Loop Quantum Gravity For Everyone 9811211957, 9789811211959
Loop quantum gravity is one of the main contenders to unify Einstein's general theory of relativity and quantum mec
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Table of contents :
Preface
Acknowledgments
Contents
1 Introduction
2 Gravitation
3 The quantum theory
4 Loop quantum gravity
5. Application: Black holes
6. Application: Cosmology
7. Further developments: Spin foams
8. Possible observable consequences?
9. Conclusions
Index
Citation preview
Loop Quantum Gravity for Everyone
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Other Related Titles from World Scientific
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KahFee - 11599 - Loop Quantum Gravity for Everyone.indd 1
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Loop Quantum Gravity for Everyone
Rodolfo Gambini University of the Republic, Uruguay
Jorge Pullin
Louisiana State University, USA
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26/8/19 2:59 PM
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Control Number: 2019954631 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
LOOP QUANTUM GRAVITY FOR EVERYONE Copyright © 2020 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 information storage and retrieval system now known or to be invented, without written permission from 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 978-981-121-195-9 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11599#t=suppl Desk Editor: Ng Kah Fee Typeset by Stallion Press Email: [email protected] Printed in Singapore
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Preface
This book grew out of an EdX.org course one of us taught (the course is in Spanish with English subtitles). It will introduce from scratch and without formulas all the necessary concepts of general relativity, quantum mechanics and will describe loop quantum gravity and two applications, in cosmology and black holes. Its style will be light and brief to make it as accessible as possible. Loop quantum gravity is no stranger to controversy. We will not shy away from it and outline why some people have problems with the theory. We hope you can enjoy the presentation and come out of it with an understanding of the current state of affairs. Rodolfo Gambini and Jorge Pullin Montevideo, April 2019
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Acknowledgments
We wish to thank Ivan Agull´o, Fernando Barbero, Guillermo Mena, Lee Smolin and Thomas Thiemann for comments. This work was supported in part by Grant No. NSF-PHY-1603630, NSFPHY-1903799, funds of the Hearne Institute for Theoretical Physics, CCT-LSU, The Foundational Questions Institute (fqxi.org), and Pedeciba.
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Contents
Preface
v
Acknowledgments
vii
1. Introduction
1
2. Gravitation
5
3. The quantum theory
15
4. Loop quantum gravity
29
5. Application: Black holes
41
6. Application: Cosmology
55
7. Further developments: Spin foams
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8. Possible observable consequences?
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9. Conclusions
85
Index
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Chapter 1
Introduction
Loop quantum gravity is one of the main contenders to unify Einstein’s general theory of relativity and quantum mechanics, therefore providing a quantum theory of gravity. If these words do not mean much to you, do not worry, we will define them in simple terms in the following chapters. The word we want to concentrate on here is contender. What does such a word mean in the context of science? It means that we are going to be discussing incomplete science. There is no consensus among physicists about how to quantize gravity. There are proposals, but they are incomplete, it is not clear if they are consistent or if they predict the correct physics. Two of the proposals are pursued by a majority of physicists. They are string theory and the one we will discuss in this book, loop quantum gravity. The number of people that pursue these approaches 1
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are in the hundreds. It is difficult to give a precise account since some people work intermittently or in more than one topic, but to give an idea, both approaches have a conference every three years that is attended by over 200 people. It should also be mentioned that at the moment we do not know of a single observable physical phenomenon or experiment that requires a quantum theory of gravity to explain it. So this raises the question: why bother? There are two reasons. The first one is unity and consistency of physics (see Fig. 1.1 for an illustration of the relation between different fields of physics). We know all the other fundamental interactions (electromagnetic, strong
Figure 1.1: How the various theories of physics fit together. Credit: Wikimedia Commons/Raidr.
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and weak) require quantum mechanics to describe correctly nature. The reason for this is that these interactions are important at the microscopic level and we know that at the microscopic level things are quantum in nature. Gravity is important only at the macro level, like in astronomical objects, and there quantum effects are negligible. As we shall see, general relativity does predict situations like black holes and the Big Bang where quantum effects are important, but we do not have direct experimental access to them. The second reason to talk about quantum gravity is that we do not know how to couple consistently classical and quantum theories. As we will see, quantum mechanics has counterintuitive properties. One of them is that physical quantities have no value until they are measured. It is not that one ignores their values until one measures, they simply do not exist. How can one therefore couple such a theory with a classical theory where physical quantities must have values all the time? One may think that not having any experiments or phenomena to explain would make it very easy to build a theory of quantum gravity. After all, one is not constrained by experiments that could rule out candidate theories. However, it has proved to be very difficult to do. The reason for this is that, as we will see, Einstein’s general theory of relativity
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describes gravity not in terms of a force like those of all the other interactions, but as a deformation of space-time. This makes gravity very different from the other three interactions. It is therefore not surprising that it presents unique challenges at the time of its quantization. And that our previous experience quantizing the other three interactions does not necessarily help. In this book we will present a self-contained description of the attempts made by loop quantum gravity to quantize gravity. In the next chapter we will review Einstein’s general theory of relativity. In Chapter 3 we will introduce quantum mechanics. In Chapter 4 we will present loop quantum gravity. Chapters 5 and 6 will be for applications, in cosmology and in black holes. Chapter 7 will talk about developments in spin foams. Chapter 8 will be the conclusion and include a discussion of the controversies surrounding the theory.
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Chapter 2
Gravitation
In this chapter we will talk about Einstein’s general theory of relativity. But to give some context, let us step back and talk about the other interactions in nature. The most well known fundamental interaction is electromagnetism. In the 19th century James Clerk Maxwell unified electric and magnetic phenomena in a single theory, that in addition predicted the existence of electromagnetic waves (see Figure 2.1). Examples of electromagnetic interactions are the magnetic forces that make a magnet stick to the refrigerator or the electric forces that make a plastic wrap stick to a pot. Electromagnetic waves were verified experimentally a few years after Maxwell’s prediction and a few years later Marconi was using them to transmit across the Atlantic. So electromagnetism has had a lot of practical applications. It was also recognized that light is an electromagnetic
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Figure 2.1: In electromagnetic waves the electric and magnetic fields are perpendicular to each other and to the direction of propagation. The wavelength characterizes the distance in which features of the wave repeat themselves. Credit: Wikimedia Commons/DECHAMMAKL.
wave and therefore the theory explains optical phenomena as well. It is an incredibly successful theory. The other fundamental interactions are the strong and the weak interactions. These do not have visible effects in everyday life. The strong interaction is responsible for keeping atomic nuclei together (see Figure 2.2). Nuclei are formed by protons and neutrons. The protons have positive electric charge and therefore repel each other. The nucleus is kept together by the strong interaction,
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Figure 2.2: The electromagnetic, weak and strong interactions and their roles within the atoms. Credit: Ng Kah Fee, composed of images from Wikimedia Commons/Yzmo, Inductiveload, Jacek rybak.
which counters the electromagnetic repulsion. The weak interaction is responsible for particle decay and does not have everyday manifestations but is very important at the astrophysical level, in particular in the production of the elements that compose matter. The electromagnetic, strong and weak interactions are all described by a class of theories known as Yang–Mills theories in honor of Chen Ning Yang and Robert Mills who introduced them in the
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1960’s. Yang–Mills theories are generalizations of Maxwell’s electromagnetism that have several electric and magnetic fields that interact with each other. We mention this since they will play a role in the development of loop quantum gravity. The theory of gravity we are all familiar with is the one that Newton introduced in 1666, which states that bodies attract each other with a force proportional to the masses and inversely proportional to the distance squared. The familiarity we have with it sometimes leads us to overlook the towering intellectual achievement Newton’s theory is. This law of universal gravitation unifies very disparate phenomena like the falling of an apple from a tree with the motion of the Moon around the Earth and the Earth around the Sun (see Figure 2.3). Newton’s gravity is immensely successful in explaining the motions of the Solar System, but Einstein realized that there was something wrong. It predicts that gravity propagates instantaneously. The formula for the force only depends on the distance to a source. If the distance changes, the force changes immediately. In his special theory of relativity, Einstein had noted that nothing can move faster than light. That required modifying Newton’s theory. Newton’s theory also failed to explain the Solar System completely. The orbit of the planets has the shape of an ellipse with the Sun in
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Figure 2.3: Artist depiction of the Solar System. The distances and sizes are not to scale.
