responsible for the fact that the macroscopic evolution is very simple. As a concrete example, consider orbits in the two-dimensional potential V (x, y) = x 2 + y2) - €X 2 y2 , holding fixed the value of the energy E and studying as a function of € the evolution of orbits initially localised in a small phase space region. For € below a critical value €1 = 1/(4E), escape is impossible energetically. For values of € slightly above €1, escape is possible energetically, but only very few particles escape on short timescales and the time of escape exhibits no striking regularities, except that the escape probability eventually appears to decay towards zero. However, for

t(

175 € above another critical value €2 (only determined numerically), one sees the onset of striking scaling behaviour: 1) After the decay of initial transients, the escape probability per unit time approaches a constant value Poo (€), which is independent of initial conditions, i.e., the location of the phase space region from which the initial ensemble was chosen. Moreover, this rate scales as Poo(€) '" (10 - €2)'" for a critical exponent a. 2) For fixed size of the initial phase space region probed by the ensemble, the time Too required to converge to Poo also depends on € and satisfies Too(e) '" (€ - €2t.6. 3) For fixed €, the convergence time Too depends on the linear size R of the phase space region that was probed initially, satisfying Too(R) '" R- o Moreover, to within statistical errors a-(3-S ~ O. In a certain sense, the qualitative change in behaviour at € = €2 is like a phase transition, complete with a critical "slowing down" as the "order parameter" e - €2 - t O. As another example in which the transient ensemble perspective is important, consider the behaviour of ensembles of orbits of fixed energy E, evolving in a twodimensional time-independent potential V(x,y), which admits both regular and sto-chastic orbits. IT V (x, y) is bounded from below and diverges at infinity, the constant energy hypersurfaces will be compact, so that the notion of "equilibrium" is well defined. One might therefore anticipate that generic ensembles of initial conditions will evolve towards an invariant distribution corresponding to a statistical equilibrium. To test this hypothesis, one can select localised ensembles of initial conditions of fixed E, corresponding to stochastic orbits initially located far from any regular phase space regions, and then evolve these initial data into the future. At least for certain potentials,33,34 one then observes that the orbits do indeed disperse in such a fashion as to exhibit a coarse-grained evolution towards a quasi-equilibrium, which is at least approximately time-independent. This approach is, moreover, exponential in time and characterised by a rate A( E) which is independent of the specific choice of initial data. The characteristic timescale t. = A-1 is typically ~ lOOter) so that, in "physical units" for a galaxy, t. ~ tHo One also observes that the rate A(E) is comparable in magnitude to the value of the Liapounov exponent X(E), which35 probes the average rate of instability exhibited by orbits of energy E. There is, moreover, a direct correlation between A and X in the sense that, e.g., both have similar curvatures when viewed as functions of E. This is particularly tantalising in view of the fact that, for the N-body problem, the timescale associated with the approach towards a statistical quasi-equilibrium is comparable in magnitude to the timescale associated with the instability towards small changes in initial conditions. Despite these regularities, there is no guarantee that this apparent equilibrium coincides with the true invariant distribution, and, in general, it will not! Astronomers are well acquainted with the fact that collisionless equilibria do not constitute true N-body equilibria, which must incorporate discreteness effects that become important on sufficiently long timescales. However, the key point here is different, and more fundamental: Even motion in a smooth two-dimensional potential

176

may not yield a uniform approach towards a ·true 8tati.5tical equilibrium. 34 ,36 The explanation of the discrepancy between the true and approximate equilibria is simple. Viewed over sufficiently long timescales, there are only two different classes of orbits, namely regular orbits, with vanishing Liapounov exponent x, and stochastic orbits, for which X > o. The distinction between these two classes is, moreover, absolute, since members of the two different classes are separated by invariant KAM tori. If, e.g., one were to compute a surface of section, plotting coordinates x and Px for successive intersections of the y = 0 hyperplane, he or she would generically find islands of regularity embedded in a surrounding stochastic sea. However, this is not the whole story. Lurking in the shallows of the stochastic sea, slightly away from the shore, are cantori,37 these corresponding to fractured KAM tori, associated with the breakdown of integrability, which contain a cantor set of holes. The point then is that these cantori serve as partial barriers that divide the stochastic orbits into two subclasses, namely confined, or sticky, stochastic orbits which are trapped near the regular islands, and unconfined, or filling, stochastic orbits which travel unimpeded throughout the rest of the stochastic sea. Because of the holes in the cantori, these barriers are not absolute, so that orbits can in fact change from one class to another via so-called intrinsic diffusion. However, this process is a slow one, requiring orbits to wend their way through a maze (cf. the "turnstile model" of MacKay, Meiss, and Percival,38 so that the characteristic timescale is typically ~ lOOter, i.e., much longer than the age of the Universe. What this implies is that, on short times, ensembles of orbits initially outside the cantori will evolve towards a near-invariant di8tribution which uniformly populates the filling regions, but avoids the confined regions. The situation is analogous to the classical effusion problem. Consider two evacuated cavities connected one with another by an extremely narrow conduit, and suppose that gas is inserted into one of the cavities. If the conduit be sufficiently narrow, the timescale on which gas effuses from one cavity to the other will be much longer than the timescale on which the gas spreads to fill the original cavity. This implies, however, that, even though the true equilibrium corresponds to a uniform density concentration throughout both cavities, one can speak meaningful of a shorter time quasi-equilibrium, in which the original cavity is populated uniformly and the other is essentially empty. Significantly, these two different populations of stochastic orbits are fundamentally dissimilar in terms of their stability properties as well as where they are located in phase space. Although both sticky and filling stochastic orbits are exponentially unstable, there is a precise sense in which the sticky orbits are less unstable overall than are the filling orbits. Specifically, if one computes local Liapounov exponents,39 X(Llt), for different ensembles of stochastic orbits, integrating for some relatively short interval .6.t, he or she will find 34 ,36 that the typical X(Llt) for a sticky orbit is substantially smaller than the typical X(Llt) for a filling orbit. Indeed, the composite distribution of local Liapounov exponents (i.e., distribution of instability timescales) generated from a sampling of the true invariant measure appears to be

177

given, at least approximately, as a sum of two different near-Gaussian distributions with unequal means. It should perhaps be noted explicitly that the general conclusions recounted in this Section have been observed for several different potentials, with rather different symmetries, including (1) the dihedral D4 potential of Armbruster, Guckenheimer, and Kim,40 (2) the sixth order truncation of the three-particle Toda 41 lattice potential, and (3) a generalised anisotropic Kepler potential of the form

V(x,y)

=-

1

(1 + x 2 + y2)

m 1/2 -

(1 + x 2 + ay2 )

1/2'

(7)

with constant m and a, for E < o. The fact that these diverse potentials, which are fundamentally different in appearance, yield similar conclusions, both qualitatively and semi-quantitatively, would suggest strongly that these conclusions are robust, depending only on such topological features as the existence of KAM tori and cantori. The existence of confined stochastic orbits is of potential importance astronomically because such orbits can help (the theorist) support various sorts of structures, e.g., bars in a spiral galaxy. It is natural to assume that, in systems like galaxies, regular orbits serve to provide the skeleton to support various structures. However, because of resonance overlap one may find that, near corotation and other resonances, the desired regular orbits do not exist, even though sticky stochastic orbits are present. Finally, it should be stressed that one can observe similar short time "zones of avoidance" in higher dimensional systems as well. The key point physically is that jU3t because a region of phase space is connected, so that orbits can pass throughout the entire region, does not mean that all of the region will be accessed on comparable times cales. 5.

Structural stability of the smooth potential approximation

The collisionless Boltzmann equation is a Hamiltonian system which neglects various realistic non-Hamiltonian irregularities that must be present in any selfgravitating system. One obvious point is that such a Vlasov description neglects entirely all discreteness effects, i.e., "collisions," by idealising the system as a continuum, rather than a collection of nearly point mass objects. Viewed in the Nparticle phase space, the statistical description of an isolated N -body evolution is of course Hamiltonian. However, when projected into the reduced one-particle phase space, any allowance for particle-particle correlations that transcend a mean field description necessarily breaks the Hamiltonian constraints. 22 Another point, perhaps less obvious but equally important, is that a Vlasov description also neglects any couplings to an external environment. In the past, astronomers have been wont oftentimes to pretend that galaxies exist in splendid isolation but, over the past several decades, it has become increasing evident that such an approximation may not be justified. 42

178

Detailed modeling of these sorts of perturbing influences may prove extremely complex. In particular, an external environment can give rise to a variety of different effects characterised by a broad range of timescales. Those influences proceeding on timescales '" tcr will be particularly complicated, in that the details of their effects may depend very sensitively on the details of the environment. However, there is a well-established paradigm in statistical physics,43,44 dating back to the beginning of the century,4S which would suggest that irregularities proceeding on shorter timescales, ~ t cr , can oftentimes be modelled as friction and noise, related via a fluctuation-dissipation theorem. This idea underlies, for example, Chandrasekhar's original formulation 46 of so-called "collisional stellar dynamics." It is therefore natural to investigate the structural stability of Hamiltonian trajectories towards the effects of friction and noise. This was done47,48 by effecting large numbers of Langevin simulations, in which the deterministic equations of motion were perturbed by allowing for (1) a dynamical friction -1]P, which serves systematically to remove energy from the orbits and (2) random kicks, modeled as white noise with temperature, or mean squared velocity, e, which serve systematically to pump energy back into the orbits. As a first simple test, 1] was assumed to be constant, in which case the fluctuation-dissipation theorem implies that the noise must be additive, rather than multiplicative.so,S} Thus, in units with particle mass m = 1, one is led explicitly to equations of motion of the form

dr

-=P

dt

and

dp dt

= -VV(r) -1]P + F,

(8)

where F is characterised completely by its statistical properties. Here, e.g., component by component,

(9) where the angular brackets denote a statistical average. The idea is to effect large numbers of different realisations of the same initial conditions, and to analyse these realisations to extract statistical properties. It is well known that even very weak friction and noise will eventually become important on sufficiently long timescales. In particular, one knows that, on the natural timescale tR '" .,.,-}, these effects will try to force the system to evolve towards a thermal state. The question of relevance here is quite different: Can the friction and noise have substantial effects already on much shorter timescales ~ tR? The conventional wisdom in astronomy is that the answer to this is: no! For example, the standard assumption that "collisionality" is irrelevant in a galaxy relies completely on the observation that the natural timescale associated with binary encounters is much longer than the age of the Universe. 46 The Langevin simulations were effected for total times t ::; 200tcTl and involved friction and noise corresponding to a broad range of characteristic timescales, 103 ::; tRiter ::; 1012 • The most significant conclusions derived from these simulations are the following:

179

When viewed in terms of the collisionless "invariants, i.e., the quantities that are conserved in the absence of the friction and noise, these perturbing influences only serve to induce a classical diffusion process, with the unperturbed and perturbed orbits diverging significantly only on a timescale tR rv ",-1. Thus, in particular,

(10) where A(E) is a slowly varying function of E with magnitude of order unity. In this sense, the conventional wisdom is confirmed. However, when viewed in configuration space or momentum space, the effects are more complicated, and actually depend on orbit class. Unperturbed and perturbed regular orbits only diverge as a power law in time, i.e., orrms,oPrms '" t P , so that, once again, one only gets macroscopic deviations after a time tR '" ",-1. However, unperturbed and perturbed stochastic orbits diverge exponentially at a rate set by the Liapounov exponent X, so that, even for very weak friction and noise, one gets macroscopic deviations within a few crossing times. In particular, when considering ensembles of stochastic initial conditions, one observes a simple scaling

(11) where X is comparable to, but slightly larger than, the Liapounov exponent X. This exponential divergence is easy to understand 52 and the functional dependence on El, 7], and X can actually be derived theoretically. 53 The obvious point is that the unperturbed deterministic trajectory is an unstable stochastic orbit, so that even the tiniest perturbing influences will tend to grow exponentially. The average rate of instability is given by X and, as such, moments like orrms should grow at a rate X '" x. That X is slightly larger than X is a reflection of the fact that, for different noisy realisations, one sees somewhat different local Liapounov exponents, and that the total Orrms will be dominated by those noisy realisations for which the rate of instability is above average. This argument might suggest that, although the unperturbed and perturbed trajectories exhibit a rapid pointwise divergence, their statistical properties should be virtually identical. Specifically, one might anticipate that, on short times, the only effect of the friction and noise is to continually displace the trajectory from one stochastic orbit to another with essentially the same statistical properties. This, however, is false. Under certain circumstances, even very weak friction and noise can also alter the statistical properties of ensembles of stochastic orbits on relatively short times ~ 100ter • Specifically, one observes that such perturbing influences can dramatically accelerate the rate of penetration through cantori by providing an additional source of extrinsic diffusion. Provided that the friction and noise are sufficiently weak, on short timescales the energy E is almost conserved, so that one can speak meaningfully of an evolution restricted to an "almost constant energy hypersurface." Suppose now that, for some energy E, this hypersurface contains large measures of both sticky and filling stochastic orbits. If, for this energy, the near-invariant distribution described

180

in Section 4 is evolved into the future, allowing for even very weak friction and noise, one then observes a rapid (t ~ lOOter) systematic evolution towards a new noisy near-invariant distribution which is (1) quite different from the deterministic near-invariant distribution and (2) much closer to the true deterministic invariant distribution. In this sense, it appears that, on times cales short compared the timescale tR on which the system would evolve towards a thermal state, the principal effect of the friction and noise is to accelerate the approach towards a deterministic invariant distribution which, in the absence of these perturbing influences, would only have been realised on much longer times cales. The key point in all of this is that friction and noise can induce changes in orbit class, from filling to confined stochastic, and vice versa. Moreover, when the deterministic invariant distribution contains large measures of both sticky and filling orbits, such transitions can happen within a time t < lOOter, even for very weak friction and noise. Visual inspection of '" 2.5 x 104 orbits in several different potentials leads to the following conclusions. Typically, for a relaxation time tR as long as I0 12 t er , not many such changes are observed within a time t '" lOOter. However, if tR be reduced to a value'" lO9 t er , transitions begin to become more frequent, and, even for tR as large as '" lO6ter , transitions are quite common, occuring for> 50% of orbits within a time t '" lOOter. If the amplitude of the friction and noise are further increased, one finds that, for tR '" I03t er , transitions are so common that the distinction between filling and confined becomes essentially meaningless. The distinction between regular and stochastic is more robust. Only for tR as small as '" I0 3 t er are significant numbers of transitions between regular and stochastic orbits observed within a time as short as t '" lOOter. The fact that even very weak friction and noise, with tR '" lO6ter - I0 9 t er , can significantly alter the statistical properties of ensembles of orbits on times cales t < tH '" lOOter has direct astronomical implications since, e.g., for galaxies, the timescale tR for discreteness effects, i.e., "collisionality" is '" 106ter! The natural timescale associated with external perturbations is less easily estimated, but may well be even shorter. To summarise, it is evident that even very weak friction and noise can alter both the pointwise and the statistical properties of stochastic orbits in a nonintegrable potential on relative short timescales ~ tR. In particular, such effects may be manifest already on time scales much shorter than the time on which numerical errors in a simulation can accumulate. This fact has direct and immediate implications for the problem of "shadowing" for numerical orbits. 54 ,55 Physicists, mathematicians, and astronomers are often worried56 ,57 about whether numerical simulations performed on a computer, which incorporate roundoff and/or truncation error, can correctly shadow the evolution of some model system described by a simple set of deterministic differential equations. However, it would also seem relevant 58 to worry about whether the "real world," replete with other sorts of irregularities, can shadow either the model system or its numerical realisations. In this regard, one final remark is in order: Rather than being an impediment to realistic modeling, in

181

certain cases numerical noise may actually be a good thing, in that it may capture, at least qualitatively, some of the effects of small perturbing influences to which real systems are always subjected.

Acknowledgments It is a pleasure to acknowledge useful collaborations with Robert A. Abernathy, Brendan O. Bradley, George Contopoulos, Salman Habib, Eric O'Neill, Haywood Smith, Jr., Christos V. Siopis, David E. Willmes, and, especially, M. Elaine Mahon. This research was supported in part by the NSF through PHY92-03333 and by NASA through the Florida Space Grant Consortium. The simulations reported herein were facilitated by time made available by the Advanced Computing Laboratoryat Los Alamos (CM5), The Parallel Computing Research Laboratory at the University of Florida (KSR), and the Northeast Regional Data Center (Florida) by the IBM Corp.

References 1. Jeans, J. H. Mon. Not. R. Astr. Soc. 76 (1915) 70. 2. Vlasov, A. A. Zh. Eksp. Teor. Fiz. 8 (1938) 291. 3. Bisnovatyi-Kogan, G. S. and Shukhman, 1. G. Zh. Eksp. Teor. Fiz. 82 (1982) 3. 4. Klimontovich, Yu. L. 1983, The Kinetic Theory of Electromagnetic Processes (Springer, Berlin, 1983). 5. Miller, R. H. Astrophys. J. 140 (1964) 250. 6. Kandrup, H. E. and Smith, H. Astrophys. J. 374 (1991) 255. 7. Kandrup, H. E. and Smith, H. Astrophys. J. 386 (1992) 635. 8. Kandrup, H. E., Smith, H., and Willmes, D. E. Astrophys. J. 399 (1992) 627. 9. Goodman, J., Heggie, D., and Hut, P.Astrophys. J. 415 (1994) 715. 10. Kandrup, H. E., Mahon, M. E., and Smith, H. Astrophys. J. 428 (1994) 458. 11. van Albada, T. S. Mon. Not. R. Astr. Soc. 201 (1982) 939. 12. Quinn, P. J. and Zurek, W. H. Astrophys. J. 332 (1988) 619. 13. Funato, Y., Makino, J., and Ebisuzaki, T. Pub. Astron. Soc. Japan 44 (1992) 291. 14. Kandrup, H. E., Mahon, M. E., and Smith, H. Astron. Astrophys. 271 (1993) 440. 15. Morrison, P. J. Phys. Lett. A80 (1980) 383. 16. Kandrup, H. E. and Morrison, P. J. Ann. Phys. (NY) 225 (1993) 114. 17. Kandrup, H. E. and O'Neill, E. Phys. Rev. D49 (1994) 5115 . 18. Marsden, J. E. and Winstein, A. Physica D4 (1982) 394. 19. Kandrup, H. E. and O'Neill, E. Phys. Rev. D48 (1993) 4534. 20. Ipser, J. R. and Thorne, K. S. Astrophys. J. 154 (1968) 251. 21. Israel, W. and Kandrup, H. E. Ann. Phys. (NY) 152 (1985) 30. 22. Kandrup, H. E. Phys. Rev. D 50 (1994) 2425.

182

23. Bloch, A., Krishnaprasad, P. S., Marsden, J. E., and Ratiu, T. S. Ann. Inst. Henri Poincare: analyse non lineaire 11 (1994) 37. 24. Kandrup, H. E. Astrophys. J. 380 (1991) 511. 25. Friedman, J. L. and Schutz, B. F. 1978, Astrophys. J. 222 (1978) 281. 26. Kandrup, H. E. and O'Neill, E. Phys. Rev. D47 (1993) 3229. 27. Sparke, L. S. and Sellwood, J. A. Mon. Not. R. Astr. Soc. 225 (1987) 663. 28. Pfenniger, D. and Friedli, D. Astron. Astrophys. 252 (1991) 75. 29. Chirikov, B. Phys. Repts. 52 (1979) 265. 30. Smilansky, U. in Lectures at Les Houches, Chaos and Quantum Physics, ed. M.-J. Giannoni, A. Voros, and J. Zinn-Justin (Elsevier, Amsterdam, 1990). 31. Contopoulos, G., Kandrup, H. E., and Kaufmann, D. Physica D64 (1993) 310. 32. Siopis, C. V., Contopoulos, G., and Kandrup, H. E. Ann. NY Acad. Sci. (1994) in press. 33. Kandrup, H. E. and Mahon, M. E. Phys. Rev. E49 (1994) 3735. 34. Mahon, M. E., Abernathy, R. A., Bradley, B. 0., and Kandrup, H. E., Mon. Not. R. Astr. Soc. (1994) submitted. 35. Bennetin, G., Galgani, L., and Strelcyn, J.-M. Phys. Rev. A14 (1976) 2338. 36. Kandrup, H. E. and Mahon, M. E. Astron. Astrophys. (1994) in press. 37. Mather, J. N. Topology 21 (1982) 457. 38. MacKay, R. S. Meiss, J. D. and Percival, 1. C. Phys. Rev. Lett. 52 (1984) 697. 39. Grassberger, P., Badii, R., Politi, A. J. Stat. Phys. 51 (1988) 135. 40. Armbruster, D., Guckenheimer, J., and Kim, S. Phy. Lett. A 140 (1989) 416. 41. Toda, M. J. Phys. Soc. Japan 22 (1967) 431. 42. Zepf, S. E. and Whitmore, B. C. Astrophys. J. 418 (1993) 72. 43. Chandrasekhar, S. Rev. Mod. Phys. 15 (1943) 1. 44. Kubo, R., Toda, M., and Hashitsume, N. 1991, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer, Berlin, 1991). 45. Einstein, A. 1905, Ann. d. Physik 17 (1905) 549. 46. Chandrasekhar, S. Principles of Stellar Dynamics (University of Chicago, Chicago, 1942). 47. Kandrup, H. E. and Mahon, M. E. Ann. NY Acad. Sci. (1994) in press .. 48. Habib, S., Kandrup, H. E., Mahon, M. E. Phys. Rev. Lett. (1994) submitted. 49. Habib, S., Kandrup, H. E., Mahon, M. E. Astrophys. J. (1994) submitted. 50. Lindenberg, K. and Seshadri, V. Physica AI09 (1981) 481. 51. Habib, S. and Kandrup, H. E. Phys. Rev. D46 (1992) 5303. 52. Pfenniger, D. Astron. Astrophys. 165 (1986) 74. 53. Kandrup, H. E. and Willmes, D. E. Astron. Astrophys. 283 (1994) 59. 54. Anosov, D. V., Tr. Mat. Inst. Steklov 90 (1982) 210. 55. Bowen, R. J. DiJJ. Eq. 18 (1975) 333. 56. Quinlan, G. D. and Tremaine, S. Mon. Not. R. Astr. Soc. 259 (1992) 505. 57. Willmes, D. E. Ann. NY Acad. Sci. (1994) in press. 58. Eubank, S and Farmer, D., in 1989 Lectures in Complex Systems, ed. E. Jen (Addison-Wesley, Redwood City, California, 1990).

183

Fully Relativistic 3D Numerical Simulations of Coalescing Binary Neutron Stars

TAKASHI NAKAMURA

Yukawa Institute for Theoretical Physics

Kyoto University, Kyoto 606, Japan

ABSTRACT

Basic equations and numerical methods of a 3D numerical relativity code developed recently are shown. For a time slice and 3-space coordinate condition, conformal time slicing and pseudo-minimal shear condition are adopted. Preliminary numerical simulations for coalescing binary neutron stars using (80)3 Cartesian grids are shown. The results of spherically symmetric collapse to black hole, formation of rotating black hole and collision of two dust spheres are also shown. Qualitatively the code works well without serious difficulties. Full scale simulations with'" (400)3 grids are urgently required for quantitative arguments on the results.

184

1. Introduction Coalescing binary neutron stars (or black holes) is one of the most important sources of gravitational waves for LIGO/VIRGO projects.:) Comparing observed wave patterns in The Last Three Minutes of coalescing binary neutron stars (or black holes) with 2

3

theoretical templates, mass and spin of neutron stars may be determined. ) Sasaki ) argued various aspects of post Newtonian calculations to obtain accurate templates in this conference. In the last three minutes, vrot/ c is a good parameter for expansion since vrot/ c rv O.l. However in the final phase of the coalescence, there is no such small parameters. So one must abandon post Newtonian expansion. Numerical relativity is one of the methods to study such nonlinear phase of coalescing binary neutron stars which I like to call it as The Last Three Milli Seconds. For this problem, there are many numerical simulations using Newtonian hydrodynamics code either with or without radiation reaction forces by gravitational waves and Post-Newtonian corrections:) From these numerical results there are many perspectives on wave patterns of gravitational waves and dynamics of coalescing events. However since Newtonian hydrodynamics is not so good approximation in this case, I think that these simulations should be considered as preludes of full general relativistic ones. Here I like to discuss recent development of my fully general relativistic code to simulate The Last Three Milli Seconds. In the workshop on numerical relativity, WilsonS) as well as Seidel

6

)

presented development of their 3D code which can be found in this

volume. As for cosmological problem Kurki-Suonio, Laguna and Matzner 7) have already published numerical results of their fully general relativistic 3D code. But here I restrict the problem only for asymptotically flat space-times for which I have already developed full 3D code for the propagation of localized gravitational waves~) Combining this with a 3D hydrodynamics code used for coalescing binary neutron stars 4) previously, I have tried to make a full 3D general relativistic code for the final phase of coalescing binary neutron stars.

2. Basic Equations A)Initial Value Equations In the (3+ 1)- formalism of the Einstein equations the line element is expressed as

(1) where ex, {3i and 'Yij are the lapse function, the shift vector and the intrinsic metric of 3-space, respectively. The initial data should satisfy the constraint equations; (3)R+ K2 - KijKij =16?rPH,

Kl;j - K;j =8?r Ji

(2) (3)

185

where (3)R, Kij, K, PH and Ji are scalar curvature, extrinsic curvature and its trace, energy density and momentum density of matter, respectively. In Eq.(3), ; means the covariant derivative with respect to metric 'Yij. Now we assume that 'Yij is conformally flat as =

'Yij

where

iij

",4'Yij,

(4)

'I'

is the flat metric. We define the conformal transformation as

k··OJ -=

",2 K· OJ,

'I'

kf0 -= PB

",6 Kf 0'

'I'

K- ij -=

",10 Kij

'I'

(5)

== PH¢6 and Ji == Ji¢6.

Now if we assume K = 0, Eq.(3) becomes

(6) where I means the covariant derivative with respect to the flat metric iij. We decompose Kij as

(7) where klT is the transverse traceless part of Kij and Wi is the longitudinal part of k ij . Assuming klT = 0 for simplicity, we rewrite Eq.(6) as

(8)

K'fl at t=O

where D.f is the flat Laplacian. In the real coalescing binary neutron stars, is needed from the results of post-Newtonian calculations or other methods. Let us define the divergence of Wi as

(9) Taking the divergence of Eq.(8), we have (10) We first solve Eq.(10) for given Ji . Using DIV, we can rewrite Eq.(8) as

(11) Although Eq.(8) was the coupled elliptic type equation of Wi originally, the above procedure enables us to decouple Eq.(8). The boundary conditions for Eq.(10) and

186

Eq.(ll) are

(12) and 1

= 0(4")' r

DIV

(13)

where Mk is the constant related to angular momentum of the system. Now Eq.(2) is rewritten as

~ ("') = _ 27rPB f

We solve Eq.(14) for given

PB

rp

'I-'

_

~",-7 k .. k ij 8'1-'

OJ

(14)

,

under the boundary condition

MG

1

rp = 1 + -2 + O( 3)' r r

(15)

where MG is the gravitational mass of the system.

B) Relativistic Hydrodynamics We assume the perfect fluid. Then the energy momentum tensor Tp.v becomes Tp.v

= (p + pc + P)Up.uv + Pgp.v,

(16)

where p, E, P and Up. are the proper mass density, specific internal energy, pressure and the four velocity of the fluid, respectively. We define the projection tensor hp.v as

(17) where np' is the normal vector to the hypersurface.

PH,

Ji and

5ij

are defined as

and

(18) Then the relativistic hydrodynamics equation (T/:;p. = 0 where; is the covariant derivative with respect to gp.v) is written as

187

{}

a

+ {}x 1(-/1O!u OpV I) =

8t(-/1O!uOp)

l' =

0,

det(-yij),

(21) (22)

and

(23) Left hand sides of Eq.(19),(20) and (21) are similar to the corresponding equations in Newtonian hydrodynamics if we define the following quantities corresponding to variables in Newtonian hydrodynamics as

== -/1O!u

PN

o

p

and

(24) Then left hand sides become as {}

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(19Y (20)'

and (21)' Using the variables PN,

uf

and

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we can apply Newtonian hydrodynamics code devel-

oped by Nakamura and Oohara 4) almost as it is. To complete relativistic hydrodynamics equations, we need the equation of state, P

= P(€,p)

C) Time evolution of the extrinsic curvatures The evolution equations of the extrinsic curvatures are

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propagation of the wave. We see the spiral wave pattern in x-y plane while in x-z and zy planes we see different pattern with main peak around z-direction. Naively quadrupole wave pattern given as

can explain this behavior. In x-y plane since () = 7rj2,PGW is constant along the spiral of r = --

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202 is the angular momentum per unit gravitational mass Mg. The value of a/Mg is 0.3 , 0.5, 0.8 and 1. for ROT1, ROT2, ROT3 and ROT4, respectively. We show the density contours and" gravitational energy" of ROT3 in Fig.3-a) to d). In the density contour, due to the centrifugal force, collapse in the direction perpendicular to the rotation axis is slowed. So we see the oblate spheroid. The gravitational wave generated by this motion has pattern shown in right figures. In x-y plane it is almost axially symmetric with two or three peaks. Since I am using the Cartesian coordinates, deviations from axial symmetry can be seen. But they are not so large. In x-z and y-z plane the wave pattern looks like quadrupole emission (ex sin 4 fJ) which is also the case in Stark and Piran. The total energy radiated at t=30 observed at r=20 are 5.7 x 10- 7 ,3.8 X 10- 6 ,1.5 X 10- 5 , and 3.6 x 10- 5 for ROT1, ROT2, ROT3 and ROT4, respectively. Stark and Piran found for Iowa, the total energy is proportional to a 4 and it levels off at a'" Mg. This same dependence can be seen also in our case. That is, the total energy is expressed approximately

Corresponding case in Stark and Piran to ROT1, the total energy is'" 1.5 X 10- 5 which is '" 20 times larger than our case. But this is consistent. Stark and Piran evolve the initial data up to t= 100 while we do only up to t=30. So the total energy we observed is just expected from the results of Stark and Piran.

D)Collision of Two Dust Sphere l3

Recently Anninos et aI. ) performed numerical simulations of two black hole collision which was first done by Smarr. To mimic the collision of two black hole we have performed collision of two dust sphere. We put two dust sphere of mass 0.5 and radius 3 just contact each other. Each dust sphere collapses to black hole and coalesce to a single black hole. Fig. 4 shows an example of such simulations. We see the quadrupole like wave pattern which is the dominant multipole in two black hole collision.

5. Summary One of the goals in numerical relativity is to perform simulations of general space times for any initial data. For this, construction of 3D code is essential. In this article I showed what I have been done for this purpose. At present my code is on a way of construction. I made some approximations to the solution of pseudo minimal shear condition (Eq.(36)) and conformal factor ¢ (Eq.(32)). The number of grids is not enough for the application of the outgoing wave boundary conditions at the numerical outermost boundary. Although all the results shown in §4 should be considered preliminary, qualitatively the code works well without serious difficulties. I think that the completion of the present 3D code will be done near future. This work is supported by the Grant-in-Aid for Scientific Research on Priority Area of Ministry Education (04234104).

203

References 1) A. Brillet, this volume. 2) C. Cutler et al., Phys. Rev. Lett. 70 (1993), 2984. C. Cutler and E.E. Flanagen, Phys. Rev. :049 (1994),2658. 3) M. Sasaki, this volume. 4) K. Oohara and T. Nakamura, Prog. Theor. Phys. 81 (1989),360; 821 (1989), 535; 82 (1989) 1060; 83 (1990), 906; 86 (1881), 73; 88 (1992), 307. M. Shibata, K. Oohara and T. Nakamura, Prog. Theor. Phys. 88 (1992), 1079; 89 (1993), 809. F. Rasio and S. L. Shapiro, Astrophys. J. 401 (1992), 226. J. Centrella and S. McMillan, Astrophys. J. 416 (1993), 719. M. B. Davies, W. Benz, T. Pi ran and F. K. Thielman, Astrophys. J. 431 (1994), 742. 5) J. Wilson, this volume. 6) E. Seidel, this volume. 7) H. Kurki-Suonio, P. Laguna and R. A. Matzner, Phys. Rev. D48 (1993), 3611. 8) T. Nakamura and K. Oohara, in "Frontiers of Numerical Relativity" ed. C. Evans (Cambridge Dniv. Press 1989), 254. T. Nakamura, in Proceedings of GR13 ed. N. Ashby ( Cambridge Dniv. Press 1989), 6l. 9) M. Shibata and T. Nakamura, Prog. Theor. Phys. 88 (1992), 317. 10) J. York and 1. Smarr, Phys. Rev. D17 (1978), 1945. 11) K. Oohara and T. Nakamura, in "Frontiers of Numerical Relativity" ed. C. Evans ( Cambridge Dniv. Press 1989), 74 . 12) R. Stark and T. Piran, Phys. Rev. Lett. 55 (1985), 891, in Dynamical Spacetimes and Numerical Relativity, ed. J. Centrella Cambridge Dniv. Press 1985, P40. 13) P. Anninos et al., Phy. Rev. Lett. 71 (1993), 2851.

Chairperson: Ray Weiss

207

THE ADVANTAGES AND DIFFICULTIES OF SPACE EXPERIMENTS FOR TESTING GENERAL RELATIVITY G.M.KEISER W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, U. S. A.

ABSTRACT The advantages and difficulties of space experiments for testing general relativity are discussed. Measurements of the gravitational redshift, the retardation of light, the parameters of the lunar orbit, the gravitational deflection of starlight, the equivalence of inertial and gravitational mass, the geodetic and frame-dragging effects, and the effects of gravitational radiation are summarized, and limits on the present experimental accuracy are presented.

1. Introduction Since the first artificial satellite was launched in 1957, a small number of gravitational experiments have been performed in space, and many more have been proposed. All of these experiments have confirmed the predictions of the General Relativity up to an accuracy of approximately 0.1 %. Recent developments in the understanding of alternate theories of gravity have provided additional stimuli for increasing the accuracy of these experiments 1,2. In an effort to increase this accuracy and to devise new experiments, it is useful to examine the advantages and difficulties of these space-based experiments. This paper discusses these advantages and difficulties, and then compares seven ground-based and space-based experiments. Both the ground-based and space-based experiments are difficult. The measured effects are small compared to potential disturbances and careful attention must be paid to possible systematic experimental errors. Performing these experiments in space is certainly no panacea, but it offers a number of advantages which must be weighed against the difficulties. No attempt is made here to summarize all of the experiments that have been performed and proposed. Thorough treatments of experimental tests of General Relativity are given in books and review articles 3-6. Instead, a number of experiments are chosen which illustrate the advantages and difficulties of experimental tests of General Relativity in space.

2. Advantages and Difficulties 2.1 Advantages In many cases, relativistic corrections to gravitational effects are proportional to the gravitational radius of an object, (GM/c 2), divided by the distance from the object. For an object of laboratory size, this dimensionless ratio has the hopelessly small value of 10-25 . At the surface of the Earth due to the mass of the Earth the ratio is 7xlO- 1O; it becomes Ix 10-8 at the surface of the Earth due to the presence of the Sun; and increases

208 to 2x1O-6 at the surface of the Sun due to the mass of the Sun. Although all of these numbers indicate that the relativistic effects will be small compared to those of classical gravitation, they emphasize the necessity for using planetary, or perhaps solar size masses in any experimental test of General Relativity. Similarly, the gravitational acceleration at the surface of the Earth due to the Earth is much larger than the gravitational acceleration due to laboratory-sized objects, other planets, or the Sun. The gravitational acceleration toward a lOO-kg mass at a distance of I meter is 7x1O-7 cm/sec2 and the gravitational acceleration due to the Sun at the surface of the Earth is 0.6 cm/sec2. Also, the angular momentum of the Earth is larger than man-made object that might be spun to its failure point, and the velocity of a satellite in near-Earth orbit exceeds velocities easily achievable on the Earth. These significantly larger magnitudes are, in themselves, no argument for performing experiments in space rather than on the ground, but in some cases, gravitational experiments in space allow one to take advantage of these larger magnitudes whereas ground-based experiments do not. Several proposals have been made to construct a satellite that would pass within several solar radii of the Sun7. In such a satellite, the magnitude of relativistic corrections to gravitational effects becomes significantly larger. One of the remarkable developments in space technology has been the increase in precision and accuracy of the Doppler and range measurements to distant satellites. Initial ranging to the Moon and planets used reflections of radar pulses from the planets. Initial radar ranging to Venus was limited to an accuracy of approximately 3 km because of the topography of the surface and the very weak returned signal, which is inversely proportional to the fourth power of the distance to the planet. Placing active radar transponders on spacecraft not only increased the magnitude of the returned signal (now inversely proportional to the square of the distance) and provided a single point as the source of the return signal but also provided accurate measurements of the velocity of the spacecraft using the Doppler shift of the return signal. These range and Doppler measurements are limited in the lower frequency S-Band (2.3 GHz) and X-Band (8.4 GHz) because of the plasma density of the solar corona and the water content of the Earth's troposphere. Measurements made at multiple frequencies can be used to monitor and remove the effects of the plasma density, and independent measurements of the water vapor content provide an estimation of the variations due to the water content of the atmosphere. The significant increase in the accuracy of planetary ephemerides is largely due to radar Doppler and range measurements. Radar systems now have the capability of measuring Doppler shifts with an error of 10- 16, corresponding to a velocity of 3x1O-6 cm/sec, and range measurements on the order of I cm. The three passive optical retroreflectors placed on the Moon during the Apollo program and one of the two retroreflectors placed on the Moon by Soviet landers 8 have also provided precise information about the lunar ephemeris. These passive reflectors have provided a point source for the reflected beam and have increased the magnitude of the reflected signal sufficiently that it is measurable. The accuracy of lunar laser ranging is now on the order of several centimeters. Significant increase in the signal-to-noise ratio and the distance over which optical ranging is possible awaits the development of space-qualified active optical transponders.

209 Additional advantages of space for experimental tests of General Relativity are the absence of a number of disturbing effects that have long plagued ground-based experimental measurements. Atmospheric turbulence has always limited the angular resolution of ground-based optical observations to approximately 0.1 arc-sec. Significant improvement in this angular resolution is possible using Very Long Baseline Interferometry (VLBI), and angular variation better than milliarc-sec has been achieved when comparing nearby sources 9. However, these measurements are also limited by the plasma density and the water vapor content of the atmosphere. Above the Earth's atmosphere, these disturbing effects are greatly dimished, and several optical interferometers have been proposed whose goal is to measure angular differences to an accuracy of several J.Larc-sec 10-12. Several other disturbing effects have long made ground-based gravitational experiments difficult. Seismic accelerations and local gravitational noise have always caused problems in measurements of the gravitational constant and in equivalence principle experiments, and have limited the low frequency sensitivity of gravitational wave detectors. Moreover, these disturbances are strongly dependent on time and location. Satellite-based experiments offer the possibility of a significant reduction in both vibration and local gravitational noise, but these advantages are limited by nongravitational accelerations and gradients in the gravitational field. In many cases, these non-gravitational accelerations may be significantly reduced by using the technology developed for drag-free satellites. For planetary masses these accelerations are a factor 107 smaller than the acclerations on satellites because the non-gravitational forces due to solar radiation pressure and high energy particles are proportional to the surface area divided by the mass13. Ground-based gravitational experiments must support test masses against the local gravitational acceleration. Traditionally the test masses have been supported using the Cavendish torsion balance which also provides a weak restoring force for rotation of the test masses. Recent experiments using torsion balances have placed limits on the nonmetric gravitational theories and the fifth force, and there is the possibility of significant improvements 14. Associated with any support is some dissipation and the noise associated with that dissipation 15 These difficulties are significantly reduced in space. At a 650 kIn altitude the acceleration on the satellite due to the atmosphere is approximately 10-7 g. Average acceleration levels smaller than 10- 11 g have been demon stated in drag-free spacecraft 16.

22 Difficulties These advantages of space for gravitational experiments must be weighed against the additional difficulties posed when any experiment is performed in space. Although space provides a benign environment for vibration, acceleration, and optical observations, the thermal and cosmic ray environments are harsh compared to those on the ground. The thermal environment is dominated by the direct radiation from the Sun with limited capability of radiating the heat to space. The variation in the thermal environment is greater if the satellite is occulted by the Earth. The effects of these

210

temperature variations may be diminished by carefully controlling the temperature of the apparatus and the electronics, and by choosing mechanical and electrical parts with small temperature coefficients. Cryogenic experiments take advantage of the decrease in temperature coefficients of expansion with decreasing temperatures, provide a natural means of shielding the apparatus from external temperature variations, and permit measurement and control of the temperature to within 10-5 K. The harsh cosmic ray environment may cause single event upsets of the digital electronics, charging of the spacecraft or test masses, or a source of heat for cryogenic experiments in space. Additional difficulties for any space-based experiments are those common to any flight program: The cost, power, weight, and launch environment place tight constraints on the design of the satellite. Difficulties associated with remote operation of any instrument add to the complexity. Finally, the time required from conception to launch of any satellite requires a significant fraction of a person's active career. These difficulties are interrelated and are the subject of many recent efforts to reduce the cost, increase the reliability, and shorten the time required to design and build satellites.

3. Measurement of the Gravitational Redshift The first measurements of the gravitational red shift were made by Pound and Rebka 17 in 1961, with additional measurements reported by Pound and Snyder 18 in 1965, using resonant absorption of gamma rays in Fe57 . The magnitude of the fractional gravitational frequency shift is expected to be L\v _ L\

-



7

10

5

II

t:, "0

10' 'Li/H=1.4x

10

3

10-10

" =

Figure 5: Contour map in the d-TJo plane for A 10 6 , aid = 0_25_ Constraints on d and 710 are obtained from the observed abundances_ If we take the so-called Spite plateau value for the 7Li abundance, the upper bound on the baryon/photon ratio is not different from that for the standard homogeneous model: 710 (3 - 4) x 10- 10 _ If we take the population-I value for the 7Li abundance, the baryon/photon ratio can be as large as 6 x 10- 10 in the narrow range of the fluctuation scale, d(T = 1 MeV) (1 - 5) x 105 cm_

=

=

valid and conclusions do not change even if we take into account the neutrino inflation and the hydrodynamical expansion of inhomogeneities_ The constraint on the baryon/photon ratio is summarized in Fig_ 5_ H we take the so-called Spite plateau54 value for the 7Li abundance, the upper bound on the baryon/photon ratio is not different from that for the standard homogeneous model: 'flo = (3-4) X 10-10 • H we take the population-I value for the 7Li abundance, the baryon/photon ratio can be as large as 6 x 10- 10 in the narrow range of the fluctuation scale, d(T = 1 MeV) = (1 - 5) X 105 cm. In conclusion, the constraints on the present density of baryon cannot be relaxed even in the inhomogeneous big bang nucleosynthesis model if we adopt the plausible halo dwarf value for the primordial 7Li abundance. Though the upper bound on the baryon density can be larger if we take the population-I value for the 7Li abundance, we should recall the fact that it is realized in the very narrow range of the fluctuation scale and that the spectrum of density fluctuations is essentially not monochromatic.

