Physics: Nature of Physical Fields and Forces [15 ed.] 9798985972412
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Physics
Nature of Physical Fields Physics and Forces Subtitle 15
th
Edition
Robert P. Massé
Lorem Dolor
Ipsum Facilisis
Also by Robert P. Massé
Copyright © 2022 Robert P. Massé
Physics: Where It Went Wrong
All rights reserved.
Vectors and Tensors of Physical Fields 15
th
Edition
Number Theory Complex Variables
ISBN: 979-8-9859724-1-2
Laplace Transforms Massé, Robert P.
by Robert P. Massé
Kay E. Massé
Physics Nature of Physical Fields and Forces
Physics: Principle of Least Action
i
Dedication This book is dedicated to my beloved wife Kay whose loving and insightful support made it possible
ii
Preface “It’s a terrible mix-up, and you might say it’s a hopeless mess physics has got itself worked into.” Richard P. Feynman QED: The Strange Theory of Light and Matter
iii
!
There are only two long-range forces known to physics:
Bay Library Consortium. I am very grateful to the personnel at
the force of gravity and the force of electromagnetism. For
the Bonita Springs, Fort Myers Beach, Tampa, and Gainesville
many years now much has been known about the laws these
Public Libraries for all their diligent efforts in obtaining research
physical forces follow. At the same time and on a more
material.
fundamental level, however, much about the true nature of these physical forces has remained a deep mystery. For example, among the entities whose true nature still remains
!
Robert P. Massé
!
First Edition – March 21, 2006
!
Second Edition – August 20, 2009
!
Third Edition – June 15, 2012
acting within atoms. Nuclear forces exhibit complications that
!
Fourth Edition – May 30, 2014
have never been explained.
!
Fifth Edition – July 20, 2015
!
It is the purpose of this book to explore the long-range and
!
Sixth Edition – January 11, 2016
short-range physical forces of our Universe together with their
!
Seventh Edition – August 29, 2019
associated physical fields. In the process, we will review some
!
Eighth Edition – January 20, 2020
!
Ninth Edition – March 16, 2021
!
10
nuclear physics, and quantum mechanics. This then is a study
!
11
of the foundations of physics.
!
12
!
The research that forms the basis of this book could not
!
13
have been accomplished without the outstanding interlibrary
!
14
!
15
virtually unknown to us are: gravity, mass, inertia, electricity, and magnetism. Even more mysterious to us than the longrange forces of physics are the short-range nuclear forces
of the important experiments that have helped reveal the laws pertaining to these physical forces. Included in the subjects covered in this book are: gravitation, light, electromagnetism,
loan system of Florida made available through the good
th
th th th th th
Edition – April 20, 2021 Edition – June 28, 2021 Edition – August 10, 2021 Edition – October 25, 2021 Edition – March 14, 2022 Edition – May 2, 2022
resources of the Lee County Library System and the Tampa iv
Contents
2!
The Nature of Mass and of Inertia
!
2.1!
Classical Concepts of Mass
!
2.2!
Nature of Mass
!
2.3!
Mass Types
1!
The Nature of Gravity
!
1.1!
Newton’s Force Law of Gravity
!
2.4!
Derivation of Newton’s Laws of Motion
!
1.2!
Poisson’s Field Equation of Gravity
!
2.5!
Derivation of Conservation Laws
!
1.3!
Einstein’s General Theory of Relativity
!
2.6!
Nature of Inertia
!
1.4!
Aether Field Equation of Gravity
!
2.7!
Absolute Motions and Galilean Relativity
!
1.5!
Continuity of Aether Density in a Vacuum
!
2.8!
Inertial Forces
!
1.6!
Force Density of Flowing Aether
!
2.9!
Dark Matter
!
1.7!
Equation of Motion for Flowing Aether
!
2.10! Applicability of Newton’s Law of Gravity
!
1.8!
Gravitational Energy
!