one of its foci, but an ellipse that rotates (see Figure 2.4). The rotation is larger as one considers planets closer and closer to the Sun. For Mercury, the closest, the rotation is around 5000 arc seconds per century. To give perspective this is slightly over a degree per century. Newton’s theory accounted for almost all that motion, it was missing around 40 arc seconds per century. Those would be explained by a new theory of gravity that Einstein introduced in 1915 called the general theory of relativity. General relativity is, at a conceptual level, radically different from Newton’s theory. It says that gravity is not a force but a deformation of spacetime. The Moon goes in an elliptical trajectory
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Figure 2.4: The motion of the ellipse of the planets, known as “precession of perihelion,” quite exaggerated in order to make it visible. The point of the orbit closest to the Sun (perihelion) rotates around the Sun. Credit: Wikimedia Commons/Rainer Zenz.
around the Earth not because there is a force tugging on it, but because the Earth deforms spacetime around it and in the deformed space-time the most natural trajectory is not a straight line but an ellipse (see Figure 2.5). When there is no gravity the most natural trajectory is a straight line. When gravity is present and space is curved, the most natural trajectory is the curve that minimizes
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Figure 2.5: The Earth deforms space-time. As a consequence the natural trajectories near it are not straight lines anymore. That explains why the Moon goes around the Earth. Credits: Wikimedia Commons/Mysid.
distances in the curved space. An analogy is to consider a mattress. If one rolls a golf ball on it, it will go in a straight line. But if one lies on the mattress bending it, the ball will go along a curve. In general relativity the mattress is space-time itself. The deformation of space-time implies that light coming from stars that passes close to the Sun will bend (see Figure 2.6). Measuring this effect is difficult, one cannot simply point a telescope in the direction of the Sun. One needs to do it when the light of the Sun is blocked by an eclipse. If one takes a picture of the sky near the Sun when it is blocked and compares it with a picture of the sky when the Sun is not there, one will see the stars slightly moved outwards. The effect is small, about one degree for the rays of light passing closest to the Sun, but it was measured in
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Figure 2.6: Gravitational lensing setup. Due to the curvature of the space-time the Sun induces in its surroundings, the position of the star in the sky appears outwards with respect to where it really is. This cannot be usually observed due to the light of the Sun being too intense, but it can be during a solar eclipse, when the Moon interposes itself between the Sun and the Earth and obscures its light, making the stars slightly behind it visible. Credit: Space Telescope Science Institute (STScI).
1919 by a British expedition led by Sir Arthur Eddington. And the measurement disagreed with the prediction of Newton’s gravity and agreed with Einstein’s. This measurement launched Einstein into super stardom, converting him into the first celebrity-scientist ever.
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Apart from these predictions, Einstein’s general relativity also predicts the existence of gravitational waves, ripples in space-time that are produced when masses are violently accelerated. These waves travel at the speed of light and their effect is so minuscule that Einstein thought we could never detect it. Gravitational waves were measured directly for the first time in 2015 coming from a collision of black holes that happened 1,500 million light years away and led to the Nobel Prize in Physics in the year 2017. Black holes are another prediction of Einstein’s theory, they are regions of space-time where gravity is so intense that nothing, not even light can escape. See Chapter 5 for more about black holes and gravitational waves. Given how different general relativity is from the theories that describe all the other interactions (all of them treat interactions as forces whereas general relativity talks about a deformation of space-time), it is not surprising that the quantization of gravity described by the general theory of relativity is more challenging than the quantization of the other interactions.
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Chapter 3
The quantum theory
At the beginning of the 20th century physicists started studying matter at a microscopic scale. The idea of atoms goes back to the Greeks, but at that time it was just a speculation. In the 20th century the fact that matter was composed of atoms and that atoms were composed of a nucleus and electrons were experimentally verified. It was also noticed that at a microscopic scale things did not behave like objects in our everyday macroscopic life. Strange things happen. To begin with, if one has an atom composed of a positive nucleus with a negative electron orbiting around it, the electron should emit radio waves since a circular motion is an accelerated motion (the velocity changes direction as it goes around) and accelerated charges radiate. This is what happens in the interior of a radio transmitter antenna, the electrons are accelerated back and forth by an
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alternating current coming from the transmitter. This emission of radio waves should lead to loss of energy for the electron in the orbit and the atom shrinking. Eventually the electron would crash onto the nucleus. The problem is that this process would happen really fast, in about a trillionth of a second! Yet atoms are stable essentially forever, we are all made of atoms. To solve this, a new theory was created known as quantum theory. This theory has a number of counterintuitive behaviors that we do not see in everyday life, because in the macro world, where we have approximately of the order of 1023 atoms such subtle effects get washed out. The name “quantum” comes from the fact that several quantities in the theory cannot take arbitrary values but must be quanta of discrete values. In the case of the atom we were discussing the orbits cannot have any given radius but their radius must take certain predetermined values (see Figure 3.1). An electron going around in one of those radii cannot emit a tiny amount of radiation and go to a slightly smaller orbit. It has to lose and fall a fixed amount. This does happen sometimes, but since there is a finite number of possible orbits, it eventually stabilizes and does not emit radiation. This has effects in our daily life. For instance if you drop salt on an open flame, the flame will excite the electrons of the salt to a higher orbit
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Figure 3.1: Quantized atomic orbits. When an electron decays from an orbit like n = 3 to a lower energy one like n = 2, it emits a photon of energy ΔE = hν. h is Planck’s constant and ν is the frequency of the photon, which is determined by the difference in energy levels of the atom. Z is the number of protons in the nucleus and e their electric charge. Credit: Wikimedia Commons/JabberWok.
and they will eventually decay to a lower one, emitting light. Because the light has a fixed amount of energy given by the difference in energies of the two possible orbits in question, the light has a distinctive color. In the case of salt (sodium) the color is yellow (see Figure 3.2). You can try this in your home’s kitchen. Other materials yield other colors. Another unusual property of the quantum theory is that it is probabilistic, it only gives probabilities for the values of quantities, not precise predictions given the initial conditions. In Newton’s mechanics given the initial positions and velocities
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Figure 3.2: A swab containing salt placed in a flame, yielding the characteristic yellow color of sodium. Credit: Wikimedia Commons/Søren Wedell Nielsen.
of particles, the theory predicts precisely their final values. In quantum theory one has that there is a probability that it be at some point and another probability that it be at a different point. We will only know where it is if we go and measure. There is a well known experiment called the double slit experiment. Suppose one has a board like a blackboard and we cut two vertical slits in it, a bit wider than a tennis ball. Now let us take one of those machines that shoot tennis balls and aim at the board. In classical mechanics the balls will either bounce off the board or make it through the slits. If the balls were covered in chalk, one would get a mark on the wall behind the board mirroring the slits as the balls make it through
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it and stain the wall. If we now were to shrink the experiment to the micro size, things would be different. Not that we would do that, but one can construct analogous situations, for instance using crystals instead of the board with the slits and using the inter-atom spaces as slits and electrons instead of tennis balls and a photographic plate instead of the wall. We would not see two vertical lines like in the macro world but an undulating range of intensity (see Figure 3.3). What is happening is that the electrons are behaving like waves. Suppose we had a tank of water and we submerge our board with the slits in it and we have a water wave impinge on the board. Parts of the waves would make it through the slits and interact with each other forming an undulating pattern in the water (see Figure 3.4). This is similar to what the electrons do.
Figure 3.3: Double slit experiment with electrons. Credit: Wikimedia Commons/NekoNekoja.
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Figure 3.4: Interference pattern in water waves passing through two slits, A and B. As sketched by 18th century British physicist Thomas Young, who discovered the phenomenon in the context of light.
However, electrons behave as particles too. If one could do the experiment one electron at a time one would see them impinge on the photographic plate as a dot. But as more and more electrons pass, the undulating pattern will form (see Figure 3.5). The quantum theory goes further in telling us that we cannot know through which of the slits the electron went through. If we try to measure it, for instance putting a detector in front of one of the slits, the undulating pattern on the photographic plate disappears. For the experiment to work we cannot know through which slit it passed. More generally, the quantum theory says that properties of the system do not take values until they are measured. It is not that we do not know their values until they are measured, as is the case in classical
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Figure 3.5: How the electrons impacting on the screen start to form the interference pattern. Credit: Wikimedia Commons/Dr. Tonomura and Belsazar.
mechanics, they simply do not have to take values for the math of the theory to work properly. All of these counterintuitive properties of the quantum theory get worse when we try to apply it to general relativity. Since there gravity is described by space-time deformations, it means we would be quantizing the space-time we live in. It will be probabilistic, sometimes may behave as waves, sometimes as particles and it is not well defined till we measure it. It is not surprising that we have problems trying to quantize gravity. That gravity had to be quantized is something we know since 1916. In his paper on gravitational waves Einstein said that like electromagnetic
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waves, gravitational waves should be quantized. Just like electromagnetic waves are quantized into photons there should be something similar for gravity: the graviton. At that point Einstein did not do any concrete quantum calculation, in fact the quantum theory was not finalized until the late 1920’s. In the 1940’s and 50’s strong and weak interactions started to be understood. In the 1960’s people attempted to apply the same techniques they had used to quantize the weak and strong interactions to gravity, but ran into problems. Infinities arose that were impossible to remove, and in nature nothing can be infinite. So there were problems and inconsistencies at the time of quantizing gravity. This led to a broad division among physicists. One group believed that the difficulties that were arising quantizing general relativity suggested that Einstein’s theory is not the right theory to quantize, that underlying Einstein’s theory there is a more fundamental theory that yields Einstein’s theory in the macro world, but that it is different and the more fundamental theory is the one to be quantized. People who work in string theory are of this persuasion. String theory is the theory to be quantized and general relativity arises from it when we consider the macro world (see Figure 3.6). There are examples of a situation like this. In the 1950’s before the weak interactions were understood as
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Figure 3.6: In string theory, ordinary matter (1) is composed of a molecular network (2) in turn composed by atoms (3) whose constituents, electrons (4) and the quarks that form protons and neutrons (5) arise as vibrations of fundamental strings. Credit: Wikimedia Commons/MissMJ.