262

Acknowledgements This work was partially supported by Grant-in-Aids for Scientific Research from the Ministry of Education, Science and Culture of Japan (05243104, 04234104 and 05640449).

References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

G. Gamow, Phys. Rev. 70 (1946) 572. RA. Alpher, H. Bethe, and G. Gamow, Phys. Rev. 73 (1948) 803. RA. Alpher, J.W. Follin, and RC. Herman, Phys. Rev. 92 (1953) 1347. RA. Alpher RC. Herman, Phys. Rev. 73 (1948) 803. R Wagoner, W.A. Fowler, and F. Hoyle, Astrophys. J. 148 (1967) 3. D.N. Schramm and RV. Wagoner, Ann. Rev. of Nuc. Sci. 27 (1977) 37. N. Terasawa and K. Sato, Prog. Theor. Phys. 80 (1988) 468. G. Steigman, D.N. Schramm, and J. Gunn, Phys. Lett. B66 (1977) 202. K. Sato and M. Kobayashi, Prog. Theor. Phys. 58 (1977) 1775. S. Miyama and K. Sato, Prog. Theor. Phys. 60 (1978) 1703. D. Dicus, E. W. Kolb and V. L. Teplitz, Phys. Rev. Lett. 39 (1977) 168. D. Dicus, E. W. Kolb and V. 1. Teplitz, Astrophys. J. 221 (1978) 327. E. Witten, Phys. Rev. D30, (1984) 272. J. H. Applegate and C. J. Hogan, Phys. Rev. D31 (1985) 3037. J. Applegate, C. Hogan, and R Scherrer, Astrophys. J. 329 (1988) 572. R Scherrer, J. Applegate, and C. Hogan, Phys. Rev. D 35 (1987) 115!. C. Hayashi, Prog. Theor. Phys.5 (1950) 224. W. Mampe, P. Ageron, C. Bates, J.M. Pendlebury, and A. Steyerl, Phys. Rev. Lett. 63A (1989) 593. M. Smith, 1. Kawano and R. Malaney, Astrophys. J. Suppl. 85 (1993) 219. C.P. Deliyanni et al, Phys. Rev. Lett. 62 (1989) 1583. L.M. Krauss and P. Romanelli, Astrophys. J. 358 (1990) 47. P. Kernan and L. Krauss, Phys. Rev. Lett. 72 (1994) 3309. Particle Data Group, Phys. Rev. D50 (1994) 1173. A. Songaila, 1.L. Cowie, C.J. Hogan, and M. Rugers, Nature 368 (1994) 599. R F. Carswell et al, MNRAS 268 (1994) L1. G. Steigman, MNRAS 269 (1994) L53. K. Olive, D. Schramm, J. Truran, R. Rood and E. V-Flam, U. Minnesota preprint (1994). N. Hata, R J. Scherrer, G. Steigman, D. Thomas and T. P. Walker, OSU-TA 26/94. G. Steigman an M. Tosi, Astrophys. J. 401 (1992) 150. D. Titler et al, private communication (1994). B E. Pagel et al, MNRAS 255 (1994) 325. T. Walker, G. Steigman, D.N. Schramm, K. Olive, and H.S. Kang, Astrophys. J. 376 (1991) 51. B.E.J. Pagel, in The Birth and Evolution of Our Universe: Proceedings of Nobel Symposium 79, eds. J.S. Nilsson et al (World Scientific, Singapore, 1991) 7.

263 34. C. Copi, D. Schramm, and M. Turner, FERMILAB-pub-94/174-A and Science, submitted (1994). 35. D.N. Schramm, in Proc. of Yamada Conf. XXXVII: Evolution of the Universe and Its Observational Quest, ed. K. Sato, Tokyo, June 1993 (Universal Academic Press, Tokyo), 61 (1994). 36. C. Alcock, G. Fuller, and G. Mathews, Astrophys. J. 320 (1987) 439. 37. H. Kurki-Suonio, R. A. Matzner, J. Centrella, T. Rothman, J. R. Wilson, Phys. Rev. D38 (1988) 1091. 38. H. Kurki-Suonio, R. A. Matzner, Phys. Rev. D42 (1990) 1047. 39. H. Kurki-Suonio, R. Matzner, K. Olive, and D.N. Schramm, Astrophys. J. 353 (1990) 406. 40. N. Terasawa and K. Sato, Prog. Theor. Phys.81 (1989) 254. 41. N. Terasawa and K. Sato, Phys. Rev. D39 (1989) 2893. 42. N. Terasawa and K. Sato, Prog. Theor. Phys.81 (1989) 1085. 43. N. Terasawa and K. Sato, Astrophys. J. 362 (1990) L47. 44. K. Sato and N. Terasawa, in The Birth and Evolution of Our Universe: Proceedings of Nobel Symposium 79, eds. J .S. Nilsson et al (World Scientific, Singapore, 1991) 60. 45. D. Thomas, D.N. Schramm, K. Olive, G. Mathews, B. Meyer, and B. Fields, Astrophys. J. 430· (1994) 291. 46. B. Meyer, C. Alcock, G. Mathews, and G. Fuller, Phys. Rev. D43 (1991) 1079. 47. R. A. Malaney and W. A. Fowler, Astrophys. J. 333 (1988) 14. 48. R. Boyd and T. Kajino, Astrophys. J. 359 (1990) 267. 49. H. Reeves and N. Terasawa, 1993, unpublished. 50. K. Jedamzik, G. Fuller, G. Mathews, and T. Kajino, Astrophys. J. 422 (1994) 423. 51. K. Jedamzik and G.M. Fuller, Astrophys. J. 423 (1994) 33. 52. K. Jedamzik, G.M. Fuller, and G.J. Mathews, Astrophys. J. 423 ( 1994) 50. 53. A. Heckler and A.C.J. Hogan, Phys. Rev. D47 (1993) 4256. 54. J. Spite and M. Spite, Astron. and Astrophys. 115 (1982) 357.

264

COSMIC RAYS FROM 1017 eV TO BEYOND 10 20 eV: EVIDENCE FROM THE FLY'S EYE EXPERIMENT AND GROUND ARRAYS PIERRE SOKOLSKY High Energy Astrophysics Institute, University of Utah, Salt Lake City, UT 84112, USA

ABSTRACT The study of cosmic rays with energies well above the knee (lOlSe V) of the spectrum has a long history. Beginning with the pioneering work of Linsley and the Volcano Ranch array, followed by the SUGAR, Haverah Park, Yakutsk, Akeno and AGASA arrays and the Fly's Eye detector, evidence has been steadily accumulating that the spectrum exhibits a change in slope near 10 EeV ( 1EeV 101SeV).1 This flattening may indicate that the spectrum in this region is largely extragalactic in origin. If the sources of this spectrum are sufficiently distant, a cut-off due to the interaction of near 100 Eev protons with black body photons should be observed. Recent data from the AGASA and Fly's Eye detectors have brought to bear a combination of information of spectral shape, composition and anisotropy to this study. In addition to clarifying the nature of the spectral structure, these new results lead to the surprising conclusion that nearby cosmic ray sources must exist that produce particles with energies well in e~cess of 100 Ee V. The recent data have spurred efforts to design and build experiments that have the enormous apertures necessary for detecting these very rare events.

=

1.

The Cosmic Ray Spectrum

Data on cosmic rays is most directly and reliably known up to energies of 1014 e V. At these energies the cosmic ray flux is high enough for balloon borne or shuttle borne experiments to be effective. Techniques such as Cherenkov and transition radiation detectors (Univ. of Chicago)2 and emulsion stacks (.JACEE)3) give reliable information on cosmic ray composition. Satellite experiments measure the total charged particle spectrum as a function of energy.4 From these experiments, we know that the total charged particle spectrum below the knee ( 101S eV) follows a power law j(E) = A· E-2.7. Both the JACEE and the Chicago experiments are in reasonable agreement with respect to the spectrum of individual components (see Figs. 1 and 2). These experiments find some evidence that the spectrum of medium and heavy nuclei are flatter than that of H and He, leading to an increasingly heavy composition near the knee. When expressed in terms of percentages, the components at 3.7.1014 eV total energy per particle are shown in the table below.3 There are roughly equal proportions of each group. The composition is thus mixed with a mean A of 10. ± 3.

The origin of these cosmic rays is very likely in diffusive supernova shock acceleration (SN). Models of such acceleration are successful in predicting the power law

265

JACEE

1-8

P r at a

It

In

or:c "~ 102

-rt++t+t 1.. _s\-rt-t t \

CJ

"lUI

....I...

1. 5

X

C-O

UI ~

'E w

~ 10

:z

1n"'C·

.,;

W

1

o.

5

Fe

X

tt-t-

.1

1

ENERGY

10

(TeV/n)

100

Figure 1: Differential energy spectrum of heavy nuclei as function of energy /a:mu from JACEE experiment- see ref. 3.

266

-

10

>

..

..

, "

o

" "

.. '"

..

'"

"

..

"

...,

o

Q)

I (I)

I t... (I)

17 and Ne-S group, while protons represent the p-He group. The middle range CNO group can be represented adequately by C nuclei. A three-component composition with the Fe fraction held at 40% gives the best fit. The resultant best fit of 40% Fe, 20% C and 40% p for the 1 to 3 EeV energy bin has essentially the same chi-square as the best two component fit. This shows that though the shape of the X ma", distribution is not very sensitive to medium weight nuclei, the Fly's Eye results are certainly perfectly consistent with a mixed composition that includes a substantial fraction from the eNO group. In fact, the above three component best fit in the 1 to 3 EeV energy bin is similar to the JACEE composition reported at 3.10 14 eV. How reliable is the conclusion that the cosmic ray composition is mixed and getting lighter with increasing energy'? The absolute position of measured Xm= in the atmosphere is certainly subject to systematic uncertainties. The Fly's Eye experiment estimates 25 gm j em 2 systematic errors. Furthermore the absolute X ma", positions in the atmosphere in MC simulations of p and Fe depends on hadronic model assumptions at the same level. The elongation rate itself is largely insensitive to these issues, however. There is also reasonable agreement between a number of ground array experiments and the Fly's Eye techniques as to its value 22 • 23 It also is clear that medium to high inelasticity hadronic models such as the QeD Pomeron and and QCD minijet models cannot reproduce the experimental elongation rate with a constant composition. It remains to be seen if other physics- based hadronic models could be found that predict such a large elongation rate with an unchanging composition and also reproduce low energy accelerator physics results. Barring the

277

800.0

o

750.0 N

E

~ 0)

.s

700.0

x

0

X

E

1150.0

1100.0 L-_ _....L..._ _--l'--_ _-'-_ _--L_ _ _........._ _--'

-1.S

-1.0

-0.5

0.0

0.5

1.0

1.5

Log (E (EeV»

Figure 8: Elongation rate from Stereo Fly's Eye: data (black circles), Fe (open circles) ,protons (open squares). predicted elongation rate from two component model (open diamonds). Error bars indicate the statistical error on the mean.

278

10

10 .,..

~

10 .,..

f~~c:~, .--'

10

E-

t.!':

~:

: +

1

1

400

600

800 1000 1200

400

~dUtfo,JII1JE.V

~~1

U.L:I:f.rrfl~~TtIT 1..

•. I .. ,

600

l..

:-.- -+t: ,:

.,

800 1000 1200

%I7IIIZ dUtlD' J

1111.0 EeV

10

1 400

600

800 1000 1200

........ dUtlD' 1.0 111 3.0 E.V

400

600

800 1000 1200

"""'" dUtfor 3.0 II> 10.0 E.V

Figure 9: Xmax distribution for data (crosses), Fe (dashes) and protons (dots) for four energy bins. Distributions are normalized in area.

279 appearance of such presently unknown mo·dels, the elongation rate alone is strong evidence that composition is changing and getting lighter. The apparent change. in elongation rate in the Fly's Eye data near .3 EeV is also very interesting. It indicates a change in either composition or hadronic physics in a very direct way. This transition is quite near the threshold of the detector however, and some caution is called for. It will be very important to independently verify this break. It is of course very interesting that this apparently increasingly light composition occurs in the energy region where the spectrum undergoes a change in slope. One picture which can explain the Fly's Eye and AGASA spectra is the dominance of a new, flatter spectrum over the lower energy one. A two component fit to the Fly's Eye stereo data can yield the slopes of each spectrum (- 3.2 for the lower energy one and -2.6 for the higher energy one). If the lower energy spectrum is primarily heavy in composition and the higher energy one primarily light, one can predict the ratio of heavy to light composition as a function of energy. Using the predicted proton and Fe elongation rates as shown in Fig 4, one can then predict the elongation rate for the two component mixture. The diamonds in Fig. 4 indicate the resultant prediction. They are in excellent agreement with the data and support this simple picture.

4·2.

Implications of Observing a Continuing Flux Past 100 EeV One of the signatures of distant extragalactic cosmic ray sources is the GZK cutoff. This effect, due to the onset of inelastic photoproduction by protons on the 2.7 deg black body radiation should produce a cutoff in the spectrum near 60 EeV, if the extragalactic sources are universally distributed. Both the Fly's Eye monocular and the AGASA spectrum would have been consistent with such a cutoff (though not statistically strong enough to prove that it existed) where it not for the observation of two events well beyond 100 EeV. To escape the GZK cutoff, these events must have sources less than 50 Mpc from us. 2i At these energies, there should be very little bending in the extragalactic magnetic fields and the particles should point back to their sources. No obvious candidates, such as AGN's or strong radio galaxies can be identified withing the error boxes at distances of less than 50 MpC. 35 Ei ther new types of sources (decay of topological defects, such as cosmic strings 36 ) or much stronger than expected extragalactic fields are required to explain these data. Since the cosmic ray flux at 100 EeV is near one event per square kilometer per century, the origin of these particles and how they relate to the bulk of the cosmic ray flux above 10 EeV and the GZK cutoff can only be settled by constructing much larger aperture detectors. It is clear that the ability to measure composition above 10 EeV is also vital for any new detector. 5.

Existing and Approved Detectors

The Fly's Eye detector has recently been decomissioned. There are three detectors presently operating or under construction: AGASA, the Yakutsk array, and the HiRes detector. In what follows we concentrate on AGASA and HiRes.

280

The HiRes Detector (Stage J) The Utah Fly's Eye group has entered into a collaboration with the University' of Adelaide, Columbia University and the University of illinois to construct a nextgeneration air-fluorescence detector.27 The first stage of construction of this detector has been funded and is underway. Two sites twelve kilometers appart have been chosen. A total of 56 two meter diameter mirrors will be installed at the sites. The mirrors are arranged in rings, each ring subtending 14 degrees of elevation. Two complete rings at one site and one partial ring at the second site will be built. Each mirror will have a 256 phototube array at its focal plane. Each phototube will view a one degree by one degree section of the sky. The signals from the phototubes will be digitized using FADC electronics so that both the amplitude and the detailed pulse shape will be available for later analysis. All events used will be recorded by both sites in stereo for good control of errors and redundancy of measurement. The aperture for this Stage I detector is optimized for greater than 10 EeV energy and approaches 7500 km 2 str at 100 EeV. With a 10 % duty factor, this detector should record 10 events per year above 100 EeV and 200 events per year above 10 EeV if the stereo spectrum observed by the Fly's Eye were to continue. This is an order of magnitude increase in statistics over the monocular Fly's Eye. The resolution in energy and Xmax is also much improved over the Fly's Eye detector, approaching 10% statistical error in energy and 15 gm/cm 2 mean error in X max • This detector can also search for a possible gamma ray and neutrino flux. The method and object of such a search will be described below. 5.1.

5.2.

The HiRes Detector (Stage II) Since the HiRes detector is modular, it can be easily expanded. IT the Stage I detector confirms a continuing UHE flux above the GZK cutoff, the aperture of the detector can be almost doubled to 12500 km 2 str by adding thirty more mirrors. This will complete the two lower rings at the second site. To increase the 100 EeV aperture further would require the construction of additional stations. A third site may in fact be constructed by the proponents of the Telescope Array (see below). 5.3.

The AGASA experiment The Akeno Giant Air Shower Array (AGASA) has been operating since the spring of March 1991. 28 111 particle density detectors are arranged with interdetector separation of about 1km covering an area of about 100 km 2 • This aperture for collecting the highest energy cosmic rays is the largest in the world at present. About 100 well reconstructed events (whose core is well within the boundary of the array, and whose zenith angle satisfy sec (} S 1.4) have been observed above 10 EeV. It is expected that more than 300 events with energies greater than 10 EeV will be accumulated by the end of 1995. This should provide futher solid evidence for the absence or presence of the GZK cutoff.

281

6.

Plans for Future Experiments

Much larger aperture detectors will be required to investigate the cosmic ray flux in detail if it continues to near 1000 EeV. This issue is adressed by two proposals: the Telescope Array, and the Giant Hybrid Array. We discuss them briefly in turn.

6.1.

The Telescope Array The Telescope Array is a proposal by a consortium of Japanese universities and the University of Utah, led by the University of Tokyo.29 The goal is to build an air fluorescence detector with a pixel size of 1/4 by 1/4 degree using multianode photomultiplier tubes. With this small a pixel size, sky noise is essentially negigible. The array is designed to also be sensitive to low energy gamma rays above 100 GeV by detecting Cherenkov light from these showers. It includes 120 fixed and 120 steerable alt-azimuth three meter diameter mirror dishes positioned at two stations 40 km appart. The aperture of such an array at 100 EeV is estimated to be approximately 40,000 km 2 str. The energy resolution is 10 % and the resolution in Xma.x is 10 gm/cm 2 A prototype mirror and pmt cluster are being tested at Dugway. There are plans to build ten prototype mirror units at a site near the HiRes detector sites. The Giant Hybrid Detector This proposal, led by groups from the University of Chicago and the University of Leeds, combines the strength of ground arrays (high statistics due to near 100% duty cycle, good sensitivity to possible anisotropies) with the advantages of the air fluorescence technique ( model independent energy measurement, measurement of cosmic ray shower development and composition).30 The main array, with 1.5 km detector spacing, is expected to have an aperture of 5000 km 2 str near 100 EeV. A densely packed 400 km 2 central array would have .75 km detector spacing and have a threshold below 1 EeV. The central portion will collect about 40,000 events per year with energy greater than 1 EeV while the full array would detect 5000 events per year above 10 EeV. Detectors would measure both the electron and muon components at the surface. Two such arrays are proposed, one in the Northern and one in the Southern hemisphere. This will guarantee good coverage for anisotropy measurements. At the center of each array will be an air fluoresence detector. Approximately ten percent of the data will thus have both ground array and longitudinal shower development information. The energy scale, as determined by the fluorescence technique, can be used to calibrate the ground array. Similarly muon information determined by the ground array can be compared to composition measurements from the longitudinal shower development measurements. 6.2.

7.

Gamma Ray and Neutrino Flux

It is a truism that when opening up a new energy region it is important to be sensitive to the unexpected as well as to resolve specific questions. An unexpectedly high flux of photons or neutrinos is one such possibility. A photon and neutrino

282 flux is predicted by the GZK mechanism though its magnitude is model dependent. Cosmic string models predict a larger neutrino flux and a photon to proton ration of. greater than one at the highest energies. 31 Detectors using atmospheric fluorescence are sensitive to near horizontal and upward going EAS. Calculation indicates that for certain assumptions about the GZK neutrino flux, a HiRes type of detector may be able to indentify a significant number of neutrino induced events. 32 Gamma rays are difficult to distinguish from protons by EAS longitudinal development for energies below 100 EeV, but their EAS may have a significantly smaller muon content at the surface. Calculation indicate that the turn-on of the LandauPomeranchuk-Migdal effect near 100 EeV causes striking elongation of gamma ray shower development. 33 Air fluorescence detectors may be able to distinguish such showers from normal hadronic showers. Observations of (or limits on) such fluxes will clearly be very important in understanding the nature and origin of the UHE cosmic ray flux. 8.

Conclusion

Recent results indicate that there is clear and striking structure in the UHE cosmic ray spectrum. Composition studies support a picture of an emerging extragalactic flux near 10 EeV. The apparent continuation of this flux beyond the expected GZK cutoff energies and the lack of obvious close-in source candidates for such a flux raises many interesting astrophysical questions. This area of astrophysics can only make significant progress by further support for construction of the sophisticated large aperture detectors that have sensitivity to all the relevant quantities of interest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14.

P. Sokolsky et al. Physics Reports 217 (1992) 227. D.Muller et al., Ap. J. 374(1991) 356. K. Asakimori et al., in Proc. 22 JCRC Vol.2, p 57,(Dublin,1991). N.L. Grigorovet al., in Proc 12 JCRC Vol.5,pI746,(Hobart,1971). A.R. Bell, Mon. Not. R. Astron. Soc 182(1978) 147. R.D. Bladford and J.P. Ostriker Ap. J. Lett.221 (1978) L29. G.F. Krymsky, Sov. Phys. Dokl.22(1977) 327. W.I. Axford et al. in Proc 15 JCRC Vol. 11, p 132,(Plovdiv,1977). D. Muller et al. in Proc. 22 ICRC Vol.2 p 25, (Dublin,1991). W.I. Axford in Astrophysical Aspects of the Most Energetic Cosmic Rays p. 406 (World Scientific, Singapore, 1991). J.R. Jokipii and G. Morfill in Astrophysical Aspects of the Most Energetic Cosmic Rays p. 261 (World Scientific, Singapore, 1991). W.H. Ip and W.I. Axford, in Particle Acceleration in Cosmic Plasmas, p.400 (Newark, 1992). J.R. Protheroe and A.P. Szabo Phys. Rev. Lett.69 (1992) 2885. J.P. Rachen and P.L.Biermann Astron. Astroph.18 (1992).

283

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

30.

31. 32. 33. 34. 35. 36.

J. Linsley in Proc. 13 lCRC Vol. 5, p 3207, (Denver,1975). A.M. Hillas in Proc. 12 lCRC Vol. 3,p 1001, (Hobart,1971). P. Sokolskyet al. Phys. Reports 217 (1992) 227. M.A. Lawrence et al. J. Phys. G 17 (1991) 733. R. Baltrusaitis et al. Nucl. lnstr. and Meth. A 240 (1985) 410. K. Greisen Phys. Rev. Lett. 16 (1966) 748; A.M. Zatsepin and V.A. Kuzmin JETP Lett. 4 (1966) 78. T.K. Gaisser et al. Phys. Rev. D47 (1993) 1919. R. Walker and A.A. Watson J. Phys. G 7 (1981) 1297. N.N. Efimovet al. in Proc 20 lCRe Vol 5. p 490 (Moscow, 1987). G.L. Cassidayet al. Ap. J. 356 (1990) 1664; T.K. Gaisser et al. Phys. Rev. D47 (1993) 1919. D. Bird et al. in Phys. Rev. Lett. 71 (1993) 3401. D.J.Bird et al. Phys. Rev. Lett 71 (1993) 3401;D.J. Bird et al. Astrophys. J. 424 (1994) 491; T.K. Gaisser et al. Physical Review D 47 (1993) 1919. Staged Construction Proposal for the High Resolution Fly's Eye (1993). S. Yoshida et al. lCRR-Report -325-94-20 (1994). M. Teshima in Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays (1993) 109. J.W. Cronin in Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays (1993) 87; B. Dawson in Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays (1993) 125. F.A. Aharonian et al. Phys. Rev. D46 (1992) 4188. Y.Ho, personal communication Y. Mitzumoto Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays (1993) 194. D.J. Bird et al., Ap. J. to be published (1995). J.W. Elbert and P. Sommers, Ap. J, to be published (1995). G. Sigl, D.N. Schramm and P. Bhattacharjee, On the Origin of Highest Energy Cosmic Rays NASA/Fermilab Astrophysics Center Preprint (1994).

PARALLEL SESSIONS

Exact Solutions

Chairperson: V. Belinsky

289

EXACT (3+1) DIMENSIONAL GRAVITATIONAL SOLITONS TO THE EINSTEIN GRAVITATIONAL FIELD EQUATION

Au a , C. Fang b , L. Z. and To c , F. T. a

Dept. of Applied Math. Hong Kong Polytechnic, Hong Kong b Dept. of Physics, University ofkrizona, Tucson, USA C Dept. of Physics, University of Hong Kong, Hong Kong

ABSTRACT We start with the diagonal space-time metric tensor in the cylindrical coordinates and by using the differential geometry approach to solve the Einstein gravitational field equation in empty space. A series of new exact cylindrical symmetry wave solutions , propagating with the light velocity, and (3 solutions have been obtained.

+ 1)

dimensional gravitational soliton

The problem of the gravitational radiation is still one of the problems in the general relativity since 1916. 1 - 6 Recently F. T. To, P. C. W. Fung and C. Au 7-8 obtained exact (1+1) I-soliton and 2-soliton solutions in vacumm. We are still searching for new (3+ 1) gravitational wave solutions. Let us consider a gravitational field having the square of the space-time interval ds 2 = e 2N(ct-z,p, F2] in terms of one and two harmonic maps. This allows us to choose the electromagnetic potential to be of monopoles, dipoles, etc. In order to compare the behavior of the different unification theories of gravitation and electromagnetism, the constant Q remains here arbitrary.

=

1. The Action

As the most important theories which unify gravitation and electromagnetism, the Einstein-Maxwell, Kaluza-Klein and String theories have been intensively studied in the last years. The action of these three theories can be written in a compact form given by (1) where R is the Ricci tensor, c]i is the scalar potential, F is the Faraday electromagnetic tensor and 0 is a coupling constant. All the three theories are contained here, the Einstein-Maxwell with Dilaton field for 0 = 0, the Low energy String theory for 0 = 1, and the Kaluza-Klein theory for 0 = J3. In this note we give a method for solving the field equations derived from action (1) for static metrics depending only on two coordinates p and (, i.e. gp.1I = gp.lI(p, (). We start with the Papapetrou form for the metric

(2) In order to solve the field equations we define four potentials in a covariant form, they are

J,

_ J",2 DX=2-DA,p

'l/; = 2At ,

(3)

p

where J is the gravitational, 'l/; the electrostatical , '" the scalar, and X the magnetostatical potential. (Ap, A" A,p, At) is the electromagnetical four vector potential. The differential operators are defined by D = (/p, jj = which implies that DD = O. The field equations in terms of the four potentials can be derived from the Lagrangian

g,),

L

* e-Mail:

= '!!""DJ2 + pK,2 (D'l/;2 + ~DX2) + ~D",2. 2

2J

tma.tos@fis.cinvesta.v.mx.

2J

",4

0 ",2

(g" - gp)'

(4)

309 We obtain four non-linear coupled partial differential equations which in general are very difficult to solve. 2. Harmonic Maps Ansatz In order to solve these differential equations we make the harmonic maps ansatz [1] supposing that the four potentials depend on a s~t of harmonic maps Ai, i = 1, ... ,p, where Ai = Ai(p, () and (pA:z),z + (pA:z),z. In the present note we assume p = 1,2. We obtain a set of six solutions of these field equations each of them depends on one or two harmonic maps [2]. Each solution represents a different class of spaceti~e.

3. Examples Let us give some examples. The first example we want to show is

K

= (bA

+ c)P'

4..fib X = bA + c

2

where, = 0'2~1 and;3 = ;2~1. This class contains the Gross-Perry-Sorkin [3] solution for A = -lr6bl , C = 2, a: =.J3. Taking A = co:o we can generate magnetic dipole solutions. The last example we have place to show is

Here we get the Gibbons-Maeda [4] solution for A = 10g(1- 2;.") = 7, ql = O. From this class we can generate dipole solutions with the same gravitational features as the Gibbons-Maeda one, assuming 7 = (r-m~Jr-~) 20. Solutions with an arbitrary -m cos electromagnetic field can be generated by choosing conveniently the harmonic map 7.

Acknowledgements This work is partially supported by CONACYT-Mexico.

References 1. Matos, T. J. Math. Phys. 35 (1994) 1302. 2. Matos, T., Quevedo, H. and Nunez, D. to be published. 3. Gross, D.J. and Perry, M.J. Nuc. Phys. B226 (1983) 29. Sorkin, R.D. Phys. Rev. Lett. 51 (1983) 87. 4. Gibbons, G.W. and Maeda, K. Nuc. Phys. B298 (1988) 741.

310

EXTENDED N-SOLITON ELECTROVAC SOLUTION

E. RUIZ, V. S. MANKO and J. MARTIN Grupo de Fisica Teorica, Universidad de Salamanca, 37008 Salamanca, Spain

In Ref. 1 special N-soliton solutions of the Einstein-Maxwell equations were considered, being only able to describe the exterior gravitational and electromagnetic fields of superextreme sources. Using Sibgatullin's integral method 2 , we have constructed the extended N-soliton solution which involves 6N arbitrary real parameters corresponding to 6N arbitrary relativistic multipole moments, hence allowing to treat both the under- and superextreme cases. The solution is defined by the Ernst complex potentials3 of the form

£ = E+/E_,

1 ±1

1

1

0

rl

r2N o.2N - /31

-1

0.1 -

±1

/31

rl /3N h 1(o.l)

0.1 -

E± = 0

0.1 -

0

/31

hN(o.t} /3N

0.1 -

f(o.t} rl 0.1 -

r2N -1 o.2N - /3N ,F= hI (o.2N) 0 o.2N - /31 hN(o.2N) o.2N - /3N

(1)

0 such that xr+1 = 0 for all X eL We found twentysix abelian two-dimensional subalgebras.

2.2.

The non-abelian case The classification in this case was done considering a tetrahedron in C4

327 where

e, = [

~1

,'2 = [

i1

,e, = [

~ 1' ~ ~ 1( =d

= [

1 )

Then according to the action of A and B E SL( 4, C) we can represent the study of the group G by exhibiting the part of the tetrahedron that G preserves. Fifteen non-abelian cases were found. For all cases abelian and non-abelian, each group G can be embedded in P12, the subgroup of SL( 4, C) that leaves fixed a point of F 12 ,3,4 possibly after sinillarity transformations. :F12 is the correspondence space between the projective twist or space and the compactified, complexified Minkowski space. In particular each group G is conjugate to a subgroup of the complex Weyl group.

3.

The real forms

In our study we also look for the simmetry groups of each algebra. This automatically provides a way of searching for the three dimensional complex algebras. A complete classification of them has already been done and results will be presented elsewhere. 5 To systematically apply the study of these symmetries to the SDYM equations we need to know which of the elements classified belong to the real algebra of the conformal group. To perform this study we need to find a pseudo hermitian form :Tof signature (2,2) such that for a given canonical form of matrices A and B (representing generators of S L( 4, C) the following equations are obeyed:

JAt = pAJ +qBJ (2)

JBt

rAJ +sBJ

This work is in progress.

References 1. Ward R. S. Nucl. Phys.B 236 (1984) 381. 2. Ward R. S. Philos.Trans. R. Soc.A 315 (1985) 451. 3. Ward R. S. and Wells R. O. Jr. Twistor Geometry and Field Theory (Cambridge University Press, Cambridge, 1990). 4. Diaz M. C. and Sparling G. A. J. in Differential Geometric Methods in Theoretical Physics Catto S. and Rocha A., editors (World Scientific, Singapore, 1991) 5. Diaz M. C. and Sparling G. A. J. (in preparation)

328

THE ROLE OF THE INTERNAL METRIC IN GENERALIZED KALUZA-KLEIN THEORIES

w.

DRECHSLER and D. HARTLEY

Max-Planck-Institut fiir Physik - Werner-Heisenberg-Institut P.O.Box 401212, 80712 Munich (Fed. Rep. Germany) ABSTRACT

=

The dimensional reduction of a Weyl space WN of N 4 + n dimensions to a principal fiber bundle P(W4 , Gn ) over a four-dimensional space-time is studied. The fibration arises from the existence of n conformal Killing vector fields of the original N-metric. The framework of a Weyl geometry is adopted in order to investigate conformal rescalings of the metric in the bundle P(W4 , Gn ) obtained. Finally, the Weyl symmetry is broken again by choosing a gauge in which the internal i.e. fiber metric is of constant Cartan-Killing form. This choice of gauge implies a conformal transformation of the metric in the space-time base of the bundle compared to the dimensional reduction of a Riemannian space VN.

1. Introduction

In the dimensional reduction 1,2,3 of a Riemannian space VN=4+n to a principal fiber bundle P(V4, Gn ) resulting from the presence of n Killing vector fields ea; a = 1, 2... n of the metric in VN with the generating the structural group of the bundle P there appear !n( n + 1) additional fields - representing the internal or fiber metric gab; a, b = 1, 2... n - the dynamical relevance of which is not apparent. One can force the internal metric to play no dynamical role in the theory by requiring that it is identical with the constant Cartan-Killing metric of the gauge group Gn on P which is the usual practice in Yang-Mills theories [compare Kerner4]. This freezing of the internal metric degrees offreedom corresponds to the "strong cylinder condition" of the original five-dimensional Kaluza-Klein theory5,6,7. On the other hand, however, these internal metric fields appear automatically in the Kaluza-Klein-type fibration by isometries yielding, for a compact group Gn , a space-time dependent volume factor upon integration over the group space. Moreover, the internal metric fields are used in the literature3,8 to conformally transform the metric in the space-time base of P implying that it is the conformally transformed metric which is the "physical metric" in the space-time base.

ea

2. Dimensional Reduction of a Weyl Space W N In order to investigate the possibitity of conformal rescalings of the metric in the bundle space obtained as the result of the presence of isometries of the original

329

N-metric we investigate the dimensional reduction of an N-dimensional space with signature (1, N - 1) to a principal bundle over four-dimensional space-time in the framework of a Weyl geometry, i.e. we start from a Weyl space WN; N = 4 + n, with metric gAB; A,B = (p.,a), (v, b) with p.,v = 0,1,2,3 and a,b = 1,2 ... n, and Weyl vector field II:A together with a Weyl connection rAB

C

-

= rAB

C

1

C

- '2(II:A..g) = 0 (EO). The Ricci tensor can be expanded in the form

=

(1)

where superscripts indicate the order in powers of hill! and its derivatives. The individual contributions to (1) are further decomposed in

352

terms of orders in

E:

(2)

(3)

RW

RW.

where [En] denotes the term of order O(En) in The vacuum field equations must be satisfied to successive orders in E. To orders E- 1 and EO, these are (4)

R~OJ (f)

+ R~~ [EO] + R~2J [EO]

= 0.

(5)

A "Brill-Hartle time averaging,,2,4 of (5) yields

(6) From eq. (6), it is clear that the high frequency approximation is a necessary condition for the existence of a gravitational geon. In fact, the time-averaged Ricci tensor component (R~~ [EO]) in eq. (6) can be an effective source for the background metric IPv only when terms bilinear in the perturbation are of the same order (EO) as the background tensor R~~ ( I). A geon is a non-trivial solution of eqs. (4) and (6) with boundary condition h pv ~ 0 as r ~ +00 . (7) BH considered the case of odd modes with quantum number m = O. They introduced the decomposition of the metric perturbations into the Einstein vacuum field equations without imposing the highfrequency constraint which would have assured the correct order of magnitude decomposition of the equations. In terms of the amplitude Q and the Regge-Wheeler coordinate r* defined in ref. 2, they find 2,4

2

d Q+ dr2 *

[w2 + ~(ZI' _ ).')ev -,\ 2r

_

1(1

+ 1) ev ] Q = O.

r2

(8)

This Schrodinger-like equation lends itself to the analogy with the dynamics of waves propagating in an effective potential. BH specified

353

a background metric with the stated intent of confining the waves to a shell of small thickness, but their actual realization was a shell of zero thickness. The BH construct harbours a singular shell with an implicit energy-momentum tensor and hence cannot be the desired singularity-free purely vacuum solution envisaged in the geon concept. We now attempt the construction of a spherical geon by incorporating the necessary requirement of the high frequency approximation. For the odd modes, it is sufficient to consider the O(l/E) Einstein equation

R~~ [E-I]

=

~ ('Y00h02,02+'YllhI2,12)

= 0,

(9)

which implies, for the relevant amplitudes ho, hI of the odd modes, we- ll h o sin(wt)

+ e-Ah~ cos(wt)

= 0 .

(10)

The only solution which satisfies (7) is ho(r) = hl(r) = O. These in turn imply that the effective source term - ( R~J) of gravitational waves in eq. (6) vanishes, leaving us with R~oJ(')') = 0, which has the Minkowski metric as its only asymptotically flat solution. The same result is obtained for the even modes. The high frequency nature of the waves, which is necessary for the viability of the geon, prevents their confinement. Thus, the system is analogous to a Newtonian galaxy made exclusively of high velocity stars, which are not trapped by the potential created by all the other stars and escape freely to infinity. The system does not satisfy the virial theorem and is not bounded. The generalization to the case of a time-dependent, spherically symmetric background metric 'YJ-lll(t, r) whose time variation scale is much greater than that of the period of the waves is straightforward. Since the essential aspect leading to the non-viability of the geon in the spherically symmetric case is the high frequency approximation, we argue that a geon with less symmetry is an unlikely prospect. Given the essential non-viability of a gravitational geon, it is natural to consider a potential connection with the question of gravitational energy. It has long been known that the energy concept in general relativity presents difficulties connected with the non-tensorial

354

energy construct. If gravitational energy were to have an independent existence as do the energies of other physical fields, then one would expect that a gravitational geon could exist, at least in principle. However, with this existence removed, one might consider the possibility that gravitational energy does not have an existence which is independent of other forms of energy, which in turn do have a true tensorial form for energy and momentum. 6

References 1. J.A. Wheeler, Phys. Rev. 97 (1955) 511; E.A. Power and J.A. Wheeler, Rev. Mod. Phys. 29 (1957) 480; R.D. Brill and J.A. Wheeler, ibidem 465; F.J. Ernst, ibidem 496; Phys. Rev. 105 (1957) 1662, 1665; J.A. Wheeler, Rev. Mod. Phys. 33 (1961) 63; J.A. Wheeler Geometrodynamics (Academic Press, New York, 1962); D. Brill, Perspectives in Geometry and Relativity, Essays in honor of V ciclav Hlavaty, ed. B. Hoffmann (Indiana Univ. Press, Bloomington, 1966) p. 38. 2. D.R. Brill and J.B. Hartle, Phys. Rev. 135 (1964) B271. 3. F.I. Cooperstock, V. Faraoni and G.P. Perry (1994), in preparation. 4. R.A. Isaacson, Phys. Rev. 166 (1968) 1263, 1272. 5. L.A. Edelstein and C.W. Vishveshwara, Phys. Rev. D 1 (1970) 3514. 6. F.I. Cooperstock, Found. Phys. 22 (1992) 1011; in Topics in Quantum Gravity and Beyond, ed. F. Mansouri and J.J. Scanio (World Scientific, Singapore, 1993) p. 201.

355

EXTRAGALACTIC BACKGROUND RADIATION AND THE TIME SYMMETRY OF THE UNIVERSE DAVID A. CRAIG Depa'l"tment of Physics, University of California, Santa Barbara, California 93106-9530, USA ABSTRACT This report describes an observable consequence for the extragalactic background radiation (EGBR) of the assumption that our universe possesses time symmetric boundary conditions. Specifically, such boundary conditions imply that the optical EGBR should be at least twice that due to galaxies on our past light cone alone, and perhaps considerably more. The minimal prediction is only barely consistent with present observations. It is therefore possible to test ezperimentally the intriguing notion that if our universe is closed, it may be in a certain sense time symmetric.