2.11! Nonexistent Higgs Particles
!
1.9!
Gravitational Energy Density Flux within a Vacuum
!
2.12! Summary
!
1.10! Irrotational Flow of Aether in a Gravitational Field
!
1.11! Vorticity
!
1.12! Circulation
!
3! The Nature of Light !
3.1!
Luminiferous Aether and Light
1.13! Generation of Vorticity in Aether
!
3.2!
Elastic Properties of Aether within Free Space
!
1.14! Nature of Mechanical Force
!
3.3!
Elastic Wave Equations for Aether within Free Space
!
1.15! Nature of Gravitational Force
!
3.4!
Light Waves
!
1.16! Nonexistent Gravitons
!
3.5!
Motion of Aether and of the Light Source or Light
!
1.17! Antigravity
!
!
Receiver
!
1.18! Summary
!
3.6!
Light Traveling through Material Media
!
3.7!
Light Waves and Rotational Motion of the Earth v
!
3.8!
Light Waves and Orbital Motion of the Earth
!
4.10! Nature of Charge and Current
!
3.9!
Gravity and the Deflection of Light
!
4.11! Historical Distinction between Electric and Magnetic
!
3.10! Gravitational Lenses
!
!
!
3.11! Time Delay of Light
!
4.12! Electromagnetic Energy Density within Material
!
3.12! Faster than Light
!
!
!
3.13! Black Holes
!
4.13! Force Density and Momentum Density of
!
3.14! Index of Refraction of a Gravitational Field
!
!
!
3.15! Summary
!
4.14! Lorentz Force Density
!
4.15! Polarization Vector and Magnetization Vector
4! The Nature of Electromagnetic Fields and Forces
!
4.16! Integral Form of Maxwell’s Field Equations for
!
4.1!
Electric and Magnetic Aethers
!
!
!
4.2!
Elastic Energy Density of Aether within Free Space
!
4.17! Scalar and Vector Potentials for Electromagnetic
!
4.3!
Continuity of Elastic Energy Density of Transverse
!
!
!
!
Elastic Waves in Aether within Free Space
!
4.18! Basic Laws of Electricity and Magnetism
!
4.4!
Poynting Vector for Transverse Elastic Waves in
!
4.19! Electromagnetic Circuit Parameters
!
!
Aether within Free Space
!
4.20! Reflection and Refraction of Electromagnetic Waves
!
4.5!
Maxwell’s Field Equations for Transverse Elastic
!
4.21! Electromagnetic Mass
!
!
Waves in Aether within Free Space
!
4.22! Acceleration of a Charged Particle by
!
4.6!
Nature of Electromagnetic Fields
!
!
!
4.7!
Maxwell’s Field Equations for Free Space
!
4.23! Einstein’s Special Theory of Relativity
!
4.8!
Electromagnetic Energy Density within Free Space
!
4.24! Electromagnetic Radiation
!
4.9!
Derivation of Maxwell’s Field Equations for Material
!
4.25! Gravitational Waves
!
!
Media
!
4.26! Nature of Electromagnetic Force
Field Intensities Media Electromagnetic Waves
Material Media Fields
Electromagnetic Force
vi
!
4.27! Unified Field Theory for Gravitational and
!
5.18! The Gravitational Constant G
!
!
!
5.19! Summary
!
4.28! Poynting Vector for Gravitational Waves
!
4.29! Summary
Electromagnetic Forces
6! The Nature of Quantum Mechanics !
6.1!
Blackbody Radiation
5! The Nature of Nuclear Fields and Forces
!
6.2!
Planck’s Light Quanta
!
5.1!
Particles
!
6.3!
The Photoelectric Effect
!
5.2!
Nature of the Neutron
!
6.4!
Nature of the Photon
!
5.3!
Nature of the Strong Nuclear Force
!
6.5!
The Compton Effect
!
5.4!
Nature of Electrons and Positrons
!
6.6!