Yang–Mills theories, Fermi had a theory for them. Fermi’s theory also had problems when one tried to quantize it and later it was understood that it appeared as a particular regime of Yang–Mills theory and the latter is the one that needs to be
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quantized and does not have the problems that Fermi’s theory had. The other group of people, and that includes people working in loop quantum gravity, believe that Einstein’s theory is probably correct, it is just that it is so different from the other interactions that the techniques we had developed to quantize them do not work and new techniques are needed. In particular, general relativity has a symmetry that the other theories do not have. Because it is a theory of geometry itself, until one solves it, that is, until one has a given geometry, the points of spacetime are all equivalent to each other and can be moved around freely. To give an analogy, consider that you are in the middle of the ocean far from the coast in a cloudy night (Figure 3.7). You cannot tell where your boat is. All points of the ocean are equivalent. You can move your boat around and everything will look the same. In that context “placing a geometry” would be being able to see the coast. Then the points of the ocean are not equivalent anymore, some are closer to the coast than others and you would notice if you move your boat around. Technically this symmetry is known as “invariance under diffeomorphisms.” “Morphism” means map and “diffeo” refers to the fact that the map is “differentiable” (in particular continuous) to avoid
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Figure 3.7: A ship in the middle of the ocean in a cloudy night would not be able to tell one point of the ocean from another. The same happens with geometric theories of gravity: until one introduces a geometry all points of the spacetime are equivalent and can be moved into each other. This is known as diffeomorphism invariance. The image is “The Gust” by Willem van de Velde the Younger.
ripping neighboring points apart from each other. This symmetry is not present in the Yang–Mills theories that describe all other interactions. There points are fixed and well defined from the beginning. Some people believe that a quantization that incorporates the invariance under diffeomorphisms
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will be crucial to develop a theory of quantum gravity. Loop quantum gravity does incorporate the invariance. As we mentioned, gravity is a theory of the macro world, where quantum effects are not relevant. At the micro world gravity is negligible. For instance, two electrons attract each other gravitationally because they have mass and repel each other because they have electric charge. The electric repulsion is 1044 times stronger than the gravitational attraction. These are the reasons we cannot come up with experimental situations where quantum gravity effects are important. This raises the question of why bother quantizing it. The most immediate answer is that physics should have coherence and if quantum mechanics underlies physics it should also underlie gravity even if its effects are negligible. Second, as we mentioned, some of the surprising properties of quantum mechanics, like that physical quantities do not take values till one measures them, make it difficult if not impossible to couple consistently quantum and classical theories. Also, as we shall see, there are situations like black holes and the Big Bang that will require the quantization of gravity. So we have set up the stage for loop quantum gravity: some people believe the problems of quantizing general relativity require a new theory
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like string theory, others believe one needs better quantization techniques, in particular they have to be compatible with diffeomorphisms. Among this group of people are the ones that work in loop quantum gravity, which we describe in the next chapter.
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Chapter 4
Loop quantum gravity
We have explained what the last two terms mean, but what is this thing with loops? To discuss it, it is good to go back briefly to electromagnetism and one of its pioneers, Michael Faraday (Figure 4.1). Faraday had the remarkable characteristic at that time, for someone who would reshape an area of physics, of being born poor. In 19th century England that meant that you did not go to school, let along college. So Faraday educated himself. He got a job in a bookstore and cut a deal with the owner that he would allow him to stay after hours, clean the place, and would be allowed to read whatever he liked. This self training handicapped him in that he was not facile with math. So to try to understand electric fields he came up with a visual tool. Suppose one has an electric field. If one places a charge in it, it will feel a force and start moving. The field lines are imaginary lines that would
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Figure 4.1: Picture of Michael Faraday in 1861.
follow the trajectories of charges that were placed in the field. The beauty of this technique is that in situations of high symmetry one can draw the field lines without doing any calculation, just by invoking the symmetry. For instance, if one has a spherical charge, it is kind of obvious that the field lines should be radial (see Figure 4.2). Any other configuration would violate spherical symmetry. Similarly if one has a very long vertical wire the lines should emanate radially in the plane perpendicular to the wire and be equally spaced vertically (any other spacing would contradict the fact
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F e2 e1
Figure 4.2: The field of a point charge e1 . A small charge e2 would feel a force along the field line.
Figure 4.3: The field of infinite line of charge.
that points along the wire for a very long wire are all equivalent; see Figure 4.3). Another property Faraday noted was that the fields were more intense where the field lines were more tightly packed.
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Let us go back to the example of the sphere. As we move outwards the density of lines diminishes. A bit of reflection shows they do so as the inverse of the distance from the sphere squared. But that is precisely the rate at which the electric field of a sphere decays! Similarly, for the wire the density diminishes as the inverse of the distance (they only open up in the plane perpendicular to the wire, not in the vertical direction). And that is precisely how the field of a wire decays. So we are computing electric fields in detail just by drawing lines. In the 1980’s it was known that three of the four interactions, the electromagnetic, weak and strong interactions were described by theories known as Yang–Mills theories. These are generalizations of Maxwell’s electromagnetism with several electric and magnetic fields that interact with each other. At that time Gambini and Trias found a way to describe these theories in terms of their Faraday lines. What does all this have to do with gravity? In 1986 Abhay Ashtekar, at the time at Syracuse University in Syracuse, NY, found a way to write Einstein’s general theory of relativity as if it were a modified Yang–Mills theory. It turns out one can rewrite the geometry of space-time in terms of the electric and magnetic fields of those theories. The Faraday lines for these types of theory
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are a bit more complicated than those of electromagnetism. In all cases the lines can intersect with each other and each segment of them has associated a “number” or “color.” This is due to the interactions and the presence of several fields. The diagrams of these lines, which are “colored” graphs with intersections are the “loops” of loop quantum gravity (initially the importance of intersections was not appreciated and people literally thought of loops). These days they are better known as “spin networks” (see Figure 4.4). In the 1960’s the well known English physicist Roger Penrose had written a prescient paper that suggested that spin networks could be relevant for quantum gravity. Twenty five years later that prediction was confirmed with precision. In 1988 Carlo Rovelli and Lee Smolin decided to take the loops seriously and build a quantum theory of gravity entirely in terms of them. In quantum 2
3 1
1 2
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Figure 4.4: An example of a spin network.
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theory an important element is the quantum state of the system that is represented by a function called “wavefunction.” In loop quantum gravity the wavefunctions are functions of spin networks. Remembering that general relativity has this symmetry where one can move points around continuously, it means that the wavefunctions have to be such that they do not change under smooth deformations of the graphs. The branch of mathematics that studies that kind of functions is known as knot theory. A novel connection between knot theory and quantum gravity was therefore born. The word knot in this context literally refers to knots because if a graph has a knot in it one cannot eliminate it by a smooth deformation of the graph, it will always be there. Whenever a new branch of mathematics is brought to bear on a physics problem, one can expect interesting things to happen. At the beginning there were a lot of expectations, it was even thought that the problem of quantum gravity had been solved. But eventually the complexities of general relativity emerged and, although we have not solved everything, interesting progress has been made. To begin with, when one has a quantum theory one needs a notion of how close or far apart the quantum states are. Technically this is known
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as a “measure” or “inner product.” This is what allows to make physical predictions. At the beginning of the 1990’s, Abhay Ashtekar and Jerzy Lewandowski realized that they could create a distance between states in loop quantum gravity that was both precise, mathematically consistent with the symmetry under diffeomorphisms and relatively simple to illustrate graphically. The idea is illustrated in Figure 4.5. Two states that are based on graphs that can be deformed to each other are essentially identical, their “distance” is zero. Two states that cannot be deformed to each other have infinite distance between them. As the diagram indicates, the graphs that have
(a)
(b)
(c)
Figure 4.5: Quantum states based on the spin networks (a) and (c) are very close using the “distance” between states of Ashtekar and Lewandowski. In fact they are the same state, as the spin networks can be deformed to each other, even though they look very different. Both are infinitely far from (b) because the knot cannot be deformed, even though in the case of (a) the loops are very similar.