Various authors have raised the possibility that the observed "arrows oftime"lare directly correlated with the fact that the universe is expanding, and would consequently reverse themselves during a recontracting phase. 2-4 Of central interest is the thermodynamic arrow of entropy increase, from which other time arrows such as the psychological arrow of perceived time or the arrow defined by the retardation of radiation are derivable. 1 However, the mere reversal of the universal expansion is insufficient to reverse the direction in which entropy increases. 5 In order to do quantum physics in a recollapsing model universe in which the thermodynamic arrow reverses itself it appears necessary to employ a time symmetric variant of quantum mechanics6, 7 in which boundary conditions are imposed near both the big bang and the big crunch. These boundary conditions take the form of "initial" and "final" density matrices which are CPT-reverses of one another. In such a model the ensemble of quantum-mechanical histories is time symmetric in the sense that each history occurs with the same probability as its CPT-reverse. 7 , Along with this quantum mechanics, the model universe considered consists in: • A fixed closed, homogeneous and isotropic background spacetime, viz. a k = +1 Friedmann-Robertson-Walker universe. The evolution of the scale factor is determined from Einstein's equations by the averaged matter content of the universe . • Boundary conditions imposed on the matter content in the form of CPT-related density matrices describing the state of matter at some small fiducial scale factor near what would in the absence of quantum-gravitational effects be the big bang and big crunch, but outside of the quantum gravity regime. The matter state described by one of these density matrices reflects the presumed state of the early universe, namely, matter fields in thermal equilibrium at some temperature appropriate to the fiducial scale factor and the total amount of matter in the universe. Spatial fluctuations should be consistent with present day large scale structure. A final key assumption is that physically interesting coarse grained histories unfold near either boundary condition {relative to the total lifetime of the universe} in a fashion insensitive to the presence of the boundary condition imposed at the other

356

end. Simple modelsEH:l suggest that this should be true whenever the "relaxation time" of the relevant process to equilibrium is short compared to the total time between the imposition of the boundary conditions, which expectation is rigorously supported in the case of Markov processes. 9 With these assumptions, this model universe might be expected to closely resemble the universe as it is observed today if the universe is close to the critical closure density, so that its lifetime is very long. a In particular, predictions regarding inflation, relative particle abundances, and the formation of large (and small) scale structure are assumed to hold. Because of the CPT-related boundary condition at the big crunch, a similar state of affairs is expected in the recollapsing era, but timereversed. As the thermodynamic arrow is caused fundamentally by gravitational collapse driving matter away from equilibrium, the arrow of entropy increase near the big crunch will run in the opposite direction to that near the big bang. Observers on planets in the recollapsing phase will find their situation indistinguishable from our own, with all time arrows aligned in the direction of increasing volume of the universe. b It is this interesting state of affairs which leads to the conclusion that observations of the EGBR can reveal the presence of the final boundary condition even if the lifetime of the universe is very great. 10,7,l1 At optical wavelengths, the isotropic bath of radiation from sources outside our galaxy is believed to be due almost exclusively to galaxies on our past light cone. 12 ,13 In a model universe with time-symmetric boundary conditions, however, there is a significant contribution correlated with the time-reversed galaxies which will exist in the recollapsing era, far to our future. 7 ,lO,l1 The reason for this is that light from our galaxies can propagate largely unabsorbed into the recollapsing phase no matter how close to open the universe is. 10,l1 This light will eventually arrive on galaxies in the recollapsing phase, or, depending on its frequency, be absorbed in the time-reversed equivalent of one of the many high column density clouds (Lymanlimit clouds and damped Lyman-a systems) present in our early universe,12 in the intergalactic medium, or if low enough energy at the time-reversed equivalent of the surface of last scattering. This will appear to observers there as emission by those sources sometime in their galaxy forming era. Since future galaxies, up to high time-reversed redshift, occupy only a small part of the sky seen by today's (on average) isotropically emitting galaxies, most of this absorption must occur in one of the other listed media. Thus the light from the entire history of galaxies in the expanding phase will constitute an isotropic bath of radiation to observers at the time-reverse of the present epoch that is in addition to the light from the galaxies to their past. By time symmetry, there will be a similar contribution to our EGBR ~ Actually, the "relaxation times" for many interesting physical processes in a recollapsing universe do not appear to be short compared to the lifetime of the universe, even if that lifetime is arbitrarily long. 10, 11 The point of view taken in reference 11 is that such difficulties provide strong arguments against the possibility that our universe possesses time symmetric boundary conditions. Here there is only space to assume that the model is consistent and see what it predicts. bThe state of the universe when it is large, which somehow must interpolate between opposed thermodynamic arrows, is discussed in reference 11.

357 correlated with galaxies which will live in the recollapsing phase, over and above that due to galaxies on our past light cone. A lower limit to this excess background can be obtained by considering how much light galaxies to our past have emitted already,14 say viv ~ 10- 5 ergcm- 2 S-l ster- 1 at 5000A. To observers at the time-reversed epoch this background will remain in the same frequency band because the size of the universe is the same. Thus, at a minimum the predicted EGBR in a universe with time symmetric boundary conditions is twice that in a universe in which the thermodynamic arrow does not reverse. C Comparing with present upper limits on the observed optical EGBR,15 viv ~ 2· 1O-5 erg cm- 2 8- 1 8ter- 1 at 5000.11, the minimal prediction taking into account only the quantity of light emitted by galaxies to our past is seen to be only marginally consistent with observation. Thus it appears unlikely on observational grounds that our universe possesses time symmetric boundary conditions; better observations and modelling may rule it out entirely.

Acknowledgements I wish to thank R. Antonucci, O. Blaes, T. Hurt, R. Geller and J. T. Whelan for useful conversations, and J. B. Hartle for suggesting this project and for provocative discussions.

References

1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12. 13.

14. 15.

H. D. Zeh, The Physical Basis of the Direction of Time (Springer, Berlin, 1992). T. Gold, Am. J. Phys. 30 (1962) 403. S. W. Hawking, Phys. Rev. 032 (1985) 2489. H. D. Zeh Phys. Lett. A126 (1988) 311. S. W. Hawking, in Proceedings of the NATO Workshop on the Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. Perez-Mercador, and W. Zurek (Cambridge University Press, Cambridge, 1994). W. J. Cocke, Phys. Rev. 160 (1967) 1165. M. Gell-Mann and J. B. Hartle, in Proceedings of the NATO Workshop on the Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. PerezMercador, and W. Zurek (Cambridge U. P. , 1994) and references therein. L. S. Schulman, J. Stat. Phys. 16 (1977) 217. L. S. Schulman, in Festschrift for M. Fisher, to appear. P. C. W. Davies and J. Twamley, Class. Quant. Grav. 10 (1993) 931. D. A. Craig, to be published. P. J. E. Peebles, Principles of Physical Cosmology (Princeton U. P. , 1993). The Galactic and Extragalactic Background Radiation, edited by S. Bowyer and C. Leinert (Kluwer, Dordrecht, 1990). S. Cole, M.-A. Treyer, J. Silk, Ap. J. 385 (1992) 9. K. Mattila, in reference 13.

.., which is adjusted to determine the correct energy, they obtain

f(r)~m(r)=7r{1-r/J-r2+2.11}, which is also displayed in figure 1.

(2)

367

2.

Approximate solutions from Merons

The proposed approximation is based on the observation that in compactified Minkowski space, meron-type solutions, when interpreted as Skyrme currents satisfy exactly Skyrme's equation, in the same space. To find the field one just integrates approximately the equation with the currents as derivatives of the fields. For the approximation we introduce in the compactified Minkowski space (XO, Xl, X2, X3) the coordinates uJ1., Jl = 0,5; in the form U

2xI' J1._,

4 _ U

1+

X2 US

---, 1"

1"

1- x 2 = - - , where 1"

U sing these coordinates we have that our Minkowski space is isomorphic to

Sl x S3/71.. 2 == S1 X SU(2)/71.. 2 • It follows 4 that biinvariant connections take the form a( V; )fi

(8)

Yi,

(3)

with a an arbitrary function of the coordinate V; in Sl. The forms ()i are the duals of the left invariant vector fields ~ifT S3. When a( V;) = 1/2 these connections solve the Yang Mills equations and are the meron solutions. To verify that (3) is a solution to Skyrme's equation recall the intrinsic formulation of Skyrme's equations

*d* (L

+ 1/4 [La, [La, L]]) =

O.

(4)

Substitution of (3) into (4) gives:

2~ (a + 2a3)Yafabed()b 1\ ()e 1\ ()3 }

*a

{ -

- *

{2~ ~m( a + 2a3)Yafabcd()m 1\ ()b 1\ ()e 1\ ()d}

1

+ * {24(a + 2a3 )Ya f abed d( ()b 1\ ()e 1\ ()d)} . The first term is zero because a depends only on V; and the field ~m depends on the S3 variables. The second term is zero because of the Maurer-Cartan identity. The field U is recovered approximately from 1 by integrating

au . (- a -=Ua(v;)B' axo

oXo

) (8)Yi.

(5)

The choice a( V;) = ~ cotv; and a path of integration above the pole Xo = gives a form:

-VI + r2 and below the pole Xo = VI + r2

368

(6) which is also displayed in figure 1. Clearly the proposed solution provides a better approximation than the one given in3 , because of the functional degree of freedom a( 'ljJ). We expect, in the future, to extend this procedure for higher baryon number solutions. Radial function

3.5

.---~--~--~--~~--~--~--~--~--~~~

(*)

3

~~'7--r---+---+---1---1--~r---r---r-~(*~--~ \

2.5

.

(e)

\.\.

\

2

'.

..... .

1.5 1 0.5

".

.. r-----+--+---t-~-.:=t_.. =-+-,---+--+-' '. .;. . :''-'-t----;.~ ----. ... '

'.

o

••:-t-----i

.

r---1----~-·-··~··-·-··-·4 ..~·~..~

o

1

2

3

4

5

6

7

8

9

10

Dimensionless radial distance Figure 1. Plots of the radial function for the Skyrmion. (*) For the numerical solution in Ref(2J. For the approximation fer) of Eq. (6). (e) For the approximation mer) of Eq. (2).

(*) 3.

References

References 1. Skyrme. A unified field theory of mesans and baryons. Nuclear Physics 31. 566 (1962). 2. Adkins Nappi Witten. Static properties of nucleans in the Skyrme model. Nuclear Physics B. 228, 552 (1983). 3. Atiyah M. Manton N. Skyrmions from instantans. Physics Letters 222.B 438 (1989). 4. A. Minzoni, J. Mucifio, M. Rosenbaum. On the structure of Yang-Mills fields in compactijied Minkouski Space. J. Math. Physics. In press (1994).

369

Effects of Topology on the Gravitational Wave Jiro SODA Department of Fundamental Sciences, FIRS Kyoto University, Kyoto 606, Japan

Abstract We have analyzed the cosmological gravitational waves which propagate on the torus universe. The topology of the universe causes the mode mL'Cing effect. This effect may give the possibility to determine the topology of the universe from the COBE's data.

It is important to consider the gravitational wave in the general background space-time. As a first step, we shall study the plain symmetric space-time with a torus compactification. I The metric can be parametrized as

d,s2 =

glJV(xA)dxlJ.dxV

+ e-P(x>')

(1)

x (eQ(x>') cosh W(xA)dx 2 - 2 sinh W(xA)dxdy

+ e-Q(x>') cosh W(xA)dy2),

where the indices fL, 1/ and ). run from 0 to 1 and gp.v, P, Q and W depend only on = t and Xl = z. The identification x + 1 '" x and y + 1 '" y implies the torus topology. This form of the metric parametrization respects the symmetry of the space-time. The Einstein-Hilbert action becomes XO

(2) where R denotes the two-dimensional scalar curvature. Thus we obtained the effective 2-dimensional field theory which is the dilaton gravity coupled to the SL(2, R) non-linear sigma model. Now, we seek the homogeneous background solutions. The solution space can be identified with the space of geodesics in the Teichmuller space. As a particular example, we consider the following metric: 1 +t2

ds 2 = -dt 2 + dz 2 + ~dx2 - 2dxdy or

Poct) = -logt, e- Q 0 =

2

..)1 + t 2 '

+ 2dy2 1

,

sinh Wo = - . t

(3)

(4)

This background space-time (3) represents the anisotropically expanding universe with a torus twisting. Let us analyze now the evolution of small perturbations lThis talk is based on the work by Yasuo Ezawa and J.S.

370

around the background solution given above. Here, we will consider the plain symmetric gravitational wave again.

gl'v(t,z) pet, z) Q(t, z) Wet, z)

T/1'V+hl'v(t,z) , Po(t) + 7r(t, z) , Qo(t) + q(t, z) , Wo(t) + wet, z) .

To solve the problem, we choose the synchronous gauge. Now we take the independent gauge invariant variables as q + 7r -

W =

1

.Jl+t2w ,

,II + t 2w

(5)

.

(6)

t at el> =

0.

(7)

a

a

(8)

7r -

From the Einstein equations, we obtain Del> Dw

1

1

a

2

+ --w = --el> . tat tat

This mixing of the modes is an interesting feature of the gravitational wave on the anisotropic space-time. The eqs.(7) and (8) can be solved using Bessel functions and Hankel functions. Notice that the perturbed metric becomes 1 + t2

1

ds 2 = -dt2 +(1+p)dz2 +--(1-7r+q+ .Jl+t2w)dx2-2(1-W)dxdy+2(1-el»dy2. 2 1 + t2 (9) Due to the anisotropic expansions of the background space-time, the polarization of the gravitational wave changes. In conclusion, we have obtained the equations for the gravitational wave propagating on the anisotropic ally expanding universe with a torus topology. The topology of the universe gives interesting effects on the cosmological gravitational waves, i.e. , a mode mixing between the plus mode and the cross mode. This result itself can be applicable to a wide range of anisotropic models. Comparing the theoretical calculation of the effects of primodial gravitational waves on the cosmic microwave background radiation with the COBE's result, we could determine to what extent the universe is anisotropic and whether the torus topology is denyed or not. This subject is under investigation.

References [1] Y. Ezawa and J. Soda, Phys. Lett. B to be published.

Mathematical Foundations

Chairperson: R. Schoen

373

GEOMETRY GRAVITATION AND MATTER IN EXTREME LIMITS LEOPOLD HALPERN Department of Physics, Florida State University, Tallahassee, FL 32306-3016, USA

ABSTRACT Arguments are presented supporting the view that the general theory of relativity requires physical modifications which preclude the formation of horizons. An approach to a gauge theory with modifications in extreme limits is outlined.

A physical discussion on the predictions of general relativity about the gravitational collapse of matter to a point requires careful consideration of all aspects of the theory-even of the mathematical axioms involved and of their relations to the physical objects. Riemann had already pointed out in his habilitation work that a continuum cannot be the ultimate mathematical fundamental of a physical theory and the points of the continuum cannot beyond all approximations be identified with physical objects. The same must hold true for lines and surfaces and thus also for solutions of partial differential equations. Quantum theory has already largely eliminated the identification of physical objects with such geometrical constructions-it arrived at the introduction of the concept of probabilities in order to make physical predictions from the solutions of its differential equations. One is hardly able to consider the prediction of the gravitational collapse of macroscopic matter to a geometrical point by general relativity as anything but an absurdity of the theory which demands a modification. Most of those dealing with the subject do however not accept such a modification to become effective before the formation of the horizon nor inside it-except perhaps very close to the bitter end. The origin of this attitude is only partly based on properties of the inertia of matter-the principle of equivalence and the absence of a localized invariant characterization of the horizon; one would even have to modify the else so convenient models of closed universes for which such a bitter end is ultimately all our fate. Mechanisms to modify the bitter end to acceptability by physicists are either not given at all or they are of a likelihood comparable to science fiction stories-even when they seem to resemble similar features evolving from elementary particle theories. The scenario of the collapse is macroscopic and must be well describable in a classical formalism. We can conclude that additional terms with derivatives of higher order have to supplement the Einstein-Hilbert equations to eliminate the absurdity. (One may even speculate from the foregoing that terms with only a finite number of derivatives can never suffice for a rigorously valid ultimate theory, in which the continuum has no place.) Quantum theory indeed gives rise to effects like Schrodinger's pair creation in gravitational fields 1 and its virtual counterpart, with the macroscopic average resulting in matter creation and Casimir forces among others. The Einstein-Hilbert equations can never describe such effects; a rigorous classical description needs thus in any case an admixture of higher order deriva-

374 tives due to averaged quantum effects. The argument that real pair creation should become of significance only very close to the formation of a horizon applies solely in a pathologically extreme case of spherically symmetric cold dust-however also there the virtual pair effects, which cannot be separated from the creation of real pairs, do modify the field equations before the horizon is formed. An estimate of the magnitude of the virtual effects would require a knowledge of all elementary particle interactions as well as an unambiguous quantum field theory. The assertion that the Einstein equations with the addition of only lower order perturbative terms due to macroscopic quantum effects are adequate to describe correctly all situations in the large may remind of the historical error that the equations of classical mechanics should apply to all domains if only thermal fluctuations are introduced pert urb atively. A different approach is to try to obtain a generalized gravitational theory from a gauge principle. C.N. Yang claimed to derive the equations: Rhi;k -

Rhk;i

= 0

(1)

(with Rhk the Ricci tensor) from a theory with gauge group GL(4,r).2 The author remarked that Eq. (1) is equivalent to

(2) which is the tensor form of the GL( 4, r) Yang-Mills analogue of Maxwell's equations -only in case that the connection is generalized to become non-Riemannian with torsion. All the solutions of the Einstein-Hilbert vacuum equations are solutions of the above equations too. The author also showed that the restriction of the gauge group to 80(3,1) results in a metric connection which can be separated into a metric part and contortion.3 (This is not possible in the case of GL(4,r).)

r\k

= { i h k } - K\k

(3)

with:

(4) the contortion term formed with the torsion tensor T. The gauge theory is presented in Ref. 3 as a 10-dimensional Kaluza-Klein theory with the group manifold of G = 80(3,2) (or in more sophisticated versions the universal covering groups) as bundle space. This choice relates the geometry to that of the group manifold and the base space to that of the anti-De Sitter universe without restriction of generality. Einstein's equations with a cosmological member are always fulfilled by the nonsingular Cartan-Killing metric of the group manifold and the latter projects to the anti-De Sitter metric on the base. With H the subgroup 80(3, 1) as typical fibre and 7r the natural projection G -+ H, the principal fibre bundle P(G,H,7r,G/H) determines the theory apart from the admission of more general solutions of Einstein's equations than the Cartan-Killing metric, which must still have a unique

375 projection onto the base. The metric, of such a solution determines a connection by choosing the horizontal vectors as perpendicular to the vertical vectors of the fibre. The connection resulting from, by projection on the base is in general not torsion free, but it allows always the decomposition of Eq. (3) because of the metric connection with vanishing covariant derivatives of the metric 9 = ?rt , on the base. The homogenous Einstein equations on G of the metric, projected on the base, yield the Einstein equations with a source term which is bilinear in the (complete) curvature tensor and of vanishing trace. This source term which corresponds to the Yang-Mills matter field source has the gravitational constant as factor. The mixed horizontal-vertical components of the Einstein equations projected on the base yield the Yang-Mills equations which correspond to Eq. (2)-however with a non-Riemannian connection and curvature. Expressed in terms of metric and torsion, we know from Eq. (2) that third derivatives of the metric occur. this fact allows for a nonminimal coupling of the metric to matter in which the Riemann tensor enters, in the averaged macroscopic theory; this appears also necessary to describe matter creation in average. Decomposing the curvature tensor which we denote by H into a purely Riemannian part B and a part Q with torsion according to Eq. (3): Haeij = Baeij + Qaeij (5) we obtain for the horizontal component of Einstein's equations on the ten-dimensional space projected on the base:

(6) with the cosmological member A and gravitational constant G. The purely vertical components of the equations are exhausted by the determination of the vertical components of the metric in terms of the Cartan-Killing metric of the typical fibre H. The projection of the mixed components becomes:

(7) where the curvature tensor H is formed with the full non-Riemannian connection The details of the derivations are presented in Ref. 3. The formalism was originally constructed to give a unified description of the dynamical variable of angular momentum and the inner quantum number spin as a Kaluza-Klein theory; its equations on the base show a remarkable combination of Riemannian and nonRiemannian terms. One must realize that no classical Kaluza-Klein theory is physically realistic in itself because it lacks description of the results of Schrodinger's pair formation and its virtual counterparts. We have in the present case the peculiar situation of a Yang-Mills field that can be decomposed into a metric and torsion terms (and of inner quantum numbers that are convertible into those of angular momentum). The author in Ref. 3 has suggested that the remaining terms of the curvature containing torsion may be related to nongravitational matter with vanishing rest mass.

r.

376 The decomposition into metric and torsion. terms is presented in the reference. (The alternative, to consider only solutions with vanishing torsion is an artificial restriction.) The model takes account of the Schrodinger phenomenon by the mixed curvature and torsion terms. The metric equations for special solutions of vanishing torsion consist of the third derivative term suggested by Yang2 as mixed components and the Einstein term plus a term bilinear in the Riemann tensor as horizontal components; this last term may be interpreted as the vacuum expectation value of the energy-momentum tensor of virtual matter fields of vanishing rest mass. This new set of special empty space equations admits the Schwarzschild solution of general relativity, whereas situations of other than spherical symmetry, or the presence of source terms like real matter sources, differ from general relativity because of the additional bilinear term. This term, due to the value of G, is negligibly small within the solar system; its value grows however with decreasing radius much faster than the matter density. Massive matter sources do because of it not follow exactly the geodesic law of motion. Such a theory is hardly accessible from an analysis of post-Newtonian classifications. Conclusions on the behaviour of highly collapsed objects, if limited to the comparison of post-Newtonian developments with general relativity alone can thus be misleading. The same equations including the torsion terms can be interpreted as the theory of a matter field of vanishing rest mass with a non minimal interaction with the metric. This field is formed from differential expression of torsion and metric alone. It can give rise to matter pair creation, for example when two gravitational wave pockets collide; this results here on a classical level. A detailed identification of the matter with known fields does not yet exist. Matter with rest mass can at present only be introduced into the theory as sources of the field equations. Both the Einstein term as well as the Yang term have then a source. The conservation laws of the sources are also given in Ref. 3.

Acknowledgements Research for this work is supported by the FSU foundations. I thank the Presidents Dr. B. Sliger and Dr. S. d'Alemberte for their continuing support.

References 1. E. Schrodinger, Physica 6 (1939) 899-912, and Proc. Roy. Irish Acad. A46 (1940) 25-47. 2. C.N. Yang, Phys. Rev. Letters 33 (1974) 445. 3. L. Halpern, Internat. J. Theor. Phys. 33 (1994) 401-423. 4. P.A.M. Dirac, AlP ConI. Proc. 74 (1981) 129-130.

377

A BLUNDER IN QUANTUM FIELD THEORY Masayoshi MIZOUCHI Faculty of Science Okrtyama University of Science. J:Jllidrti-cho. Ok(tyrtma, Japan ABSTRACT The prest'nt. met.hod for solving t.he basic equat.ion of quant.ulll field t.heory is incorrt'ct. The correct method is shown hert' for the case of the intt'raction picture. To pt'rform this, a nt'w transformation of frequency part exchange is introduced. It implit's t.hat every field operator appearing in a t.heory is transformed int.o its opposit.e frequency part. Moreover, it requires the introduction of an adequate transformation of Fock space. Thus, the basic equation has two kinds of solution, i.e. the solution of Fock space and of counter Fock space. We must clearly distinguish between these t.wo kinds of solution. It. is at t.his point where up till now, we have failed t.o construct a correct quantum field t.heory. After correcting this point, it will be shown as an example that the self energy of a fermion is not a divergent quantity, but simply indefinit.e. Thus, finally it is shown that. the quantum field t.heory for the point particle model can not predict the self energy quantitat.ively.

1.

The definition of frequency part exchange

Every field operator appearing in a theory is interchanged with its opposite frequency part. For example. ill the case of QED. it implies that A

(±) ---+ A(=F) PI"

Normal order products are transformed into reversed normal order products, that is, N~n)( 0, then requires that M2 > O. Due to the conformal invariance of the theory, this result guarantees that M is real in general t . An advantage of the geometric model, if nonlocal interactions prove inescapable, is that it allows different locality requirements in the interior and exterior spaces. As well, curved spaces accommodate a richer scope of causal properties than Minkowski space. An interesting possibility is that closed timelike curves 7 and/or topology change ll may play some role in the causal interpretation. Whatever the final verdict is on the issue of nonlocality, is seems clear that, if a causal interpretation proves viable, there are several good reasons to develop it in terms of a geometric model in Weyl space.


-

R" (31'>.9 -'P

= - 2Z'" (3["( !>.]9-6p -

2Z" (311' p!>.]9 -.6

(5)

is of great physical importance, for it is the only rule needed for the Einstein equations to be averaged out. The result is the macroscopic field equations 2,3

G"O M 0(3 - ~S"G/'v M /'v -- _ Itr:;(macro) 2 (3 (3

(6)

where the macroscopic stress tensor T;(macro) is of the form It

Ta(macro) _ < t,,(micro) > _(za /lv(3 _ ~6"Q )-IJV + uae M .(3 _ ~saU/lv M IJV , (7) (3 - 1t.B 2 (3 /lV 9 2 (3

1 Note

the change in the sign in the definition of Q" f!-r). as compared with the papers."

3, 5

396 the stress tensor being conserved, T~f:acro) = O. The equation of motion for the averaged matter has been shown to read It

_ (ZE .. 1Q )-jJv < t ",(micro) {3 >11",jJv{3I1E - 2 IJVII{3 9

.

(8)

Here Z'" jJv{3 = 2Z'" jJ[/!!(3] is a Ricci-tensor like object for the correlation tensor, QjJv = QE JJVE> and < t~(micro) > is the averaged energy-momentum tensor. The algebraic structure of the covariantly constant tensor U",{3 = g",{3 - G",{3,

(9)

U",{3I1, = 0,

is determined due to the theorem. 3 Though somewhat unexpected, the result is natural: a space-time averaging out of the Einstein equations brings the field equations which can be written in the form of the Einstein equations for the induction Ricci tensor defined through the macroscopic metric, and the macroscopic stress tensor (7) includes, in addition to the averaged matter, the correlation tensor terms as the geometric correction of averaged matter. But, in their geometrical meaning, Eqs. (6) are not Riemannian, which reflects the fact of another underlying macroscopic space-time geometry. The macrovacuum equations of the theory (10) states the Ricci non-flat character of the macroscopic gravitation in the absence of averaged matter, which is a non-trivial geometric and physical fact. In the case of all correlation functions vanishing when also U",{3 = 0 (the absence of metric correlations) Eqs. (6) become the usual Einstein equations for the macroscopic metric G",{3 with < t~(micro) > in the right-hand side that is usually taken as the perfect fluid tensor in cosmological tasks. This reveals the physical meaning and the range of validity of using Einstein's equations in cosmology. The correspondence principle for the theory has been formulated 3 ,5 which states that the macrovacuum equations (10) become Isaacson's equations 7

(11) in the high-frequency limit. As a result, the correlation tensor QjJv = -2Z0 jJ [v Eh: 0.06 for large Zc. Therefore it exists a new region of stable solutions which extends at even large Zc up to infinite central densities, with temperature T less than 0.06. This stable equilibrium configurations present a regular center without singularities even for very large density. However there is a very sharp separation between core and envelope, the core being up to only 10-4 times the radius of the cluster! The core is in gravitational equilibrium with the external region: there is no possibility of existence for a so dense core without the envelop which permits to the system to be stable as

450

a whole. Moreover the value of the ratio 2G M / Rc2 is small and of the order of the ones relevant for Newtonian configurations. More details will be presented in two forthcoming paper (Merafina & Ruffini 1994 and Bisnovatyi-Kogan et aI. 1995) References Bisnovatyi-Kogan G.S., Merafina M., Ruffini R., Vesperini E. 1993, ApJ, 414, 187 (BMRV) Bisnovatyi-Kogan G.S., Merafina M., Ruffini R., Vesperini E. 1995, in preparation Ipser J.R. 1969, ApJ, 158, 17 Merafina M., Ruffini R. 1990, A&A, 227,415 Merafina M., Ruffini R. 1994, in preparation Oppenheimer J.R., Volkoff G.M. 1939, Phys. Rev., 55, 374 Suffern K.G., Fackerell E.D. 1976, ApJ, 203, 477 Zel'dovich Ya.B. 1963, Voprosy Kosmogonii, 9, 157 (Russian) Zel'dovich Ya.B., Podurets M.A. 1965, AZh, 42, 963 [tr. 1966, Sov. Astron. - A.J., 9,742]

451

LANDAU DAMPING IN SEMI-DEGENERATE GRAVITATING SYSTEMS M. CAPALBI, S. FILIPPI, J.G. GAOl, R. RUFFINI and L.A. SANCHEZ 2 IeRA, Dipartimento di Fisica, University of Rome, 1-00185 Roma, ITALY ABSTRACT Collisionless self-gravitating systems of fermions are investigated using the methods of kinetic theory. The influence of the degeneracy parameter on the evolution of perturbations is analyzed and the Landau damping time for the simply exponentially decaying solutions is studied. The effects of the degeneracy on Jeans wavenumber and on stability could have important implications on galaxy formation in a universe dominated by fermionic dark matter.

In a collisionless self-gravitating system the mean gravitational field is more important than the fields of the individual nearby particles and therefore the role of collective effects in the dynamics of such a system is decisive. Rarefied plasmas have the same property so that many of the techniques of plasma physics l ,2 can be used to study the dynamics of collisionless gravitating systems. 3 ,4 Gravitating systems never form static homogeneous equilibria; however, it is useful to analyze an infinite homogeneous self-gravitating static mediumS to obtain an understanding of more complicated and realistic models. The perturbation modes of infinite homogeneous self-gravitating systems have been extensively investigated in the case of a Maxwellian unperturbed distribution function. 6 In this work we consider an infinite collisionless self-gravitating system of semidegenerate fermions and the limiting cases of fully degenerate and Maxwellian systems, studying in particular the relevance of Landau damping for the stability on small scales. Our aim is to determine whether the introduction of the degeneracy generates some modifications in the behavior of the perturbations. We assume that the Newtonian theory of gravitational interaction is valid for our system and also that the characteristic distance of variation of the field (1/ k) is much greater than the de Broglie wavelength of the particles (h/mv), so that we can describe the system using the classical collisionless Boltzmann equation and the Poisson equation. The equilibrium state is assumed to be homogeneous and timeindependent. We consider small perturbations, and then we expand the perturbed equations up to the first order:

(1)

(2) lpermanent address: Physics group, Xinjiang Institute of Technology, Urumqi 830008, P.R. CHINA. 'Permanent address: Department of Physics, Universidad Nacional de Colombia, A.A. 3840, Medellin,

COLOMBIA.

452

From (1) and (2), using the "Jeans swindle,,5 and the methods of integral transforms, we obtain that the evolution of perturbations is described by a sum of terms each one proportional to a factor exp( -iwpt) where Wp = Wr + iWi is the p-th solution of the following dispersion relation 47rGm 1+-2

k

Jk.v-w a-~ _ k· v

d 3V 0. - =

(3)

The presence of an imaginary part Wi gives a perturbation exponentially growing in time (Wi> 0) or exponentially decaying (Wi < 0). We choose the Fermi-Dirac distribution function with an arbitrary degeneracy parameter as the unperturbed distribution function fo. For a semidegenerate system eq. (3) becomes

(4) with W

I( -k) =

j+ex>

U

-ex>

U -

du

1 2 2 k'" e (u /2u )-'1

+1

'

(5)

and the Jeans wavenumber is

(6) Solving the dispersion relation we find i) for k

< kJ: exponentially growing solutions (Jeans instability),

ii) there are no overstable modes (Wi = 0, Wr #- 0),

(Wi

>

0,

Wr

#-

0) and no undamped waves

iii) for k > kJ: damped modes (the phenomenon called Landau damping in analogy to the case of plasma physics). Let us consider the damped modes. From eq. (4) with Wi < 0 we can numerically deduce the characteristic damping time T = Iwill as a function of k. In Fig. 1 the damping time for solutions with Wi < 0, Wr = 0 is plotted and different curves for some selected values of Tf are shown. We can see that T strongly increases when the wavenumber decreases towards the value kJ, then wavelengths much smaller than the Jeans wavelength are more rapidly damped. Moreover the influence of the degeneracy parameter on the damping time is evident; in fact, for a fixed k we observe that T increases for increasing values of Tf. Thus the effect of an increasing degeneracy of the system is to make the damping of a perturbation slower. For Tf ~ 0 and Tf » 0 the limiting cases of Maxwellian and fully-degenerate systems respectively are recovered. It might be interesting to extend these results to the study of galaxy formation in an expanding dark matter dominated universe.

453 0 0

0

7)=7)=-10

7)=

Cll

a E= tID

,

ci

-,;

a

0

0

ci

j,

1'1

's. OJ and Cb, in which the path crosses Zn = O. If we define the reflection of ~ through the plane Zn = 0 by ~c = ~-2znn, it is possible to show by the method of images S that Kl(z", ~') = 0( -~~) [~F(~"-~') - ~F(~"-~~)l 0( -~~) and Kr(z",~') = 0(~~) [~F(~"-~') - ~F(Z"-~~)J 0(~~)j the third restricted propagator can be found by superposition: Kb = tJ.F - Kl - K r . If we choose the initial state ,p(~') to be antisymmetric about ~n = 0, (i.e., 'I/;(~c) = -'I/;(~)), the branch wave functions take the particularly simple forms 'l/;l(~") = 0( -~~)'I/;+(z"),

462

1/;r(Z") = e(z~)1/;+(z"), and 1/;&(z") = o. If the initial state is normalized so that 1/;+ o 1/;+ = 1, the decoherence functional is then D(l,l) = ~ + ~D D(l,r) D(r,l) = -~D D(r,r) ( D(b,l) = 0 D(b,r) where ~D

= -21/;i o1/;i =

J

WI

~D =

2

-21/;; o 1/;;

=

-~D

= ~ + ~D =0

D(l,b) = 0 ) D(r,b) = 0 D(b,b) = 0

= 21/;i o 1/;; = 21/;; o1/;i

dkIndk2nrPkl. + W2 ~+(k2)*~+(kd e(271")2 2JWIW2

i

W (Wl- 2)t"

kIn - k2n

in

(5)

is given bya

(WI - kIn) . W2 - k2n

(6)

Note that whenever the alternatives do decohere (~D :::i 0), the probabilities are given by p(l) :::i 1/2:::i p(r), p(b) = O. One special initial condition is for ~+ to be peaked with a small width 8k around a single wavenumber k (and its reflection k c ). Then ~D becomes

~D = -4~ [~_ ~ln (wo - kOn )] + O([c5kJ2), (271" )3/2 Wo

kOn

WOl.

(7)

and we have approximate decoherence to lowest order in c5k. Acknowledgements

The author wishes to thank K. V. Kuchar, N. Yamada, R. S. Tate, and D. A. Craig; the Isaac Newton Institue in Cambridge, England; and especially J. B. Hartle for advice, direction, and encouragement. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. This work was also supported by NSF grant PHY90-08502. References

1. R. P. Feynman, Rev. Mod. Phys. 20 (1948) 367. 2. N. Yamada and S. Takagi, Prog. Theor. Phys. 87 (1992) 77. See also N. Yamada and S. Takagi, Prog. Theor. Phys. 85 (1991) 985; 86 (1991) 599. 3. J. B. Hartle, Phys. Rev. D44 (1991) 3173. 4. See, for example, J. B. Hartle, in Quantum Cosmology and Baby Universes, Proceedings of the 7th Jerusalem Winter School, 1989, edited by S. Coleman, J. Hartle, T. Piran, and S. Weinberg (World Scientific, Singapore, 1991). 5. J. T. Whelan, "Spacetime alternatives in the quantum mechanics of a relativistic particle", to appear in Phys. Rev. D (1994); preprint gr-qc/9406029. 6. For more details, see J. B. Hartle, in Gravitation and Quantizations, Proceedings of the 1992 Les Houches Summer School, edited by B. Julia and J. Zinn-Justin (North Holland, Amsterdam, 1994).

=

=

"We use here several pieces of notation, namely V.l. V - v"n and W.l. Jkl + m 2 , and also that ~+ is the Fourier transform of the positive energy part of..p. We are also working in a reference frame where n has no time component and with a final surface (Til which is a surface of constant time til.

Alternate Gravitational Theories

Chairperson: Y. Fujii

465

Summary of the Session; Alternative Gravitational Theories Yasunori Fujii The subjects of the talks in this session are so diversified that it is not an easy task to give an extensive summary. I only give an overall brief guide. The talks presented may be divided into three categories; (1) Extensions of some of the existing theories, (2) Attempts to build entirely new theories, and (3) Others. Included in (1) are the talks of Kallingas, Hirai, Chauvet, Mannheim, Zhytnikov, Hammond, Rankin, N akariki and a combined presentation by Lam, Hong, Fung and Lee. Kallingas discussed how the primordial nucleosynthesis constrains the scalar (dilaton)-tensor theory with the added ingredient of the principle of asymptotic decoupling, which was also discussed by Damour and by Nordtvedt in the plenary talks. The scalar-tensor theory was also discussed by Hirai, who focused upon its possibly favorable influence on the structure formation, which has plagued the standard theory. Chauvet took up the same type of theory, reporting the analysis of the cosmological solutions. Mannheim showed that the conformally invariant theory with its characteristic term '" r in the static potential provides another way of understanding galactic rotation curves. In the proceedings paper, however, he discusses another but related question on the mass in this theory. The same issue of the rotation curves was discussed by Zhytnikov who tried to see if the fields of very light mass, much lighter than anticipated for the fifth force, can replace the dark matter. Torsion was the subject of the papers by Hammond, Rankin and by Lam et al. Hammond emphasized that the spins can be promoted to the same level of mass and charge in the dynamical roles, though I point out that essentially the same idea had been also proposed by Hayashi and Shirafuji in the late 70's in the frame work of Poincare gauge theory. He, however, proposed a new phenomenological approach by looking at solar neutrinos. Rankin talked about the quantum effect to the Schrodinger equation in terms of spacetime geometry. I asked Lam, Hong, Fung and Lee to combine their talks to a single title, "Various Investigations of the (n, Al'v)-Field Theory." They developed a new theory with torsion represented by a single scalar field called n, and a skew symmetric field Al'v' They presented an extensive studies including cosmology. Also related to the Poincare gauge theory, Nakariki studied a new exact solution, showing the presence of the complex gauge field; the complex nature comes from the fact that the Lorentz group has a complex subgroup SU(3). Three talks by Noyes, Santilli and Vera may be classified into the category (2). Noyes talked about part of his ambitious attempt to have a complete theory of particles and gravitation. He put a particular emphasis on the prediction that an antiproton should fall up on the earth, as will be checked in the forthcoming LEAR experiment in CERN.

466

Gravity of antiparticles was also a subject of Santilli who, due to the time limitation, gave only a flavor of his theoretical scheme based on which he has worked on more general framework of gravitation by extending conventional geometry in terms of his isotopic methods. An attempt in another direction was made by Vera who based his argument of particle theory on the standing-wave solution in a box. The third category may include the talks by Harper and by Mazilu. Harper pointed out that the current progress of gravity theories, notably the PPN fittings, is precisely along Newton's ideal of empirical success in his theory of universal gravity, accurate measurements of theory's parameters by the experiments. Mazilu revisited Gustav Mie's work of 1913, showing that the Lorentz invariant action principle for a massive particle formulated by modern mathematical tools results in the equivalence between inertial mass and passive mass, thus shedding a new light on the historical development toward curved spacetime. I close this summery by adding a remark. I suspect that some of the papers which might have fitted to this session were presented in other sessions based on the nature of their results. I wished if we could have more time to attend other related sessions. Taking advantage of writing this report, may I advertise that I myself gave a talk at the session of Equivalence Principle and Inertial Frames, reporting that the parameter w in the Brans-Dicke theory should be constrained much more strongly than widely accepted if I include quantum corrections and compare the result with the null experiments on Weak Equivalence Principle.

467 A BIANCHI-TYPE IX COSMOLOGICAL SOLUTION IN BRANS-DICKE THEORY

P. AG. CHAUVET and J.L. CERVANTES-COTA Departamento de Ffsica Universidad A utonoma M etropo/itana-Iztapalapa P. O. Box. 55-534, Mexico D. F. C. P. 09340 MEXICO

ABSTRACT We present the Bianchi-Type IX solutions for perfect, barotropic, fluids that can be obtained in scalar-tensor theories of which the Jordan-Brans- Dicke Theory is taken as a concrete example. These solutions are given in terms of reduced variables.

1. Bianchi-Type IX Field Equations Our Jordan-Brans-Dicke field equations for a barotropic, perfect fluid, p = f3p, -1 < f3 < 1 (the f3 ~ equation of state, for incoherent radiation or ultrarelativistic matter, is excluded) Mp, withMp = const.,are where pa 3 (l+P)

=

=

87rMp

mp

= 3+2w

(1)

The dynamic equation, which is equal to the sum of the three equations that describe the evolution of the scale factors a, , a" and a3 :

('f/JH;)' = [1 + (1- (3)w]mp +t/; a6P * RIX,

i = 1,2,3

(2,3,4)

is,

(5) and the constraint equation is 2

(t/;") _ _1_ ('f/J') _ (1-3f3) (1-3f3)mP17+17o) ('f/J') t/; (1 - (3)2 t/; (1 - (3)2 'f/J t/; [2 - 3f3 + ~(1- (3)2w] (1 - 3(3)mp17 + 170) 2 [2 + (1 - (3)(1 + 3(3)w]mp + (1 - (3)2 t/; + 2( 1 - (3)'f/J = -(Hi - H2)2 - (H2 - H3)2 - (H3 - Hi? == U(1)) ,

_3_ 2(1 - (3)

(6)

468 where 1/1 == ¢>a3 (1-P), 4> is the JBD scalar field, w is the coupling parameter of the theory, a is the mean scale factor, a 3 = ala2a3, and the Hi's i = 1,2,3 are the Hubble expansion rates, Hi = aUai. 3

• R rx

= L •Rj is the "spatial three-curvature".

The "partial curvature" terms are, respectively

i=l

• Rl

= [at -

a~ - a~ + 2a~a5J/( -2a 6 )

,

(7)

• R2

= [a~ - a~ - ai + 2a~a~/( _2a 6 )

,

(8)

and

(9)

=

=

I] is the "cosmic time parameter", where dt a3P dTJ , and ()' QI], TJo is an integration constant'. 2 Finally, (T is the "shear" ( (J' = 0, is a necessary condition to obtain a FRW cosmology since it H2 H3). implies Hl

=

=

2. Analitic solutions (10)

is a solution to the above equations. A, B, and C are constants and ~ 1a

31

==

B2 - 4AC its determinant.

hi

H·---+• - 3 a3 1/1' The functions hi's obey the condition hl + h2 the solutions. From Eq. (6) one obtains that

h~ + h~ + h5 == K

2

(11)

+ h3 = 0 and determine the w3

2

= - 2(1 _ ,8)2 [PTJ + QI] + s]

anisotropic character of

,

(12)

and, from Eq. (2,3, and 4) that (13,14,15) The h;'s can then be given as (16)

(17) and

h _ [2K:2 + 2K: 3 -

1] K 3(K:2 +K:+ 1)

(18)

Finally, P, Q, and S are given in terms of the A, B, and C constants: P = X A - [4A - Y](1 - 3,8)2mp

,

(19)

469

Q = XB - (4A710 - 2Ym,6710 + 2(1- 3,8)B](I- 3,8)m,6 ,

(20)

and

s =XC where X

[2~ + 2(1- 3,8)m,671oB - Ym~7151

=3(1 + 3,8)(1- ,8)2wm,6 + 6(1 -

(21)

,

,8)mp - 2(1 + 3,8)A ,

(22)

and (23)

=

The isotropic model solution for the above Bianchi-Type is obtained if ~ 0, where one has h 1 (71) h2(71) h 3 (71) O. For ~ :I 0, by solving Eq. (13, 14, 15) one can obtain the explicit functional dependence of the "is = ''is(71), i 1,2,3 on 71.

=

=

=

=

3. References 1. Chauvet, P. and Pimentel, O. Gen. ReI. Grav. 24 (1992) 243. Chauvet, P. Astrophys. Space Sci. 90 (1983) 51. 2. Chauvet, P. Cervantes-Cota, J. and Nunez-Yepez, H. N. in Proceedings of the 7 th

Latin American Symposium on General Relativity and Gravitation, SILARG VII, World Scientific, 1991 p. 487.