Two-Slit Experiments
!
5.5!
Nature of the Proton
!
6.7!
de Broglie Waves
!
5.6!
Proton Mass
!
6.8!
Schrödinger’s Wave Equation
!
5.7!
Nature of the Weak Nuclear Force
!
6.9!
Quantum Numbers
!
5.8!
Antiparticles
!
6.10! Heisenberg Uncertainty Principle
!
5.9!
Nature of Neutrinos
!
6.11! Delayed-Choice Single Photon Experiment for Light
!
5.10! Nature of the Atomic Nucleus
!
6.12! EPR Experiments and Quantum Nonlocality
!
5.11! Mass Defect
!
6.13! The Stern-Gerlach Experiment
!
5.12! Radioactivity
!
6.14! Nature of Quantum Mechanics
!
5.13! Nature of Atoms
!
6.15! Exchange Forces/Mediating Particles
!
5.14! Unstable Particles and Aether Disturbance
!
6.16! Absence of Compressional Wave Generation in
!
5.15! Time Dilation and the Lifetime of Unstable Particles
!
!
!
5.16! Nonexistent Quarks
!
6.17! Summary
!
5.17! Unified Field Theory for Nuclear Forces
Aether
vii
7! Tests of Gravitational Theories
8! The Nature of Space and Time
!
7.1!
The Galileo Falling Body Experiments
!
8.1!
Newton’s Space and Time
!
7.2!
Accelerated versus Unaccelerated Motion
!
8.2!
Einstein’s Space and Time of Special Relativity
!
7.3!
Newton’s Laws of Motion
!
8.3!
Minkowski’s Space-time
!
7.4!
Inertial Forces
!
8.4!
Einstein’s Space-time of General Relativity
!
7.5!
Dark Matter
!
8.5!
Nature of Space
!
7.6!
Light Source Speed
!
8.6!
Nature and Direction of Time
!
7.7!
Gravitational Frequency Shift
!
8.7!
Summary
!
7.8!
Solar Deflection of Light
!
7.9!
Time Delay of Light
!
7.10! Without Infinities
!
9.1!
Nature of the Big Bang
!
7.11! Derivation of Maxwell’s Field Equations for
!
9.2!
Before the Big Bang
!
!
!
9.3!
Unification of Fields and Forces in Our Universe
!
7.12! Advance of Mercury’s Perihelion
!
9.4!
Gravity and the Future of Our Universe
!
7.13! Strong and Weak Nuclear Forces
!
9.5!
Aether and Our Universe
!
7.14! Nature of Quantum Mechanics
!
9.6!
Physics of Material Body Structure
!
7.15! Unified Field Theory
!
9.7!
Universal Constants of Nature
!
7.16! Why any Correct Results from the General Theory of
!
9.8!
Conservation Laws of Our Universe
!
!
!
9.9!
Origin of Our Universe
!
7.17! Summary
!
9.10! Summary
Electromagnetic Waves in Free Space
Relativity?
9! The Nature of Our Universe
viii
10! The Nature of Life !
10.1! Characteristics of Life
!
10.2! Basic Unit of Life in Our Universe
!
10.3! Definition of Life
!
10.4! Fundamental Levels of Life
!
10.5! Summary
Appendix A! Vector Field Operations Appendix B! The Greek Alphabet Appendix C! Vector Identities Appendix D! Helmholtz’s Vortex Theorems Appendix E! Physical Constants Appendix F! Reynold’s Transport Theorem for Aether ! in Free Space Appendix G! Earnshaw’s Theorem References
ix
Chapter 1 The Nature of Gravity
“But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; . . . And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.” Isaac Newton Principia
10
The search for an understanding of the physical force
gravitational forces based upon his studies of pendulums,
known as gravity has occupied philosophers and scientists for
falling bodies, the orbits of planets, and the orbits of their
many centuries. Early concepts of force and matter developed
moons. He found, for example, “that the forces by which the
in antiquity by Aristotle and other philosophers have come
primary planets are continually drawn off from rectilinear motions,
down through the ages and influenced the work of scientists
and retained in their proper orbits, tend to the Sun” and that these
such as Kepler, Galileo, Huygens, and Newton (see Jammer,
forces “are reciprocally as the squares of the distances of the places of
1957, 1961). Advancement in science is, after all, an incremental
those planets from the Sun’s center.” He also found that: “there is a
process with new discoveries resting on the foundation of
power of gravity tending to all bodies, proportional to the several
previous work.