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infinite distance between them can be very close in shape, but the little knot guarantees that they cannot be deformed into each other. The explanation is simple, but the technical details (nonlinear spaces modulo gauge transformations) behind this “distance” are quite challenging. Even more interesting, a few years later Lewandowski, Okolow, Sahlmann and Thiemann and independently Fleischhack proved a theorem (known by their initials as “LOST-F”) that indicates that under some assumptions the distance of Ashtekar and Lewandowski is essentially unique. So it is interesting that math is guiding us towards the theory, it is picking the distance for us. A key assumption is the invariance under diffeomorphisms and this is interesting because in other approaches to quantum gravity this symmetry is not necessarily built in from the beginning as it is here. Based on the space of wavefunctions endowed with the distance we discussed, Rovelli and Smolin and Ashtekar and Lewandowski started building physical quantities at a quantum level (known as “quantum operators”) that correspond to classical physical quantities that one can observe. For example, the area of a surface or the volume of a region of space. In the case of the area of a surface, it is related to how many lines of the graph of the
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j
Figure 4.6: A surface acquires a quantum of area when pierced by a line of a spin network.
quantum state pierce the surface (see Figure 4.6). That means that the area of a surface cannot take an arbitrary value because the number of lines is a discrete number. The area is “quantized.” For surfaces of our everyday life, like if we measure the area of a table, the number of lines that cross them is enormous (about 1066 per square inch), so it appears the area can take any value. But if one were to consider a microscopic table, its area could only take certain values. In fact, one could be in a situation that a table has zero area if no line of the quantum state pierces it. How could a table have zero area? That would be a “very quantum” situation, quite different from what we experience in our everyday life. Similar considerations apply to the volume of a region. There it is proportional to how many intersections does the graph of the quantum state have within the region in question (see Figure 4.7).
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Figure 4.7: The elementary bricks that construct a spacetime based on spin networks. The vertices generate quanta of volume.
It had been speculated for some time that areas should be quantized in quantum gravity but here the quantization is not speculated, it emerges precisely from the framework and with a specific formula that differs from the ones people had conjectured in the past. In particular it implies that the weird things that happen in the micro world disappear rather quickly as one considers bigger situations and our everyday world rapidly emerges. The image of quantum space-time that arises from spin networks is constituted by bricks of a given volume determined by the number and properties of intersections bounded by areas. Loop quantum gravity will assign probabilities to the different spin networks.
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Finally, in 1996, Thomas Thiemann was able to write a quantum version of the Einstein equations in a precise way using those quantum operators we discussed. He showed that the equations were mathematically well defined, they were consistent with each other and no infinities arise. He had constructed a well defined, non-trivial, consistent theory of quantum gravity. So you might ask, are we done? The only remaining step is to check that the theory predicts the correct physics. The problem is that the theory that resulted is quite hard to study. This is reasonable; already classical general relativity is quite hard to study. There has been little progress made with this theory in general, so we do not know if it is correct. So what people have tried to do is to concentrate on simple situations with a lot of symmetry where the calculations can be brought under better control. In Chapters 5 and 6 we are going to study two applications of Thiemann’s theory to situations with a lot of symmetry. One will be black holes and their thermodynamics. The other will be cosmology, the study of the universe as a whole.
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Chapter 5
Application: Black holes
We will introduce black holes and discuss their thermodynamics. This will require defining both terms. Thermodynamics is the branch of physics that studies systems with incomplete information. A prototypical example of a thermodynamical system is a gas. Typically it is constituted by approximately 1023 molecules. Obviously we cannot keep track of all of their positions and velocities. So we resort to “macro” variables to characterize its state, like the pressure and temperature. Apart from these well known macro variables, there is another one that is less well known in everyday life that is called the entropy. It gives a measure of our level of ignorance about the system. It has the property that when systems interact the total entropy always grows. It may go down in some subsystem, but overall it grows. An example of a subsystem that lowers its entropy is the interior of a
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refrigerator, where lower temperature equates more order and less ignorance. But in the back of the refrigerator there is a radiator that gets hot, and therefore the entropy of the whole kitchen, including the refrigerator, grows. The law that states that the entropy always increases is known as the second law of thermodynamics. The first law of thermodynamics is just the conservation of energy, and it has a formula that relates the changes of energy with changes in entropy, temperature, volume and pressure. We will not need it in detail, but it is good to know it exists. So that was thermodynamics, what about black holes? Black holes are regions of space-time where gravity is so intense that nothing, not even light can get out (see Figure 5.1). If one throws an object up in the Earth it will reach a certain height and come down. If one were to throw it at about 25,000 miles an hour, it would not come down; it would escape into space. That is how spaceships escape Earth’s gravity. If the Earth were denser, one would require a higher speed in order to escape its gravity. What if it were so dense that the “escape speed” is equal or higher than that of light? Then, since Einstein’s theory of relativity states that nothing can go faster than light, nothing could escape. The concept of black object is ancient, Bishop Mitchell in England and Pierre Simon de Laplace in France were
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Figure 5.1: A depiction of how a black hole placed in front of the Large Magellanic Cloud would look like. The streaks above and below the hole are highly distorted images of the Magellanic Cloud. Credit: Wikimedia Commons/Alain r.
discussing them in the 17th and 18th centuries. What they were missing was the concept that the speed of light is the maximum speed, so they were not aware that nothing could escape that region. Black holes are limited by a surface called the event horizon. Any object that ventures beyond the event horizon becomes trapped inside the black hole and cannot get out. Black holes are predicted by Einstein’s general relativity. In fact the solution that describes the simplest of them (the ones that do not rotate) was found already in 1916. But it was not properly understood until the 1960’s. The concept of black hole eluded some of the best minds of 20th century physics. Today they are well
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understood and accepted as part of the astronomical landscape. How can we observe black holes if they are black? Due to effects they create in their surroundings. Many black holes get matter dumped on them by a nearby star. The matter settles quickly via collisions into a Saturn-like disk around the black hole called accretion disk. The intensity of gravity is so great there that matter gets heated up and emits light. That light, since it is produced outside of the event horizon, can make it to us. This was captured in a rather dramatic fashion recently by the Event Horizon Telescope (see Figure 5.2). Moreover, with the LIGO gravitational wave detectors we can now detect directly gravitational waves that come from
Figure 5.2: The first image of a black hole obtained by the Event Horizon Telescope — the galaxy M87. Credit: Event Horizon Telescope Collaboration.
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Figure 5.3: Artist conception of the gravitational waves produced by the collision of two black holes. Credit: R. Hurt/Caltech-JPL.
collisions of black holes (see Figure 5.3). This has opened a new “golden era” of experimental verification of general relativity with unprecedented precision. So black holes are now accepted as regular astronomical objects, but they have this property that if something falls in them, it can never go out. What does thermodynamics have to do with black holes? At first it appears not much. A black hole being black, it does not have associated with it anything like a temperature. Contrary to a gas, it is a very simple object. Black holes are characterized by their mass and angular momentum (and electric charge if they are not neutral). They have no other distinguishing characteristic. This is very different from, say, a star, of which there are many different kinds (see Figure 5.4). Yet we believe that
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Figure 5.4: The Hertzprung–Russel diagram of stars catalogs them according to their luminosity in the vertical axis and the color of its light in the horizontal one. Credit: European Southern Observatory (ESO).