470 GENERAL RELATIVITY AND EMPIRICAL SUCCESS R. DiSalle, W.L.Harper and S.R.Valluri Depts. of Philosophy and Physics, University of Western Ontario GR meets a standard of empirical success which goes beyond predictive success. According to this standard, parameters of a successful theory are accurately measured by the phenomena it explains. Measurement of a parameter by a phenomenon requires systematic dependencies that make alternatives to the phenomenon carry information about alternative values of the theoretical parameter (Harper 1991). A classic illustration is Newton's use of stability of an orbit to measure inverse square variation of a centripetal force. According to Cor. 1 Prop. 45 Principia Bk.1, precession

= ndegrees/revolution

iff

f oc r",

where z = [360/(360 + n)]2 - 3. Zero precession measures z = -2, so that Newton could use this result to measure inverse square variation of gravitation toward the sun, for the planets whose precessions are accounted for by perturbations. Newton made such measurements from phenomena the basis for his sharp distinction between scientifically established facts and mere hypotheses. To the extent that background assumptions generating the systematic dependencies can be regarded as secure, one can reason from phenomena to theoretical claims that count as measurement outcomes. The PPN formalism, developed by Nordtvedt and Will (Nordtvedt 1968, Will 1971, Nordtvedt and Will 1972) from work initiated by Eddington (1922), compares metrical theories of gravity by generating systematic dependencies in weak-field (solar system) approximations(Disalie, Harper and Valluri 1994). For example, the PPN parameter'Y (curvature per unit rest mass) is measured by the post-Newtonian difference in time delay of light. For signals transmitted from the earth to another planet, the round trip delay should be affected by the solar mass according to the formula:
~

37!'m

7!'m 2

-r + . 4r2

(3)

2. Circular Geodesics and Gyroscopic Precession in Schwarzschild Spacetime

The condition governing a circular geodesics; the proper rate of rotation and the Fokker de Sitter precession in the Schwarzschild spacetime calculated by W.Rindler and V.Perlick are respectively 2

m

w =r3

(Kepler third law);

I r21=w;

(4) (5)

and (6)

3. Conclusion

When comparing Eq.(3) with Eq.(6), the Fokker de Sitter precessions (geodetic precessions) in (O,A!'v)-field and Schwarzschild spacetime are equal in magnitude to the first order approximation. 4. Reference

1. W.Rindler and V. Perlick, Geneeral Relativity and Gravitation 22 no. 9 (1990) 1067. 2. A.Yu(Xin Yu), Astrophysics and Space Science 154 (1989) 321. 3. A.Yu(Xin Yu), Astrophysics and Space Science 202 (1993) 237.

474 Gravitation and Spin, Minimal Coupling and Beyond

Richard T. Hammond Physics Department, North Dakota State University Fargo, ND 58105, U.S.A.

ABSTRACT The theoretical foundations and some experimental implications of a new theory of gravitation with torsion are presented. The torsion is defined as the exterior derivative of a potential.

1. Introduction

Three fundamental characteristics of a particle are its mass, charge, and spin. Although mass and charge enjoy a similar status of being the origin of a field, spin suffers the affiiction of being a simple parameter. However, from investigations as early as Cartan to a myriad of recent work, there have been attempts to bring spin to an equal footing with charge and mass. In the following, spin aquires such a status via the torsion field. 2. New Theory

Suppose one were to develop a theory of spin, as described above, from basic principles. How to proceed? In developing a theoretical description of nature, a useful, appealing, and successful method consists of identifying key fundamental aspects of existing, correct theories and extending or generalizing them to develop new theories. Consider the two highly successful classical theories, gravity and electromagnetism. One of the features they share is that the source of the field is characterized by a scalar parameter- mass or charge. The source tensor is built up from that by using the four velocity. To generalize things, consider a particle that is described by an intrinsic vector quantity, which is what we expect if the source is spin. In the rest frame the source would be of the form (I'. In electromagnetism a particle is described by its scalar charge density (T in its rest frame, but becomes (TVI' in general. In the same way, assume that (I' is the source tensor in the rest frame and that in a general frame the source tensor becomes JI'V = (I'vv. Still using electromagnetism as a guide, assume that the source tensor JI'V (without gravity present only the anti symmetric part of this source tensor is used) is coupled to a potential tPl'V and that there exists a field defined as the exterior derivative of the the potential, SI'VU = tP[I'V,u]' This gives a gauge invariance under the transformation tPl'V -+ tPl'V +e[I',v]' Following electromagnetism's example further, we expect a variational principle with Sl'vuSI'VU + Jl'vtPl'V as the Lagrangian. This formulation uses the fundamental building blocks of theoretical physics, but the question is, is it a toy model or does it correspond to Nature? I will argue that not only does this describe Nature (the spin of a particle), but that it is already contained in gravity! Already contained, that is, with one key assumption: The torsion, which is defined by SI'Vu = f[l'v]u is assumed to be derived from a potential according to SI'VU = tP[I'V,u]' In fact with this, the gravitational Lagrangian R can be broken into oR - Sl'vuSI'VU where oRis the torsion free part of the Lagrangian. Without gravity, the Lagrangian is precisely what was guessed

475

above. Even more compelling is the fact that 'here, SOllh enters like the Kalb-Ramond anti symmetric field in the low energy limit of string theory. Thus, the variation principle becomes 8I = 0 where I = J FYd4 x( R + LM) and LM is the material Lagrangian. The field equations are obtained by considering independent variations of gl'v and tPI'V' and are

GI'V - 3SI'V~" - 2SVOl /3 SJl.Ol /3 = kTJl.v

(1)

(k = 87rG/c4 ) and the anti symmetric part is SJl.V"." = -(K/2)JJl.v where K is the new coupling constant. ' The question persists, what is the physical significance of all this? It has been shown that, in the absence of forces, the equation of angular momentum, is!

L Ol /3

+ SOl/3 + yOl~ _

pOly/3

= constant,

(2)

In this equation, the first term on the left side represents the macroscopic rotational angular momentum. The last two terms on the left side are represent orbital angular momentum. The term SOl/3 is constructed in terms of the intrinsic vector (JI. introduced above and, according to Eq. (2) must be interpreted as the spin of the particle. With (JI. = 0, Eq. 2) reduces to the Papapetrou result and the usual law for conservation of angular momentum when intrinsic spin is neglected. Thus, the physical interpretation of torsion and its source is evident and non-negotiable. 3. Dirac: Minimal and Nonminimal Coupling The variational principle for the Dirac equation is DJ e (~

+ L) d4 x = 0 where (3)

and the covariant derivative of the Dirac spinor is given by

Dl'tP = tP,JI. -

~r Jl.ab/a'ltP.

(4)

Variations with respect to If, e~, and tPJl.V yield, using Planck length Lp = JnG/c3, I

aD

.1. zmc .1. a2). If we choose a positive coupling constant as usual, the gravitational constant blows up at a critical point, where ¢>C'r = (81rGO- 1/ 2, and it changes its sign beyond the point. Fig.2 illustrates the general behavior of density perturbations in this theory. In the calculation, we have again considered flat dust universe and set ~ = 10, m = 10-31eV and f!d"st,O = 0.1. We can see a significant enhancement of growth rate of density perturbation just after starting point. In fact, our calculation have started from near the critical point ¢>C'r' That implies an unusual evolution of the universe near ¢>cr accelerates the growth of density perturbations. Then, to confirm this numerical result near ¢>C'r' we have searched for and found an asymptotic solutions near ¢>cr in a flat dust universe with a massless scalar field. 3 Around ¢>cr, the scale factor and the Hubble parameter of the background universe behave a ,..., constant and H "" O. Energy density of dust /-L also takes a finite value at ¢>C'r' As for density perturbations, the growing mode varies as 6. ex (1 - 81rGe¢>2)2 in the positive G eff region. It has verified that the rapid growth of 6. in the numerical analysis is consistent with this

e

e

479

solution. From the behavior of the background universe, the reason for the big enhancement of the growth rate is understood clearly. a '" constant and H rv 0 around 4

(1)

where gi; denote the metric coefficients of Minkowski space E and Gi(x;) i,j = 1, 2, 3, 4 denote the components of a vector field defined on E and subjected to the continuity condition. By using Lagrange multiplier it follows that there is a scalar field U(Xi) such

490 that Gi

= au/ax i holds.

The action (1) then reads

S(U)

1

r g';.. axi au au r-:a,4 axi y-g x

(2)

= "2 JE

3. Singular Mass Motion In the special case when a second scalar field U' generated by a mass charge having at rest the singular density distribution mS(x') is superposed over a given gravity field U(xi) then (2) reduces to

S(x'(t))

= mc

JU V1-

V 2 /C 2

dt ,

t

= 1,

(3)

2, 3

where t = x 4 /c, x'(t), t = 1,2,3 denote the trajectory of the singular gravity mass and v its velocity. The variation of (3) supplies the equations of motion

d(

dt

m

c2

U

}1 -

V 2 /C 2

-d (

dt

dX') = -m y/1- v /c2 -;:;au

dt

m

V1-v2/c2

2

,

L

uX'

) =m . / 1-v2 y

= 1,

2, 3

au /c2-. at

(4)

(5)

One notes that only eqs. (4) is independent. In these eqs. the ratio of inertial and passive masses depends only on the scalar field U and on the velocity v. Let us denote by Uo the value of the scalar field U corresponding to a presumptive previous homogeneous state of the universum (before the "Big-Bang") and let us split the current field into U(x', t) = Uo- U*(x', t). If one introduces (6) and neglect U* /c2 and v 2/c 2 , then Newton's gravity model follows when Uo = c2. This supplies a first value of Uo and gives an idea how the remote and old mass-energy distribution can govern the inertia here at the present time. These results have confirmed Mie's statement and have encouraged to develop the appropriate mathematical tools for non-flat space (cf. [7]). By this it becomes possible to link the Lorentz-invariant gravity approach and general relativity, and by this to understand better the equivalence and Mach's principles. The author acknowledges the financial support by the Deutsche Forschungsgemeinschaft (DFG). 4. References 1. A.J. Kox, Archive for History of Exact Sciences 40 (1988) 67.

2. H. Dittus and P. Mazilu, DFG-Projekt Di 527/1-1, ZARM Bremen (1993). 3. M. Abraham, Jahrbuch der Radioaktivitiit und Elektronik 11 (1914) 470. 4. G. Mie, Annalen der Physik 40 Nr. 1, IV. Folge, (1913) l. 5. O. Bergmann, Amer. J. Phys. 24 (1956) 38. 6. M. Mazilu, 4-D Fundamental Principle of Scalar Field Theory as Theory of Gravitation, Bremen (1992) (private communication). 7. M. Mazilu, Scalar Field Theory of Gravitiy in Non-Flat 4-D Spaces (in preparation).

491

GRAVITY COUPLED TO VECTOR COULOMB FIELDS Shin-ichi NAKARIKI, Ka7.umi FUKUMA, Tt'tsuo FUKUI and ~Iasayoshi ~IIZOUCHI Dt:]Jt. of App/icll Phy.~ics. OklLYlLmlL Unil'er,~ity of Science. Ridai-cho 1-1. OkcLyama 700. JAPAN. TlLi..-lLmcL NationcLl Col/egc! of Technology. TaJ.'lLl1UL-Gho. Mitoyo-g'ILn. KCLglL'Ula 769-11. JAPAN. Mukoga'WcL Women's Uni·lII!rsity. /kebimki-cho. Ni..hinomiya. Hyo'ILg n 663, JAPAN. cmcl Dept. of Ap1J/ieti Physics. Okayama University of Science. Ridai-cho 1-1. Okayama 700. JAPAN and TE'ruya OHTA~l KC£1LSCLi Gaidai University. Himkata 573. JAPAN ABSTRACT 'vVe treat the Poincare gaugE' theory with ninE' independent parameters. In some conditions the field equations can be rt'duced to the camp/ex Einstein-Yaug-Milb equations. We here research a sE'ries of gravitat.ional fields generated by "a complex vect.or Coulomb field" which is a solution for the complex Yang-Mills equations. As a result., we find the fields being classifiE'd by t.wo real 'vector' charges. 02, 01 which are the gaugE' charges of "Yang-Mill~" field and its partner. respectivt'ly.

Poincare gauge theory (PGT) was first formulated by Utiyama l and Kibble. 2 and later extended by HayashL 3 We here adopt the Lagrangians proposed by Hayashi*, and we research an exact solution by a method using complex quatities. Sinee Hayashi has proposed the most general model with 9 independent parameters. there exist various models which can he obtained by selecting the special choice of parameters." We here make the following choice to propose a model and to solve the equations given by it: 0: + ~ a = O. f3 - ~ a = 0, I + ~ a = 0: 4al = 3a2 = 2a4 = 24(J.6. (J.5 = 2a;j. In this choice we get the complex Einstein-Yang-Mills equations: GilV -- ~ 2a T.(L) I'" • jhtv · - 'iA x jl'"

(2)

jtltv. v -

(3)

v

= 0, -iAv X jtl'" = O. v

(1)

We research special solutions in the form A,. = SAil' where S is a constant vector in 3-dilllensional complex space. Then we are led to complex Einstein-Maxwell equations. From the equations we can easily find a solution for complex Yang-Mills equations. Le., "complex vector Coulomb field" (CVCF) .Ao = SAo = ~ with a complex vector charge Q.(= QI + ·i(2). This result shows that T

al'

Q.)

= -bl.o r

for the "Yang-Mills" field.

(4)

"'Vl' Itere usl' t.Itl' same not.at.ions as t.hosl' of Ref:' 4 t Although ahove condit.ions for the parameters is different. ±i.·om ones in Ref.. the imaginal'r part. of A" ret.ains to be interl'rrted as "Yang-Mills" fidd and tit .. real part. a,. its partner field.

492

aud 'lilt

= -(215: r

UltO

lor t IlC partner fill e(.

C

(") v

Ou the other han 0,

in a region of

~

< O. (7)

As we can see easily from the relation between ~ and 1', the spacetime is classified by the signature of squared charge q. 2 as follows: The spacetime is (I) just Sehwarzschild if q. 2 = O. (II) Reissner-Nordstrom if q. 2 > O. and (III) Schwarzshildlikc if q. 2 < O. The differencc between case (I) and (III) can, for example, be clarified by considering relativistic Kepler orbits. From the orbits the advance in the perih("l.. lion per revolution, under post-Newtonian approximation It « 1. ). « 1, J;,« 1. is calculated to be ~cp '" 27r{3Il+th:+3(1+~e2)).}. where It = '~, ). =~, J;, =~. Finally. we consider a Dirac field in CVCF. In a region ~ > 0 Dirac equation . 'tt [kn iA kllll1 I'~(I ·j,T. 0 WI'thA {A IS wn ,en as 1 Uk + 4i...l. mn kl a - 'i'Lvi'I 15 + 'Lm, 'J!' = i...l.kml1 = i...l.121 =

-"J,

~1:11 = CO,~O' ~2:12

= 1.

~200

= -'~~IJ!.2.

other zero}. From this equation

02

we can easily see that the Dirac field directly interacts with the gauge charge of "Yang-Mills field". but the field interacts with the gauge charge Ql of the partner only through gravity, so that the direction of has no effect on Dirac field.

01

References 1. 2. 3. 4. 5. 6.

R. Utiyama. Phys. Rev. 101, (1956) 1597. T.W.B. Kibble, .J. Math. Phys. 2, (1961) 212. K. Hayashi. Prog. Theor. Phys. 39, (1968) 494. S. Nakariki . .J. Math. Phys. 32, (1991) 1612. K. Hayashi and T. Shirafuji. Prog. Theor. Phys. 64, (1980) 883. S. Chandrasekhar, The Mathematical The01'Y of Black Holes (Oxford Univ. Press, New York, 1983).

493

CROSSING SYMMETRY IS INCOMPATIBLE WITH GENERAL RELATIVITY· H. PIERRE NOYES Stanford Linear Accelerator Center, Stanford Univeristy, Stanford, CA 94309

Consider a proton moving in empty space past a positively charged earth with GmpME. In the absence of further information, we expect it to move past the earth without being deflected. The CPT theorem asserts that an anti-proton must move past a negatively charged anti-earth in precisely the same way, so gives us no new information. Conventional theories predict that an anti-proton moving past a positively charged earth at distance r would experience an acceleration toward the earth of twice the amount Gm p ME/r'2 = Q2/r2 that a neutral object of mass mp would experience. However, if we are correct in asserting that "crossing symmetry" requires all "forces" (Le., accelerations per unit inertial mass) on a particle to reverse when that particle is replaced by its anti-particle, then the fact that we can use electromagnetic forces to balance a particle in a gravitational field plus crossing symmetry predict that the electromagnetic fields would have to be reversed in order to balance an antiparticle in the same configuration. Our argument from "crossing symmetry" is unconventional in that we use the observable phenomenon of "acceleration per unit mass", or in relativistic S-matrix theory the change in the space-components of 4-velocity, in our definition. The conventional second-quantized relativistic field theory starts, instead, from an interaction Lagrangian expressed in terms of a "gauge potential" which is not observable. Such theories are not problem-free in the Newtonian and Coulombic limits. To quote Weinbergl "The most general covariant fields .... cannot represent real photon and graviton interactions because they give amplitudes for emission and absorption of massless particles [of spin i] which vanish as pi for momentum p -> 0." One is permitted to question whether the "gauge invariant" prescription which Weinberg uses to meet this problem is mathematically well defined in a class of theories which (a) do not have a well defined "correspondence limit" in either nonrelativistic quantum mechanics or the classical, relativistic (Maxwell and Einstein) field theories of electromagnetism and gravitation, (b) necessarily give, as Oppenheimer put it, "nonsensical [Le., infinite] answers to sensible questions", and (a) have not reached consensus on how to formulate "quantum gravity" , In contrast, our finite and discrete reconciliation between quantum mechanics and relativity meets problem (a) by deriving the Maxwell and Einstein fields from scaleinvariant measurement accuracy bounded from below in a manner reminiscent of Bohr and Rosenfeld's analysis of the measurability of electric and magnetic fields. 2 We claim that this analysis removes the physical "paradox" in the Feynman 1948 derivation reported by Dyson3 and extended to gravitation by Tanimura. 4 Our finite particle

Q1 =

• Work supported by Department of Energy contract DE-AC03-76SF00515.

494

number S-matrix theory conserves (relativistic) 3-momentum at 3-vertices, but is off energy shell by a finite amount. Our finite and discrete kinematics satisfies discrete conservation laws for physically realizable multi-leg diagrams; it fits comfortably into the practice of high energy elementary particle physics. Because our theory is finite and discrete, and hence can identify c as the maximum velocity at which information can be transferred between distinct locations, problem (b) never arises. As to problem (c), gravitation and electromagnetism are reconciled at the bound state level by a common treatment of both which is formally equivalent to the Bohr's relativistic calculation of the energy levels of the hydrogen atom, using combinatorially calculated coupling constants; the method can be extended to yield the Sommerfeld formula for the fine structure of hydrogen. 5 and its gravitational equivalent. The classic tests of GR are met, and we believe that the time tests of the pulsar data can be met as well. The quantum number structure predicts the observed particles of the standard model of quarks and leptons, and yields no unobserved particles. Given Ii, c and mp as empirical input, we compute G,G F ,e2 ,mp /m.,m1(/me ,ml'./m. to an accuracy of a part in ::::: 104 - 107 , leaving enough room for improvement to make the theory interesting to pursue, or to produce a crucial conflict with experiment. Results are given in the table. As we explain in the first paragraph, the most dramatic prediction of our theory to date, which is currently under direct experimental scrutiny at the CERN Low Energy Anti-proton Ring, and can be tested by still more sensitive techniques,6 is that ANTI-MATTER "FALLS" UP. REFERENCES 1. S.Weinberg, Phys. Rev. 138 B, 988 (1965). 2. H.P.Noyes, in Marshak Memorial Volume, E.G.Sudarshan, ed., World Scientific (in press). 3. F.J.Dyson, Amer. J. Phys., 58, 209 (1990). 4. S. Tanimura, Annals of Physics. 220, 229 (1992). 5. D.O.McGoveran and H.P.Noyes, Physics Essays, 4, 115 (1991). 6. V.Lagomarisino, V.Lia, G.Manuzio and G.Testera, Phys.Rev. A, 50, 977 (1994).

495 Table. Coupling constants and mass ratios predicted by the finite and discrete unification of quantum mechanics and rela~ivity. Empirical Input: c,1i and mp as understood in the "Review of Particle Properties", Particle Data Group, Physics Letters, B 239, 12 April 1990. COUPLING CONSTANTS Coupling Constant G- 1

/ic

~

Calculated

Observed

[2127 + 136] x [1 - 3.7\0] = 1.693 31 ... x 1fr38 [1.69358(21) x 1fr38]

GFm;/rtc

[256 2v'2]-1 x [1 - 3:7] = 1.02 758 ... X 10- 5

[1.02 682(2) x 10- 5]

sin28Weak

0.25[1 - 3\]2 = 0.2267 ...

[0.2259(46)]

a-I (me)

137 x [1- 30;127 1 = 137.0359 674 ...

[137.0359 895(61)]

G!NN

[e~:? -1]! = [195]! = 13.96..

[13,3(3), > 13.97]

t

MASS RATIOS Mass ratio mp/me

Calculated 3 ( 1;7"4) Ii 1+7'+49

4

= 1836.15 1497 ...

Observed [1836.15 2701(37)]

5

m~/me

275[1 - 2.3~7.7] = 273.12 92 ...

[273.12 67(4)]

m 1ro/me

274[1- 2.337.2]= 264.2 143 ...

[264.1 373(6)]

m,,/me

3 . 7· 10[1 - 3}1O] = 207

[206.76826(13)]

496

QUANTUM BEHAVIOR IN ASYMMETRIC, WEYL-LIKE CARTAN GEOMETRIES J .E. RANKIN" Rankin Consulting, 527 Third Avenue, Ste. 298, New York, NY 10016, USA ABSTRACT This discussion examines recent developments in the theory of a Weyl-like, Cartan geometry with natural Schrodinger field behavior proposed previously. In that model, very nearly exactly a coupled Einstein-Maxwell-Schrodinger, classical field theory emerges from a gauge invariant, purely geometric action based solely on variations of the electromagnetic potentials and the metric. In spite of this, only slight differences appear between the resulting Schr6dinger part, and the conventional theory of the Schrodinger field. Close examination of the differences reveals that most are general relativistic effects which are unobservable in flat spacetime, and which are estimated to interact significantly only via their gravitational fields, or on scales comparable with neutrino interaction cross sections. The only remaining difference is that the wavefunction obeying the conjugate wave equation is not always restricted to be exactly the complex conjugate ofthe primary wavefunction. Generalizations of the model lead naturally to spinlike phenomena, a possible new mechanism for a theory of rest mass, and spinor connections containing the form of an SU(2) potential.

1.

Introduction

In 1927, London published his famous paper on the quantum mechanical interpretation of Weyl's classical geometry.l In the last 15 years, a flurry of new articles has appeared, reviving interest in intrinsic quantum properties of both Weyl geometries, and closely related geometric structures. 2- i In this particular discussion, I will elaborate on a few points mentioned in a recent paper on such a case, a purely field theoretic model cast in a Weyl-like Cartan geometry with intrinsic Schrodinger field behavior. i Indeed, this discussion should be viewed as an additional section of comments in that paper. One reason this model is of interest is because it yields essentially a full, coupled Einstein-Maxwell-Schrodinger set of fields from a purely geometric action in which only the Weyl four vector and the metric are varied. Yet the Schrodinger fieldS also emerges from the results, correctly obeying the Klein-Gordan equation and a charge conjugate, complementary equation, and producing the correct Schrodinger current and stress tensor as sources of the electromagnetic and gravitational fields. Additional terms do appear in the equations, but they will be noted to produce vanishing effects in the limit of a flat spacetime of special relativity. Thus, they are essentially general relativistic modifications to special relativistic Schrodinger theory. Addition of further geometric fields to the model produces spinlike phenomena, and terms which have the form of an SU(2) potential. "E-mail address:(Internet)jrankin@panix.com

497

2.

General Relativistic Nature of Additional Terms

In the earlier reference (~ = the wave equation

7f'

case),7 the Schrodinger wave function

(1/ R)( R gjJ.v'f,v ),jJ. + 2gjJ.v vjJ.'f,v +(1/ R)( R gjJ.vvv),jJ.'f + gjJ.vvjJ.vv'f = -~ (1

+ R)'f

'f obeys

(1)

where v/-' is the Weyl vector, and is proportional to the standard Maxwell potential through an imaginary coefficient. A conjugate wave equation (minus vjJ.) governs the conjugate wavefunction~. The formalism does not require ~ always to be the complex conjugate of 'f. Except for that, the Schrodinger current and stress tensor contain the standard terms. However, the stress tensor also contains additional terms ~lhll'YgjJ.v-~IIjJ.lIv, where ~ = ~'f, and the II derivative is the covariant derivative using the Christoffel symbols formed from gjJ.v' These terms are in addition to manifestly general relativistic modifications, such as the R term in Eq. (1). However, terms in the stress tensor must interact with their environment in some way in order to be observable. In that regard, even though ~1I'YlbgjJ.v - ~1IjJ.llv appear capable of carrying energy and momentum magnitudes that might be comparable to the conventional Schrodinger terms, these terms can only be detected insofar as their divergence is nonvanishing (allowing energy exchange with other terms), or via the gravitational field they produce. But the gravitational field is a manifestly general relativistic effect. And, the divergence of the terms is

(2) Thus, the interaction of these terms with the rest of the stress tensor is also a manifestly general relativistic effect (more precisely, it is a higher order effect). The terms are therefore completely unobservable in the limit of the flat spacetime of special relativity. But given that these terms are therefore general relativistic, when might they, or other general relativistic effects be important? Of course, they might contribute significantly to overall gravitational fields in any almost any configuration. But beyond that, examination of Eq. (2) suggests that whenever the magnitude of RjJ.v approaches or exceeds unity, these modifications to standard Schrodinger theory may become as significant as the usual kinetic energy and rest mass terms in the stress tensor. To obtain some crude estimate of when such cases arise, assume the ReissnerNordstrom metric9 gives at least reasonable magnitude estimates of RjJ.v in the vicinity of electrons, quarks, and other nearly pointlike, fundamental particles. Further assume that the natural charge magnitude associated with the problem is the electronic charge, and that the natural length scale (all quantities in these equations are kept dimensionless) is 'h/( J6 moc), where ma is the electron rest mass. This appears to be the natural rest mass to use here rather than the Planck mass because later extensions of this structure make it easy to generate masses larger than

498 some reference mass, but hard to generate masses smaller than that mass. 7 Thus, the natural reference mass to introduce is one of the very smallest, nonzero masses known, in this case, the electron rest mass. Using the Reissner-Nordstrom solution then as a very rough guide, one finds that to lowest order, the largest components of R!J. v '" 12[( Gm6)/(lic )][e 2 / (lie )](1/r4 ), and that this approaches unity for a radius ro = 5.6· 10- 23 em .. While this is noticeably smaller than the current resolution of the pointlike nature of electrons or quarks,I° which is about 10- 17 em., it is still huge compared to the Planck scale. Indeed, if ro is translated into a very crude estimate of a cross section 11 by forming 7rr6, this is found to be '" 10- 44 cm. 2 , a value actually comparable to the low end of observed neutrino cross sections.lO And, some additional general relativistic terms in the equations will also begin to be important under the same conditions.

3.

Comments on an Extended Formalism

The previous paper also proposes enhancements to the proposed model via inclusion of a self dual antisymmetric part to the metric, o'!J.v (for spinlike phenomena), and a tracefree part to the torsion. 7 For such cases, the results suggest that the effective rest mass in the wave equations is determined via m6 '" -e- i ¢(1 + ~ 0,), where ;p is a constant angle, and 0, = O,!J. v o'!J.v. This form illustrates why it would be difficult to produce rest masses smaller than some reference mass, since that would require 0, ~ -4. On the other hand, 0, tends to be complex, which should mimic the effects of a complex scattering potential,l1 creating wavefunction sources and sinks (but without violating conservation laws here). It remains to be seen if this can be controlled and used to advantage to require real values for 0, by forbidding pathological behavior. If not, it may be necessary to simply constrain a to be real. But when o'!J.v =f. 0, previous results also suggest that it generates the tracefree part to the torsion (the vector Vi" is 2/3 the trace of the torsion). Indeed, if the tracefree part of the torsion is constrained to be totally antisymmetric, the forms show a similarity to a theory of torsion and spin developed by Hammond. 12 Another point associated with a tracefree part to the torsion (denoted by Qvcr!J.) concerns the form of the spinor connection associated with the affine connection of the four space. In this case, the affine connection is presumed to have a complex, non-Christoffel (tensor) portion, so the spinor connection can be formed from either the regular four space connection, or from its complex conjugate. In practice, it appears necessary to mix both forms into some expressions in order to have consistent results. However, rather than elaborate on that here, simply assume that the spinor connection is defined via the usual relation 13

(3) While space will not permit much detail here, the above expression can be reworked by reexpressing 2aE,Ba -rE'A as the sum of two terms, gv-r8~ symmetric in v and /, and sv-r/ antisymmetric (and self dual) in the same index pair. Utilizing

499

properties of self dual expressions,s Eq. (3) becomes

A = {A} r Bcx Bcx A

-

21 S cx'YBA( v'Y A

-

·3WA) 1 SIJ.'Y AQA 'Y + 2 B IJ.'YCX

Z

(4)

where W'Y is the dual of the totally antisymmetric part of QIJ.'Ycx. Thus, a portion of the tracefree part of the torsion is seen to mix directly with the four vector, Vw Furthermore, since the term SIJ.'Y/ is self dual in the tensor indices, the term . h 21 SIJ.'YB AQA Jl.'YCX can b e reWrItten as 4"1 SIJ.'YB A(QA IJ.'YCX + z. *QA Jl.'YCX ) were t h e d ua1·IS t a ken on the first two indices of QIJ.'Ycx. But this is now the inner product of two self dual forms, so that it can be expressed as the three space dot product of two complex three vectors (with additional free spinor and tensor indices in this case).s But the three vector corresponding to slJ.'Yl is just the Pauli matrices. The overall result then is that

(5) n

where the Tn are the Pauli matrices. But the last term now is in the usual form for an SV(2) potentialY The fact that the preceding term is not in the standard form for the V(l) potential in electroweak theory does not appear so serious, since the purely V (1) case here is already the cleanest form of this model. Acknowledgments

I would like to thank Jim Nester and Jim Wheeler for helpful discussions. References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

F. London, Z. Phys. 42,375 (1927). J.E. Rankin, Int. J. Theor. Phys. 20,231 (1981). Daniel C. Galehouse, Int. J. Theor. Phys. 20,457 (1981) and 787 (1981). James T. Wheeler, Phys. Rev. D 41, 431 (1990). J .E. Rankin, Class. Quantum Grav. 9, 1045 (1992). W. R. Wood, G. Papini, Phys. Rev. D 45, 3617 (1992), Found. Phys. Lett. 6, 207 (1993). J. E. Rankin, Preprint RCTP9401, gr-qc/9404023 (1994). P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953). See chap. 3. R. Adler, M. Bazin, M. Schiffer, Introduction to General Relativity (McGrawHill, 1965). See chap. 13. Articles by Gross and Totsuka, Texas/PASCaS '92: Relativistic Astrophysics And Particle Cosmology, Ann. N.Y. Acad. Sci. Vol. 688, (1993). See pp. 160 (Gross), and 348 (Totsuka). L. 1. Schiff, Quantum Mechanics (McGraw-Hill, 1968). See chap. 5. R. Hammond, Gen. Rel. Grav. 26,247 (1994). M. Carmeli, Classical Fields: General Relativity and Gauge Theory (Wiley, 1982).

500

Dedicated to REMO RUFFINI in recognition of his efforts for the organization of the Marcel Grossmann Meetings on General Relativity ISOTOPIC QUANTIZAnON OF GRAVITY AND ITS UNIVERSAL ISOPOINCARE'SYMMETRY

RUGGERO MARIA SANTILLI The Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USA ABSTRACT We propose for the first time a novel isotopic quantization of gravity without Hamiltonian; we identify its universal symmetry as being isomorphic the Poincare' symmetry; and we point out a number of intriguing implications.

We here outline a novel quantization of gravity without Hamiltonian first presented at the MG7 which is based on the so-calIed isotopiC methods introduced by this author back in 1978 1, worked out in detail in monographs 2 and independently studied in ref .s3. The main idea is to lift the conventional associative product AB among generic quantities A, B, into the form A*B = ATB, where T is a fixed positive-definite quantity calIed isotopic element, while jointly lifting the original unit I of an amount equal but inverse of the deformation of the product, in which case 1 = T- l is the correct left and right new unit, l*A = A*l '" A. calIed isounit. Such dual lifting is isotopic in the sense of preserving the original axioms la, e.g~ the isotopic images of fields, vector spaces, algebras, geometries, etc., remain isomorphic to the original structures. Isotopic liftings are physicalIy non-trivial because the functional dependence of the isounit 1 remains unrestricted. As a result, the isotopic image of a linear-local-Lagrangian theory is given by a theory which is: a) arbitrarily nonlinear in the space-time coordinates x, wavefunctions IjI(x), their derivatives of arbitrary order, X, x, Clljl, ClClljl, ... , interior local denSity J.1., temperature T, etc.; b) arbitrarily nonlocaJ-integral; and c) non-(first-orderlLagrangian (see ref.2b for the local-differential Birkhoffian/second-order-Lagrangian mechanic and ref. 2d for the more general, nonlocal-integral isobirkhoffian mechanics) . The lifting is also mathematicalIy nontrivial because it requires the consequential isotopies of the totality of the mathematical structure of the original theory into a simple yet unique and nontrivial form admitting 1 as the new unit. This includes the lifting of: numbers; angles; fields; vector, metric and Hilbert spaces; trigonometry; functional analysis; Lie algebras, groups and symmetries; Euclidean, Minkowskian and Riemannian geometries; classical and quantum mechanics; etc. 2 The above methods permit a new quantization of gravity hereon calIed quantum isogravity. Its carrier space is the isominkowski space introduced by this author back in I983"a. Let M(x,T],R) be a conventional Minkowski space in the chart x = {xli) = {r,x4}, x4 = cot, where Co is the speed of light in vacuum, " =diag. (I, I, I, -Il, with invariant x2 = xt"x on the field R(n,+,x) of real numbers n with conventional sum n+m and multiplication nXm = nm. The lifting 11 -+ fJ = T(x, x, x, ljI, aq" aaq" J.1., T, ..JTJ, where T is a 4X4 positivedefinite matrix, while jointly lifting the unit I -+ 1 = y-l, evidently preserves the original

501

axioms of M, including flatness, resulting in the isospace ~(x,1l.ru over the isofield ~(n,+,*) of isonumbers n = n1 with sum n+m = (n+mll and isomuItiplication n*m =nTm = (nmn. The lifting M(x,'fl,Rl -+ ~(x,1l.~) is geometrically nontrivial because the separation has the most general possible nonlinear integral form, e.g., of the diagonal type x"2. = [xl Tll(x, X, .J xl + x2 Tzix, X, ... ) x2 + x3 T33(x, X, .•. ) x3 - x4 T4ix, X, .J x4 11

E

~(ft, +,*) (I)

The primary application of the isominkowskian geometry is for the so-called interior problem (motion of extended relativistic particles or electromagnetic waves within inhomogeneous and anisotropic phYSical media such as planetary atmospheres or astrophysical chromospheres) studied in detail in ref.2d with a considerable number of exact-numerical representations of astrophysical data on quasars cosmological redshifts, internal redshifts and blueshifts, etc. In this note we use for. the first time the isominkowskian geometry for the gravitational characterization of the exterior problem (motion of point-like test bodies or electromagnetic waves within the homogeneous and isotropic vacuum). Its most fundamental implication is that curvature is not necessary for the characterization of gravity because Riemannian metrics and equations are identically admitted by the isominkowskian geometry. Let)R(x,g,R) be a conventional (3+Il-Riemannian space with symmetric and real-valued metric g(xl and separation x2 = xtgx over the reals R. It is easy to see that g(xl is identically admitted as a particular case of the isominkowskian metric i){x, X, ~ .J resulting in the local isomorphism 9\(x,g,R) ... ~(x,1l.ru, g(x) a i){x). The main idea of quantum isogravity is to embed gravitation in the unit of a conventional relativistic quantum field theory (RQIT). This is permitted by the isotopic methods via the factorization of any given Riemannian metric in the form g(x) = T(xl'fl, where T(x) is always positive-definite from the locally Minkowskian character of 3l, and the lifting of the unit I = diag. (1, I, I, I) of any given RQFT into the gravitational isounit 1 = [T{xlr l which evidently contains all the essential elements of the original curvature. Note that T can always be diagonalized from its positive-definiteness, the metriC for raising and lowering the indices in ~ is i){xl a g(x), and 1 = 011v) = 0 IlII) = 0 iJ.J = 01111). A consistent isoquantization of gravity then requires the lifting of the totality of the mathematical structure of RQIT into that of the iso-RQFT, also known as relativistic hadronic mechanics. 2d We here recall: the Iiftings R(n,+,x) -+ ~(n,+,*) and M(x,'fl,R) -+ ~(x,1l,ru outlined above; the lifting of the enveloping operator algebra ~ over the field of complex numbers dc,+,x) with generic product AB into the isotope ~ with isoproduct A*B = ATB over t(c,+,*); the lifting of the original Hilbert space:JC with inner product < I > E C into the isohilbert space with isoinner product < r> = < I T I >1 E t under which originally Hermitean-observable quantities remain Hermitean-observable; the lifting of eigenvalue equations H1 > = Eol > into the isotopiC form H * I> = HTI > = £ * I > = ElTi > a EI >, E ¢ Eo (necessary for isolinearity) indicating that the final numbers of the theory are the conventional ones; the lifting of the operator four-momentum pJ > = -i ~I > into the isoform Pll* I > = -i ~Ill >, where ~Il = 1a1aJll is the isoderivative, and the compatible Iiftings of the remaining aspects of RQIT (see ref.2d for brevity). Most important are the following properties: Il the isotopic image of the original RQFT is invariant under its own time evolution; 2) the iso-RQFT admits the conventional theory as a particular case for 1 a I; and 3) iso-RQIT and RQFT coincide at the abstract

502

e.

level in which (from T > 0) ~(n.+.*) == R(n.+.x). ~(x,1l.ru == M(x,l1.Rl. ~ == Jt == JC. etc. In turn, these abstract identities assure the mathematical and physical consistency of iso-RQFT. In conclusion. the main conjecture submitted in this note is that a consistent operator form of gravity already exists. It did creep in un-noticed until now because it is embedded in the unit of conventional RQFT. As an illustration. the embedding of gravity in Dirac's equation for a diagonal isounit (which is assumed hereon) can be written ( I'll * PI1 + i m ) * I > = [).I1(x) T(x) ~I1Jx) pV - i m 1 I T(x) I> = 0 • { I'll :1'V}

= 'YI1 T1'v

+

1'v T 'YI1 = 2 fJl1V == 2 gIlv.

'YI1

(2a)

= T I-ijJ. 112 1'111 •

(2b)

where I'll are the conventional gammas and 'YI1 are cal\ed isogamma matrices. The important point is that at the abstract level the conventional and isogravitational Dirac's equations coincide. (1'I1PI1 + im)1 > == ('YI1*PI1 + iml*j >. Note that the anticommutator of the isogamma matriCes yields (twice) the Riemannian metric g(x). thus confirming the fuIl embedding of gravitation. A similar isotopic realization of gravity can be formulated for any other RQIT. As an example. the Dirac-SchwartzschiId equation (here presented for the first time) is given by Eq.s (2) with h = (I - 2M/rr 112yk1 and 1'4 = (J - 2M/r)l/2.y 41. Similarly one can construct the Dirac-Krasner equation and others. By no means the above quantum gravity is a mere curiosity because it carries rather deep geometrical. theoretical and experimental implications. such as:

Consequence 1: Quantum gravity permits the introduction for the first time of a universal symmetry for gravitation called isopoincare' symmetry P(3.l),which results to be locally isomorphic to the conventional symmetry P (3. Il. The isosymmetry can be readily constructed via the Lie-isotopic theoryl-£ and consists in the reconstruction of P(3.l) for the gravitational isounits 1 = [T(x)JI. g(x) = T(x)T]. Since 1 > O. one can see from the inception that P{3.l) '" P(3.l). Under the the lifting P(3.l) -+ P(3.l) the original generators X = ~l = {MIlV> Pal. MIlv = xlipv - xVPI1 k = I. 2, ...• 10. Il, v = I, 2, 3, 4, remain unchanged while the original parameters w = {wk} = {(e, vOl, aO} E R become isonumbers, w = w1 E ~. The connected component Po(3.il of the isopoincare symmetry P(3.l) can then be written A

Po(3.l): .A.(w) = TIkeiX*w

e

=

(TIkeiXTw)1,

(3)

where A = (eATn = 1(eTA) is the isoexponentiation as characterized by the isotopic POincare-Birkhoff-Witt Theorem original\y derived in ref. la, while the preservation of the original dimension is ensured by the the isotopic Baker-Campbell-Hausdorff Theorem also originally derived in ref. la (see the ref.s3a,3b,3c for algebraiC, geometric and historical aspects, respectively). It is easy to see that structure (3) forms a connected Lie-isotopic transformation group with isogroup laws .A.(w)*A(w') = A(W')*A(W) = A(W+W'), A(w)*Ahv) = A(o) =1 = [T(x)J I. Note that P(3.Il acts isotransitively in ~(x,~,~), Le., x' = A(W}ox, because the preservation of the original action Ax would now violate isolinearity. To identify the isoalgebra PoCS.U of Po(3.Il, we note that the canonical isocommutation rules are [xIJ.;Pvl * I> = (xl!- Pv - ~* xV) * I> = i ~l1v * I > = i 111a aav * I >. The isotopic lifting of the conventional transition from a Lie group to a Lie algebra (see the recent study5) then yields the isocommutation rules of Po(3.I)

503 [WL v ;Ma~l [WL v ; Pal

= d1 va WLj3 = dlllapv

W· a Mvj3

-

- l va

-lvj3M1la + W·~Mav)'

[Pa;~l

I\L),

= 0,

(4a) (4b)

where [A,ABl = AT(x)B - BT(x)A is the Lie-isotopic product originally proposed in ref. la which does indeed verify the Lie axioms as one can see. Since the elements 111 are positive-definite and 111v = 0 for 11 "¢ v, rules (4) confirm the local isomorphism Po~.l)" Po(S.I). Note that momentum operators become commutative in their iSominkowskian representation (while they are notoriously noncommutative in their Riemannian representation). This confirms the achievement of a representation of gravitation in a flat space. The isocasimir invariants are

dO) =1 = [T(x)r l ,

c(I) = P~ = PI!*p.r = PI! * Tfv PV'

d3) = wI! * WIJ., WI! = E I!~ ~ * {p. (5)

Under sufficient boundedness and continuity properties of the T ~ elements, the original convergence of Po(3.!) into finite transforms ensures the convergence of their isotopic images which can then be readily computed from Eq.s (3). The space components S0(3), called lsorotations, were first computed in ref. 6a and can be written for a rotation in the (x, y)-plane x'

= XCOS(TltT~t63)

-

yTII-tT2/sin(TlltT22!63)'

(6a)

y' = x Til! T 22-t sin ( Til! T22 t 63 ) + Y cos ( Til! T 22! 63 ) ,

(6b)

(see ref. 2d for general isorotations in the three Euler angles). Isotransforms· (6) leave invariant all eIlipsoidical deformations of the sphere in the Euclidean space E(r,8,R), r = {x, y, z}, 8 = diag. (I, I, Il. Such ellipsoids become perfect spheres r~ = (rt~r)ls in isoeuclidean spaces £(r,~,lU, ~ = TsB, Ts = diag. (Til, T22> T3:v,ls = T s- I, called isospheres, because of the jOint lifting of the semiaxes Ik -+ Tkk and of the related units Ik -+ Tkk-I. This perfect isosphericity is the geometriC origin of the isomorphism 0(3) '" 0(3) The space-time isosymmetry S()(3. Il is characterized by the above isorotations and the isolorentz boosts originally derived in ref.4a which can be written say, in the (z, t)plane, in terms of the conventional parameter v z' = z sinh (T 33! T 4i v) - t T33-t coT«! cosh (T33t T4i v) = l' ( x3- i3 x4 ), t' = z T33!Co-IT44-tsinh (T33 t T «t v) + t cosh (T33! T4i v) =

i3 = v 1 co' ~

== vI< Tkk!

i,

I Co T 4

l'

= 11 -

l' (x4- ~ x3) 1 co'

~21-I12.