black holes are formed when a star collapses under its own gravity. Stars are essentially balls of fluid that tend to contract due to their own gravity. As they do so they heat up and nuclear reactions take place in
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them. These reactions generate pressure and that halts the collapse and stabilizes the star. It also produces the light that the star emits. That light takes away energy and the star consumes its nuclear fuel. When it is exhausted, nothing stops the collapse anymore, and the star shrinks. Depending on the mass that shrinkage may stop when electrons and protons fuse into neutrons that repel each other via a mechanism known as the Pauli exclusion principle, and a neutron star is formed. A neutron star is a star of one or two solar masses and the size of a city. So they are very dense objects. But if the mass of the initial star is higher, the collapse continues and a black hole forms. This led Jacob Bekenstein to theorize that there is a sense in which there is ignorance in a black hole. It is the ignorance about the star that formed it. Irrespective of what kind of star formed the black hole, the final object only has mass and angular momentum, we have lost information about the many characteristics of the progenitor star. Our level of ignorance is bigger the bigger the black hole. This led Bekenstein to theorize that the area of the event horizon played the role of an entropy, since it characterized the level of ignorance about the system. If you couple that with the fact that when black holes interact the area of the event horizon always grows (the black hole swallows stuff and
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becomes bigger, it cannot let something out), then we have a second law for the black hole entropy. He even wrote something that looked like a first law of thermodynamics for black holes, relating the changes of energy (in relativity energy and mass are interchangeable since E = mc2 ) with the changes of area and some other variables. But people were skeptical of this analogy since there was no concept of temperature for a black hole. This changed in 1974. Stephen Hawking studied the effect of a quantum field living on the geometry of a black hole. He did not quantize gravity nor the black hole, in his calculation the black hole just sits there providing a background. The quantum field did not “backreact” on it. He noted that the quantum field emitted thermal radiation (as if it were hot) with a given temperature. And the temperature was given by the gravity at the surface of the horizon, which was precisely the quantity that appeared in Bekenstein’s first law where the temperature was supposed to be. So the pieces of the puzzle fit perfectly! So summarizing, a classical black hole is not a thermal system, but when one turns on quantum mechanics, even only for quantum fields living outside the black hole, it becomes thermal with a given temperature. So the entropy is related to the area and to the ignorance of what is inside the black hole and is
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quantum in nature. In that case a theory of quantum gravity may shed light on the connection of the entropy and the area of the horizon. We have already argued how areas work in loop quantum gravity, so let’s see if it works. A surface acquires an area when it gets pierced by the lines of the spin network of the quantum state corresponding to a given space-time the surface is in. A surface of a given area could stem from many different quantum states in which the spin networks pierce the surface in different ways but at the end of the day give the same area (see Figure 5.5). When we talk about a horizon of a given area we do not know which quantum state is giving rise to it. That is the ignorance that quantum gravity claims we have about a classical event horizon. And a detailed calculation in
Figure 5.5: A given area of the horizon can be realized by different spin networks piercing through it. Our ignorance of which one it is, is represented by the entropy of the black hole.
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loop quantum gravity shows that it is proportional to the area. This was first suggested by Krasnov and Rovelli and later refined by Ashtekar, Baez, Corichi and Krasnov. So this gives a fundamental explanation to why the entropy of a black hole is proportional to its area. There is a wrinkle in the calculation in loop quantum gravity, which is that the theory has a free parameter known as the Immirzi parameter and it appears in the calculation of the entropy. So the result only gives a proportionality to the area, not an exact coefficient. But it is satisfying that the calculation has been repeated for black holes of different kinds and the factor that appears is always the same, so at least there is consistency. You might ask, why can they do this calculation and not others? Because this calculation only depends on how the lines of the spin network pierce the surface; it does not require solving for the full theory outside or inside the black hole. That makes things much more tractable. Apart from the entropy of black holes, loop quantum gravity is beginning to give some other answers concerning black holes. As we mentioned, black holes are formed when stars collapse. After the formation, the matter of the star keeps on contracting inside the black hole until it gets all concentrated in a point with infinite density
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(technically called “singularity”). Of course, nothing can be infinite in nature. The expectation is that when densities become large, quantum effects become important and Einstein’s classical general relativity is no longer valid. Unfortunately, we are not there yet in loop quantum gravity in a position to study a quantum gravitational collapse. But if we assume that the black hole is static and has been there forever and is not rotating, there is enough symmetry to solve the problem. The authors of this book did it in 2013. We find that the singularity is eliminated and replaced by a region of large curvature through which one can pass into another region of space-time into the future. Again, this calculation is very limited since we are not studying an actual collapse of a star. But it is a calculation that shows that what was infinite in Einstein’s theory can be replaced by something finite in loop quantum gravity. It is the beginning of our understanding of the situation, but not a full calculation. The big calculation that neither loop quantum gravity nor string theory can do up to now has to do with the following: we saw that Hawking’s partial calculation (because it does not backreact) says that a black hole radiates thermal radiation. The temperature he gets is inversely proportional to the mass of the black hole. That temperature implies the black hole emits radiation that takes energy
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away from the black hole. Therefore it should shrink. As it shrinks, it has less mass and therefore the temperature rises (inversely proportional to the mass!), and the black hole radiates even more. This leads to a runaway process in which the black hole eventually “evaporates.” For astrophysical black holes this process is incredibly slow. It would take many times the age of the universe for an astrophysical black hole to evaporate, so it is really not very important. But it nevertheless raises this important conceptual question: what happened to all the information about how the black hole was formed and that was shrouded behind the event horizon? One of the fundamental properties of quantum mechanics is known as “unitarity,” and it means that the theory preserves information. If at the end of the day we are just left with thermal radiation characterized by a single number (the temperature), where did all the information that went into the black hole go? This is known as the information loss paradox, and in our opinion it is perhaps the central problem of fundamental theoretical physics, since it challenges general relativity, quantum mechanics and thermodynamics in their most extreme regimes. Notice that the above process is speculative: Hawking’s calculation does not include “backreaction,” that is, how the emitted radiation affects the
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black hole. So in particular, in that calculation, the black hole does not shrink. We do not know yet how to do a calculation that would account for the shrinking. That is what everyone is after. The previous calculation we mentioned we did in loop quantum gravity for an eternal black hole that connects into a region in the future suggests a possible way out for the information to escape, but it lacks too many elements (there is no Hawking radiation) to be considered an explanation.
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Chapter 6
Application: Cosmology
This is the second application we wish to discuss, and it is called cosmology or the study of the universe as a whole. You may ask: how can they study the whole universe? The answer is: very coarsely. One ignores most details of it, and concentrates on a few of them. In the simplest example, the only degree of freedom considered is the scale of the universe. This is actually not a bad approximation: if one looks at things at a large scale, the universe is pretty homogeneous and isotropic (it looks the same in all places and in all directions; see Figure 6.1). That is not the case on a small scale: obviously the Solar System is different close to the Sun than close to Pluto, for example. But if one goes at the scales of galaxies or, even further, clusters of galaxies, things look pretty uniform. We know that the universe expands. We learned this by noting that the farther away galaxies are
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Figure 6.1: The deep field view of the universe by the Hubble Space Telescope. Credit: NASA, ESA, H. Teplitz and M. Rafelski (IPAC/Caltech), A. Koekemoer (STScI), R. Windhorst (Arizona State University), and Z. Levay (STScI).
from us, the faster they move away from us. An analogy here would be a raisin bread in the oven. As it bakes it expands and the raisins move away from each other. The ones that are further away move faster apart. Einstein’s theory predicts that the universe expands, he noted it already in 1917. Except that at that time it was thought the universe was
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static. Einstein did what any good theorist does: he tampered with the equations to make it static. This turned out to be a terrible mistake; the universe he got was unstable. And twelve years later, Edwin Hubble, an experimental astronomer, would measure that the universe was expanding. Einstein had missed an incredible opportunity to make a striking prediction that would have been confirmed. Some say he called this “the biggest blunder of my life,” although it is not confirmed he actually said that. Since the universe expands towards the future, if we run it backwards in time, it contracts. Einstein’s theory tells us that eventually it shrinks to a point where everything is concentrated, pretty much as we encountered in the interior of black holes, and the density becomes infinite; there is again what is technically known as a “singularity.” This is known as the “Big Bang” that gave rise to the universe. As we stated, nothing can be infinite in nature, so the expectation is that close to the Big Bang quantum effects will kick in and alter things. People have studied this in loop quantum gravity. Of course doing the full calculation would entail solving Thiemann’s equation for a quantum state that is homogeneous and isotropic. We do not quite know how to do this, although recently some progress has been made. What people do is work
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within an approximation. First we eliminate all degrees of freedom except the scale. The resulting system is very simple, but you cannot apply loop quantum gravity to it, because due to the simplification there are not Faraday lines anymore. But what one can do is to introduce an analogue of the Ashtekar–Lewandowski distance between states. That gives rise to what is known as loop quantum cosmology. Loop quantum cosmology was first discussed by Martin Bojowald and later refined by Ashtekar, Pawlowski and Singh and many other authors. It is supposed to be an approximation to the full theory. How good an approximation it is we do not know. To know how good an approximation is one needs to go beyond the approximation and show that indeed it works, but we cannot do that yet in this case. What this approximation tells us is that when the universe is big, everything is very well described by Einstein’s theory. But when the universe becomes very small, instead of concentrating in one point with infinite density, the universe reaches a minimum size and running it back in time it starts to re-expand in the past into a large pre-Big-Bang universe. That is, our universe started large, went through a period of contraction, reached a very high density, very quantum, and then re-expanded again into the universe we
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Figure 6.2: In the Big Bounce scenario the universe contracts into a singularity before expanding to its current size. Credit: Ng Kah Fee, modified from Wikimedia Commons image by Fredrik.
have today. So the Big Bang, which was that initial moment, has been replaced by a “Big Bounce” (see Figure 6.2). This is something people had expected would happen, but when they tried to do it using traditional techniques in the 1960’s it failed to materialize. The Ashtekar–Lewandowski distance between states is the key new ingredient that leads to this. In fact, other models of the universe have been analyzed, with anisotropies and other properties, and the “bounce” appears as a robust feature. So this is nice because it is something that exhibits some level of robustness and had not been achieved before.
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But you may ask: are we ever going to be able to probe if there was a bounce instead of a bang? Does not all that happen so early there is no chance we can probe it? Those moments when the density is so high, protons and electrons merge into a primordial soup that light cannot go through. It is only later when things expand and cool off that atoms form and the universe becomes transparent to light. Therefore, if we look at the sky we cannot see anything prior to the period when atoms form, because light cannot make it through. The light that managed to escape when the atoms form can reach us, but it “cools down” as the universe expands. For light “cooling down” means the wavelength gets longer. The fact that something expanding cools down is well known: if one takes an aerosol can and lets the gas out, one will feel the can cooling as the gas expands into the atmosphere. By the time it reaches us, that light has wavelengths so long that it is not light anymore; it is a microwave. It is what is known as the “cosmic microwave background,” and it is the first light that can reach us after the Big Bang. In the 1960’s two astrophysicists in the Bell Telephone Company were dealing with “noise” in a microwave antenna. Eventually they realized it was not noise, but the antenna was picking up this light from the Big Bang that had been lowered
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in frequency due to the expansion of the universe, until it becomes a microwave. The two Bell astrophysicists, Arno Penzias and Robert Wilson, would go on to win the 1978 Nobel Prize for their discovery. That background of microwave radiation has been measured by now with great precision using satellites to avoid the interference of the atmosphere. The cosmic microwave background has helped us confirm that at the largest scale the universe is homogeneous and isotropic. If one looks in one direction and then in another direction, the “temperatures” (wavelengths) of the microwaves agree to one part in one hundred thousand. Figure 6.3 shows the “celestial sphere” (the sky mapped into
Figure 6.3: The colors illustrate the differences in temperature in the cosmic microwave background in different directions of the celestial sphere (amplified 100,000 times).
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an oval). What they have done is multiply the variations in temperature by one hundred thousand and color coded them. The different colors represent the different values of the temperature (again multiplied times one hundred thousand). Looking at the picture you might think “this looks random, right?” There does not seem to be much structure. But actually there is. In order to see the structure we need to do a bit of math, fortunately not much. Take a point in the figure, then take another point at an angular distance from it, say, 30 degrees removed. Remember the oval represents the celestial sphere so moving in it is done in angles. Now take all the points removed the same angular distance from the first one and average the temperature over all of them. If the diagram were truly random and one plotted this quantity for all possible angles one would get always the same value: a constant.a But that is not what one gets. Figure 6.4 shows the result and it has a lot of structure. This structure implies that the field is not random, it has what are called “correlations,” and they have a specific form. The interesting thing is that we can predict this structure. Before discussing how, we must talk a
Technically since in the diagram they subtract the average the constant would be zero.
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Figure 6.4: The structure of the angular dependence of the cosmic microwave background. From P. Ade et al. (Planck Collaboration), Astron. Astrophys. 571, A15 (2014). The multipole moments are another way of characterizing angular separations.
about a small problem in the expanding cosmological model predicted by Einstein’s theory (without a bounce). If one looks into the distance towards a point in the celestial sphere, and then looks into the distance towards another point in the celestial sphere, one can encounter points that are so far apart from each other that there was not enough time in the whole history of the universe for light (or anything else) to go from one to the other. So the question is: how could the cosmic microwave background for those two points have almost exactly the same temperature if there is no
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way for them to communicate with each other? This requires an explanation and that explanation is a process known as cosmic inflation (just to clarify: it has nothing to do with the inflation of money). Inflation says that for a period the universe expanded in an accelerated form, in such a way that those points that are today so far apart indeed had a chance to communicate with each other in the past (see Figure 6.5). One carries out a calculation of the same type that Hawking did for black holes, putting a quantum field to live on top of a universe that
Figure 6.5: A visual depiction of the history of the universe, starting with the Big Bang, the inflationary period and the later cooling of the universe, which first creates protons and eventually hydrogen. When the latter forms the universe stops scattering light and becomes transparent. There is where the cosmic microwave background we observe today was formed. Credit: Wikimedia Commons/Drbogdan, Yinwichen.
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inflates and expands, and one assumes that at the beginning the quantum field is in its most basic state called the “vacuum.” That state is not “empty,” as there are always quantum fluctuations in a quantum theory. If one evolves the state through the inflationary period, one finds out that the resulting state is not the vacuum anymore and that it has the type of “correlations” that one observes in the cosmic microwave background. So the quantum field leaves an imprint on the latter that then cools off throughout the rest of the expansion and reaches us in the form we observe it today. If we plot out the correlation by computing the quantity we described above as a function of the angle between two points, one obtains a curve as shown in Figure 6.4. The green curve is the theoretical prediction with the best fit for the free parameters and the red dots are the measured points of the cosmic microwave background. These measurements are done today with satellites. As you can see the agreement is astonishing. And one needs to take into account that the model we described that explains this is incredibly simple: a quantum field starting in the simplest possible state, evolving through inflation and the subsequent expansion of the universe. It is difficult to think of something simpler and with fewer assumptions. As we see in
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the picture, when one considers large angles, that is large separations in the points considered, the experimental data have large error bars. What does loop quantum gravity say about all this? People have repeated the calculation mentioned above but with the bouncing cosmological model of loop quantum cosmology supplemented by a period of inflation, in particular Agull´o, Ashtekar and Nelson and other authors like Barrau, Mena and Olmedo. Now, in a universe like this there is no reason to place the quantum field in a vacuum initial state at the beginning of inflation, since there is all the dynamics before it. The natural thing is to put it in the vacuum state when the universe started in the distant past, before collapsing to the bounce and starting to re-expand again. But then the quantum field is not in a vacuum anymore at the beginning of inflation. There are small and subtle changes. Those changes leave an imprint in the cosmic microwave background at large angular separations. There the prediction of loop quantum cosmology differs from that of traditional inflation. Unfortunately, that is the region where the experimental data have large error bars, so at the moment we cannot say which agrees better with the data: loop quantum cosmology or traditional inflation. But it opens the hope that as the data of the satellites get better and
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better in a not-too-distant future we would be in a position to see which theory fits better the data. As in the case of black hole entropy, there is a free parameter in this calculation. It is difficult to do a calculation for the whole universe without making assumptions about initial conditions. In this case it is the value of the “inflaton,” the field that gives rise to the inflationary behavior at the beginning of the universe. Tweaking the value one can make loop quantum gravity almost agree with the results of traditional inflation, but it requires artificially large initial values. So if the experimental data were to agree with traditional inflation, loop quantum cosmology can be tweaked to agree with it too, but in a rather unnatural way. On the other hand, if the experimental results were to agree better with loop quantum cosmology, more complicated inflationary models can be constructed to also match things, but again it will be a less natural choice. This is the closest we are to having a “prediction” of loop quantum gravity that could be measured. Again we have to remark how tantalizing is that with such a simple model a complicated structure of the correlations can be fit so well!
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Chapter 7
Further developments: Spin foams
We have discussed some of the difficulties encountered in handling Thiemann’s quantum version of the Einstein equations. This has led to the development of an alternative approach to the dynamics of the theory, known as spin foams. Many of the results in this area are highly technical and cannot be well covered in a book for the general public, here we will just give a general idea of what is being attempted. There is a technical book on the subject by Rovelli and Vidotto.a Spin foams are based on an alternative approach to the construction of quantum theories known as path integral approach. This was pioneered by Feynman and it has found wide applicability in many quantum settings, including particle physics and condensed matter physics. The approach consists a
Carlo Rovelli and Francesca Vidotto, Covariant Loop Quantum Gravity (Cambridge University Press, 2014).