(7al (7b) (7cl

Note that the above isotransforms are nonlinear, as expected for a correct symmetry of gravitation, and are formally Similar to the Lorentz transforms, as expected from their isotopic character. For TIlIJ. = I/nll2 one can introduce the "isogravitational speed of light" c =cO /n4' lsotransforms (7) then characterize the gravitational isolight cone, Le., the perfect cone in isospace ~(x;TJ,~), including the conventional characteristic angle (the derivation of the latter property requires the isotrigonometry2b and it is omitted for brevity), which is the geometriC ori~in of the isomorphis~ ()(3.l) ... ()(3.Il. The isotranslations can be written x' = (elpa).X = x + aOA(x), p' = (e1pa;*p = p, where ~= T111/2 + ao2)],

with 0 = molr; mo = mass of the central gravitating body. For convenience, we have taken c = G = 1. 2. The Equation of Motion 2.1. The Spin Equation The spin equation5 ,6,7 is written in the form DsofJ

U

O

DSfJO

ufJ DS oO

-Ds -+- - - -u-O -Ds = 0. u O Ds Here,

U

O

= d;s" where the coordinates are denoted by xO, and DsofJ - - - := up. SOfJ.

Ds

,1-'

(1)

507

as usual. Eq.(1) gives, after some calculations, the following

812 + (~ + 3D')rS 12 + r(rD' + 1) sin2 ()¢S23 r

sin() cos ()¢S13 = 0

8 13 + [r(3D' + ~) + cot ()O]S13 - (rD' + 1)rOS23 + cot ()¢S12 r

·23 S + [2r(D'

(2a) (2b)

= 0

1 . +1 -) + cot· ()()]S23 + (2D' + _)()S13 -

1 . 12 (2D' + -)4>S = 0, (2c) r r r where a dot denotes differentiation with respect to the parameter s, and a prime denotes differentiation with respect to r. 2.2. The Orbital Equation The equation of motion of the spinning test particle, known as the orbital equation, has the following form 5 ,6,7 d DSOlfj DSJi-fj Ol -(mu + u f j - - ) +r~vuV(muJi- +ufj--)

ds

Ds

Ds

r

+ sp.vu.2 + (D' _ 2D,2 +2D")rS 12 -rD'(D' +r+3) sin2 ()¢S23 = 0 ds r (4c) (4d) Here, >. °l =

r~ v uJi-UV,

which has the following explicit expressions >.0 = -2D'ri

>.1 = _e- 4o D'i 2 + D'r 2 _ r(rD' >.

2

= 2(D' + -r1 )r(). -

>.3 == 2¢[(D' and (m

+ 1)0 2 _ r(rD' + 1) sin2 ()¢2 ·2

sin () cos ()4>

+ ~)r + cot()O], r

+ ms) is an effective mass of the particle (m s is defined as ~ Dg:"').

508

3. Integrals of Motion 3.1. The General Case In considering the particular cases of motion of a spinning test particle in the r2-field, we have to employ some of the results from previous works 6 ,7. They are the Cartesian components of the spin Sx, Sy and Sz. Also we need to have the general form of the three components of the total angular momentum J x , J y , and Jz. These equations are indeed those in the standard Schwarzschild metric and in the Vaidya gravitational field. This fact is related to the spherical symmetry of the background gravitational field 6 , 7 . 3.2. Six Particular Cases Now with these results, we can consider the particular cases of motion of a spinning test particle in the r2-field. These special cases are obtained by demanding that various components of the spin tensor vanish, thus leading to restrictions on the possible types of orbital motion. The results are summarized below: Case A B C D E F

Components of Spin Sik S13 # 0, S12 = S23 = 0 S12 # 0, S13 = S23 = 0 S23 # 0, S12 = S13 = 0 S13 = 0, S12 # 0, S23 # 0 S12 = 0, S13 # 0, S23 # 0 S23 = 0, S12 # 0, S13 # 0

Restriction on the Orbit 0 = f,8 = 0 ~= 0 8= ~ = 0

o. =

f

,bI12,bI13,bV) are the perturbations of the diagonal part of the metric. 6 is the Lagrangian displacement and bp is the perturbed pressure. They can both be expressed in terms of the metric perturbations. Eq. (19) has been formally derived from the equations describing the polar perturbations by considering a solution (b7,/>, b112' 5J.L3, bv, 6, bp), and the complex conjugate solution of the same equations, for real wand real R. If we put the fluid variables equal to zero in Eq. (19), separate the variables, impose an eiwt time dependence, and integrate over the angular variable, Eq. (15) reduces to the constancy of the Wronskian of the independent solutions zt and of the Zerilli equation

Zr

[zt, Z.t']r. =

const .

(20)

Thus the conservation law (15) is the exact generalization of the constancy of the Wronskian for a Schwarzshild perturbed black hole to the case when a fluid source is present. However, there is a problem. While in the case of black holes zt and are independent solutions of the Zerilli equation for real w ) 5v and bV* , complex conjugates of each other with respect to real frequency, are not independent solutions of the perturbed equations (and similarly for the other functions). Therefore Eq. (19) cannot be used as it stands. The problem can be overcome if we assume that the functions that describe the polar perturbations are analytic in the complex R-plane, and consider their analytic extension in this plane. For example

Zr

Nc = N(r; O'o,R) + if. [:RN(r; O'o,R)] ,

(21)

where N is the radial part of bv ,and N(r; 0'0, R) is a solution of the perturbed s integrated for real 0' = 0'0 and real R , and similarly for the other functions.

550

Analytic continuation in the complex .eo plane is the basic idea underlying the Regge theory,63 which is applicable to problems of potential scattering when the second order wave equation is separable and the wave function can be written in terms of a radial function and a Legendre polynomial Pt ( cos B) . Since we are dealing with a fifth-order system, the generalization of the Regge theory to our present case is nontrivial. In Ref. 64 we have shown that this extension is indeed possible, and that for any pole (0"0, O"i) there exists a corresponding pole (.eo, .ei) belonging to the same quasi-normal mode, though the mapping between the two has been determined only numerically. Under these premises, the flux of gravitational energy through the star can finally be evaluated. It should be mentioned that an analytic extension of the variables in the complex w-plane would not work. In fact, since the solution is required to be damped in time, the resulting flux integral would blow up at radial infinity. The relations between the flux integral for static spacetimes and the Einstein pseudo-tensor have been discussed in Ref. 65.

The axial perturbations The theory of nonradial oscillations of stars was first formulated in 1967. Since then the axial perturbations have not received much attention in the literature because they do not excite pulsations in the fluid composing the star. They have been recently reconsidered, however, and it has been shown that separating the variables in terms of Gegenbauer polynomials and assuming the usual e iwt time dependence, the equations governing the radial part of the perturbation can be reduced to a Scroedinger-type equation66

(22) where

V =

e2v -3

r

[.e(.e + 1)r + r3(E - p) - 6m(r)] ,

(23)

e2v is the goo-component of the unperturbed metric (found by solving the equilibrium equations for the selected model of star) and m(r) is the mass contained in a sphere of radius r . The potential barrier depends on how the energy-density E and the pressure p are distributed in the interior of the unperturbed star. In the exterior m(r) = M , E = P = 0 , and the potential reduces to the Regge-Wheeler potential (2). It is important to stress that the axial perturbations are purely gravitational, and that they do not have a Newtonian counterpart. The Schroedinger-type equation (22) is valid from r = 0 to radial infinity, and it allows one to study the behaviour of the star against axial perturbations in terms of the scattering of gravitational waves by a central potential barrier. An interesting question immediately arises: can the scattering of axial gravitational waves be resonant? The problem is of interest because if the scattering is resonant

551 Table 2: The first l = 2 axial resonances for homogeneous star with the energy density (= 1 ( M and ware measured in the units d and C ~ .)

(~) 2.26 2.28 2.30 2.40

M

Wo

,



0.509798 0.213863874 0.23 ·10 .~ 0.12.10- 5 0.503105 0.3689962 0.26.10- 4 0.496557 0.473525 0.92.10- 2 0.465848 0.7767

a set of complex eigenfrequencies exists (axial quasi-normal modes) at which the star can vibrate and emit axial gravitational waves. If it is not resonant, then the star will behave as a center of elastic scattering for the incident radiation. In order to answer this question, Eq. (22) has been numerically integrated for a model of star with a uniform energy density distribution. Although this is a clearly unrealistic model, it presents the advantage that the equilibrium configuration is known as an exact solution of Einstein's equations (the Schwarzschild solution). Moreover, in such models it is possible to test the effects of general relativity in a regime where they are stronger than in any other stellar model. It is known that homogeneous stars can exist only if their radius R exceeds 9/8 times the Schwarz schild radius Rs , or RIM> 2.25 . The models we have considered have been labeled by the parameter (RIM). We find that if (RIM) > 2.6 the axial modes are not resonant, but if (RIM) is smaller than 2.6 one or more resonances appear. The reason can be understood by plotting the potential barrier as a function of the radial coordinate, for different values of (RIM) . Inside the star the potential is a well which becomes deeper as the value of (RIM) decreases and the star shrinks. In the exterior the potential is a barrier and coincides with the Regge-Wheeler potential that has a maximum at r = 3M . When (RIM) < 2.6 the potential well in the interior becomes deep enough to allow, as atomic physics has taught us, the existence of one or more quasi-stationary states. In Table 2 the values of the first R = 2 quasi-normal frequencies for different values of (RIM) are shown. It is interesting to note that there is a progressive increasing of the damping time T = 27r IWi as the star tends to the limiting configuration. This means that the lowest quasi-stationary states are effectively trapped, and that the star cannot emit radiation at those frequencies. In conclusion, though the axial perturbations do not excite any pulsation in the fluid, in extremely compact stars they can become resonant. These modes are purely gravitational.

8.

Perturbations of rotating stars

Rotating stars are far more complicated objects than those we have previously analyzed. Models of rotating stars can be constructed only by numerical integration and no exact solutions are known. Moreover, contrary to what happens in the static

552

spherically symmetric case, we do not have an exact form of the metric appropriate to describe the spacetime in the exterior of the fluid configuration. The Kerr metric cannot be used for this purpose and the problem of finding an interior source for the Kerr metric is still unresolved. Even in the Newtonian case, the frequencies of the normal non-axisymmetric modes of a rotating inhomogeneous star have been computed only recently by Ipser and Lindblom,67 and the first post-Newtonian corrections have been subsequently determined by Cutler and Lindblom. 68 The stability of rotating stars, however, has been thoroughly investigated and conclusive results have emerged: in 1970 Chandrasekhar 69 showed that rapidly rotating stars are unstable to the non-axisymmetric perturbations due to the gravitational radiation reaction, and in 1878 Friedman and Schutz70 showed that the result is generic for all rotating stars. However, viscosity can counteract this instability and consequently only sufficiently rapidly rotating stars are unstable.71 The perturbations of rotating stars can be studied following the same approach used for the perturbations of nonrotating stars if some approximation is made. If one assumes that the stltr rotates with an angular velocity n so slow that the distortion of its figure from spherical symmetry is of order n2 and can be ignored, the theory of nonradial oscillations developed in the previous section can be generalized ,72 and interesting results emerge. For compact objects, small angular velocity means nR « 1 ,a condition which is satisfied by most neutron stars. We have restricted our analysis to the axial perturbations of slowly rotating stars, but the work has been extended and completed by Kojima73 who has derived the equations for the polar perturbations and studied the characteristic frequencies associated with both kinds of perturbations. 74 He will report on his work in this session. The metric for the unperturbed spacetime, derived by Hartle in 196775 and subsequently investigated by Chandrasekhar and Miller76 in 1974, can be written as ds 2 = e2V (dt)2 - e27f;(dr.p - wdt)2 _ e2JJ2 (dx 2? - e2/J3(dx3?, (24) where v, '1/;, /12, /13 differ from those of a spherical non-rotating star by quantities of order n2 (which will be neglected), and w ,that is zero in the non-rotating case, is now of order n . The equation determining w is

r:v, , + ±r:v r' TT

T -

(/12 + v)" T (r:v T + ±r:v) r

=0

,

(25)

where

(26)

In the vacuum outside the star, /12 + v = 0 and the solution of Eq. (25) can be written as r:v = n - 2Jr- 3 , (27) where J is the angular momentum of the star. It should be stressed that the function r:v is responsible for the dragging of inertial frames predicted by the LenseThirring effect. Let us now consider the perturbations of this slowly rotating star.

553

We may expand all perturbed quantities in terms of n , say 81/; = 81/;°+n81/;1+ ... , etc., where 81/;° is a solution of the perturbation equations of a nonrotating star that we have considered so far, and 81/;1 are the first order corrections to be considered because of the slow rotation. We shall concentrate on the equations describing the axial perturbations. We have seen that if the star does not rotate they reduce to the Schroedinger-type equation (22) for a function Z suitably defined. In the present context, we should indicate that function as ZO . The axial equations for the next term in the expansion with respect to n are (28)

(in this equation a is the frequency), where S~ = tv,r[(2WtO +

N2 + 5L~ + 2n Vf.0 PI,p + 211 Vf.0 Pt,p,p] + 2tv W2( Q -

1)//,rPt,p , (29)

3

'Y is the adiabatic exponent, and C;+~(p) and Pt(p) are respectively the Gegenbauer and the Legendre polynomials. The meanings of the various functions are as follows. Zl is the function which describes the axial perturbation to first order in Wl, N2, L~, Vf.0 are the radial parts of the polar perturbations of zero order in ,i.e., those computed for a nonrotating star. Thus Eqs. (28) and (29) show that if a star is slowly rotating the polar and the axial perturbations are no longer independent. They couple because of the rotation in the manner indicated by Eq. (28), and the coupling function is tv which is responsible for the dragging of inertial frames. This is a new purely relativistic effect. To further clarify the nature of this coupling one can eliminate the angular

Q=

(€"!;:) ,

n. n

+ and integrating over the range P =

_2

dependence by multiplying Eq. (28) by Cm _2

+

2

cos () = (-1,1) . Since C m 2 , Pt,p and I1Pt,p,p are of opposite parities, it follows that the polar perturbations belonging to even £. can couple only with the axial perturbations belonging to odd £., and conversely, so that the following must hold

£.

= m + 1,

or

£.=m-1.

(30)

Moreover, a propensity rule is true. Due to the behaviour of the source term S2 near the origin (for details see Ref. 72), the transition I --+ £. + 1 is strongly favoured over the transition I --+ £. - 1 . It is interesting to note that these 'coupling rules' are known in atomic theory: the first is the Laporte rule, while the propensity rule has been formulated by Fano in 198577 in the context of light absorption. The problem which we have formulated is essentially a two-channel problem, the two channels being the axial and the polar perturbations, and it is clear that a whole range of problems with different initial conditions can be formulated. For example in Ref. 72 we have considered a model of star for which, in the nonrotating case, only

554

the polar perturbations were resonant, and asked the question whether the coupling between the axial and the polar perturbations induced by the rotation can generate resonances in the axial perturbations. By solving for the minima, (28) we have found that there indeed exist frequencies for which axial purely outgoing waves are prevailing at infinity, showing that the star does radiate axial gravitational waves at those frequencies. They are different from those of the forcing polar perturbations, and in particular, the damping times are considerably longer (a hundred times longer in the example we have considered). Different boundary conditions have been considered by Kojima in his work.

9.

A new family of strongly damped quasi-normal modes

The polar modes of nonrotating stars are the relativistic counterpart of the Newtonian tidal normal modes. The main difference with the Newtonian case resides in the complex nature of the relativistic eigenfrequencies which is due to the emission of gravitational waves. These modes can be further classified by using the Newtonian terminology: f-p-g- and r- modes (see for example Ref. 78). Conversely, the axial perturbations do not have a Newtonian analogue, and we have shown that under appropriate circumstances they can become resonant and that the excitation of the corresponding modes results in the emission of axial gravitational waves. Another family of modes exists which does not have a Newtonian counterpart, the so-called W-modes, first discovered by Kokkotas and Schutz in 1885. 79 These modes exist both for the polar and for the axial perturbations. 8o Contrary to what happens for the eigenfrequencies of the modes we have so far analyzed, where the imaginary part of the frequency was much smaller than the real part (slow damping), in the W-modes the imaginary part is comparable to the real part, and the corresponding modes are strongly damped. An interesting feature of these modes is that the lowest frequency of each sequence has a value similar to that of the lowest order mode of a Schwarz schild black hole. For higher modes however the spectrum is different from that of black holes. The W-modes are thought to be associated mainly with the gravitational field, and this is the reason why they radiate energy away very quickly. However, if the star is highly relativistic the coupling between the fluid and the gravitational field is stronger and much energy can be transferred from the fluid to the gravitational field and radiated away through these modes. A new branch of strongly damped quasi-normal modes for polar perturbations, the W II-modes, has been recently identified by Leins, Nollert and Soffel,81 in which the frequencies are much more similar to the black hole quasi-normal frequencies. A characteristic of these modes is that near the surface of the star they have a much larger amplitude than near the center. However, the frequencies of the Wmodes and those of the W II-modes do not seem to approach the eigenfrequencies of a black hole when the star becomes more compact. This can be understood by remembering that the boundary conditions imposed on the problem are different in the two cases, and that there is no smooth transition from the mathematical

555 structure of the equations describing the perturbations of stars to those describing the perturbations of black holes.

10.

The Kundu effect

In previous sections we have considered perturbations of isolated sources. What happens if a perturbed black hole or an emitting star are inside a galaxy, as they actually are? Might the gravitational field of the galaxy suppress the emergence of quadrupole gravitational waves generated inside it or nearby? An earlier calculation by Kundu 82 seemed to suggest that this is indeed the case. In particular, since the information about the outgoing gravitational radiation is encoded in the shear (31) where u is the retarded time, and in the Bondi news function da / du ,he computed a when the background gravitational field is that of a Schwarzschild spacetime of mass M ,and found 2 2

a(u) _ ± (.e - 2) r w ~o - (.e + 2) [1 ± t~~;l12iwM] o·

(32)

~g is the radial part of the Weyl scalar % = wgr- 5 + O(r- 6 ) which contains the information about the axial and polar outgoing perturbations of the emitting source. Eq. (32) suggests a dramatically strong suppression of radiation, considering for example that for a typical galaxy M rv l016 cm and that w rv 1O-7 cm- l for radiation coming from the sources of greatest interest. IT that were true we could abandon any hope of detecting gravitational waves. A subsequent investigation by Kundu, Price and Pullin83 showed that the same factor is contained in ~g ,which they explicitly computed, and therefore it fortunately cancels the bad term in Eq. (32), at least unless

Rsource

rv

M

rv ). •

(33)

where ). is the wavelength. In other words, we may see no radiation only if the emitting source is very close to a black hole.

11.

Concluding remarks

The study of perturbations of stars and black holes has brought to light the enormous richness of the theory of general relativity. In the case of black holes since the perturbed equations reduce to a Schroedinger-type equation, it is possible to analyze the perturbation in terms of the scattering of gravitational waves by the spacetime curvature. This approach proved extremely powerful in the study of how a black hole reacts to an external perturbation, and clarified the manner in which incident gravitational waves are absorbed and reflected, and gravitational energy is converted into electromagnetic energy and viceversa whenever the black hole is

556 charged. Many interesting results have emerged, like the existence of quasi-normal modes, the super-radiant nature of the scattering of waves of appropriate frequency by rotating black holes, and perhaps most importantly, the proof that black holes are stable. In the case of stars the harvest has been comparably rich and the scattering approach has proven likewise fruitful. Purely relativistic new phenomena have been found that do not have a Newtonian counterpart, such as: the existence of dynamical instabilities, of crucial importance for the theory of stellar evolution; the resonant nature of the axial perturbations in some particular circumstances; the coupling between polar and axial perturbations in slowly rotating stars; the existence of the W- and the W II-modes and of a conservation law for the energy flux. About the W - and the W II-modes one comment should be added. Since they are strongly damped modes one may argue that their relevance is marginal. However, it has been suggested 79 that in the first milliseconds of a gravitational collapse after the formation of a neutron star, apart from the radiation coming from the pulsating fluid source, there would be a large amount of radiation produced by the gravitational initial data the collapse itself generates. This part of the signal should be essentially a superposition of W- and WII-modes which would then be present in the burst of gravitational waves the detectors are hunting for. Furthermore, all classes of modes are essential in any proof that one would provide about the completeness of the quasi-normal mode spectrum and in any stability analysis. At the end of this marathon one might get the impression that so much has been accomplished in the field that nothing remains to be done. There are aspects of the theory of perturbations of stars and black holes, however, that deserve further investigation. The study of perturbations of Kerr-N ewman black holes, for example, is still at a very unsatisfactory stage, and the stability of these objects remains to be demonstrated. The process of capture of masses by black holes has been studied in great detail, and much is known about the energy, the spectrum and the waveforms that the emitted radiation should exhibit. Conversely, nothing similar has been done for compact stars, and the comparison between the two cases might be very interesting. 85 The problem of finding an internal source for the Kerr metric is still unresolved, and due to the difficulties these investigations involve, not too much is known about the behaviour of rapidly rotating stars when perturbed. The completeness of the quasi-normal mode spectrum has never been ascertained and the mechanisms that excite these modes in connection with given initial data has been so far poorly investigated (see for example Ref. 84). Furthermore, how far can one trust the perturbation approach in the study of the process of collision of black holes? What is the role of the radiation reaction and how much does it alter the picture provided by the theory of perturbations in various astrophysical situations like capture of masses, gravitational collapse, colliding black holes, etc.? (The same questions apply to stars.) These are some of the examples regarding the issues which remain to be clarified, and further investigations are auspicable in this domain.

557

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9. 10. 11.

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26. 27. 28. 29. 30.

T. Regge, J.A. Wheeler, Phys. Rev. 108, 1063 (1957). S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford: Clare don Press, (1984). J. Weber, Phys. Rev. Lett. 22, 1320 (1969). J. Weber, Phys. Rev. Lett. 24,276 (1970). J.F. Zerilli, Phys. Rev. D2, 2141 (1970). J.F. Zerilli, Phys. Rev. Lett. 24, 737 (1970). S. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972). S. Teukolsky, Ap. J. 185, 635 (1973). V. Moncrief, Phys. Rev. D9, 2707 (1974). V. Moncrief, Phys. Rev. DID, 1507 (1974). V. Moncrief, Phys. Rev. D12, 1526 (1975). F.J. Zerilli, Phys. Rev. D9, 860 (1974). S. Chandrasekhar, Proc. R. Soc. Land. A365, 453 (1979). C.H. Lee, J. Math. Phys. 17, 1226 (1976). D.M. Chitre, Phys. Rev. D13, 2713 (1976). S. Chandrasekhar, Proc. R. Soc. Land. A358, 421 (1978). E. Bellezza, V. Ferrari, J. Math. Phys. 25 n.6, 1985 (1984). C.H. Lee, Prog. Theor. Phys. 66 n.1, 180 (1981). S. Chandrasekhar, S.L. Detweiler Proc. R. Soc. Land. A344,441 (1975). S. Chandrasekhar, Proc. R. Soc. Land. A369, 425 (1980). C.V. Vishveshwara, Phys. Rev. D1, 2870 (1970). W.H. Press, Ap. J. 170 (1971). V. Ferrari, Phys. Lett. Al71, 271 (1992). J.M. Bardeen, W.H.Press, J. Math. Phys. 14, 7 (1973). S. Chandrasekhar, Proc. R. Soc. Land. A343, 289 (1975). S.L. Detweiler, Proc. R. Soc. Land. A352, 381 (1977). V. Ferrari, B. Mashoon, Phys. Rev. Lett. 52 n.16, 1361 (1984). V. Ferrari, B. Mashoon, Phys. Rev. D30, 295 (1984). A.A. Starobinski, S.M. Churilov, Soviet JETP 38,1 (1973). W.H. Press, S. Teukolsky, Ap. J. 185, 649 (1973). B.F. Schutz, C.M. Will, Ap. J. Lett. 291, L33 (1985). S. Iyer, C.M. Will, Phys. Rev. D35, 3621 (1987). S. Iyer, Phys. Rev. D35, 3632 (1987). K.D. Kokkotas, B.F. Schutz, Phys. Rev. D37, 12 (1988). J.W. Guinn, C.M. Will, Y. Kojima, B.F. Schutz, Class. Quantum Grav. 7, L47 (1990). N. Andersson, Class. Quantum Grav. 10, L61 (1993). M.E. Araujo, M.A.H.MacCallum, preprint. D. Gunter, Phil. Trans. R. Soc. Land. A296, 457 (1980). D. Gunter, Phil. Trans. R. Soc. Land. A301, 705 (1981). N. Andersson, Proc. R. Soc. Land. A442, 427 (1993).

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31. S.L. Detweiler, in Sources of Gravitati-onal Radiation, edited by L. Smarr, Cambridge, England, 211 (1979). S.L. Detweiler, Ap. J. 225, 687 (1978). 32. S.L. Detweiler, Ap. J. 239, 292 (1980). 33. E.W. Leaver, Proc. R. Soc. Lond. A402, 285 (1985). 34. E. Seidel, S. Iyer, Phys. Rev., D 41, 374 (1990). 35. KD. Kokkotas, Class. Quantum Grav. 8, 2217 (1991). 36. KD. Kokkotas, n Nuovo Cim. B108, 991 (1993). 37. M. Davis, R. Ruffini, W.H. Press, RH. Price, Phys. Rev. Lett. 27, 1466 (1971). M. Davis, R Ruffini, J. Tiomno, Phys. Rev. D5, 2932 (1972). V. Ferrari, R Ruffini, Phys. Lett. B98, 381 (1984). S.L. Detweiler, E. Szedenits, Ap. J. 231, 211 (1979). K Oohara, T. Nakamura, Phys. Lett. 94A, 349 (1983). K Oohara, T. Nakamura, Prog. Theor. Phys. 70, 757 (1983). K Oohara, T. Nakamura, Phys. Lett. 98A, 407 (1983). K Oohara, T. Nakamura, Prog. Theor. Phys. 71, 91 (1984). T. Nakamura, M. Sasaki, Phys. Lett. 106B, 1627 (1981). S. Shapiro, 1. Wasserman, Ap. J. 260, 838 (1982). 38. T. Nakamura, M. Sasaki, Phys. Lett. 89A, 68 (1981). T. Nakamura, M. Sasaki, Prog. Theor. Phys. 67, 1788 (1982). 39. T. Nakamura, M. Sasaki, Phys. Lett. 89A, 185 (1981). T. Nakamura, M. Haugan, Ap. J. 269,292 (1983). 40. Y. Kojima, T. Nakamura, Phys. Lett. 96A, 335 (1983). 41. Y. Kojima, T.Nakamura, Prog. Theor. Phys. 71, 79 (1984). 42. Y. Kojima, T. Nakamura, Prog. Theor. Phys. 72,494 (1984). 43. S.L. Detweiler, Ap. J. 225, 687 (1978). E. Poisson, Phys. Rev. D47, 1497 (1993). C. Cutler, L.S. Finn, E. Poisson, G.J. Sussman, Phys. Rev. D47, 1511 (1993). 44. V. Moncrief, Ann. Phys. 88,323 (1974). 45. RM. Wald, J. Math. Phys. 20, 1056 (1979). B.S. Kay, RM. Wald, Class. Quantum Grav. 4,893 (1987). 46. S. Chandrasekhar, S. Detweiler, Proc. R. Soc. Lond. A345, 145 (1975). S. Chandrasekhar, S. Detweiler, Proc. R. Soc. Lond. A350, 165 (1976). 47. B.F. Whiting, J. Math. Phys. 30, 1301 (1989). 48. B.F. Whiting, Proceedings of the Fifth Marcel Grossmann Meeting, Perth, Western Australia 1988, edited by D.G. Blair and M.J. Buckingham, Cambridge U.P., Cambridge (1989). 49. S. Chandrasekhar, Phys. Rev. Lett. 12, 114, (1964). S. Chandrasekhar, Ap. J. 140,417 (1964). 50. KS. Thorne, A. Campolattaro Ap. J. 149, 591 (1967). 51. R. Price, KS. Thorne, Ap. J. 155, 163 (1969). KS. Thorne, Phys. Rev. Lett. 21,320 (1968). KS. Thorne, Ap. J. 158, 1 (1969). A. Campolattaro, K.S. Thorne, Ap. J. 159,847 (1970). J.R Ipser, KS. Thorne, Ap. J. 181,181 (1973).

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L. Lindblom, S. Detweiler, Ap. J. 292, 12 (1985). KS. Thorne, Ap. J. 158, 997 (1969). S. Detweiler, J.R Ipser, Ap. J. 185, 685 (1973). L. Lindblom, S. Detweiler, Ap. J. Suppl. 53, 73 (1983). C. Cutler, 1. Lindblom, Ap. J. 314, 234 (1987). C. Cutler, 1. Lindblom, RJ. Splinter, Ap. J. 363, 603 (1990). V. Ferrari, Phil. Trans. R. Soc. Lond. A340, 423 (1992). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A428, 325 (1990). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A432, 247 (1990). RH. Price, J.R Ipser, Phys. Rev. D44, 307 (1991). RH. Price, J.R Ipser, Phys. Rev. D43, 1768 (1991). S. Chandrasekhar, V. Ferrari, R. Winston, Proc. R. Soc. Lond. 434,635 (1991). V. Alfaro, T. Regge, Potential scattering, Amsterdam: North Holland Press, (1963). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A437, 133 (1992). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A435, 645 (1991). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A434, 119 (1991). J.R Ipser, L. Lindblom, Phys. Rev. Lett. 62, 2777 (1989). J.R Ipser, 1. Lindblom, Ap. J. 355,226 (1990). C. Cutler, Ap. J. 374,248 (1991). C. Cutler, 1. Lindblom, Ap. J. 389, 392 (1992). S. Chandrasekhar, Phys. Rev. Lett. 24, 611, (1970). J.1. Friedman, Comm. Math. Phys. 62,247 (1978). J.L. Friedman, B.F. Schutz, Ap. J. 222,281 (1978). L. Lindblom,S. Detweiler, Ap. J. , 211,565 (1977). L. Lindblom, W.A. Hiscock, Ap. J. , 267, 384 (1983). S. Chandrasekhar, V. Ferrari, Proc. R. Soc. Lond. A433, 423 (1991). Y. Kojima, Phys. Rev. D46, 4289 (1992). Y. Kojima, Ap. J. 414,247 (1993). Y. Kojima, Prog. Theor. Phys. 90, 977 (1993). J.B. Hartle, Ap. J. , 150, 1005 (1967). S. Chandrasekhar, J.C. Miller, Mon. Not. R. Astr. Soc. 167,63 (1974). U. Fano, Phys. Rev. A32, 617 (1985). V. Ferrari, M. Germano, Proc. R. Soc. Lond. A444, 389 (1994). KD. Kokkotas, B.F. Schutz, Mon. Not. R. Astr. Soc. 225, 119 (1992). KD. Kokkotas, B.F. Schutz, Gen. Relativ. Grav. 18,913 (1986). KD. Kokkot as , Mon. Not. R. Astr. Soc. 268, 1015 (1994). M. Leins, H.P. Noellert, M. Soffel, Phys. Rev. D48, 3467 (1993). P.K Kundu, Proc. R. Soc. Lond. A431, 337 (1990). RH. Price, J. Pullin, Phys. Rev. D46, 2497 (1992). RH. Price, J. Pullin, P.K Kundu, Phys. Rev. Lett. 70, 1572 (1993). N. Andersson, On the excitation of Schwarzschild black hole quasinormal-modes preprint. A. Borrelli, V. Ferrari, in preparation.

560

ASPECTS OF BLACK-HOLE SCATTERING NILS ANDERSSON Department of Physics and Astronomy, University of Wales College of Cardiff, Cardiff CF2 3YB, United Kingdom ABSTRACT Two descriptions of scattering of monochromatic scalar waves from a Schwarzschild black hole are discussed. Some results related to the deflection function in the standard partial-wave picture do not seem to have been discussed before. Calculations relevant for a complex angular momentum (CAM) description of black-hole scattering are presented for the first time. The interpretative power of the CAM representation is very appealing: All features of the scattering cross sections can be understood in terms of interference between a few surface waves (represented by a sum over Regge poles) and a single optical ray (a smooth background integral).

1.

Standard scattering picture

Much can be learned about a physical system by letting waves of a known character be scattered off it. Studies of scattering problems have led to an improved understanding of many of nature's most beautiful phenomena. It is therefore not surprising that a considerable effort has gone into studies of waves scattered off black holes. l The basic scattering problem is simple: One assumes that an asymptotically plane wave impinges on the object of interest. The scattering amplitude then tells us what effect the central object had on the original wave. If we, for simplicity, focus on massless scalar waves in the Schwarzschild geometry, the scattering amplitude feB) can be extracted from the field at infinity

(1) As Matzner showed more than 25 years ago,2 the scattering amplitude follows from the partial-wave expansion 1

feB)

00

= 2iw E(2C + 1) [S(C) -lJ Pe(cosfJ)

(2)

A massless scalar field in the Schwarzschild background is governed by the well-known Regge-Wheeler equation. If one assumes that the field has a harmonic dependence on time this equation becomes a second order differential equation with an effective potential that corresponds to a single barrier. The potential vanishes at both the event horizon and spatial infinity. A physically acceptable solution agrees with the fact that the horizon of the black hole should act as a one-way membrane. In general, such a solution corresponds to a linear combination of out- and ingoing waves at infinity. In order to evaluate the scattering amplitude at a given frequency w one must extract the ratio of the asymptotic amplitudes. Then the phase-shifts

561

o£ -

and the "scattering matrix" S(C) - follow from

(3) The calculation of scattering phase-shifts is more or less straightforward3 (a detailed discussion of previous work in this field can be found in Futterman et aZ.!). 100

·300

o

20

40

60

80

100 120

140 160 180 200

Angular momentum Figure 1: The deflection function for a plane scalar wave scattered from a Schwarzschild black hole. The frequency of the scattered wave is wM 10. The deflection function as obtained from the phase-shifts (solid line) is compared to two approximations (dashed lines): 1) Einstein deflection which should be accurate for large £. - and 2) the predictions of Darwin's formula - which is reliable for impact parameters close to that associated with the unstable photon orbit (£. ~ 52 in this case).

=

One can use the celebrated semiclassical description of scattering (see Ford and Wheeler 4 ) also in the case of black holes. The deflection function (the angle by which a certain partial wave is scattered) then plays a central role. It is related to the phase-shifts by

(4) The deflection function can be approximated in two important cases. The first regards waves that pass far away from the black hole, i.e., for which the impact parameter b = (C + 1/2)/w is large. Then Einstein's familiar result suggests that e ~ -4Mlb. On general grounds one expects a singularity in the deflection function for impact parameters close to the peak of potential barrier (which is analogous to the unstable photon orbit at r = 3M). In 1959 Darwin5 suggested the following approximation;

(5) These two approximations are compared to the result obtained from approximate phase-shifts 3 in Figure 1. An example of the corresponding cross sections is given in Figure 2. That figure exhibits a beautiful example of glory oscillations in the backward direction. This feature arises because of interference between the few partial-waves that are scattered into large angles! (cf. Figure 1).

562

2.

CAM paradigm

Although calculations in the standard scattering picture are straightforward it does suffer from deficiencies. The partial-wave sum (2) is formally divergent. Moreover, it rapidly gets difficult to interpret the results as the frequency increases. This is especially true in the case of scattering from Kerr black holes. 1 An alternative description of black-hole scattering that makes use of complex angular momenta (CAM) has recently been developed. 6 In this picture the scattering amplitude splits naturally into two terms. The first is a sum over the so-called Regge poles; i 7r 00 ( en + ~) r n (6) fp(O) = - Pfn ( - cosO) , W n=O cos 7r en + 2

L

(

1)

and the second is a smooth background integral that can be approached by asymptotic methods. 6 [Pl ( - cos 0) represents a Legendre function of complex degree here.] For references and a discussion of the use of complex angular momenta in atomic and molecular physics see Connor's review article. i An entertaining account of the technique can be found in Nussenzveig's recent book. s The Regge poles a.re singularities of the S-matrix, i.e., are solutions to the Regge-Wheeler equation that correspond to purely outgoing waves falling across the event horizon, and at the same time behave as purely outgoing waves at spatial infinity, cf (3). This means that the Regge poles are complex-angular-momentum analogues to the so-called quasinormal modes of a black hole. 9 Many of the methods used to compute quasinormal-mode frequencies can consequently be used also to determine the Regge pole positions en. However, in order to make use of the CAM description of scattering one must also extract the corresponding residues rn. These are simply defined by (7) in the vicinity of the nth pole. We have recently shown that well established phaseintegral formulae can be used to compute both .en and rn in the context of scalarwave scattering from Schwarzschild black holes. 10 From Figure 2 it is clear that, the pole sum is dominated by the first Regge pole for large angles (0 ~ 40°). Each consecutive pole then gives a contribution that is roughly two orders of magnitude smaller than that of the preceding pole. This means that only one Regge pole need be included in a reasonably accurate description of the black-hole glory. Figure 2 provides an impressive example of the usefulness of complex angular momenta for black holes. But equally important is the interpretative power of the CAM representation. Using an asymptotic expression for the Legendre function in (6) one can show that each Regge state should be interpreted as two surface waves that travel around the black hole in opposite directions.lO At the same time the waves decay at a rate corresponding to the inverse of the imaginary part of Moreover, it is easy to show that the anticipated diffraction oscillations in

en.

563

...............---.

.~--......~....-.............-~.J. ~O~~~~~~~~OO~~8~O~lOO~~1~~~1~~~lOO~~lW

Scattering angle

=

Figure 2: The differential cross section for wM 2 as obtained from approximate phase-shifts (solid line) is compared to the individual contribution from each of the first three Regge poles (dashed lines). Note that a one-pole approximation should be sufficient to describe the black-hole glory.

the backward direction (the black-hole glory) will have a period of 7r/Re (.en + 1/2). One can also infer from the standard localization principle that the real part of each Regge pole position is associated with the distance from the black hole at which angular decay occurs, i.e., Re (en + 1/2) ~ wR,.. In the case of a Schwarzschild black hole one expects such surface waves to be localized close to the unstable photon orbit at r = 3M (or, strictly speaking, the maximum of the Regge-Wheeler potential). This would correspond to R,. = 3.J3M ::::::J 5.196M, and it turns out that the first Regge pole for various frequencies leads to R,. remarkably close to this value. References 1. J.A.H. Futterman, F.A. Handler and R.A. Matzner Scattering from Black Holes (Cambridge Univ. Press 1988). 2. R.A. Matzner J. Math. Phys. 9 163-170 (1968). 3. N. Andersson Scattering of scalar waves by a Schwarzschild black hole in preparation (1994). 4. K.W. Ford and J.A. Wheeler Ann. Phys. 7259-286 (1959). 5. C. Darwin Proc. R. Soc. London A 249 180-194 (1959). 6. N. Andersson and K-E. Thylwe Complex angular momentum approach to blackhole scattering to appear in Class. Quantum Grav. (1994). 7. J.N.L. Connor J. Chem. Soc. Faraday Trans. 861627-1640 (1990) . 8. H.M. Nussenzveig Diffraction Effects in Semiclassical Scattering (Cambridge Univ. Press 1992). 9. N. Andersson, M.E. Araujo and B.F. Schutz Class. Quantum Grav. 10735-755

(1993). 10. N. Andersson Complex angular momenta and the black-hole glory to appear in Class. Quantum Grav. (1994).

564

TRANSFORMATIONS BETWEEN THE REGGE-WHEELER AND BARDEEN-PRESS EQUATIONS J.F.Q.Fernandes and A.W.-C.Lun Department of Mathematics, Monash University, Clayton, Victoria 9168, Australia ABSTRACT We examine how gauge invariant quantities, which satisfy the Regge-Wheeler equation, arise in the Bianchi identities. We demonstrate that these quantities are related, in a gauge independent way, to the Newman-Penrose quantities !/loB and !/I4B, which satisfy the Bardeen-Press equations. Finally we show how the relationships between the Bardeen-Press equations and the Regge-Wheeler equation arise in the integrability conditions for the Bianchi identities.