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Figure 7.1: In the path integral approach to quantum mechanics all possible paths between an initial state (A) and a final one (B) are considered, including those not allowed by classical physics and probabilities are assigned to each of them. Credit: Wikimedia Commons/Matt McIrvin.
in studying all possible trajectories of a system between a given initial and final state (see Figure 7.1). This includes trajectories not allowed by the classical equations of motion of the system. One then assigns probabilities to the trajectories using a specific recipe involving a function of the variables of the system known as the action. With these probabilities one can assign “expectation values” to physical quantities, that is, predictions for the values the quantities would have if measured. By focusing on the space-time trajectories of the system, the approach does not make a distinction between space and time, something desirable in
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the context of theories like general relativity, where space and time are on the same footing. When applying this approach to loop quantum gravity the initial and final states are given by spin networks, and the sweeping trajectory connecting them looks like a “foam,” as if one had made the spin networks out of wire and submerged them in soapy water. That gives rise to the “spin foam” name. As in the case of Thiemann, capturing the full dynamics of general relativity is also challenging in the case of spin foams. As the spin networks sweep forward in time, they can develop new vertices and links, giving new richness to the foam. These processes would capture the dynamics of the theory. There are several proposals for how these processes occur, and they are technically known as “vertices.” Some of the original proposals were shown to fail to capture the dynamics of general relativity. This has led to more sophisticated proposals but it is still unclear if the full dynamics of Einstein’s theory is really captured. An interesting aspect is that this approach to quantum gravity has proved fruitful for gravity in three space-time dimensions. Einstein’s gravity in three space-time dimensions is a much simpler theory than the four-dimensional version. All space-times are flat except at a finite number of points. This makes the dynamics of the theory
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much simpler. In fact, the theory had been successfully quantized in 1968 by Giorgio Ponzano and Tullio Regge. Remarkably, when people applied spin foam techniques to three-dimensional spacetimes they noted that the resulting quantization coincided with that of Regge and Ponzano! Spin foams are a particular example of a more general class of theories known as tensor networks that are encountering important applications in condensed matter physics and in string theory. Some conjecture that they could constitute a bridge between loop quantum gravity and string theory. Another potential connection is that Daniele Oriti and others have been constructing special quantum field theories like the ones that describe elementary particles, known as group field theories. When studying the dynamics of these theories using a technique known as Feynman diagrams (a sort of graphical way to represent the interactions of the theory; see Figure 7.2), the resulting diagrams are actually spin foams! There exists precedent for this type of constructions in the context of twodimensional space-times, there they are known as matrix models and they were the subject of intense activity by string theorists in the 1990’s. Path integral approaches are ideal for studying scattering processes like those in particle physics where one has well defined initial and final states
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Figure 7.2: A positron (e+ ) and electron (e− ) annihilate each other emitting a photon (γ) which in turn emits a quark/antiquark pair and one of the quarks emits a gluon (particle of the strong interaction). Credit: Wikimedia Commons/Joel Holdsworth.
and a complicated dynamics in between. Rovelli and collaborators have been spearheading the use of spin foam calculations in this spirit to try to understand the final fate of black holes. One starts the calculation with a black hole and what they are finding is that the final state the spin foams lead to is a “white hole,” a sort of time-reversed black hole where stuff instead of getting trapped in it actually emerges (popularly called “fireworks”). This could potentially offer a solution to the information loss paradox we mentioned before. The calculations are at the moment only preliminary and are actively being worked upon.
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Chapter 8
Possible observable consequences?
As we have mentioned, we do not know of any experiment that we can perform that requires quantum gravity to explain it, although we noted in Chapter 6 that some observational imprint could be left on the cosmic microwave background. The fundamental constants involved in quantum gravity are Newton’s constant G, the Planck constant of quantum mechanics and the speed of light c. Out of the three of them one can construct a length, called the Planck length that is 10−33 cm. That is 20 orders of magnitude smaller than the radius of a proton! That is the kind of scale where one expects quantum gravity to be relevant. Compared to it, macroscopic scales are enormous. The kind of granularity of space-time that is implied by the quantization of areas and volumes that we mentioned in loop quantum gravity has such a scale. That is why areas and volumes of macroscopic objects appear as smooth. 75
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Arrival time of gamma ray bursts
When light travels through granular media, different colors of the light travel at different speeds. Therefore if one considers a pulse of white light, which is the superposition of many colors, the pulse will disperse, as the different colors travel at different speeds. This is known as dispersion (see Figure 8.1). So in principle, light traveling through a quantum space-time could feel a phenomenon like this. The magnitude of the effect is of the order of the Planck length divided by the wavelength of
Figure 8.1: White light is dispersed when passing through a prism. Credit: Wikimedia Commons/Spigget.
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the light. For visible light, their wavelengths are 10−5 cm. Compared to the Planck length this is huge, so the effect is incredibly tiny. The effect is, however, cumulative, the longer the light travels the bigger it is. In the 1960’s, satellites set up by the US to monitor the Nuclear Test Ban Treaty started picking up flashes of gamma radiation. It was quickly ruled out that they originated in nuclear tests. Further analysis showed that they actually were not terrestrial in origin, they came from outer space. Eventually it was determined that their origin was extragalactic. They were named gamma ray bursts. Finally, in 2017 the LIGO experiment picked up gravitational waves associated with one such burst and determined that it originated in the collision of two neutron stars. Not all gamma ray bursts originate in collisions of neutron stars, there exist other models for their origin. The reason we bring up gamma ray bursts in this context is twofold. On the one hand, they happen really far away. On the other hand, the wavelength of gamma rays is considerably shorter than that of visible light, around 10−9 cm. These two factors help make the effects of granularity of space-time more likely to be detectable. Somewhat detailed models of the propagation of gamma rays have
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been worked out both in string theory, by Giovanni Amelino-Camelia and collaborators, and in loop quantum gravity by the authors of this book and others. In loop quantum gravity it was observed that for the effect to be in the range that was possible to observe with gamma ray satellites, one had to assume a rather artificial type of quantum state which is left/right asymmetric. The Compton Gamma Ray Observatory is a satellite and one of the experiments onboard can measure gamma rays of different frequencies. If the time of arrival of the bursts in different frequencies were different, that could be viewed as evidence of the dispersion. Unfortunately, up to now no differences have been observed. Worse, the rather unnatural symmetry of the quantum state needed in loop quantum gravity to have an observable effect would have also implications for the arrival of radio waves from the universe. Such waves are studied with great precision and the effect that loop quantum gravity with the unnatural quantum state is ruled out. Some people have inferred that this rules out the theory, but that is not true, it just rules out the quantum state that had been considered. As we mentioned, for more reasonable quantum states the effects are very small, but it could be that in future experiments there is a chance for an observation.
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Measuring times and distances
In 1958, Nobel prizewinner Eugene Wigner, together with H. Salecker, asked the question: what is the most accuracy one can get in a clock? To address this they considered a highly idealized clock consisting of two mirrors between which light bounced back and forth. Every time light hit a mirror it would be a “tick” of the clock. To try to keep things as undisturbed as possible they assumed the clock was isolated from the rest of the world. Even such a clock, however, would have inaccuracies. In quantum theory, objects spread without subjecting to any interaction, they become fuzzy. That means the mirrors too. Ordinary everyday mirrors do not spread because they are not isolated: they are constantly bombarded by air molecules. As the mirrors they considered spread, the “tick” of the clock becomes less accurate. They determined the inaccuracy of the clock and it is inversely proportional to its mass. So if one wishes a more accurate clock, one needs to make it more massive. But then came North Carolina physicists Jack Ng and Hank van Dam and they argued that one cannot make clocks as massive as one wished, because they would turn into a black hole. This creates a sweet spot in terms of how accurate a clock
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can be. Because the effect is due to quantum theory and gravity, one again hast the three basic constants we mentioned in the previous section. Out of them, in addition to forming a length, one can form a time called the Planck time, it is 10−44 seconds. This is incredibly short. For reference, the best projected atomic clocks have inaccuracies of 10−18 seconds. What Ng and van Dam observed is that the inaccuracy they found indeed was proportional to the Planck time, but the proportionality factor was rather large, about 1015 . That is not enough to put it in the range of atomic clocks. But measuring times is tantamount to measuring distances. If one translates the time inaccuracy into a distance inaccuracy simply multiplying by the speed of light, one gets a value that is in the range of those that interferometers like LIGO measure in their quest to detect gravitational waves. The story is, however, more complicated. LIGO involves high powers of light, which means many photons. The effect of the inaccuracy gets averaged over all of them and gets washed out. So one cannot use LIGO to see the inaccuracy. The fact that nevertheless one is in the ballpark perhaps may suggest other experiments to test these fundamental limitations to the measurement of times and distances.