1. Introduction

In his analysis of the nonspherical perturbations of the Schwarzschild black hole, Price1 ,2 found that Im(!/I2B) satisfies the Regge-Wheeler (RW) equation. (In this article we will follow Price's convention of denoting a perturbation quantity by a subscript B.) Subsequently, Lun and Fackere1l3 and Lun4 were able to show that !/I2B itself can be made to satisfy the RW equation, upon the adoption of a suitable gauge. More recently, Chandrasekhar5 has investigated the perturbation equations extensively, and showed how the Bardeen-Press (BP) equations can be transformed to the RW equation. His analysis is based on the transformations of differential operators (see also Sasaki and Nakamura6 ). Our aim here will be to show how the perturbed gauge invariant integrability conditions on the Newman-Penrose Bianchi identies give rise to the BP and RW equations. We present a gauge invariant RW field quantity, which is related to !/I2B, and show how the integrability conditions provide the transformations between the BP and RW equations. Our anaylsis will be presented in a coordinate free fashion.

2. Regge-Wheeler and Bardeen-Press equations In this article we use the null tretrad provided by Chandrasekhar5 and the modified Newman-Penrose (GHP) formalism (see Penrose and Rindler 7 ). In particular, the Bianchi identities, Ricci identities and commutators are as given in Penrose and RindIer. Only half of the equations are considered here. The results for the other half can be obtained by using the GHP prime operator. For a thorough analysis and proof of our results, readers are referred to Fernandes and Lun (in preparation). From the integrability conditions on the perturbed Bianchi identities we obtain two "wave" equations:

565

[0; + 51l)(p -

U) -

6' 5 - 31/12]1/I4B = 0

(2)

and two gauge invariant (transformation) identities: (p - 3U)(p - U)1/I4B

= 5' 5' 1/I2B + (5' 5~ -

3UAB - 31lUB)1/12

(3)

= 551/14B.

(4)

(p' + 51l)(p' + 31l)1/I2B + [(p' + 51l~p~ - 3p~Il]1/I2

(With the stated choice of tetrad, (2) is the usual spin -2 BP equation.) Define two weighted quantities (see Penrose and Rindler 7 ) eo and e-2 of (p,q) type (0,0) and (-4,0), respectively (5) (6)

(1;::: 2 for graviational radiation). It can be proved that our transformation identities (3) and (4) become (p - 3u)(p -U)1/I4B = 5'5' (1/I2B (p' + 51l)(p'

+ eo)

(7)

(8)

+ 31l)(1/I2B + eo) = 55(1/I4B + e-2)

and the equation (1) simplifies greatly to give the gauge invariant RW equation (9)

The quantity e-2 can be shown to satisfy the spin -2 BP equation: (10)

This comes as no surprise, indeed one can prove the following commutation relation for quantities of type (-4,0):

[(p' +51l)(p-u)-5' 5-31/12] [r 41/12(UP' +IlP-21/12)] = [r 41/12 (up' +IlP-21/12)] [(p' +51l)(p-u)-6' 5-31/12] (11)

where the operator r 41/12 (up' + IlP - 21/12) corresponds to the at Killing vector of the background space-time. Taking (7), we have, after resolution into spin-weighted spherical harmonics, 1/I2B + eo

4r 4

= (/- 1)/(1 + 1)(1 + 2) M(p -

3U)(p - U)1/I4B.

(12)

Applying the RWoperator [(p-3u)(p' +31l)-M' +31/12] to this quantity, we derive a commutation relation for quantities of type (-4,0): [(p-3U)(p' +31l)-55' +31/12] [r455(p-3u)(p-u)]

= [r4M(p-5u)(p-3u)] [(p' +51l)(p-u)-5' 5-31/12]' (13)

566

Hence assuming the spin -2 BP equation (2), we have the RW equation

[(» - 3e)(»' + 3p) -

66'

+ 31/>¥rt>2B +eo) = o.

Conversely, from (8) 4r4 "t" 1/>4B +e-2 = (/- 1)/(1 + 1)(1 + 2) 0 (S (»

,

+ 5p)(» + 3p)(1/>2B + eo)

(14)

and we may derive the spin -2 BP equation for 1/>4B + e-2 from the RW equation, by operating on (14) with the spin -2 BP operator [(»' + 5p)(J> -e) - (S' (S - 31/>2]' Similar results follow by applying the GHP prime operator to (7) and (8). They give the relationships between the spin +2 BP equation and the RWequation. Since 1/>2 is real then one can show (see Lun4) that Im(1/>2B) and Im(eo) are gauge invariant, and these purely imaginary quantities also satisfy the RW equation in a gauge independent way. This is reminiscent of Price's 2 early result. Futhermore, the radial parts of the identities (12) and (14), when written in coordinates, agree with the transformations which Chandrasekhar5 and Sasaki and Nakamura6 derived through a consideration of their differential equations (except that the "constant" [(/-1)/(I+1)(/+2)+12Miu] arises naturally in our equation (14)). 3. Discussion

In the past, much attention has been focussed on attempts to unify the approach to the perturbation equations and to provide a theory of transformation between the various forms of the master perturbation equations and various perturbed NP field quantities (Chandrasekhar5 , Sasaki and Nakamura6 ). These have invariably been coordinate approaches, and as such have obscured how some of the results also arise naturally from the integrability conditions on the perturbed Bianchi identities. Specifically, as we have shown, the RW equation occurs naturally in the Bianchi identities, in a gauge invariant way. Furthermore, the Bianchi identities also give rise to gauge invariant transformation identities relating RW and BP field variables. We are prompted to ask whether the approach outlined in this article can be generalized to other vacuum Petrov type D background metrics. Our preliminary results suggest that this is very likely, however the NP commutators become much more difficult to handle. 4. References 1. R.H.Price, Phys. Rev. D 5(1972)2419.

2. 3. 4. 5.

R.H.Price, Phys. Rev. D 5(1972)2439. A.W.-C.Lun and E.D.Fackerell, Lett. Nuovo Cimento 9(1974)599. A.W.-C.Lun, PhD thesis (Monash University, 1976). S. Chandrasekhar, 'The Mathematical Theory of Black Holes' (Oxford University Press, New York, 1983), pp 134-136, 174-188. 6. M.Sasaki and T.Nakamura, Prog. Theor. Phys. 67(1982)1788. 7. R.Penrose and W.Rindler, 'Spinors and Space-time, Volume l' (Cambridge University Press, Cambridge, 1984), pp 250-260.

567 DISCHARGE OF TIlE ELECTROMAGNETIC FIELD OF A REISSNERNORDS1ROM BLACK HOLE RHEIT HERMAN and WILLIAM A. HISCOCK Department if Physics, Montana State Universiry Bozeman, MT 59717, USA ABSTRAcr Treating the electromagnetic field of the Reissner-Nordstrom spacetime as a dynamical entity leads to the discharge of both the exterior and interior of the black hole by the process of pair creation. Possible effects of this phenomenon are discussed. Past treatments of this process and an outline of current work are also presented.

1. Introduction The interior spacetime region of charged black holes has mainly been studied with the presumption of a background geometry described by the Reissner-Nordstrom metric and with the subsequent physics being allowed to occur on that background. Within this context, the eternal electromagnetic field present gives rise to such exotic and unstable structures as Cauchy horizons, timelike singularities, and an apparent tunnel to other asymptotically flat spacetimes. 1 Investigations have begun to address this assumption of an unchanging electromagnetic field, leading to preliminary models for the dynamical evolution of this field and possible backreaction effects on the spacetime geometry.l.2

2. Pair creation by the Schwinger process The Schwinger formula gives the number of pairs created per unit four-volume r by the electric field E: :n: m

2

E =-C Ii e

(1)

for particles with mass m and charge e, to lowest order in (e 3 Elm 2 ).3 The original derivation of this formula required the electric field E to be uniform and static, and the background spacetime to be flat. The pairs produced would subsequently move in response to the external electric field. Outside of the horizon, the current due to these pairs discharges the electric field of

568

the black hole as seen by an external observer. Within the horizon, the discharge eliminates the opposing charges of the two timelike singularities. Working within this regime, and with further simplifying assumptions, Novikov and Starobinskii were able to derive approximate expressions for the dynamical evolution of the electric field within the outer horizon. The solutions to their equations indicate the electric field serves as the source of its own demise through the production of electron-positron pairs. The effect of these pairs on the interior spacetime geometry suggests that a spacetime with an initially R-N geometry evolves towards a final, uncharged, Schwarzschild-like spacetime. 3. Semiclassical treatment of the initial value problem The previous treatments of the problem of pair creation by the R-N electric field were not performed within the context of a fully relativistic, back-reaction formulation. Many simplifying assumptions made were in addition to those of Schwinger. No explicit equations were used to evolve the relevant metric functions and discern the true geometry within an outer R-N geometry. To move towards a more realistic model, we have begun work necessary to evolve an initial spacelike R-N surface into the future using semi-classical treatments of both the Einstein Field Equation and the Maxwell equation:

(2) We will use point-splitting4 to renormalize these equations and formulate them into a computationally-tractable form. We first study the effects of charged pair creation by the electromagnetic field in the presence of a complex scalar field. The infinities of the Feynman Green function Gp(x,x') may be found using the gauge-invariant differential equation J P

(-g)2{(v p -ieA,.)(v -ieA P ) iGF(x,x')

=[8(x

-

(m2+~R)}G~x,x')

=

-li(x-x')

(3)

-x' )q{x) cp • (x') + 8(x ,0 -x o )cp· (x')q{x)]

00

and that information may be used to isolate the infinities of the Hadamard function of the scalar field G(l)(x,x') with the relation GF(x,x') =

G (x,x')

- (J /2) iG(l)(x,x') .

(4)

569

To describe the geodesic action, we use the Lagrangian L

= (112) g /Hi d1" Id)')( dx" Id)') + eA,. (dx" Id)') ,

(5)

where A. is the affine parameter along a geodesic, AIL is the electromagnetic vector potential, and e is the coupling constant between the scalar and electromagnetic fields. The point-splitting recursion relations become v I'a( vI' -ieAI')a o = 0 I

v I'a( v" -ieAI')a,,+l +(n+l)an+l

I ,

= .t7( v I' -ieAI')( vI' -ieAI')( .1 T a,,) -~a"

(6)

where v denotes covariant differentiation, aEa (x,x')- (J 12)( v I'a)( v" a) is one-half the square of the geodesic distance between x and x' (the properties of a(x,x') are appropriately unaffected by the presence of the gauge field), and 4 E 4 (x,x') = g-l12(X) D(x,x') g-l!2(X') ,with D(x,x') the VanVleck-Morette determinant. The first of these allows the solution which reduces to the expected value of unity with the gauge field absent, and carries the gauge information to all subsequent coefficients: (7)

Acknowledgements The work of R H. was supported under Cooperative Agreement #DEFC02-9IER75681, Amendment >A002, with the U.S. Dept. of Energy and the Montana University system. That of W. A. H. was supported in part by National Science Foundation Grant No. PHY92-07903.

References 1. R Herman and W. A. Hiscock, Phys. Rev. D49, (1994) 3946. A more complete list of references is given here. 2. 1. D. Novikov and A. A. Starobinskii, Sov. Phys. JETP 51, (1980) I. 3. J. S. Schwinger, Phys. Rev. 82 (1951) 664. 4. S. M. Christensen, Phys. Rev. D14 (1976) 2490.

570

CLASSICALLY UNSTABLE MODES OF AN ISOLATED BLACK HOLE OSAMU KABURAKI A,tronomicallnliitute, Faculty of Science, Tohoku Univer,ity Sendai 980-77, Japan We have systematically examined the thermodynamic stability of Kerr holes l and more general Kerr-Newman holes 2 based on the turning point method. However, this method is valid only for continuous parts of an equilibrium sequence. For this reason, some of the results of our previous studies have had to remain inconclusive. We propose here that this defect can be overcome, at least in the case of isolated black holes, by assuming a continuous off-equilibrium entropy function. The equilibrium states of a general Kerr-Newman hole is completely specified in terms of three quantities: its mass M, angular momentum J and electric charge Q. H such a hole is isolated from the surroundings, its entropy (whose intrinsic independent variables are M, J and Q) serves as the proper thermodynamic function from which alone all the possible knowledge about its stability is extracted3 • It is convenient in such discussions to introduce a parameter defined as h = ../a2 + Q2/1'H in addition to various thermodynamic variables, where a = J / M, and l'H represents the radius of outer and inner horizons which are, respectively, the larger and smaller roots of a = 0 in Boyer-Lindquist coordinates. This parameter specifies the location of a state along an equilibrium sequence. Generally speaking, h in the ranges 0 ::; h ::; 1 and 1 ::; h refer to the outer and inner horizons, and the two limiting cases of h = 0 (and h = 00) and of h = 1 correspond to a Schwarzschild (J = Q = 0) and an extreme (M2 = a2 + Q2) holes, respectively2. Plotting S against M, we have the curves shown in Fig. I. The doUed curve indicates the sequence of Schwarzschild holes with different masses and the full curve does that of Kerr-Newman holes with non-zero but fixed values of J and Q. On the laUer, the change in M is reflected in the change of h. The discontinuous transition from a sequence of the holes with non-zero rotation and/or charge to that of Schwarzschild holes should be noted first. The isolation of the Schwarzschild sequence from that of general KerrNewmann holes is very interesting to see when one reminds that the spacetime nature of the singularities at l' = 0 of both types of black holes differ essentially from each other. The essential feature of a non-Scwarzschild sequence is the two-valuedness of the entropy as a function of hole's mass, with the upper and lower branches representing their outer and inner horizons. Although these branches merge smoothly at the point where h = 1, they do not in fact do so in the three-dimensional space (M, (3, S), where

571

13 is the inverse temperature, as seen from

Fig. 3(a) of Ref. 2. In this space, there are two separate curves each of which corresponds to a locus of extrema of the extended entropy function 3 a, a function of the conjugate variable (in this case, 13). We already know that there is no change of stability along each curve 2 • The problem is, therefore, whether these sequnces of extremxa are in fact those of maxima or minima. The answer can be reached easily. H there does not exist any other sequence of equilibrium states, the sequence of larger entropy is the locus of maxima and vice versa. This can be shown by assuming the inverse (i.e., the larger is minima and lower is maxima). Then, we cannot avoid to have at least two more sequences of extrema in between along the 13 axis. Thus we have the conclusion that the outer horizons are stable but the inner ones are unstable to temperature (i.e. surface gravity) fluctuations. Although we cannot discuss here the details of the stability for the mechanical and electrostatic modes, similar procedures lead us to the same conclusions also for these modes. The results of our analysis may now be summarized as follows: the outer horizon, of i,olated Kerr·Newman hole, are alway, ,table while their inner horizon, are alway, unltable in the whole range of possible rotation and charging-up except the maximal cases, and eztreme hole, are merginally ,table. Schwarz schild holes are included in this conclusion as a limiting case of h ~ O. Finally we add a speculation related to the above conclusions. Allen 4 has examined the stability of Schwarzschild holes by calculating the one-loop quantum corrections to the Euclidean effective action. It has been shown that, under the reflecting boundary condition, a single gravitational mode is always unstable irrespective of the boundary radius. Here, we suggest that his negative mode might be reflecting the existence of an unstable inner horizon and hence is unimportant for an observer on the outside of the outer horizon. In this interpretation, we have regarded a Schwarz schild hole not as a hole with strictly zero angular momentum and charge, but as a h ~ 0 limit of Kerr-Newman holes. This seems to be natural when one does not ignore the quantum fluctuations in J and Q. 1. O. Kaburaki, I. Okamoto and J Katz, Phy,. Rev. D47 (1993) 2234. 2. J. Katz, I. Okamoto and O. Kaburaki, Clall. Quantum Grav. 10 (1993) 1323. 3. O. Kaburaki, Phy,. Lett. A185 (1994) 21.

s

4. B. Allen, Phy,. Rev. D30 (1984) 1153.

o

11

572 NON-RADIAL OSCILLATIONS OF A SLOWLY ROTATING RELATIVISTIC STAR YASUFUMI KOJIMA Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-03, Japan ABSTRACT The non-radial perturbation equations of a slowly rotating star are derived, and the rotational effects on the normal frequencies are examined in the framework of general relativity. The stellar rotation is assumed to be slow and the first-order rotational effects are included to the non-rotating case. The dependence of eigenfrequencies on m, and rotational coupling between I-polar mode and (1 + 1)-axial mode are studied.

1.

Introduction

The calculation of the normal modes in a relativistic rotating star is one of the unresolved problems. Recently, the non-radial pulsations of non-rotating stars are reexamined and simple sets of basic equations are derived in the diagonal gauge (Chandrasekhar and Ferrari [1)) and in the Regge-Wheeler gauge (Ipser and Price (2)). Chandrasekhar and Ferrari [3] have extended their work to the pulsations of a slowly rotating star and found interesting results caused by the mode coupling. However, their analysis is incomplete in two points. (i) They have only showed how the odd-parity modes in a spherical star are affected by the coupling with the even-parity modes. The opposite problem remains, that is, how the even-parity modes are affected by the odd-parity modes. (ii) They have limited to azimuthally symmetric perturbations, because their coordinate system is valid only for such perturbations. These drawbacks are removed here in the Regge-Wheeler gauge. 2.

Equations of non-radial oscillations [4]

We shall introduce two small quantities, i.e., the dimensionless pulsation amplitude", and rotation parameter g = fl/JGM/R3. The Einstein equations are expanded in powers of e and ",. Equations in order gO",O and gl",O describe the nonrotating equilibrium states and its rotational correction, respectively. Equations in order eO",l determine the linear pulsations of the non-rotating star. The perturbations can be decomposed by e- itTt and appropriate combinations of spherical harmonics, Yim. The oscillation modes are completely separated by the spherical harmonic indices, (l, m) and the parity. The structure of the equations in order glr/ to determine the linear pulsations of a slowly rotating star is the same as that of order gO",l, but there are source terms. The basic equations are schematically given by ,. [(1)

(1)]

[

(0)]

(0)] + m£'m [(0) H"m' K"m , £o[X,~~] = .1i±l,m[H,~Lm' K~Lm] + mN'i,m[X,~?l]

'--e H"m, K"m = D'±l,m X ±l,m '

(1)

(2)

573

(·) K(') ) · components 0 f t h e metnc . perturbatlon · h H I,m' were I,m an d X I,m ( are·certaIn 0f 1 order e'1l (s = 0,1).

3.

Shift of the normal eigenfrequency_ [5]

In the non-rotating star, a mode with 1 is independent of m and hence 21 + 1 fold degenerate. The degeneracy is removed in the presence of rotation. The eigenfrequency of the slowly rotating star is given by

(3) where (TR and (Tl are the frequency and the damping rate of the non-rotating star, and (TR and (T[ are the rotational corrections. The dependence of eigen-frequencies on m is crucial to the gravitational radiation reaction instability. The corrections are numerically calculated for the polytropic stellar models. The results show that the counter-rotating modes become unstable and that the instability sets in for smaller angular velocity as the system becomes more relativistic. 4.

Coupled pulsations between polar and axial modes [6]

The coupling between different parity modes is also important. The coupling is subject to the selection rule: the even-parity mode with (I,m) is coupled with the odd-parity mode with (I ± I,m) and vice versa. Rotational coupling between I-polar mode and (1 + 1 )-axial mode of axially symmetric pulsations is studied. Chandrasekhar and Ferrari [3] showed that the axial ("odd-parity") wave modes are induced by the in-corning polar ("even-parity") gravitational waves, and that the out-going wave condition in the axial mode is satisfied only for a class of frequencies. Using the correct boundary condition, we can show that the out-going wave condition is satisfied for any frequencies, hence there is no characteristic frequency in the induced oscillations. If the out-going wave condition is imposed on both axial and polar modes, the coupled system shows discrete eigen-frequencies, which are exactly the same values as in the non-rotating star. That is, the eigen-frequencies for axisymmetric perturbations are not affected by the first-order rotational coupling between the different parity modes. References

1. 2. 3. 4. 5. 6.

S. Chandrasekhar and V. Ferrari, Proc. R. Soc. Land. A432, (1991) 247. J.R. Ipser and R.B. Price, Phys. Rev. D43, (1991) 1768; ibid. D44, (1991) 307. S. Chandrasekhar and V. Ferrari, Proc. R. Soc. Land. A433, (1991) 423. Y. Kojima, Phys. Rev. D46, (1992) 4289. Y. Kojima, Astrophys. J. 414, (1993) 247. Y. Kojima, Prog. Theor. Phys. 90, (1993) 977.

574

QUASINORMAL FREQUENCIES OF STEP POTENTIALS HANS-PETER NOLLERT

Theoretische Astrophysik, Computational Physics, Universitiit Tii.bingen ABSTRACT It has recently been suggested that completeness of quasinormal modes is related to a discontinuity in the potential. We construct step potentials with an increasing number of steps such that they approximate the Regge-Wheeler potential with arbitrary accuracy. We find that the quasinormal frequencies of the step potentials do not approach those of the smooth potential with increasing number of steps.

1. Introduction

Quasinormal modes are single frequency modes dominating the time evolution of perturbations of systems which are subject to damping, either by internal dissipation or by radiating away energy. Examples are oscillations of stars damped by internal friction. In general relativity, damping occurs even without friction, since energy may be radiated away towards infinity by gravitational waves. Even linearized perturbations of black holes exhibit quasinormal oscillations 1 , despite the absence of an oscillating fluid. Quasinormal modes of Schwarzschild black holes satisfy the Schrodinger-like differential equation 2

7f;"(x)

+ (w 2 -

V(x) )7f;(x) = 0,

(1)

with purely outgoing boundary conditions at infinity, i.e. and

0"( ) x-t+oo 0/ X ---'+ e -iwx ,

(2)

if the time dependence is chosen to be eiwt . V (x) is the Regge-Wheeler potential

V(x) = (1 _

~) r

(l(l r2+ 1) _ ~) . r3

(3)

1 originates from the .expansion of the perturbation in terms of spherical harmonics, r is a function of x such that x = r + In(r - 1), G = 1 and c = 1, and 2M = 1, where M is the mass of the black hole. An interesting question is whether quasinormal modes completely describe the behavior of a perturbation of the black hole, i.e. the time evolution of some initial data for the perturbation equation. By this we mean that the time dependence of a solution at a given point in space is obtained exactly as a sum over quasinormal modes. A model system with this property has been studied by Price and Hussain3 . However, Eqs. (1), (2), and (3) do not define a selfadjoint problem, and in general, quasinormal modes do not form such a complete system. Ching et a1. 4 have argued that any discontinuity in the potential or any of its derivatives will guarantee that the quasinormal modes of such a system must be

575

V(x)

0.5 0.4 0.3 0.2 0.1

0.0 -5

0

5

10

15

20

25

30

35

40

Figure 1: Step potentials with 4 and 16 steps on either side of the maximum, and the smooth Regge- Wheeler potential for comparison. complete. On the other hand, quasinormal modes of a black hole cannot be complete, since there are no modes with short oscillation periods. This implies that changing the Regge-Wheeler potential slightyly such that a disconinutity is introduced will make its quasinormal mode spectrum become complete. On the other hand, a small change in the potential should not significantly affect the physical response to a perturbation, i.e. the time evolution of some initial data. This appears to be a contradiction, since the quasinormal mode spectrum is believed to represent crucial aspects of this time evolution. We construct potentials which are piecewise constant, such that they approximate the continuous Regge-Wheeler potential, studying the quasinormal mode spectrum of these step potentials as they are allowed to approximate the continuous potential better and better. 2. Procedure and Results

In the following, we will always use the Regge-Wheeler potential (3) with 1 = 2. The step potentials are constructed such that the difference in potential between steps is (roughly) constant. In order to achieve this, the length of the steps is variable and depends essentially on the derivative of the smooth potential. Figure 1 shows the resulting step potentials for different numbers of steps. The quasinormal frequencies of the step potentials are shown in Fig. 2. The spacing between the frequencies becomes closer as the number of steps increases, but they generally occupy the same part of the complex plane. In particular, they remain 'lined up' more or less parallel to the real axis, rather than to the imaginary axis as the quasinormal frequencies of the smooth Regge-Wheeler potential. There is no indication that the frequencies cease to reach arbitrarily large real parts, i.e. arbitrarily small oscillation periods, as the number of steps becomes very large. As the number of steps increases, individual frequencies 'escape' from the line of frequencies towards the imaginary axis. Even these frequencies, however, do not seem to approach the quasinormal frequencies of the Regge-Wheeler potential.

576

8'(w) 0.5 0.4

t Do

• N st = 1 o Ns t = 64

I

• 0

0.3

10

II

0.2

• Ns t o Ns t

=4 = 256

• Ns t

= 16

* Smooth

•••••

•••••••

0.1 0.0

=.,.------r-----.----~---____, ~(w)

0 1 2 4 5 3 Figure 2: Quasinormal frequencies of step potentials with Ns t steps on either side of the maximum, and the frequencies of the smooth Regge- Wheeler potential. 3. Discussion

The presence of discontinuities in the potential has a significant influence on the quasinormal modes of a system, even if the jumps become very small. For the step potentials, the behaviour of the frequencies indicates that the quasinormal modes might form a complete set, even if the difference between the step potential and the smooth potential, whose quasinormal modes are incomplete, becomes very small. It is generally assumed that at least the fundamental quasinormal frequencies of a system have physical meaning in the sense that they will dominate the time evolution of a perturbation of a system. This is true for the quasinormal frequencies, e.g., of the continuous Regge-Wheeler potential. The quasinormal spectrum of a very similar step potential, on the other hand, might actually lead to a complete system of quasinormal modes, but there is no single mode or frequency which has an obvious relationship to the time dependence of the perturbation. How, then, can we know that the quasinormal spectrum of a given system does indeed tell us something about the time evolution of a perturbation of this systems? What are the criteria which distinguish a "meaningful" from a "meaningless" spectrum? Acknowledgments

We wish to thank Richard Price for bringing the completeness problem to our attention, and for enlightening discussions. References 1. W. H. Press, Ap. J. 170, L105 (1971)

2. S. Chandrasekhar and S. Detweiler, Proc. R. Soc. London A 344, 441 (1975) 3. R. H. Price and V. Husain, Phys. Rev. Lett. 68, 1973 (1992) 4. E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, preprint (1994)

577

QUASINORMAL MODES OF NEUTRON STARS AND BLACK HOLES: KIN OR STRANGERS? HANS-PETER NOLLERT, MARKUS LEINS, MICHAEL SOFFEL Theoretische Astrophysik, Computatio"nat Physics, Universitiit Tii.bingen ABSTRACT We study small, nonradial oscillations of neutron stars in a general relativistic perturbation treatment, considering different values for the central density of the star. We determine the strongly damped complex normal modes of the star and discuss their relationship to quasinormal modes of black holes.

1.

Introduction

The equations describing small, nonradial oscillations of neutron stars using a general relativistic perturbation treatment were first derived by Thorne and Campolattaro1 and later refined by Lindblom and Detweiler. 2 ,3 They describe the fluid oscillations of the star as well as the emitted gravitational waves, thus leading to a damping of the star's oscillation. The normal modes of the coupled system are defined as those oscillations which lead to purely outgoing waves at spatial infinity. The real parts of their eigenfrequencies correspond to the oscillation rate, the imaginary parts describe the damping due to radiative energy loss. Lindblom and Detweiler2,3 have determined normal modes with small damping (imaginary parts at least 1000 times smaller than real parts, f- and p-modes). Kokkotas and Schutz4 have found a family of strongly damped normal modes (imaginary parts comparable to real parts), which they call w-modes. The weakly damped modes may be regarded as the realivistic generalization of the Newtonian modes of a neutron star. The strongly damped modes have no Newtonian analogue, they are a purely relativistic effect. Do they correspond to quasinormal modes of black holes, which are strongly damped as well? 2.

Method and Results

We assume the unperturbed spherically symmetric equilibrium state of a neutron star to be given by a solution of the Tolman-Oppenheimer-Volkhoff (TOV) equations. For comparison with Kojima 5 and Kokkotas and Schutz4 we use a polytropic equation of state: p = K pl+1/n, where K = 100 km2 and n = 1, unless noted otherwise. The neutron star models we use are characterised by their central densities Pc; they are summarized in Tab. 1. Models 34, 4, and 5 are unstable against radial perturbations, but no instability has been found so far for I ~ 2. For the perturbations of the metric and the fluid we integrate the system of four ordinary differential equations obtained by Lindblom and Detweiler,2 using the appropriate boundary conditions at the center and the surface of the star. The boundary condition at infinity is treated using two methods adapted from the computation of quasinormal frequencies of black holes. 6

578

Ml M2 M23 M3

Pc [g/cm3] 1.0.10 15 3.0.1015 4.5.10 15 6.0.10 15

M/M0 R[km] 2M/R Pc [g/cm3] M/M0 0.802 10.81 0.219 M34 8.0.10 15 1.331 1.266 8.862 0.422 M4 1.0.10 16 1.300 1.341 8.008 0.494 M5 3.0.10 16 1.097 1.351 7.414 0.538 Table 1: The stellar models (Ml-M5).

R[kmJ 2M/R 6.858 6.466 5.211

0.573 0.594 0.622

All numerical result are for l = 2. We recover the weakly damped f-modes (cf. Lindblom and Detweiler2 ), which we will not discuss here any further. Fig. 1 shows the strongly damped modes for model 2 of Tab. 1. We generally confirm the w-modes found by Kokkotas and Schutz4 for the same model. We find a similar picture (general agreement, relatively large difference for the first w-mode, absence of a kink at the higher w-modes) for the other models (not shown here). In addition to the w-modes, there is a new branch of frequencies located near the imaginary axis. We will call these wn-modes. Also included in Fig. 1 are the results of Kojima5 for the same stellar model. Figure 2 shows the first two w-modes and the first three wn-modes for a sequence of stellar models with different central densities (see Tab. 1). Note that the Wnmodes trace a different path than the w-modes. Q(2M~w~:r----r_-r--r-.....,.._""---'.----r---' Kojima's results 0 w-modes. Kokkotas & Schutz -8--

w-modes, our results

~

tvwmodes +Qnm', of. black hole __

1.5

w-modes 0&wn-modes -$Qnm's of a black hole

2.5

*

0.5 0.5

4

5

!!l(2Mw}

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

!!l(2Mw}

Figure 1: Quasinormal frequencies for Figure 2: Quasinormal frequencies of model 2 and l = 2, and for a Schwarz- stellar models with increasing central schild black hole. densities (Ml-M5), and of a black hole. Changing the polytropic index, we realize stellar models even more compact than those shown in Fig. 2. Table 2 lists the frequencies of the lowest wrr-mode: They do not approach any quasinormal frequency of a Schwarzschild black hole, even if the compactness of the star reaches the limit allowed by general relativity. We have integrated the absolute values of the amplitudes of the metric coefficients K, HI, and the fluid perturbation W from the center of the star to its surface (Tab. 3). As Kokkotas and Schutz have already found,4 the excitation of HI is comparable to that of K for both the f-mode and the w-modes. The excitation of the fluid perturbation W, however, is much weaker for the w-modes as compared to that of the f-mode. The w-modes mainly excite oscillations of the metric, rather than the fluid of the star. The wn-modes show a similar tendency, most pronounced for the fundamental wn-mode.

579

n R./R 0.622 1 0.75 0.697 0.50 0.769 0.25 0.836 0.10 0.870 0.095 0.871 0.05 0.880 0.02 0.885 0.01 0.888 BH 1.000

~(2Mw)

S(2Mw)

1.236 1.322 1.368 1.380 1.341 1.346 1.347 1.347 1.340 0.747

0.757 0.642 0.507 0.354 0.284 0.265 0.239 0.231 0.221 0.178

Table 2: Frequencies of the lowest wn-mode for an increasingly compact star with polytropic index n, and the fundamental frequency of a black hole. 3.

Mode f Wi W2 W3 W4 W5 w 1ll w 2II wF

Model 2 Model 4 fW fW f Hi f Hi 136.0 1.215 46.79 1.507 6.328 0.732 1.592 0.676 6.414 0.791 1.395 0.710 6.500 0.828 1.494 0.754 6.698 0.854 1.499 0.777 6.881 0.872 1.516 0.792 4.438 0.645 3.793 0.804 13.41 0.945 11.36 1.321 11.25 1.324

Table 3: The relative amplitude integrals

f W = 10- 10 • fOR IW(r)1 dr / foR IK(r)1 dr and f Hi = fOR IH1 (r)1 dr / foR IK(r)1 dr for a model 2 and a model 4 neutron star.

Discussion

We have studied strongly damped quasinormal modes of neutron stars. These modes consist mainly of oscillations of the metric outside the star, rather than the fluid inside the star, making them comparable to quasinormal modes of black holes. However, the frequencies of these modes do not seem to approach the quasinormal frequencies of a black hole, even if the star is made as compact as general relativity allows. This is not surprising: The mathematical structure of the differential equations and of the boundary conditions for a neutron star is rather different from that of a black hole, it does not seem possible to define a 'continuous' transition between them. Physically, there is no sequence of static states providing a smooth transformation of a neutron star into a black hole. Acknowledgements This work was partly supported by the Deutsche Forschungsgemeinschaft. We thank H. Herold, H. Ruder, and B. G. Schmidt for inspiring discussions. References 1. 2. 3. 4. 5. 6.

Thorne K. S., Campolattaro A., Ap. J. 149, 591 (1967) Lindblom L., Detweiler S. L., Ap. J. Suppl. 53, 73 (1983) Detweiler S. L., Lindblom L., Ap. J. 292, 12 (1985) Kokkotas K. D., Schutz B. F., MNRAS 255, 119 (1992) Kojima, Y., Prog. Theor. Phys. 79, 665 (1988) M. Leins, H.-P. Nollert, M.H. Soffel, Phys. Rev. D 48, 3467 (1993)

580

QUASI-NORMAL MODES AND LATE TIME BEHAVIOR OF WAVES IN GRAVITATIONAL SYSTEMS E. S. C. CHING1 , P. T. LEUNG 1 , W. M. SUEN 2 , and K. YOUNG

1

1 Department

of Physics, The Chinese University of Hong Kong, Hong Kong 2 Department of Physics, Washington University, St. Louis, MO 63130, USA The evolution of wa.ves in bla.ck hole spa.cetimes is described by the Klein-Gordon equa.tion. The boundary conditions are out-going wa.ves a.t both the outer (infinite) and inner (bla.ck hole horizon) bounda.ries, so tha.t the system is non-Hermitian, and ha.s a. spectrum of qua.sinormal modes (instea.d of normal modes) with complex frequencies. We show tha.t under suita.ble conditions such non-Hermitia.n systems described by the Klein-Gordon equa.tion can ha.ve a. complete set of qua.si-normal modes, in complete analogy with Hermitian systems. In the ca.se where the qua.si-normal modes do not form a. complete set, the la.te time beha.vior of the wa.ve depends crucially on the large ra.dius a.symptotic structure of the potential in the Klein-Gordon equa.tion. We ha.ve determined this rela.tion, and find tha.t the la.te time deca.y of the wa.ve ma.y or ma.y not be a. power la.w in time. Our work extends, pla.ces in context, and provides understanding for the known results for the Schwarzschild spa.cetime.

Radiation from optical cavities is often analysed in terms of the "modes" of the cavity.l Because the waves escape, causing the total energy in the cavity to decrease, these "modes" are quasi-normal modes (QNM's). The observation of the QNM's from the outside immediately gives information on the spatial structure of the cavity (but not the spatial structure of the source of radiation inside the cavity), e.g., in obvious notation, Wj '" j7rc/ L, where L is the length of a simple I-dimensional optical cavity. Similarly, gravitational radiations from relativistic systems can be described by QNM's. Although "cavities" in gravitational systems, i.e., regions of spacetimes which scatter waves significantly, tend to be more leaky than optical cavities, we argued that QNM analysis and the optical analogy can still be very useful. Indeed the QNM's of black holes 2 and relativistic stars 3 have been subjects of much study. In numerical simulations, it is often found 4 ,5 that the radiation observed in many black hole processes is dominated by the QNM's of the hole. For processes as violent and non-linear as the head-on collision of two black holes, the domination of the radiation by QNM's may seem surprising. 5 However, it is easy to understand with the optical analogy-the distant observer sees only the QNM's of the "cavity", but not the details of the radiation generation mechanism. Gravitational QNM radiation may be observed by LIGO and VIRG0 6 in the next decade, and may therefore reveal spacetime structures of various gravitational systems, e.g., the strong field region around a black hole-in much the same way as the spectrum of a laser permits a distant observer to infer some characteristics of the cavity. This exciting possibility calls for the study of the general properties of QNM's of gravitational systems. For the interpretation and extraction of observational data, one would like to know, for example, how the QNM frequencies of a black hole are perturbed by a massive accretion disk around it. A fundamental question, raised by Price and Husain,7 is whether QNM's can form complete sets. This question is not only of theoretical interest, but also of technical importance, e.g., the application of the usual Rayleigh perturbation theory to calculate frequency shifts hinges on the existence of a complete set. The answer to this question is thought to be in general negative, as the system giving rise to QNM's is non-Hermitian. Yet Price and Husain 7 have given a model of relativistic stellar oscillations that does have a complete set of QNM's.

581

In Ref. 8 , we considered this question in. a. general context, and give conditions for completeness to hold. The crucial condition is the existence of spatial discontinuities. (In the model of Ref. 7, this discontinuity corresponds to the stellar surface9 .) Such a condition is not surprising: a discrete sum of QNM's can at best be complete over a finite interval; the discontinuities provide a natural demarcation of this interval, analogous to the boundaries of an optical cavity. The propagation of waves in curved space is often modeled by the Klein-Gordon (KG) equation 2 (1) D4>(x, t) == [8; - 8; +V(x )]4>(x, t) = 0 , with the outgoing wave boundary condition at infinity. The potential V(x), assumed to be positive, bounded and vanishing at infinity, describes the scattering of wave by the background geometry. The retarded Green's function can be expressed in terms of its Fourier transform G(w). By distorting the contour to a large semicircle in the lower half w plane, one sees that

G(x Y· t) "

= i '"' h(x)fj(Y) e-iw;t + I + I 2~ J

Wj

~

hlh ~

e

s

(2)

with

(3) where h represents the wavefunction of the jth QNM, assuming each QNM is a simple pole. 10 The sum comes from the QNM contributions, while Ie is the integral along a semicircle at infinity, and Is comes from singularities of G(w) other than poles in the lower half w plane. We showed8 that the QNM's of (1) form a complete set, in the sense that the Green's function can be expressed purely as a sum over QNM's, with both Is and Ie in (2) being zero, provided the following three conditions hold: (i) V(x) is everywhere finite, and vanishes faster than any exponential; (ii) there are spatial discontinuities demarcating a finite interval; (ill) consideration is limited to certain domains of spacetime. In particular, for propagation from a source point Y to an observation point x in a time t, the QNM's give a complete description for (a) Y inside the interval, x outside the interval, and t > t p ( x, y), for a certain tp; or (b) both y and x inside the interval, and t > 0 (with the retarded Green's function being zero for t ::; 0). Case (b) is analogous to the completeness of normal modes in Hermitian systems defined on a finite interval. The above conditions allow dispersion, backscatter, as well as differences in the damping times, the absence of which had been conjectured to be important. 7 The results are valid to all orders in the rate of leakage. (Lowest order results would be trivial, since the system becomes hermitian in that limit.) This result settles a question in the literature. 7 In so far as the Green's function G provides the solution to all the dynamics, the QNM expansion of G will lead to a variety of physical applications, much as the normal mode expansion of Hermitian systems. Next we turn to potentials which decay slower than an exponential at large distance. Then in general Is is nonzero, and often leads to the phenomena of "tail". It is well known that waves propagating on curved spacetime develop "tails". A pulse of gravitational waves

582 (or other massless fields) travels not only along the light cone, but also spreads out behind it, and slowly dies off in "tails" ,n-16 It is also known that for asymptotically late times this tail often has a particularly simple behavior, namely, it decays as an inverse power in t. Detailed analyses of this late time tail phenomenon had been carried out for the Schwarzschild geometry, using both analytic ll ,13,15,16 and numerical techniquesy,14 These works were based on the ReggeWheeler perturbation formulation, in which the propagation of linearized gravitational, electromagnetic and Klein-Gordon scalar waves, is described by (1) above with a particular V(x). One would like to ask, for a general potential V, the following questions: (i) Does the late time tail always decay as a power in t? In principle, any decay law slower than exponential is possible (see below). What then determines whether or not it decays in a power law form? (ii) How far can we generalize the understanding obtained in the Schwarzschild perturbation analysis? For a general spacetime, is the late time tail (time-dependence and magnitude) determined completely by the asymptotic form of the potential? More fundamentally, how are the two related, and how do they determine the magnitude and the time dependence of the tail? Can the magnitude and time dependence be affected by local geometry and/or the presence of a horizon? In Ref. 17 we studied these questions and developed a formulation for determining the late time tail behavior for a general V(x). For example, for the case of power law potential V(x) = l(l + 1)/x2 + V(x) for x > a, where I is an integer and V(x) ~ x~-2 /x a with a> 2 as x -> 00, we found that, G(w) has a branch cut along the negative imaginary axis, and consequently as t -> 00

G(x,y;t) ~ _C(l,a)F(a/(0,x)f(0'2y )t-(21+a) go 1-1

C(l, a)

= IT a j=O

(4)

.

2) -~ a + 1 + 2)

1= 1,2,...

(5)

where C(l,a) = 1 for I = 0, and F(a) = 2(2x o)a- 2r(21 +a)/[r(a)]. f(w,x) is the Fourier transform of the solution to (5) satisfying f(w,O) = 0 and f'(w,O) = 1. Correspondingly we define g(w, x) as the solution satisfying the out-going wave boundary condition at x -> 00 and go == limw-+o[(iw)l W(g,f)], with W being the Wronskian. This implies i!l general at late times ¢>(x, t) '" t-(21+a) or '" t-(21+a+l) depending on whether or not ¢>(y, t = 0) :/: o. But there is an exceptional case: when a is an odd integer less than 21 + 3, then C(l, a) = o. In this case the late time tail vanishes in the leading order and higher order approximations have to be considered. The next term goes as '" r(21+2-2) . Another example is for the logarithmic potentiall(l + 1)/x2 + (x~-2/x 00

(6) Hence the leading term in ¢> at late times is r(21+ jut)! is the value of '1£4> jut at the fast magnetosonic point. References l.E.S.Phynney, in A,trophy,ical Jet" ed. A.Ferrari & A.G.Pacholczyk (Dordrecht, North-Holland, 1983). 2.M.Takahashi, s.Nitta , Y.Tatematu and A.Tomimatsu, AP.J.363 (1990). 206.

Groups in General Relativity

Chairperson: M. P. Ryan, Jr.

589

SPACETIMES ADMITTING A 3-PARAMETER GROUP OF HOMOTHETIES JAUME CAROT and ALICIA M. SINTES Departament de Fisica, Universitat mes Balears, E-07071 Palma de Mallorca, SPAIN ABSTRACT We shall be concerned with spacetimes admitting a 3-parameter group of homotheties acting on non-null orbits, providing a classification of all possible Lie algebra structures (in terms of the Bianchi type of7t 3 ), and giving in each case the form of the metric as well as that of the proper HVF and the two KV's, in terms of local coordinates.

1.

Introduction

A global vector field X on M is called homothetic if the condition .c x gab = 2>'gab holds on a local chart, where>. is a constant on M. If >. #- 0, X is called proper homothetic and it can always be scaled so as to have>. = 1, if >. = 0 then X is a KV on M. A necessary condition that X be homothetic is xa;be = Rabcdx d where Ra bed are the components of the Riemann tensor; thus, an HVF is a particular case of affine collineation1 and therefore it will satisfy .cXRa bed = .cXRa.b = .cXCabed = 0 where Rab and Cabed stand, respectively, for the components of the Ricci and the Conformal Weyl tensor. It can easily be shown that whenever a proper HVF exists in a Lie algebra of HVF's 1lr, this necessarily contains an (r -I)-dimensional Lie sub algebra of KV 9r-l; therefore one can always choose a basis for 1lr in such a way that it contains at most one proper HVF, the r - 1 remaining ones thus being KV's. If these vector fields in the basis of 1lr are all complete vector fields, then 1lr gives rise in a well known way to a Lie group of homotheties; otherwise, it gives rise to a local group of local homothetic transformations of M. The case r = 3 has an associated Killing subalgebra 92 and the respective dimensions of their orbits are 3 and 2. In this case one can classify the Lie algebras 113 according to their Bianchi type;2 the only possible types being those corresponding to soluble groups, as it follows from the fact that 113 must contain a 2-dimensional subalgebra 92, which in all cases but one, turns out to be abelian. In this case (abelian 92), there are only two different topologies possible for the (non-null) orbits V2 ; namely: V2 diffeomorphic to m?, and V2 diffeomorphic to SI x IR.; and it follows that in the latter case3 the only Bianchi type possible for 113 is I; as for the case V2 ~ IR.2, all seven types can, in principle, occur. (For more information on HVF, see. 4 )

2.

Bianchi types of 113

We shall restrict ourselves to the case of non-null orbits, assuming that the Killing orbits V2 admit orthogonal 2-surfaces. We shall denote the KV's spanning 92 by ~ and 1/, and the proper HVF in the basis of 113 as X.

590

Case 02 abelian Under the above assumptions, the possible Bianchi types of 1£3 containing an abelian 02 are: 2 2.1.

[e,1]] = [e,X] = [1], X] = 0 [e,1]] = [e,X] = 0 [1],X] = e [e,1]] = 0 [e,X] = e [1],X] = 0 [e,1]] = 0 [e,X] = e [1],X] = e + 1] [e,1]] = 0 [e,X] = e [1],X] = 1] [e,1]] = 0 [e,X] = e [1],X] = q1] [e,1]] = 0 [e,X] = 1] [1],X] = -e + q1] (q2 < 4)

(I) (II) (III) (IV) (V) (VI) (VII)

Assume now that the Killing orbits V2 are spacelike and diffeomorphic to ill?; since and 1] commute, we locally have = 1] = Taking now two more coordinates, t and z, the line element associated to the metric 9 can be written as

e

e

;x'

;y.

(1) where \}f, s, b and P are all functions of t and z alone, their fundional dependence on these coordinates to be determined (to some extent) by the HVF X in each case. The case V2 ~ 8 1 X JR, spacelike (cylindrical symmetry) is easily obtained by simply changing y to cp, angular coordinate. In what is to follow, and for the sake of simplicity, we shall assume that the Killing orbits V2 are spacelike and diffeomorphic to JR2 • The case of timelike Killing orbits can be formally obtained by changing (z, y) into (t, z) for the KV's and 1]; and t -----+ iz, y -----+ z, P -----+ iP in Eq. (1), to get the line dement in this case. With all the above assumptions we come to \}f = e-tJ(z)

e

(I) (II) (III) (IV) (V) (VI)

(VII)

X=8t s=s(z) b=b(z) P=p(z) (2) X = 8t + y8x s = s(z) b = b(z) P = p(z) - t (3) X = 8t + z8x s = s(z) b = e-tb(z) P = etp(z) (4) X = 8t + (z + y)8x + y8y s = e-ts(z) b = e-tb(z) P = p(z) - t(5) X = 8t + z8x + y8y s = e-ts(z) b = e-tb(z) P = p(z) (6) X = 8t + z8x + qy8y (q # 0,1) (7) s = e-qts(z) b = e-tb(z) P = e(l- q)tp(z) (8) X = 8t - y8x + (z + qy)8y (l < 4)

.Do = Ja(z)2

+ c(z)2 + g(z)2 + c(z) cos( y'4 - q2t) + g(z) sin( y'4 - q2t) 4 - q2 ()A-1/2 b 2 2 .u. s=e _9.tV az.u. =e -9.tA1/2 2

P= ~

q2

+ y42- (-g( z) cos( y'4 - q2t) + c(z) sin( V4 - q2t))j.Do

591

2.2.

Case (h non-abelian The Lie algebra structure in this case is

(9) Assuming that the theorem of Bilyalov 5 and Defrise-Carter6 holds one can show that the KV, HVF and the metric tensor can be written as (10)

(11)

where A"'13 = A"'I3(Z4) and € = ±1. The case € = +1 corresponds to the 3dimensional homothetic orbits being timelike, whereas € = -1 corresponds to spacelike homothetic orbits. Acknowledgements

Financial support from DGICYT Research project PB 91-0335 is acknowledged. References 1. 2. 3. 4. 5. 6.

G.S. Hall and J. da Costa, J. Math. Phys. 29 (1988) 2465. A.Z. Petrov, Einstein Spaces (Pergamon Press, 1969) p.63. M. Mars and J.M.M. Senovilla, Class. quantum Grav. 10 (1993) 1633. G.S. Hall and J,D. Steele, Gen. ReI. Grav. 22 (1990) 457. R.F. Bilyalov, Sov. Phys. 8 (1964) 878. L. Defrise-Carter, Commun Math. Pys. 40 (1975) 273.

592 SO(2,4) GAUGE SYMMETRY AND THE EINSTEIN-CARTAN-MAXWELL EQUATIONS

D.M. KERRICK

Department of Mathematics and Physics, Philadelphia College of Phannacy and Science, 600 S. 43rd St., Philadelphia, PA 19104

ABSTRACT Einstein-Canan-Maxwell theory is reformulated as an SO(2,4) gauge theory in which the vierbein, Lorentz connection and electromagnetic vector potential emerge from a single SO(2,4) gauge field one-form.

1. Introduction In recent years the idea the gravitational field, represented by a metric or vierbein, along with the Lorentz connection may correspond to different pieces of a single gauge field associated with one of the linear groups (usually some sub-group of GL(5,R» has met with some succesS I •2 • Since the non-gravitational interactions are already described by gauge fields it seems natural to attempt a unification scheme in which all the interactions are contained in a single gauge field (i.e.bundle connection). The purpose of this brief report is to show that such a unification is, in fact, possible for the gravitational and electromagnetic field for a certain type of gauge theory based on the group SO(2,4). 2. An SO(2,4) Invariant Action Consider the following SO(2,4) gauge invariant action

J

I= iJABI\R AB ,

(1)

where ~ = tfP4B + J'AcA r / are the fifteen (p4B = - r BA ; A,B,C =1,2,3,4,5,6,) SO(2,4) curvature two-forms and the two-forms /lAB are given by

593

where SO(2,4) indices are manipulated via the metric TJAB = ( 1,-1,-1,-1,1,-1), V/ and qI are constrained fields; V/I/;A = _A2, qlCPA = m2, V/CPA = 0, where A and m are constants with the dimensions of inverse length, €ABCDEF is the SO(2,4) alternating symbol, D denotes the gauge covariant derivative, and cP =:(if/"rhABCDmDqI A Dt?)cpCD where cpCD = _cpDc are Lagrange multiplier fields. Break the original SO(2,4) symmetry down to the Lorentz group SO(1,3) by choosing a gauge for which 'PA = (O,O,O,O,m,O) and I/;A = (O,O,O,O,O,A) and let m = (4(11")112 lpt 1 where lp is the Planck length. !fthe field equations for ril'6 (i,j,k =1,2,3,4,) and CPy (the Lorentz part of CPAB) obtained from Eq.(l) in the special gauge described above, are put back into Eq. (1) we get

The action, Eq.(3), with y = drY + ik A rig, ~ =: f 12m- 1 i"s dxl', A =f1l2r561' dx" and f =: AIm, is just the Einstein-Cartan-Maxwell action with an undetermined cosmological term Ac 1-'-. Here Ac =: 12rlm4 and I-'- is the n = 4-form volume element.

a

r

r

3. References 1. S.W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739. 2. Heinz R.Pagels, Phys. Rev. D29 (1984) 1690.

594 CONFORMAL MOTIONS IN BIANCHIISPACET~E

DB

LORTAN and

S D MAHARAJ Department of Mathematics and Applied Mathematics, University of Natal, King George V Avenue, Durban 4001, South Africa

ABSTRACT We obtain a class of solutions to the conformal Killing vector equation in the Bianchi I spacetime. This solution is subject to integrability conditions which restrict the gravitational potentials. By imposing the Einstein field equations we solve the integrability conditions in a special case. This corresponds to a homothetic vector which generates the Kasner solution.

The conformal Killing vector equation is given byl X(a;b) = 1j;gab

(1)

where X is the conformal vector and 1j;(x a) is the conformal factor. The Bianchi I spacetime has line element

(2) in comoving coordinates. This spacetime admits a G3 Lie algebra {Xi, i = 1,2, 3} of Killing vectors

which are generated from (1) with 1j; = O. The spacetime (2) generalises the k = 0 Robertson-Walker models. The maximal GIS Lie algebra of conformal Killing vectors in Robertson-Walker spacetimes was obtained by Maartens and Maharaj2. We seek the conformal vectors in the generalised Bianchi I model with line element (2). It is difficult to obtain the general solution of (1) for the spacetime (2). However we can obtain a particular family of solutions with interesting properties. This conformal Killing vector is given by

595

-

X

(3)

~ gY(X, y) + i1-£Z(X, Z) + alx + a2

(4)

2 =

_~gX(x,y)_ ~P(Y'Z)+f3lY+f32

(5)

3

-

~1iX(x, z) + ~P(y, z) + lIZ + 12

(6)

Xl = X

B2(~}FZY(y,z) + p(t)

=

with the conformal factor

~

=

[A2 (~)lgyX(x, y) + [A2 (~)l1iZX(x, z) -

[B2(~)lFZY(y,z) + p(t)

(7)

We have used the notation that superscripts denote integration and subscripts denote differentiation with respect to the spacetime variables, The quantities F, g, 1i and p are functions of integration and aI, f31 and 11 are constants, The solution (3)-(7) is subject to the following restrictions: gxx -

A2 B

(B)'J' 9 = 0

[A A

B2 [A2 gYY+A B (B) A

'J' g=O

[~ (~)lg=o 1ixx

-

A2[ (C)']' C A A 1i=O 2

(C) A 'J' 1i = 0 A2 (C)']' 1i=0 [-B -A

C [A2 1izz + A C

al =

A(~}

f31 =

B(~}

11

= C(~}

which have to be satisfied in addition to the field equations, It is possible to integrate the above integrability conditions in special cases, For example we may generate the Kasner solution3 by requiring that ~,a = 0 =I-~, This corresponds to a homothetic vector, We set F = 9 = 1i = 0 and p = ~t + () where () is a constant, This generates the conformal Killing vector

x=

o

(~t + ()) at

a a 0 + (al x + (2) ax + (f31Y + 132) ay + bIZ + 12) oz

(8)

596

subject to the integrability conditions

A A

13 B (} C

=

'lj; -al 'lj;t+O

=

'lj; - fil 'lj;t+O

'lj;-71 'lj;t+O

on the metric functions. These conditions are easily integrated and we obtain the line element

where 01 , O2 , 03 are constants and we have set PI

'lj; - al

= --'lj;

'lj; -71 P3= - - 'Ij;

This line element satisfies the vacuum Einstein field equations Rab = 0 if the conditions PI + P2 + P3 = 1 hold. We have demonstrated that the conformal Killing vector (8) generates the line element (9). This is the familiar Kasner solution3 . Note that the Kasner solution may be obtained directly from the field equations. We have shown that the Kasner solution may also be obtained by imposing a conformal symmetry condition on the manifold. This may be viewed as an alternate derivation of this solution. There may be other solutions which may be obtained in this way; this will be a subject forfurther investigation. 1. Choquet-Bruhat Y, Dewitt-Morrette C and Dillard-Bleick M Analysis, Manifolds and Physics (Amsterdam: North-Holland, 1977) 2. Maartens Rand Maharaj S D Class. Quantum Grav. 3 (1986) 1005 3. Kramer D, Stephani H, MacCallum M A Hand Herlt E Exact Solutions of Einstein's Field Equations (Cambridge: Cambridge University Press, 1980)

597

A GENERALIZATION OF THE NOETHER THEOREM Nikolai V. MITSKIEVICH Departamento de Flsica, Universidad de Guadalajara, Apdo. Postal 1-2011, C.P. 44100, Guadalajara, Jalisco, Mexico. and Sarnir A. SIDAWI Department of Theoretical Physics, Peoples' Friendship University, Moscow, Russia We consider here the Second Noether Theorem connected with infiniteparametric groups of transformations, in our case: general transformations of space-time coordinates. The standard Noether Theorem (cf. Refs. 1, 2) is based then on invariance property of the action integral of a physical system under consideration, in the sense that the action integral is a scalar under these transformations. If an arbitrary four-dimensional integration region is considered (while quite naturally transformations do not reduce to the identity transformation on the boundary of this region), this invariance becomes equivalent to scalar density property of the corresponding Lagrangian density.c (this assumption being of course more restrictive than that of general covariance of the dynamical equations for the same physical system). We propose now a generalization of the Second Noether Theorem from the scalar density property of.c ,

to the tensor density property of the covariantly conserved symmetric stressenergy tensor density'! J1.V (the latter, as well as its conservation law, results from the standard Second Noether Theorem),

£e'! Ot{J = ('I OtPe - '! Ot-ye - '! -YPeOt);-y. Here 'I ~v reduces to a variational derivative with respect to the metric tensor, of.c (in the case of the gravitational .c , this being simply a partial derivative with respect to g~v, if the Palatini approach3 is used). Hence our generalized Noether relations involve second derivatives of the Lagrangian g~v,

598

density. The number of free space-time indices in these relations is naturally higher than in the standard Noether relations. Non-triviality of the standard Second Noether Theorem is obvious from the conservation laws following from it (in particular, vanishing of covariant divergence of the stress-energy tensor, otherwise often called the Bianchi identity). As to the generalized Second Noether Theorem, its non-triviality is revealed by deduction from it of a new definition of the Bel-Robinson superenergy tensor (for the standard one, see Ref. 4) and its covariant conservation for a vacuum gravitational field:

Here we have used as..c the Gauss-Bonnet invariant (multiplied by R), the first partial derivative of which with respect to the metric coefficients, vanishes identically in the four-dimensional space-time if a Palatini type approach is considered, due to the Lanczos identities 5 quadratic in the curvature tensor. The same..c leads to the quasi-Maxwellian equations of the gravitational field - in the sense of Bianchi identities - when a variation with respect to the Christoffel symbols a la Palatini is performed. Thus this application of our generalization of the Second Noether Theorem has much in common with the issue of analogy between gravitation and electromagnetism (see a paper by N.V. Mitskievich in these Proceedings). Naturally, the use of the higher Lovelock Lagrangians 6 as..c should yield in the both standard and generalized Noether Theorem (the Palatini approach being obviously still applicable), further generalizations of the Lanczos identities together with those of the superenergy tensor (cubic, quartic, etc., constructions with respect to the curvature tensor). 1. N. V. Mitskievich, Ann. Phys. (Leipzig), 1 (1958) 319. 2. N. V. Mitskievich, Physical Fields in General Relativity (Nauka, Moscow, 1969). (In Russian). 3. A. Palatini, Rend. circolo mat. Palermo 43 (1919) 203. 4. L. Bel, Compt. Rend. Acad. Sci. Paris 248 (1959) 1297. 5. C. Lanczos, Ann. Math. 39 (1938) 842. 6. D. Lovelock, J. Math. Phys. 12 (1971) 498.

599 A GENERALIZATION OF ASHTEKAR AND THE

MACDOWELL-MANSOURI PROPOSALS J. A. NIETO· Escuela de Ciencias F{sico-Matematicas de la Universidad Michoacana de San Nicoltis de Hidalgo, A.P. 749, C.P. 58000, Morelia, Michoacan, MEXICO and O. OBREGONt and J. SOCORRO* Instituto de Fisica de la Universidad de Guanajuato, Apdo Postal E-143, 37150 Leon, Guanajuato, MEXICO, and Universidad Autonoma Metropolitana-Iztapalapa Departamento de Fisica, P. O. Box. 55-534 D.F. Mexico, MEXICO ABSTRACT An action is proposed that can be reduced to that of Ashtekar in addition to the Euler and the Pontrjagin topological terms and a cosmological term. Also the MacDowell and Mansouri action in addition to the Pontrjagin topological term are obtained from the action of this gauge theory of gravity.

Considering as the fundamental fields in the action the gauge fields of an appropriate group G MacDowell and Mansouri! and other authors2 introduced field theories for which the action is completely independent of the metric and the connection. The gravitational gauge fields WI" AB = -wI" BA, A,B = 0,1,2,3,4. are subsequently identified as the six components of the connection WI" ab and the four WI" 4a with the vierbein el" a, /L, a = 0, 1,2,3. Let us begin by reviewing the MacDowell-Mansouri theory for the group 80(3,2), which is given by the action 8MM

=

J

,"cd d4 X € I"vOt/3 €abcd -nab , ....I"V ""a/3'

(l.a)

where

(l.b) ab RI"V =

a

I"WV

ab

-

a

vWI"

ab+

WI"

ca

W vc

b

-

Wv

ca

WI"C

b

,

(l.e) (l.d)

It was then shown that this theory is equivalent to three action terms: the Euler topological term, the Einstein-Hilbert action, and the usual cosmological term . •janieto@zeus.ccu.umich.mx t octavio@ifug.ugto.mx Isocorro@ifug.ugto.mx

600

The theory proposed here is S=

J

+w

ab -

d 4 x t;JJ.VOt(3 '-abed c +nab +ned JJ.V ,,(3,

(2.a)

where

1

JJ.

-

-(w ab 2 JJ.

. _

':t;ab

2

wed)

ed JJ.

,

(2.d)

+wJJ. ab is the usual self-dual quantity, +~:~, and +R:~ are constructed correspondingly. Introducing Eq. (2.b) into Eq. (2.a) we find

S

=

(3)

The second term gives the Ashtekar action in the form given by SamueP and J acobson and Smolin4 and the last expression is the usual cosmological term. Using +R~~ it can be shown that the first term in Eq. (3) reduces to (4)

We recognize the first term as the Euler topological term and the second is the Pontrjagin topological term. So, the first term in (3) is a topological term. By considering the sum of the second and third terms in (3) and the first term in (4) we get the MacDowell-Mansouri formulation!, since it is known that the second term in (3) gives half the Einstein-Hilbert action3 •4 • However, we get here also the second term in (4), the Pontrjagin topological term. Then our theory reduces to the MacDowell-Mansouri proposal by removing the Pontrjagin topological term and yields Ashtekar action with a cosmological constant by taking away both topological terms, the first term in Eq. (3), which is equivalent to Eq. (4). Acknowledgments

This work has been supported in part by CONACyT, through a Catedra Patrimonial and Grants Nos. 1683-E9209 and F246-E9207 and by Coordinaci6n de Investigaci6n Cientifica de la UMSNH. References 1. S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739. 2. H. R. Pagels, Phys. Rev. D29 (1984) 1690. P. C. West, Phys. Lett. B76 (1978) 569; A.H. Chamseddine, Ann. Phys. 113 (1978) 212. 3. J. Samuel, Pramana J. Phys. 28 (1987) L429. 4. T. Jacobson and L. Smolin, Class. Quantum Grav. 5 (1988) 583.

601

BUBBLES WITH AN 0(3) SYMMETRIC SCALAR FIELD IN CURVED SPACETIME NOBUYUKI SAKAI(l), YOONBAI KIM(2),., and KEI-ICHI MAEDA(l) (1) Department of Physics, Waseda University, Shinjuku-ku, Tokyo 169, Japan (2) Department of Physics, Nagoya University, Nagoya 464-01, Japan

ABSTRACT We study the first-order phase transition in a model of scalar field with 0(3) symmetry coupled to gravity, and, in high temperature limit, discuss the existence of new bubble solution with a global monopole at the center of the bubble.

Since the first-order phase transition was formulated in the context of Euclidean path integral 1 , it has attracted much attention as a possible resolution of cosmological problems. It is widely believed that process of the first-order phase transition is described by formation and growth of bubbles in which no matter lumps remain. In this note, we shall discuss a possibility of a new bubble solution that the global monopole is supported at the center of it 2,3. Such a new type of solution was first discovered in a flat spacetime by one of the present authors2. The 0(3)symmetric scalar field at high temperature in the presence of gravity is described by the action, 1 R 1 SE = dr d ifg {-167rG + 2gp.v 8p.a811 a + V( A, t/J A. Use is made of two linearly independent spinors satisfying

(2) The three dimensional complex space of two index spinors of the type XA B where X AA = 0, (A,B = 1,2) has a basis XiAB (i = 0,1,2) where

XOAB

= t/>At/>B;

2X1 AB = ¢At/JB + t/JA¢B; X 2 AB = t/JAtjJB

(3)

where ¢A and tjJA satisfy (2). The Xi satisfy (4)

where aijXiX j = 2 X OX2 - (1/2)(x l )2; aijajl< = 6f; a = detllaijll; and fijI< is the LeviCivita alternating tensor density. The Xi are the components of a basis for the Lie algebra of 0(3, C). By choosing the weights of the spinors ,,..A B appropriately, one can determine a spin connection KAB" such that KA A" = and that the covariant derivatives of ,,..AB' £AB and ~ vanish. The curvature two form of the spin space is then

°

RAB""

1

= "4P""A BRptI"" = (K",,, -

K",,, + [K",K"DAB

where Rptl/J" is the curvature tensor of the space-time with metric g"".

(5)

605

It was shown in

I

that

(6) where

Aov = 4>A4>Aw; Aiv = t/JA rPAW = 4> At/JAW; A2v = t/JA4>Aw· The Ail' are determined by the Newman-Penrose spin coefficients, i.e. by the Ricci rotation coefficients. It is a consequence of the definitions of Xi AB and Ail' that X i AB;# XkB A --

1n2

v~aim

EmklA1#

(7)

where

X/ B;# = X;AB.# + KAc#XCiB - XiACKcB# = wki#XkAB (8) and Wki# is a connection in the bundle with the structure group 0(3, C). It is a consequence of the last two equations that

Wki# =

v2aimemlk AI,..

(9)

Let Ki# = KAB#X? A. Then equation (5) may be written as In "k -v2E' In "kl ] R.#II = -Kiv.#+Ki#.v-v2Ei;kK!Kv ali [(Ak# + Kk#)K;v -(Akv+Kkll)K;# , (10) after expressing X;A B.# in terms of X;A B;# by means of equations (8) and (7). On comparing equations (10) and (6) it follows that

Ail' = -Ki,.. References 1. Taub, A. H., Curvature invariants, characteristic classes, and Petro v classification of space-times: differential geometry and relativity, Cohen and Flato, editors (D. Reidel Publishing Co., Dordrecht, Holland, 1976), pp. 277-289.

Appendix Let S = IISABII be a 2 x 2 complex matrix of determinant one and IIsABI! its inverse matrix. Then if • A _ S-A C D A LP "'/ # B c'Y# DS B = ",/p B # then

LP #gptTL" II = gI'll i.e. L#II defines a Lorentz transformation. Similarly, if

X*;A B = sAcxFDSoB then

"

k

= X/Bd i

0' ia;kO I = ail i.e. oi; defines a three dimensional orthogonal transformation.

606

TENSOR MULTINOMIALS AND EINSTEIN-MAXWELL THEORY CHRIS VUILLE Department of Physics, Embry-Riddle Aeronautical University Daytona Beach, FL 32114 ABSTRACT Using tensor multinomials, it is possible to obtain a unification of electromagnetism and general relativity on a four-manifold. In this theory, the electromagnetic potentials manifest themselves as the direct sum of two identical covector fields.

1. Background and Theory The concept of tensor multinomials on manifolds has been only recently developed. 1 For the purposes of this article, the basic geometric object is truncated, taken to be the direct sum of a scalar field Q and a vector field P a on a four-manifold:

VA = Q EB pa

(1)

The group of transition functions for these objects is isomorphic to GL(4,R), so the operator given above is a function of four physical dimensions, while being fivedimensional over the space of real-valued functions. The field containing the gravitational and electromagnetic potentials is given by

~=~EB~EB~

m

Particle trajectories can be obtained from variation of

t>(fGUVAVBdA)

=

0

(3)

with the parameter constrained so that the norm of the 4-velocities are -1 for massive particles and 0 for photons. The variational calculation in Eq.(3) leads to the definition of the nonlinear connection (subsequently, +=EB is understood):

(4) This connection, therefore, when operating on a multinomial as given in Eq.(I), gives rise to the direct sum of the usual tensor field of type (1,1) and various other fields. In the following, the scalar field in Eq. (1) will be taken as proportional to the charge of the particle, with the vector field being the momentum. The equation for the motion of a charged particle is given by:

607 (5)

where (6)

For the corresponding covector operator, we can deduce that

-

aw

awb

).

VeWB = + - - - rebW)' axe axe

).

rb W).

).

(7)

- reW).

Calculating the commutator of the covector field results in:

(8) where the first term on the right is the Riemann Tensor and the second is given by (9)

Raising and lowering indices and contraction operations for these multinomials is accomplished by using 1'\AB=l EB gob • Defining the analog of the Ricci curvature by HAC=Hfu ,we can write down the invariant integral corresponding to this theory: (10)

Variation of this expression yields the Einstein-Maxwell equations in free space. Completing the theory requires. only the addition of various stress-energy terms.

2. Conclusions Using the analog of multinomials, it is possible to create a theory of gravity and electromagnetism on a manifold of four dimensions. This has the obvious advantage of avoiding the compactification problem. Classical general relativity and electromagnetism, therefore, can be grasped as the action of a single symmetric operator, which can be written as the direct sum of a symmetric tensor of type (0,2) and two covector fields.

3. References 1. C. Vuille, Annals of the New York Academy of Sciences, 631, (1991) p. 246-253.

Numerical Relativity

Chairperson: M. Choptuik

611

NUMERICAL INSTABILITIES ASSOCIATED WITH GAUGE MODES MIGUEL ALCUBIERRE and BERNARD F. SCHUTZ Department of Physics and Astronomy, University of Wales College of Cardiff, Cardiff, UK. ABSTRACT Using linear gravitational waves as a model problem, it is shown how the presence of non-dynamical degrees of freedom in the evolution equations of general relativity can give rise to numerical instabilities. Some ways of dealing with these instabilities are discussed, such as the introduction of a correction term in the finite difference equations, and the use of a low pass filter. Finally, the advice is given numerical relativists to keep in mind the existence of such non-dynamical solutions and the numerical problems that they can cause.

Introduction In the 3+1 formalism of general relativity, spacetime is separated into a foliation of spacelike hypersurfaces. 1 This foliation allows for a clean separation of the components of the metric tensor into four "gauge" variables, known as the lapse function and the shift vector, that relate the coordinate systems on nearby hypersurfaces, plus the six independent components of the metric tensor on each hypersurface. In this formalism, Einstein's equations can be written as a system of six second order evolution equations for the components of the spatial metric, plus four constraint equations that these components must satisfy at all times. The form of the evolution equations guarantees that if the constraints are satisfied in the initial hypersurface, they will be satisfied during the whole evolution. From the existence of the constraints, plus the freedom that we have of specifying the initial hypersurface, and the fact that different choices of initial data on that hypersurface can still represent the same geometry, one can show that the six evolution equations have only two dynamical degrees of freedom. 2 Notice that imposing the constraints on the evolution does not eliminate all non-dynamical degrees of freedom. One can still have solutions that are pure "gauge modes", i.e. solutions that satisfy the constraints but only represent a non-trivial evolution of the coordinate system. When one tries to solve the evolution equations numerically things are still more complicated. Small numerical errors in the initial data can easily excite both the pure gauge modes and some extra modes that don't even satisfy the constraints. If the numerical scheme is stable, this should not represent an important problem. However, since these extra degrees of freedom are not necessarily wavelike, one might find that numerical methods that propagate waves in a stable way can become unstable due to the presence of these modes. 1.

2.

Model problem: linear gravitational waves on a flat background Consider a small perturbation of a fiat 3D metric: (1)

612

Let us also assume that the lapse a and the shift a

f3i are such that:

= 1,

(2)

To first order, the evolution equations for the perturbation hii take the form:

(omn_f3mf3n)OmOnh;i+2f3mOmOthij-tfth;j = omn(OmOjhin+OmOihjn-OiOjhmn). (3) Notice how, if one chooses the transverse-traceless (TT) in which om" h mn = 0 and om"Om h,,; = 0, then the r.h.s of these equations vanishes, and one is left with six uncoupled wave equations. Let us now assume that we use the TT gauge only for the initial data, but we evolve the full set of equations. Will the numerical evolution preserve the gauge? To answer this, we have considered the special case of plane waves traveling along the x - y plane. IT we assume that the shift vanishes, the system of equations (3) simplifies considerably. The resulting equations have been solved using two different numerical techniques: a time-symmetric AD! (alternating direction implicit) method,3 and an AD! version of the fully averaged implicit scheme. 4 As initial data we have used a plane wave with a gaussian profile. The results of the numerical simulations show a remarkable behavior: even though the TT gauge conditions are satisfied exactly by our initial data, the numerical solution rapidly violates such conditions. This can be seen clearly from the evolution of the h:z::z:. As the pulse moves away with the correct wave speed, a small perturbation is left behind. The perturbation then grows linearly with time (see Figure 1). That this perturbation is not a numerical artifact can be seen from the fact that in this special case our equations have a solution of the form:

(4) with f an arbitrary function, and the other three components equal to zero. * This solution doesn't satisfy the TT gauge conditions, but is nevertheless excited because even though our initial data correspond exactly to a purely traveling wave, from the point of view of the numerical scheme they are combination of a traveling wave plus a small contribution of the form (4). The linearly growing mode can be thought of as having zero "wave speed". This suggests a way of controlling its effects numerically: one can add a "correction" term to the numerical scheme that effectively introduces a small wave speed for this mode. The small contribution present in the initial data will then propagate away without growing. It is a remarkable fact that this can be achieved by adding a term to the original finite difference equations that is of the same order in ~x and ~t as the truncation error, and therefore does not degrade the accuracy of the scheme. "This is not the only non-dynamical solution contained in our equations. There are three other independent non-dynamical solutions, two that also grow linearly with time, and one that behaves like a wave.

613 COMPONENT

h,."

TIme otep - 0

TIme

otep - 110

Figure 1: Evolution of the h",,,, component of a plane wave with a gaussian profile. The figure on the left shows the initial conditions, while the figure on the right shows the situation after 60 time steps. The wave packet has moved with the correct wave speed, and has been somewhat dispersed. However, a well defined perturbation has remained behind.

However, as soon as we introduce a non-vanishing shift the situation changes radically: an instability develops rapidly even when using the correction term. It is easy to identify the origin of this instability: the linearly growing mode has a zero wave speed (or a small wave speed when using the correction term), which means that even a small shift will force the numerical scheme to propagate information in a non-causal way, a situation that is well known to lead to numerical instabilities. 3 ,s We have tried to deal with these instabilities by adding some numerical dissipation to our evolution scheme. However, we find that the presence of a shift can easily turn what looks like a dissipative term in the differential equation into an extra source of instabilities. It is much cleaner to apply a low pass filter to our solution every time step (i.e. a form of averaging that produces a smoother function). When this is done, we find that the instabilities can be eliminated when the shift is small, but for larger values of the shift they are only delayed for some time. At the moment, it would seem that there are only two ways out of the problem: 1. Impose a gauge condition that will eliminate the unwanted non-dynamical modes. In the case of linear gravitational waves the appropriate gauge choice is obvious: the TT gauge itself. In the general case, however, the question of what gauge condition to impose is less clear. Nevertheless, it is quite possible that almost any condition that is geometric in origin will do, as long as the condition is enforced during the whole evolution.

2. Make a local transformation of coordinates that will eliminate the shift from our equations. This is precisely the idea behind two closely related numerical techniques: causal reconnection3 and causal differencing. s Causal reconnection has the disadvantage that it can only reduce the value of the shift by

614

integer multiples of the "grid speed" /}.x/D..t. Thus, in general, the shift can not be completely eliminated, and the presence of zero wave speed modes will still represent a problem. Causal differencing, on the other hand, does eliminate the shift completely from the equations, and thus offers a real possibility of curing the instabilities that we have found here. This, however, is still a matter for future research. 3.

Conclusions

The presence of non-dynamical degrees of freedom in the evolution equations of general relativity can have important effects in the numerical study of dynamical spacetimes. We have shown how, in the case of linear gravitational waves, the existence of such solutions can give rise to numerical instabilities. These instabilities arise because the extra degrees of freedom do not necessarily behave like waves, and numerical schemes designed to deal with the propagation of waves can therefore become unstable in their presence. A number of ways of dealing with these instabilities has been put forward, though at present none of these techniques has eliminated the problem completely. Even though we have studied the effects of non-dynamical degrees of freedom on a very simple model problem, we believe that a similar situation should arise when dealing with the full Einstein's equations. Because of this, we would like to finish with the following advice for people working in numerical relativity: keep in mind the existence of the non-dynamical degrees of freedom, they might represent an important problem for the 3D evolution codes that are presently being developed around the world. References 1.

2. 3. 4. 5.

R. Arnowitt, S. Deser, and C.W. Misner, "The Dynamics of General Relativity", in L. Witten, editor, Gravitation, An Introduction to Current Research, John Wiley and Sons, 1962. R.M. Wald, General Relativity, The University of Chicago Press, 1984. M. Alcubierre and B.F. Schutz, J. Compo Phys. 112,44, (1994). M. Alcubierre, submitted to J. Compo Phys., (1994). E. Seidel and W.M. Suen, Phys. Rev. Lett. 69 (13), 1845 (1992).

615

MODELLING MOVING BLACK HOLES GABRIELLE ALLEN, MIGUEL ALCUBIERRE, SIMON FARRAR, BERNARD F. SCHUTZ and LEE ASHTON WILD Department of Physics and Astronomy, University of Wales College of Cardiff, Cardiff, UK ABSTRACT A prerequisite for the simulation of the coalescence of black holes is the ability to move the holes across a background coordinate grid. A model problem is described which will provide a framework in which to develop new coordinate gauges for this purpose. Preliminary results from a new 3D Cartesian code for a boosted black hole are presented. A numerical instability is shown to arise from the form of the second order evolution equations, and methods of removing it are discussed. Finally a stable evolution for a limited scenario is described.

1.

Introduction

The collision of two black holes in an inspiralling binary system will provide one of the strongest astrophysical sources of gravitational radiation, and should be one of the first systems observed with the new generations of gravitational wave detectors. The simulation of such a coalescence is a current goal in numerical relativity. The numerical solution will provide new information about the mechanics of such a collision, in addition to predictions of gravitational waveforms. This article describes the progress in developing a code to study a single moving black hole in 3D, which we believe to be an important step towards solving the full coalescence problem. In the past, numerical simulations of the 2D axisymmetric collision of holes have used coordinate systems in which the holes are comoving. 1,2 The holes are then stationary on the numerical grid, with the metric functions containing the decrease in proper distance between them. We believe that for the coalescence problem the holes should move in a 'natural' manner across some 'background' coordinate system. This will lead to a robust formalism, which can handle general initial data, e.g. additional black holes, without problem. Moving a black hole in this manner presents two notable difficulties. First, new dynamic coordinate conditions are needed to position the spacetime coordinates in an appropriate manner. Second, since the gridpoints must move tachyonically to travel from inside to outside the black hole, numerical schemes are needed which can cope with this non-causal behaviour. The two black hole problem will require evolution timescales of many thousands of M, thus it is important to develop numerical schemes which remain stable and accurate for these periods. A single boosted Schwarzschild black hole was chosen as an initial model problem, since it possesses (at least in some coordinate systems) an analytic solution. This solution is used for many purposes: to provide initial data; to calculate numerical errors; to locate horizons; to fix boundary conditions and to facilitate theoretical insight into the problem.

616

2.

Formalism

The standard 3+ 1 formalism3 is used to split spacetime into a foliation of spacelike hypersurfaces. The ten metric components are separated into four kinematical or gauge variables, a (the lapse function) and f3i (the shift vector), and six dynamical variables 'Yij (the spatial metric on each hypersurface). The field equations can then be written as four constraint equations, containing only first time derivatives of 'Yij, and six evolution equations which contain second time derivatives of 'Yij. These evolution equations can be manipulated into a form which mimics wave equations with a coupled 'source' term, involving derivatives of a, f3i and 'Yij,

These evolution equations are written in a different form to those usually seen conventionally the evolution equations are written as twelve first order equations for the spatial metric, 'Yij, and the extrinsic curvature, K ij . Although both the first and second order form of the evolution equations are, of course, equivalent analytically, their numerical treatment will be different. The evolution equations (1) are finite differenced and used to evolve all six metric variables 'Yij, no a priori assumption of coordinate gauge or symmetry is used to modify the equations. The constraint equations are used only to monitor the evolution.

3.

Coordinate Gauge and Numerical Methods

The choice of coordinate gauge (a and f3i) is crucial for moving the black hole through the numerical grid. The lapse function must be chosen so that the slices penetrate the event horizon in a regular manner, ideally preventing large coordinate shears in the metric variables. The shift vector must be chosen to move the grid points through the black hole at some approximately constant rate. Eddington-Finkelstein coordinates satisfy these requirements for a Schwarzschild black hole. These coordinates can be boosted along the direction of movement to give a suitable coordinate gauge for a moving black hole. A dynamic coordinate gauge is required for the general problem which approximates, or ideally coincides with, boosted Eddington-Finkelstein in this special case. For this preliminary work, the coordinates are fixed using these analytic boosted Eddinton-Finkelstein conditions, allowing us to concentrate on the numerical treatment of the evolution equations. Note that using analytic coordinate conditions in this way can be expected to cause unexpected growth in the metric variables since the coordinates cannot move dynamically to 'smooth out' small errors. The Eddington-Finkelstein slicing is not singularity avoiding, and an inner boundary condition is placed inside the apparent horizon to remove the singular region from the computational domain. The regular behaviour of the slices across the event horizon is achieved by demanding that radially ingoing light rays have unit coordinate speed. A similar condition can be demanded to achieve a regular

617

slicing for a general metric. The requirement that light rays travelling in the spatial direction a i move with unit coordinate speed implies the algebraic slicing condition,

(2) Note this this gives a linear relationship between all the metric components. IT ai is pointing towards the black hole then this gives us a suitable condition for a. The appropriate values for ai can straightforwardly be found for a boosted Schwarzschild black hole, although deciding how a i should be found in general is not a trivial matter. The spatial metric components are evolved on a rectangular grid, (t,x,y,z). The evolution equations (1) are integrated using finite difference methods. The wave operator part is converted to a second order implicit finite difference scheme, and the right-hand-side of the equation is treated as a source term, using fourth order extrapolation to find any unknown quantities on the advanced timelevel. Two different implicit schemes have been used. The first is a time-symmetric alternating direction implicit (ADI) method4-6 which it stable for tachyonic grid shifts when combined with causal reconnection. 5 • 6 The second method is an ADI version of the fully averaged implicit schemes. 7 which is stable for any grid shift. A low pass filter is used at each timestep to smooth the highest frequency components of Iii' This completely removes two mild numerical instabilities which affect only the high frequencies. The first instability is due to a technicality of the ADI methods, and the second is due to a cancellation occuring when f3i = 1. 4.