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Bell’s theorem for temporal order
Bell’s theorem is a result in quantum mechanics that tests one of the “weird” aspects of the theory: entanglement. In quantum mechanics one can have states of systems composed by subsystems in which the whole is more than the sums of the parts. Even further: some of the properties of the parts cannot be determined without knowing the properties of the whole. Bell’s theorem involves a series of inequalities that entangled quantum systems satisfy and non-entangled systems as classical ones do not. Checking that they are satisfied, one can determine that entanglement took place. Entanglement is deeply related to the probabilistic nature of quantum mechanics we mentioned in Chapter 3, the theory does not predict a definitive answer. One can have states of systems in which it appears “superposed” in two possible classical states. It is the origin of the famous thought experiment of Schr¨odinger where a cat ends up in a quantum state in which it is both dead and alive (see Figure 8.2). In practice that does not happen with cats because, like the mirrors we discussed in the Salecker and Wigner experiments, cats interact and entangle themselves with the environment (air molecules, etc.). Zych, Costa, Pikowski and Brukner have suggested that in quantum gravity, one can have a
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Figure 8.2: The Schr¨ odinger’s cat thought experiment: A cat, a flask of poison, and a radioactive source are placed in a sealed box. If radiation is detected, the flask is shattered, releasing the poison, which kills the cat. In the Copenhagen interpretation of quantum mechanics, since we don’t know whether the source has radiated, the cat is in a quantum state in which it is simultaneously dead and alive. Credit: Wikimedia Commons/Dhatfield.
mass in a quantum state where its position is not well defined. This can lead to entanglement. A key element in general relativity is the notion of causality: certain processes in a given space-time can be the cause of others whereas other cannot. The reason is nothing can travel faster than light. So if I have a process that occurs at noon at a certain point, it cannot influence one at noon plus one second if the second process is farther away than 300, 000 kilometers. It is too far away to be reached even by light in the time alloted. This
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becomes blurred when quantum effects are brought into account. So given two processes A and B, the theory gives us as final result some probability of “A can influence B” and “A cannot influence B”! This property of the final quantum state can be tested by seeing if it satisfies Bell’s inequalities. Tabletop experiments in which tiny masses are placed in quantum states where the position of the mass has a probability of being at different places could provide examples of fuzzy quantum spacetimes. The mentioned authors have suggested a way to turn this into an experiment about the quantization of space-time, but they acknowledge that with present technology it will be a very challenging experiment. This and other tabletop experiments are created to explore a new frontier in which gravity and quantum mechanics exhibit some of their properties without having to resort to the tiny granularity implied by the Planck length. As we mentioned in the double slit experiment of Chapter 3, particles in quantum mechanics behave both as particles and waves. The wavelength of their wavy behavior is inversely proportional to their energy. So a ballpark estimate indicates that probing the Planck length would require particles of a very large energy. Such energy is known as the Planck energy, and it is many many orders of magnitudes higher than those attainable by the
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large particle accelerators of today like the Large Hadron Collider at the European Nuclear Research Center (CERN, for its initials in French). By comparison, the tabletop experiments are considered “low energy.” Time will tell if these types of experiments can unravel some of the mysteries of the fuzzy space-times of quantum gravity.
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Chapter 9
Conclusions
We have reviewed how the application of new techniques, based on the field lines that originated with Faraday long time ago and applied to gravity rewritten as Abhay Ashtekar taught us, has allowed to build a quantum theory of gravity. We still do not have complete control over it, but it has been applied to a couple of interesting situations and has yielded attractive results. There is much left to be done. But we hope the reader can appreciate that this mixture of the Faraday lines, the spin networks of Penrose, and the notion of diffeomorphism invariance have led to new insights on the problem of quantum gravity. A comment we should make is that loop quantum gravity, being over 30 years old, is a large field and here we have covered only a small sliver of results, which we consider to be the most relevant to the general public.
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As we mentioned in the introduction, there is controversy. If one goes on the internet one can find blogs and other pages that are very critical of loop quantum gravity. We do not recommend to follow those sources. Many are not very well informed and jump to conclusions based on prejudices. For those who wish to dive deeper in the controversy there are two articles, one by Nicolai and collaborators,a which discusses some of the criticisms and a reply by Thiemann.b Unfortunately the articles are technical. If we were to briefly summarize, the main critical points can be boiled down to two. The first one concerns the “distance” introduced by Ashtekar and Lewandowski. It differs dramatically from those used in other quantum theories. The criticism therefore is that it is “too revolutionary.” Insiders reply by saying that more traditional distances were tried in the past and failed. Something had to change and this is what this approach changes. Others ask “have you tried distances like these in a
Hermann Nicolai et al., Classical and Quantum Gravity 22, R193 (2005); “Loop and spin foam quantum gravity: A brief guide for beginners,” in Approaches to Fundamental Physics, eds. I.-O. Stamatescu and E. Seiler (Springer, 2007), pp. 151–184. b Thomas Thiemann, “Loop quantum gravity: An inside view,” in Approaches to Fundamental Physics, eds. I.-O. Stamatescu and E. Seiler (Springer, 2007), pp. 185–263.
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other theories.” The answer is yes, and it has been shown that one can make things work, but things are not necessarily pretty. The distance was clearly built with diffeomorphism invariance in mind and other theories do not have that invariance. It is as if one tries to run the Dakar Rally using a Ferrari. It will not work well because it was designed to do other things. But the reader can appreciate that this criticism is a valid difference of opinion. As we mentioned the distance used is unique if one assumes diffeomorphism invariance, but that does not mean it is right. The second criticism has to do with the emergence of discrete structures. These structures may conflict with high precision experiments like those done in particle physics. In physics there is a symmetry known as Lorentz invariance that relates how two observers moving with respect to each other perceive physics. This invariance is present in electromagnetism and the theories of the strong and weak interactions, and is also present in general relativity but at a “local level” (i.e. in a small region). This invariance has been tested to a high degree of precision. Discrete structures have the potential to break the symmetry, whose intrinsic nature is continuous
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(this is associated with the fact that the relative velocity of two observers is a continuous variable). People have tried to build arguments showing that the discrete structures conflict with this invariance, but up to now always later it was realized how to reconcile things. But that does not mean that true problems could not be found in the future. In our view this is one of the issues that most concerns string theorists about loop quantum gravity. Finally, some people have concerns that Thiemann’s theory does not capture the dynamics of general relativity well. As we mentioned, the theory is quite non-trivial, but perhaps it is not complex enough to capture all the content of Einstein’s theory. There are some hints that this could be true but there is no definitive proof of it up to present. Until that happens we need to continue working with the theory to get out physical results from it and see if a contradiction appears. Ultimately what will make or break the theory is to find some experimental prediction that could be tested. That is what physics is all about. We discussed some possibilities but we are not there yet. Someone once mentioned to the Pakistani Nobel Prizewinner Abdus Salam “there are no experimental predictions of quantum gravity.” His reply was
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something along the lines of “how do you know that a theory does not have experimental predictions when you do not have a theory.” It is not that there is no possibility for significant changes in our understanding of nature. Just to mention one point, we know that 96% of the matter in the universe is not ordinary matter, but two strange forms of matter that interact almost only gravitationally, known as dark matter and dark energy (see Figure 9.1). So we are saying that we do not understand 96% of the universe. We do not know if quantum gravity has to do with this or not, but it might. Perhaps when we unravel more features of loop quantum gravity it will have something to say about this. Only more work will tell.
Figure 9.1: The different current constituents of the universe, according to observations. Ordinary matter accounts for less than 5% of all matter. Credit: Wikimedia Commons/ Ben Finney.
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If this book has piqued your curiosity for more detail, we also have a bookc at the undergraduate level that assumes little knowledge of physics, but the learning curve in it is steep! Have fun!
c
R. Gambini and J. Pullin, A First Course in Loop Quantum Gravity (Oxford University Press, 2013).
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b3737-index
Index accretion disk, 44 area operator, 37 Ashtekar variables, 32 Ashtekar–Lewandowski measure, 34
Faraday lines, 29 fireworks, 73
Big Bang, 57 Big Bounce, 59 black hole entropy, 48 black hole evaporation, 51 black holes, 42
Hawking radiation, 48 homogeneous and isotropic universe, 55
general relativity, 9 gravitational waves, 12
Immirzi parameter, 50 infinities, 22 inflation, 66 information paradox, 53
controversy, 85 cosmic microwave background, 61 cosmological constant, 55
LIGO, 45 loop quantum cosmology, 57 loop quantum gravity, 29 loop representation, 34 LOST-F theorem, 36
dark energy, 89 dark matter, 89 deflection of light by the Sun, 12 diffeomorphisms, 26 double slit experiment, 18
matrix models, 72 Newtonian gravity, 8
electromagnetism, 5 entropy, 42 event horizon telescope, 45
path integral, 69 precession of perihelion, 9
91
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quantization of atomic orbits, 17 quantum theory, 17
tensor networks, 72 thermodynamics, 41 Thiemann’s quantum equations, 38
Regge–Ponzano model, 72 singularity (black holes), 51 singularity (cosmology), 57 spin foams, 69 spin networks, 33 string theory, 22 strong interaction, 6
weak interaction, 7 white holes, 73 Yang–Mills theories, 8
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