Results and Conclusions

The numerical evolution of the six variables Iii using a finite difference version of Equations (1) are seen to exhibit an exponentially growing numerical instability. The cause of this instability becomes apparent when the full version of the evolution equations (1) is carefully examined. The tensor notation used in (1) (and in the numerical code) conceals the fact that in each of the evolution equations some of the spatial derivatives in the wave operator cancel with corresponding terms in the 'source' term. For example, in the evolution equation for IZZ the term a2/zza~/ZZ cancels on both sides (an obvious fact when we remember that (3)R zzzz = 0). This cancellation implies that, when f3i = 0, the wavespeed in the z-direction is zero. As soon as f3z #- 0, the coordinates are shifting faster than the wavespeed, and for this special case of a zero wavespeed the numerical schemes described above become unstable. This behaviour should be compared to the gauge mode instabilities described elsewhere in these proceedings. 8 That it is the cancellation of these second derivatives which causes the instability can be demonstrated by artificially preventing them from cancelling. For example, if the analytic solution is used for the troublesome derivatives in the source term, all six components show a stable and second order convergent evolution. Alternatively, if the algebraic gauge condition (2) is used to replace e.g. second derivatives of IZZ with second derivatives of the other five metric components, a totally stable and

618

convergent evolution (to > lOOOM) is seen when the other five components are fixed analytically. Unfortunately, an unstable evolution is again seen when all six variables are treated in this way. There are two immediate strategies for removing this numerical instability. First, the coordinate freedom in choosing a and pi could be used to amend the evolution equations into a form which can be integrated, using our current numerical schemes, in a stable fashion. For example, setting pi = 0 would provide a stable, if short, evolution. However, this is not a preferred choice, since our viewpoint is that the gauge freedom should be used to construct hypersurfaces with approximr.tely regular proper time and proper distance intervals between gridpoints, over which the black holes can move without developing large coordinate shears. A second approach is to adapt the numerical schemes to be stable for all grid shifts, even when the light cone has closed in any direction. One course which currently appears promising is to use causal differencin!/ to numerically remove the shift. Causal differencing is particularly favourable since it uses interpolation to entirely remove the shift from the finite difference scheme. This would allow us to choose whatever shift is needed to treat the black hole properly, without having to worry about its numerical effects. Once this instability has been overcome, we will have a general 3D code which can be used to develop coordinate gauge conditions for moving black holes. The code will also allow us to compare the use of the second order formalism and implicit numerical methods with the more traditional first order formalism and explicit numerical methods. References 1. Smarr, A. Cadez, B. DeWitt, K. Eppley, "Collision of Two Black Holes: Theoretical Framework" , Phys. Rev. D 14, 2443, (1976). 2. P. Anninos, D. Bernstein, D. Hobill, E. Seidel & L. Smarr, "Collision of Two Black Holes", Phys. Rev. Lett. 71, 2851, (1993). 3. R. Arnowitt, S. Deser, and C.W. Misner, "The Dynamics of General Relativity", in L. Witten, editor, Gravitation, An Introduction to Current Research, John Wiley and Sons, 1962. 4. G.D. Allen, PhD thesis, University of Wales, 1993. 5. M. Alcubierre, PhD thesis, University of Wales, 1994. 6. M. Alcubierre and B.F. Schutz, "Time-Symmetric AD! and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids", J. Compo Phys. 112,44, (1994). 7. M. Alcubierre, "Stable Numerical Integration of the Wave Equation on Arbitrarily Rapidly Moving Coordinate Systems", submitted to J. Compo Phys., (1994). 8. M. Alcubierre and B. F. Schutz, "Numerical Instabilities Associated with Gauge Modes" , this volume. 9. E. Seidel and W.M. Suen, "Towards a Singularity-Proof Scheme in Numerical Relativity", Phys. Rev. Lett. 69 (13), 1845 (1992). 1.

619

ROTATING BLACK HOLE SPACETIMES Steven R. Brandt and Edward Seidel National Center for Supercomputing Applicatio1l.$ 605 E. Springfield Ave., Ch.ampaign, IL 61820 ABSTRACT We have developed a numerical code to study the evolution of distorted, rotating black holes in axisymmetry. This code is used to evolve a new family of black hole initial data sets corresponding to distorted "Kerr" holes with moderate rotation parameters. The gauge conditions and methods used build upon the wisdom gained from the non-rotating codes.

1. New Initial Data Sets We have developed a new class of rotating black hole data sets 1,2,3. They describe vacuum rotating black holes with a gravity wave imposed upon them. These are a generalization of previous data sets which include Kerr, Bowen and York rotating black holes 4 , and the NCSA distorted black holes 5 . As a consequence of our analysis, we have found the functional form of the gravity wave added to the Kerr data set to make it conformally flat. It is roughly similar in form to one of the NCSA Brill waves studied previously, but the distortion is smeared out over a larger fraction of the spacetime than those typically studied. We are using two new solutions to the momentum constraint equation that we discovered2 ,3. Each provides odd-parity radiation to a non-rotating black hole. One has equatorial boundary conditions like the Kerr spacetime; the other has equatorial boundary conditions like the "cosmic-screw" spacetime (on-axis collision of two counter-rotating black holes). We plan to use the latter form to model the end state of the "cosmic-screw" spacetime.

2. Evolution We have evolved these data sets with a new code developed to handle rotating black holes 1 ,2. We were able to extract even and odd parity radiation waveforms from the spacetime using the gauge invariant techniques developed by Abrahams and Evans 7 . We compute energy radiated during the evolution by integrating these extracted waveforms and can verify this energy loss by comparing the mass of the horizon at later times in the evolution to the ADM mass on the initial slice. As in previous calculations (which had only even parity modes) we find the radiation is dominated by the quasi-normal mode excitation (QNM). The 1 = 5 QNM had not been published at the time. We made a prediction which was later verified by the code of Leaver for the real part of the frequency of the 1 = 5 mode. We plot the waveforms from several modes below as measured at a distance of r = 15MADM from the hole. We have some data sets in which virtually all of the radiation energy comes out in the odd parity modes.

620

We have also studied the apparent horizons of the black holes. For rotating holes, the equilibrium (Kerr) geometric shape of the apparent horizon is oblate. By measuring the geometry of these distorted rotating black holes we are able to detect the quasi-normal mode oscillations on the apparent horizon of the black hole l . Because the horizon settles down to a non-spherical shape its polar to equatorial circumference ratio (CR) will settle to a value less than one. If we plot CR vs. time and fit the curve to the 1 = 2 QNM plus a constant we are able to calculate the rotation parameter aim from that constant. In the figures below aim = .43. Smarr6 showed that the flat-space embedding would begin to disappear for black holes in which the rotation parameter is large enough. We measure the point at which the embedding vanishes along the horizon and find results in accord with Smarr's calculation. This work is described in more detail in l ,2,3. Acknowledgements

We are pleased to acknowledge helpful conversations with Larry Smarr, David Bernstein, Wai-Mo Suen, and Edward Leaver. The calculations were performed at NCSA and at the Pittsburgh Supercomputing center. The work has been supported by NCSA and by NSF grants PHY 94-07882 and ASC93-18152. References 1. P. Anninos, D. Bernstein, S. Brandt, D. Hobil, E. Seidel and L. Smarr, Phys. Rev. D 50, 3801 (1994). 2. S. Brandt and E. Seidel, "The Evolution of Distorted Rotating Black Holes", to be submitted to Phys Rev D., (1994). 3. S. Brandt and E. Seidel, "The Initial Value Problem for Distorted Rotating Black Holes" , in preparation. 4. J. Bowen and J. W. York, Phys. Rev. D 21, 2047 (1980). 5. P. Anninos et. al., in Computational Astrophysics: Gas Dynamics and Particle Methods, edited by W. Benz, J. Barnes, E. Muller, and M. Norman (SpringerVerlag, New York, 1994), to appear. 6. L. L. Smarr, Phys, Rev. D 7,289 (1973). 7. A. Abrahams and C. Evans, Phys. Rev. D 42, 2585 (1990).

621

1=5 Wave Mode Extraction Fit to Fundamental and First Harmonic of 1=5

1=3 Wave Mode Extraction Fit to Fundamental and First Harmonic of 1=3 0.2 ,--r-----c:;--r-,----..--,---..---,

0.1 0.03

o -0.02 -0.1

-0.07 '-----'--"'---'--"'-----'---'----'---" 0.0 20.0 40.0 60.0 80.0

-0.2 '----'---"'---'--"'-----'---'-----'-----' 20.0 40.0 60.0 80.0 0.0

TIme(M...,..)

Time(M...,..)

CR vs. Time

1=2 Wave Mode Extraction

cR = crlce

Fit to Fundamental and First Harmonic of 1=2 0.8

1.1&+00

-Data -

- -

- Equilibrium

1=2 Mode Fit

0.3

1.0e+00 -0.2

9.8e-01

,::.;::---0.7

9.4e-01 9.0e-01 '-----'--"'---'--"'-----'---'----'---" 0.0 20.0 60.0 80.0 40.0 TIme(M...,..)

-1

L---'--_L--'-_"'------'-_~____'_----'

0.0

20.0

40.0 TIme(M...,..)

60.0

80.0

622

MASSES OF SPINDLE AND CYLINDRICAL NAKED SINGULARITIES TAKESHI CHIBA and KEN-ICHI NAKAO Department of Physics, Kyoto University, Kyoto 606-01, JAPAN and TAKASHI NAKAMURA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, JAPAN ABSTRACT In relation to the possibilities of formation of naked singularities from regular initial data, we consider how these naked singularities are characterized and suggest that they are characterized by their masslessness.

1. Introd uction

Recent numerical simulations l suggest that spindle naked singularities may be formed from regular initial data implying the possible violation of cosmic censorship hypothesis 2 • It is important to investigate the physical properties of these naked singularities (for example, their masses, strength of the curvature invariants, causal structure, etc). Here, let us pay attention to the masses of these spindle naked singularities.

2. Circumstantial Evidence of Massless Naked Singularities There are some circumstantial evidence that naked singularities are massless.

2.1. Spherical Symmetry In a spherical symmetric space-time, a naked singularity is massless in general, which is proved by Lake and Hayward 3 . See ref.3 for details. 2.2. Cylindrical Dust Shell To get some insights into the nature of the spindle naked singularity, we consider cylindrical dust shell model investigated by Apostolatos and Thorne 4 . We find that with the proper rest line-mass density fixed, as the radius of the shell goes to zero (in this limit a naked singularity appears), the C-energy of the system vanishes 5 • 2.3. Critical Behaviour in Gravitational Collapse Recently, critical phenomena in gravitational collapse are found numerically 6 in which the mass M of a formed black home plays the role of the order parameter and shows the power law dependence on the arbitral initial parameter p: M ex: Ip- p*I.B. This implies the formation of a massless naked singularity at the critical value p*. 3. Gravitational Waves emitted during Prolate Collapse

623

These examples suggest formation of a massless naked singularity rather than a massive one. However, why a naked singularity can be massless? In order to answer this question, we attempt a possible explanation. Estimation of energy emitted by gravitatinal waves might be the key to the answer. 3.1. Newtonian Analysis We estimate the energy emitted by collapse of a homogeneous dust spheroid7 . Remarkably, using the quadrupole formula, we find that infinite energy is radiated during the collapse. 3.2. Perturbative General Relativistic Treatment We also attempt an approximate solution of the Einstein equation starting from a momentarily static initial data of dust spheroid7,8. Taking the geodesic slice, we seek a perturbative solution of the following equation _(3) R·· K"·· ') ')'

of the form: iij =

i~) + i~)t2

'Y"

-

I') -

+ ... ,

-

2K·· ')'

Kij = Kg)t

+ ....

We find that when a naked singularity may appear, the energy EG emitted as gravitational waves until free fall time is roughly equal to the total energy, which again suggests that almost all the total energy is converted into gravitational waves. Of cource, these are only speculation and the detailed numerical simulation is necessary which is now under investigation. 4. Weakest Cosmic Censorship?

Penrose's weak cosmic censorship hypothesis 2 may be just modified as " does there exists a cosmic censor who forbids the appearance of massive naked singularity?" . Distinction between massless and massive should be an essential part of the precise formulation of the cosmic censorship. References 1. S.L. Shapiro and S.A. Teukolsky, Phys. Rev. Lett. 66 (1991) 994; T. Nakamura and H. Sato, Prog. Theor. Phys. 67 (1982) 1396.

2. R. Penrose, Riv. Nuovo. Cim. 1 (1969) 252. 3. K. Lake, Phys. Rev. Lett. 68 (1992) 3129; S.A. Hayward, preprint, gr-qc 9408002. 4. A.T. Apostolatos and KS. Thorne, Phys. Rev. D46 (1992) 2435. 5. T. Chiba, KNakao, and T.Nakamura, unpublished (1994). 6. M.W. Choptuik, Phys. Rev. Lett. 70 (1992) 9; A.M. Abrahams and C.R. Evans, Phys. Rev. Lett. 70 (1992) 2980; C.R. Evans and J.S. Coleman, Phys. Rev. Lett. 72 (1993) 1782. 7. T. Nakamura, M. Shibata, and K Nakao, Prog. Theor. Phys. 89 (1993) 821.

624

INITIAL DATA FOR QUASI-CIRCULAR ORBITS OF BLACK-HOLE BINARIES GREGORY B. COOK Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853

ABSTRACT A method for identifying initial-data sets that represent binary black-hole systems in quasi-circular orbits is described. For the case of equal-mass nonrotating black holes, a sequence of quasi-circular orbits, including the innermost quasi-circular orbit, is explicitly located.

1.

Initial data

Initial data containing black holes can be constructed in a variety of ways. The conformal-imaging formalism represents one well-explored approach l and methods for constructing general binary configurations have been explored recently.2 The parameter space of possible three-dimensional binary-black-hole initial-data sets is enormous, and one must decide which initial-data sets are interesting and worth evolving. Because of the effects of gravitational-radiation damping, we expect most tight binary systems to have very small eccentricities. Therefore, it makes sense to search for initial-data sets that represent a pair of black holes in a quasi-circular orbit. One approach for locating stable quasi-circular orbits is to construct an appropriate effective potential and then search for minima.

2.

The effective-potential method

In general, an effective potential describing quasi-circular orbits of black holes will be a function of their orbital angular momentum J 111m, the separation of the holes lim, the spins of the holes SdM; and SdMI, the ratio of the masses of the holes Md M 2 , and the radiation content of the system. m == Ml + M2 and 11 == M l M 2 /m, respectively, define the standard total and reduced masses for a binary system. The radiation content of the system is determined by the assumptions and procedures of the conformal-imaging approach for constructing black-hole initial-data sets and cannot be varied. The effective potential requires the notion of a mass for each hole which is not a well-defined quantity for binary black holes. Since there is no unique definition to assume, we take the mass to be defined by the area of the marginally outer-trapped surface of each hole and the magnitude of its spin via the Christodoulou formula3 M2 = Mi~ + S2/4M;", where MiT == JA1161r. For the effective potential, we use the binding energy Ebll1, which is related to the ADM mass E of the system via Elm = l+Tf(Ebll1), with Tf == 111m. Note that due to our definition of the mass of a black hole, Ebl p. contains both the binding energy between the two black holes and their orbital kinetic energy but not the rotational kinetic energy of the individual black holes. In terms of these definitions, stable quasi-circular orbits can be located by searching for minima in

625

Ebl J.l as a function of lIm, keeping all other quantities constant. The method for constructing effective-potential curves from initial-data sets is described in a recent paper.4 3.

Equal-mass nonrotating black holes

The construction of effective-potential curves is, in general, a large computational task. In the case of equal-mass nonrotating black holes, the problem simplifies somewhat, and the construction of high accuracy effective-potential sequences has been accomplished. 4 Figure 1 shows this set of effective-potential curves, each at a different value of J I J.lm. Also shown is the sequence of quasi-circular orbits, plotted as a bold line. Note that this sequence terminates at small separation at an innermost stable quasi-circular orbit. Table 1 displays the physical parameters characterizing the quasi-circular orbits along this sequence. In this table, m!1 represents the orbital angular velocity as measured at infinity. Figure 1. Effective-potential sequences for equal-mass non rotating black holes. Also shown is the sequence of quasi-circular orbits.

Table L Physical parameters characterizing configurations along the sequence of stable quasi-circular orbits for a pair of equal-mass nonrotating black holes.

ljm

Eb/J.l

JIJ.lm

m!1

4.880 5.365 5.735 6.535 7.700 8.695 9.800 10.96 12.02 13.16 14.07

-0.09030 -0.08890 -0.08684 -0.08112 -0.07226 -0.06534 -0.05862 -0.05270 -0.04810 -0.04388 -0.04104

2.976 2.985 3.000 3.050 3.150 3.250 3.370 3.500 3.620 3.750 3.850

0.172 0.145 0.130 0.104 0.0774 0.0622 0.0504 0.0414 0.0352 0.0300 0.0270

11m

References 1. J. M. Bowen and J. W. York, Jr., Phys. Rev. D 21, 2047 (1980). A. D. Kulkarni, L. C. Shepley, and J. W. York, Jr., Phys. Lett. 96A, 228 (1983). G. B. Cook, Phys. Rev. D 44,2983 (1991). 2. G. B. Cook et al., Phys. Rev. D 47, 1471 (1993). 3. D. Christodoulou, Phys. Rev. Lett. 25, 1596 (1970). 4. G. B. Cook, Phys. Rev. D in press, (1994).

626 FINITE DIFFERENCING ON THE SPHERE

ROBERTO GOMEZ PHILIPP OS PAPADOPOULOS and JEFFREY WINICOUR Department of Pk1lllicil and Alltronom1l, Univerllitll of Pittllburgk Pittllburgk, PA 15!60, USA

ABSTRACT We describe a finite difference version of the eth formalism, which allows use of spherical coordinates in 3-dimensional systems with global second order accuracy. We briefly present the application of the formalism to the evolution of linear scalar waves and to the calculation of the curvature scalar of a curved geometry on a topolOgically spherical manifold.

1. Introduction

Spherical coordinates and spherical harmonics are standard analytic tools in the description of radiation. The eth formalism 1,2 and the associated spin-weighted spherical harmonics 1,3 allow a simple and unified extension of these analytic techniques to vector and tensor fields. In computational work, spherical coordinates have mainly been used in axisymmetric systems, where the polar singularities may be regularized by standard tricks. In the absence of symmetry, these techniques do not easily generalize and they would be especially prohibitive to develop for tensor fields. Here we present a finite difference version of the eth formalism, which allows use of spherical coordinates in computational relativity with global second order accuracy. 2. The Eth Formalism We introduce two stereographic coordinate patches (North and South) covering the sphere, (N = tan(8/2) eit/> and (s = 1/(N. Let (qs,ps) and (qN,PN) be the real and imaginary parts of (s and (N, respectively. Both patches extend between -1 :::; q :::; 1 and -1 :::; P :::; 1. A uniform, square, numerical grid is introduced in each patch. Functions and their derivatives are represented with the usual second order difference method. At the boundaries, the derivatives are obtained using functional values supplied by interpolation from the opposite patch. For second order accurate numerical differentiations a fourth order interpolation scheme is necessary and has been developed.

627

We next introduce a complex vector basis in each patch.In the S patch we make the choice e~ = (1 + (s(s)(6~ + i6(2)/2, so that its real and imaginary parts line up with the S axes. Similarly, in the N patch, eN = (1 + (N(N)(6~ + i62)/2. Tensor objects are now contracted with various combinations of basis vectors and reduce to spin weighted scalars 1. A spin weighted scalar on the sphere is represented by a set of grid values on the two patches and an integer spin value. A covariant (unit sphere metric) derivative of a tensor field is reduced to derivatives of scalar fields via the introduction of the eth and eth-bar operators. Their action on spin weighted scalars is given by

ov av

= qa oav + n(v = ifoav - n(v,

(1)

where n is the spin weight of the scalar. With the above prescription, a tensor equation on the sphere is reduced to scalar equations involving fields of different spin weights. All derivatives are reduced to eth and eth-bar operators which have a simple, everywhere regular, finite difference representation. 3. Applications and Tests A crucial first implementation of the scheme is the discretization of the Laplace operator on the sphere. In terms of the complex coordinate (= (q,p) we have

(2) The centered finite difference approximation of Eq. (2) is now standard since the operator is conformal to the cartesian form. At the boundary, a virtual grid point is implied and acquires a value through interpolation from the opposite patch. The discretization is confirmed to be globally second order accurate. A complete implementation of the formalism, involving repeated evaluations of the Laplace operator on numerical data, is the numerical solution of the 3-D wave equation in spherical coordinates. An algorithm for computing the solution of the wave equation in the characteristic initial value formulation is known. 4 In retarded time coordinates the wave equation takes the form D2g 2g ,ur - 9rr , + -r2 - =

o.

(3)

The main accuracy problem in Eq. (3) is now treated, since the angular momentum term is regular throughout each patch. The linearity of the problem allows a thorough accuracy check since sufficiently general exact solutions are easily identified. Using exact multipole solutions we verified second order global convergence for large harmonic values. As a tensorial illustration of the forgoing methods, we consider a problem which arises in many different contexts in general relativity: Given the metric hab of a

628

topological sphere, calculate the scalar curvature. The metric is uniquely determined by its unit sphere dyad components K = hab eae!' /2 and J = hab eae b/2. The scalar curvature corresponding to hab is given by

R = 2K - oaK + ~[a2 J + 02Jj + 4~[aJoJ - aJoJj.

(4)

The comparison of the numerical evaluation of R with exact calculations provides a first test of accuracy. A strong global test is suggested by the Gauss-Bonnet theorem for spherical topologies, namely

f RdS =

81r.

(5)

Starting from an arbitrary metric we compute the numerical curvature and then integrate over the sphere using a second order integration scheme. The integration must take into account the overlap between the coordinate patches. A simple and natural choice is to use the equator (( = 1 as the smooth and symmetric boundary of the integration within each patch. We checked the convergence of the GaussBonnet integral over a wide range of curvature radii and verified the remarkable robustness of the method. In summary, the finite difference implementation of the eth formalism we presented offers a robust and accurate method for developing numerical schemes in 3-D. The traditional reliance on cartesian coordinates can be relaxed and spherical coordinate topologies can be used whenever the nature of the problem renders them suitable. 4. Acknowledgements We are thankful for research support under NSF Grants PHY92-08349 and PHY93-18152/ASC93-18152 (ARPA supplemented) and for computer time made available by the Pittsburgh Supercomputing Center. 5. References 1. R. Penrose and W. Rindler, Spinors and Space-Time (Cambridge University Press, Cambridge, 1984). 2. E.T. Newman and R. Penrose, J. Math. Phys. 7 (1966) 863. 3. J. N.. Goldberg and A.J. Macfarlane and E.T. Newman and F. Rohrlich and E.C.G. Sudarshan J. Math. Phys. 8 (1967) 2155. 4. R. Gomez and R. Isaacson and J. Winicour, J. Compo Phys. 98 (1992) 11.

629

CRITICAL TRAPPING OF A SPHERICAL SCALAR FIELD SEAN A. HAYWARD Department of Physics, Kyoto University, Kyoto 606-01, Japan hayward@murasaki.scphys.kyoto-u.ac.jp ABSTRACT On the trapping horizon of a black hole, there is a "fIrst law" detennining the rate of change of area 4m2• For a scalar fIeld in spherical symmetry, this integrates to r = 'If r where 'If is an invariant involving the scalar fIeld andy

I

I

= I/.J2it "" 0.3989.

black hole has a fInal area 47tR2, then R - c P - Po r as R behaved measure P on the trapping horizon.

-7

If the

0 for any suitably well

A notable success in numerical relativity is the discovery of certain critical phenomena in black-hole fonnation by Choptuik 1. The idea here is to study how this might be explained in terms of the dynamics of the trapping horizon 2. Consider a minimally coupled scalar field with arbitrary potential V( 0, marginal if Vr ·Vr = trapped if Vr ·Vr < 0. A black hole may be defmed by the existence of a future trapped region bounded by a marginal hypersurface H of a certain type, referred to as a trapping horizon2. Introduce an affine parameter ~ along H, with ~ = at the centre r = where the horizon first forms. Denote the fmal radius of an asymptotically flat black hole by R=~~..,r. Such black-hole space-times can be generated from initial data on an asymptotically flat spatial hypersurface L. By varying the initial data, one can investigate the limit R --7 in the space of parameters on L. One of the remarkable observations of Choptuik was that as R--70, there is an asymptotic power-law relation R - c(P - Po)'Y for all tested choices of parameter P, where y"" 0.37 appears to be a universal constant, independent of P. This can be phrased eq,uivalently in terms of the mass- energy3 E = (l/2)r(l- Vr·Vr), since E IH=(I/2)rl H. In the limit R--70, the trapping horizon and trapped region shrink away to nothing, presumably leaving a directional conformal singularity at t, which being at infmite distance would not be a physical singUlarity. (Imagine the conformal diagrams). Intuitively one can see that as R --70, almost all information crossing H is squeezed out, leaving the asymptotic behaviour determined by behaviour at the moment S= of formation. Three relevant properties of general trapping horizons were derived previously2. (i) The signature of the horizon (or of alaS) is spatial or null, in this case if Vr .V is non-zero or zero respectively. In the spatial case, fix Sto be arc length. (ii) The second law: the area is non-decreasing,

°

°

°

°

a

-41tr2 ~ 0. a~

°

630 (iii) The fIrst law: 1C

~41t r2 = 8x cp a~

where the trapping gravity is 1C = 1I2r and the energy flux is cp =

.fii IVr· Vel> I r

in this case. The fIrst law can be rewritten as

dr

-

r

~

= 2,,21t IVr· Vel> Id~

which shows how the radius develops as detennined by the invariant I Vr·Vel> I. This may be integrated to log r = (l / .fii) log'll, where

I

2 log'll = *r- IVr · Vel> I = 41t IIVr. Vel> I d~ H

Thus there is the exact power-law relation r =

'II 1IJ'fi

where

1/.fii = 0.3989 ...

is a little above the quoted value of 'Y. One also has R = 'PIIJ'fi where 'P=lin~ __'II. One can think of 'I' as a natural measure of data on H. The question now is whether other measures P on H also give the same power-law relation as R~ O. Consider a family of spacetimes labelled by A. with trapping horizons Hi.. such that limi..~oRi..= O. Consider any family of functions Pi.. on Hi.. such that (i) there exists Pi.. = ~~.. Pi.., i.e. there is a fInite measure Pi..; (ii) there exists Po = limi..~ ~~ Pi.., i.e. the measures converge to Po = limi..~ Pi..; (iii) there exists non-zero a.=limi..~~pA.ld'll , i.e. the limiting measure is diffeomorphically related to 'II (Imagine a graph of Pi.. against'll. Then since Pi.. ('I'i..)- Pi.. (0) -I· Pi.. - Po a. -I· - Im ......o - Im ......o 'P.. 'Pi.. Thus any such measure P has a power-law relation to R as R~ 0 with the same exponent as the power-law relation that 'P has for all R. This falls short of a full explanation of the critical phenomenon, since it concerns measures on the trapping horizon only, whereas the. observed phenomenon is for data on an apparently arbitrary spatial hypersurface~. The two main ideas here are (i) that the universality of 'Y may be due to mathematical properties of limits of measures, and (ii) that the underlying dynamical reason for the critical phenomenon may be the exact relation, referred to as the fIrst law, between the radius r and the invariant Vr·Vel> on the trapping horizon.

References 1. M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). 2. S. A. Hayward, Phys. Rev. 049, 6467 (1994). 3. S. A. Hayward, Gravitational energy in spherical symmetry, gr-qc/9408002.

631

A 3D APPARENT HORIZON FINDER JOSEPH LIBSON1, JOAN MASS01,2, EDWARD SEIDELl, WAI-MO SUEN'3 1 National Center for Supercomputing Applications 605 E. Springfield A"e., Champaign, Illinois 61820 2 Departament de Fisica, Uni"ersitat de les Dies Balears, E-07071 Palma de Mallorca, Spain 3 McDonnell Center for the Space Sciences, Department of Physics Washington Uni"ersity, St. Louis, Missouri, 63130 ABSTRACT We report on an efficient method for locating the apparent horizon in numerically constructed dynamical 3D black hole spacetimes. Instead of solving the zero expansion partial differential equation, our method uses a minimization procedure. Converting the PDE problem to minimization eliminates the difficulty ofimposing suitable boundary conditions for the PDE. We demonstrate the effectiveness of this method in both 2D and 3D cases. The method is also highly parallelizable for implementation in massively parallel computers.

1.

Introduction

The apparent horizon (AH) is a crucial characteristic of a black hole spacetime. For numerically constructed black hole spacetimes, while there are efficient methods to find the AH in a wide variety of 2 dimensional situations,! there is as yet no satisfactory method for the 3D cases. The by now standard method of finding AH, at least for lower dimensional cases, is to solve the zero expansion equation:

(1) where si is the outward spatial normal vector of the AH on the constant time slice at which the AH is to be found, and K is the trace of the extrinsic curvature Kij of that slice. This equation can be easily integrated to determine the AH in spacetimes with symmetries, as the symmetries often provide boundary conditions at the edges of the computational domain (e.g., the axis and the equator). However, for a general 3D spacetime, as the AH is topologically. a 2-sphere, one immediately faces the difficulty of imposing suitable boundary conditions even for the starting of the integration, let alone the devising of an efficient method for solving this nonlinear partial differential equation (PDE). Our method for handling this problem of finding the AH surface in a general 3D spacetime is to convert the PDE problem to a minimization problem, the solution of which is well understood. This can be achieved by: 1. Express a surface F(Xi) = 0, which is topologically a 2-sphere living on a constant time slice, in terms of the symmetric trace free (STF) tensors,2 or

632 other suitable complete set of basis functions, denoted by Pn: N

F(Xi)

=L

AnPn

=0

.

(2)

n=O

As our 3D codes for evolving the black hole spacetimes are written in cartesian coordinates, STF tensors are most convenient for our purpose. For relatively smooth surfaces, as one expects the AH to be, only the first few lower rank tensors (i.e., N being finite and not too large, see below) are needed. 2. A surface is hence represented by the set of coefficients An. The expansion e at each point of the surface is a function of An. The condition for the trial surface to be the apparent horizon becomes

(3) As e2 (An) is semi-positive definite, there are efficient optimization algorithms to search the Ai-space for the surface closest to the AH among all test surfaces so parameterized.

2.

Implementation and Results

We first applied our AH finder based on this method to 2D testbeds. We used as our background spacetime data obtained from a code developed by Bernstein et al. 3 This code evolves a black hole distorted by an axisymmetric distribution of gravitational waves (Brill wave). The black holes can be highly distorted by the incoming wave. We compare the results obtained with our new AH finder with the results from the AH finders constructed using the standard PDE method. For test surfaces with 16 coefficients (with the spherical harmonics Ytm as basis functions), we find that both the new method and the old methods coincide within the given accuracy of the PDE solvers. The new method takes less than 5 minimization iterations to converge to the correct surface in typical situations where we use the previous solution as a guess on the present time slice. For the 3D test beds, we use the symetric trace free (STF) tensors as basis functions. For the Schwarzschild spacetimes constructed in our 3D cartesian codes,4 we find that out method can converge to the correct Schwarzschild horizon location within a few iterations and using only a 6 coefficient expansion. The horizon is correctly located to within a small fraction of a grid zone. Fig. 1 shows how the 3D AH finder converges. We are presently testing the AH finder on evolved Schwarzschild black holes and also on two black hole data sets in 3D cartesian coordinates. Results of this work will be reported elsewhere.

3.

Conclusion

We have developed a promising method of finding the AH in a numerically constructed spacetime by a minimization procedure. This method should be an efficient and accurate tool for use in 3D numerical relativity.

633

Figure 1: (A) Initial trial surface. (B) After 4 minimization iterations, we converge to the spherical Apparent Horizon for a Schwarzschild black hole in a 3D cartesian grid.

Acknowledgements

This research is supported by the NCSA, the Pittsburgh Supercomputing Center, and NSF grants Nos. PHY91-16682, PHY94-04788, PHY94-07882 and PHY / ASC9318152 (arpa supplemented). J.M. also acknowledges a Fellowship (P.F.P.I.) from Ministerio de Educacion y Ciencia of Spain. References 1. G. Cook and J. W. York, Phys. Rev. D, 41, 1077 (1990). 2. K.S. Thorne, Rev. Mod. Phys. 52, 299 (1980). 3. D. Bernstein et al., Phys. Rev. D, 50, 5000 (1994). 4. P. Anninos, K. Camarda, J. Masso, E. Seidel, W.-M. Suen, M. Tobias, J. Towns. "3D Numerical Relativity at NCSA", in these proceedings.

634

ADAPTATIVE MESH REFINEMENT .IN NUMERICAL RELATIVITY JOAN MAss6 l ,2, EDWARD SEIDELl, PAUL WALKERl 1 National Cente7' fo7' Supercomputer Applications, 605 East Springfield A1Ienue, Champaign,IL 61280, USA 2 Departament de FIBica, Uni1lerBitat de leB nleB BalearB, E-07071 Palma de Mallo1'Ca, Spain

ABSTRACT We discuss the use of Adaptative Mesh Refinement (AMR) techniques in dynamical black hole spacetimes. We compare results between traditional fixed grid methods and a new AMR application for the I-D Schwarzschild case.

1.

Introduction

AMR was introduced in the last decade l as a very promising technique for numerical treatment of partial differential equations involving strong dynamic ranges. The basis of AMR methods begins by defining a coarse mesh that covers the entire computational domain. Refined grids of higher resolution are added to regions of the domain where additional resolution is required. This process of adding finer and finer meshes continues to some prespecified level of accuracy. In addition, this hierarchical structure is dynamic so that the algorithms are capable of adapting themselves to arbitrary problems by automatically refining and moving meshes to resolve small scale features as they develop and evolve. The result is a tremendous savings of computer memory and a reduction in execution time over large fine grid simulations. Unfortunately, until recently, AMR has not had major impact in the field of Numerical Relativity. The work by Choptuik 2 ,3 can be considered pioneering and his recent discovery of critical phenomena in scalar field collapse3 would not have been possible without the use of AMR techniques. Black hole numerical spacetimes lead to extreme dynamic ranges in length and time, making it difficult to maintain accuracy and stability for long periods. In Figs. la-b we show the results of evolving a dynamically sliced Schwarzschild spacetime with a standard ID code developed by Bernstein, Hobill and Smarr.4 The lack of resolution to resolve accurately the growth of the peak in the radial metric function A (see Ref. 4 for details) eventually causes the code to crash. Increasing the resolution of the grid allows longer and more accurate evolutions (note the reduced error of the apparent horizon mass) but does not avoid the final fate, as the growth of the peak is unbounded (but should always be smooth). In these proceedings, we investigate the use of AMR techniques to obtain more accurate evolutions of black hole spacetimes, as they will provide the necessary resolution in the sharp regions. This does not mean that AMR is the best way to solve this unphysical problem. In this case, AMR could be considered the "Computer Science" approach to the problem. The "Relativistic" approach would use the special causal nature of the horizon to excise the region of sharp gradients; this is the idea behind the Apparent Horizon Boundary Condition techniques and recent progress is

635 300

4.0 I /1 /1 /1 / I

250 200

_ _ n=200 _._. n=400 __ . n=800

3.5 00 yields Uepoch-era -> 0 and that this value of U max is the envelope of values expected on an orbit of infinite duration for the map (7) if it is ergodic. 38 ,29 The question then became whether or not the numerical results of zero LE could be reconciled with the chaotic or weakly (non-?)chaotic behavior of the maps (4) and (7). It is now generally accepted that the traditional definition of the LE is inappropriate for Mixmaster dynamics particularly due to the arbitrariness of the choice of time in general relativity. The standard relationship between LE and chaos assumes a finite phase space volume and a natural concept of time and distance. None of these apply here. As seen in Fig. 2, the length in the ,8±-plane of successive Kasner epochs increases. Since 'H = 0 for Kasner implies (.6.,8)2 = (.6.0)2 where (.6.,8)2 = (,8+ - ,80)2 + (,8- - ,80)2 and .6.0 = 0 - 0 0, it can be seen from Fig. 2 that epoch durations steadily increase their 1.6.01 as the singularity is approached. (The

686 subscript 0 refers to values at the start of an epoch.) This increase in duration of the Kasner epochs can be shown analytically40 to be essentially exponential in Inl (which we recall is the logarithm of the scale factor). It thus appears that the LE is driven to zero by the epoch duration rather than by any lack of sensitivity to initial conditions. 38 ,39 In their computations, Hobill et af33 used a time variable T defined by23 dt = e3fl dT (for comoving proper time t) while Burd et af35 used Inl. It should be noted that Pullin,41 using T ~ In Inl within Misner-Chitre coordinates,42 obtained a positive LE. Perhaps a natural time variable for Mixmaster dynamics is MSS proper time A given by38

(11) The behavior of A mimics that of the iteration number n of the map (7) since A remains constant during the Kasner-like part of the evolution and changes by the fixed amount 7r / 4V3 at each bounce. With)' as a time variable, a positive LE is found. Although the behavior of the LE in Mixmaster is now understood, the main question of whether or not it is chaotic is till unresolved although this may be a semantic issue. Recently, Contopoulos et al43 and Cotsakis and Leach44 have claimed that the Mixmaster equations satisfy the Painleve criterion (PC)45 also satisfied by all known integrable systems. Although they could not find any integrals of the motion (other than the Hamiltonian constraint), they conjectured that Mixmaster dynamics was not only non-chaotic but was in fact integrable. (The PC examines the structure of the singularities in the equations of motion.) However, Rugh et a[l6 pointed out that the PC claims held for all values of 1t which plays the role of energy integral not just for the vacuum Einstein case 1t = O. Previous studies33 had shown apparent chaotic behavior for 1t < O. Rugh et al numerically obtained orbits of the motion in this case that were counterexamples to integrability. Most recently, Contopoulos et a[l7 and Latifi et a[l8 have shown that previous PC calculations had not included all the singularities in the Mixmaster equations. When the calculation is performed correctly, the PC is violated. This means that the integrability of Mixmaster remains open. Since the BKL map's properties are more easily understood than are those of the solution to Einstein's equations, a remaining question becomes the extent to which the BKL approximation (leading to the map) is valid. This approximation assumes the potential is approximated by the first three terms on the right hand side of (5), that the bounce is instantaneous in n, that the dominant term in the potential is precisely unity at the bounce, and that the interbounce behavior is Kasner. For these purposes, one wishes to be able to compute the map parameters from the numerically evolved trajectories. An algorithm that does this is known. 32 One also wishes to compute the trajectory for a large number of epochs. We note that the 24 epochs of Fig. 2 correspond to 0 ~ Inl ~ 108. It is convenient therefore to consider new variables which do not grow with n as n - -00. Such rescaled variables due to Bogoyavlensky49 were first used to describe Mixmaster dynamics

687

by Ma and Wainwright. 50 Hobill and Creighton are now using these variables to examine the validity of the BKL approximation. 51 We note that in any system with sensitivity to initial conditions there must be concern that numerical inaccuracies will drive the numerical evolution away from the true solution. One way to test this would be to compare results for various ODE solvers. However, there exists a mathematics literature on this very issue which is called "shadowing." Techniques exist to determine how well a numerical solution shadows a true solution. 52 So far, these methods have not been applied to Mixmaster dynamics.

4.

Inhomogeneous Cosmologies

One of the most potentially fruitful areas of numerical cosmology is the study of spatially inhomgeneous universes. Here there are three major questions: (1) What is the nature of the generic cosmological singularity? This requires evolution toward the singularity. (2) Can (classical) inhomogeneous cosmologies shed light on the origin and growth of structure in our Universe? This requires evolution away from the singularity. (3) What is the phenomenology to be expected in spatially inhomogeneous cosmologies? Of course, the spatial inhomogeneity and the time dependence of cosmology mean that Einstein's equations are PDE's rather than the ODE's of homogeneous models. Thus one might apply "standard" numerical relativity (SNR) methods developed for colliding black holes in the cosmological arena. 53 The first attempt to do this was made by Centrella and Matzner54 as part of a long program discussed most recently by Anninos et ai. 55 Their attention was restricted to polarized plane symmetric cosmologies. They employed a general coordinate system within constant-mean-curvature (tr K = canst) slicing and used SNR methods to compute a lapse and shift. In the final paper of this series, they were able to construct the Newman-Penrose parameters and the Bel- Robinson tensor to show that the nonllnearities were in the Coulomb (i.e. background) part of the metric and not (since this is a polarized system) in the gravitational wave degrees of freedom. However, there is a more straightforward approach to plane symmetric cosmologies. 56 To be specific, we shall consider the (vacuum) Gowdy cosmology on T3 x R.57 The metric has the form (note the change in sign of >. which was given incorrectly in Ref. 56)

e- A/ 2eT/ 2 ( _e- 2T dT2 + d(P) , +e-T [e P d0- 2 + 2eP Q do- d8 + (e P Q2 + e- P ) d8 2] ,

(12)

where >., P, and Q are functions of 0, T only and 0 S 0, 0-, 8 S 27r. The time variable has been chosen to measure the area in the 0--8 symmetry plane with T -+ 00 at the singularity. It has been shown analytically that regular initial conditions imposed at T = TO become singular simultaneously only at T = 00. 58 Einstein's equations are

688

P ,TT -e -2Tp,00 -e 2P (Q 'T2- e-2TQ , O2) = Q'TT -e- 2T Q,00 +2 (P'T Q'T _e- 2T P,o Q,o) =

0,

A'T- [p,;+e-2Tp,~+e2P (Q,;+e- 2T Q,m 2P A,O -2 (p,o P'T +e Q,0 Q'T)

0,

0,

o.

(13)

(14)

We note the following features of these equations which make them particularly useful as a numerical test case. (1) For Q =1= 0, there is a nonlinear coupling between the gravitational wave mode amplitudes P and Q that can generate interesting effects. (2) The wave equations (13) for P and Q do not involve the background variable A. (3) Eqs. (14) are respectively the Hamiltonian and O-momentum constraints. They may be solved by using the solution to (13) to construct A. As a numerical problem, this formulation has eliminated two primary difficulties of numerical relativity: To solve the initial value problem (IVP), choose arbitrary values of P, Q, and their first time derivatives. (Construct a consistent initial value of A.) The constraints are automatically preserved (which is never guaranteed numerically) for all T trivially. Finally, the assumed T3 spatial topology may be implemented by periodic boundary conditions. Any method for PDE's applicable to hyperbolic equations can be used to solve (13). Examples of solutions to the Gowdy equations using standard staggered leap frog methods are given by Swift 59 and Berger et al. 60 Perhaps more interesting, because it can also be applied to axially symmetric cosmologies, is a symplectic operator splitting formulation of (13).56 The principal advantage of the symplectic method is, as we shall see, that it splits off the expected singularity behavior of these models. The approach begins with the recognition that the wave equations (13) can be obtained by the variation of the Hamiltonian

H = HI

+ H2 = ~

f

dO

f

(7r~ + e- 2P 7r~) + ~e-2T (p,~ +e2PQ,~)

(15)

7rQ

where 7rp, are canonically conjugate to P, Q. The symplectic approach represents the evolution operator b y 61,62 eHA.T = eH2A.r/2eHIA.T e H2 A.r/2 + O(~T3). (16) It is easy to construct higher order representations of the evolution operator. 63 This operator splitting is advantageous when the subhamiltonians HI and H2 are each exactly solvable. The exact solutions are then used in the numerical evolution. Variation of HI yields the terms in (13) containing time derivatives which have the exact solution

P

Q

In [ae- i3T (1 (e 2i3T a (1

+ (2e 2i3T )]

+