quantities of matter which they contain.”
Newton’s approach to the study of gravity was to use “the
!
Newton had discovered the force law of gravity.
phenomena of motions to investigate the forces of nature, and then
Moreover, he had discovered that this force law of gravity
from these forces to demonstrate other phenomena.” From various
applies not only to objects on Earth, but also to objects in the
observations
in
heavens. For example, he found that the Earth’s gravitational
determining particular propositions that: “are inferred from the
force keeps the moon orbiting the Earth rather than flying off
phenomena and afterwards rendered general by induction.” This type
into deep space.
and
experiments,
Newton
succeeded
of methodology works well in physics. !
In this chapter we will determine the nature of gravity. We
1.1.1!
LAWS OF PHYSICS
will begin by examining the gravitational force law of Newton,
!
A law in physics is a relation between physical entities
and then proceed to consider some field theories of gravity.
that has been abstracted from empirical observations of these entities. In formulating laws of physics, the assumption is made
1.1!
NEWTON’S FORCE LAW OF GRAVITY
that the existence of identical physical circumstances will
Isaac Newton’s great Philosophiae Naturalis Principia
always result in the occurrence of identical phenomena
Mathematica (known as Principia) was first published in 1687.
(Poincaré, 1905b). A law in physics then denotes neither a
In this work Newton derived mathematical expressions for
requirement nor a prohibition, but only an observed pattern. In
!
11
other words, a law in physics does not govern nature; it only
(see Clotfelter, 1987; and Falconer, 1999). The first estimate of a
describes nature. A law in physics, therefore, is perhaps better
numerical value for G was made almost one hundred years
considered an effect rather than a cause.
later by Boys (1894a, b, c, and d) using empirical procedures developed by Cavendish. Since then many measurements of G
1.1.2! !
MATHEMATICAL EXPRESSION
have been made. The current estimate of G is:
Bodies containing matter are, by definition, material
bodies. Newton (1687) defined mass to be a measure of the quantity of matter in a material body. Newton thus introduced the concept of mass into physics (Cohen, 2002). Note that Newton did not state that mass is the same thing as matter, but only that mass is a measure of the quantity of matter. !
Matter is the only entity in our Universe that possesses
gravity. Mass, being only a measure of the quantity of matter, cannot possess gravity. !
Newton’s force law of gravity for two material bodies of
masses m and M can be expressed in the form: !
mM F = −G 2 ! r
! !
G = 6.67430 × 10 −8 cm 3 sec-2 gm -1 !
(1.1-2)
Considering the force F on a material body of mass m
resulting from the gravitational pull of a material body of mass
M , we can rewrite equation (1.1-1) as: !
F M = −G 2 ! m r
In vector form, this can be written as: ! F M ! = − G 2 rˆ ! m r
(1.1-3)
(1.1-4)
where rˆ is a unit line vector that has its coordinate origin at the (1.1-1)
where F is the gravitational attractive force between the bodies, r is the distance between the bodies, and G is the gravitational constant of proportionality. The constant G was first measured implicitly by Cavendish (1798) using a torsion
center of the body of mass M , and that is directed along the line from this body to the body of mass m (see Figure 1-1). The minus sign indicates that the force F acting on the body of mass m is in the opposite direction of rˆ and so is directed towards the body of mass M .
balance to determine the density of the Earth by weighing the world. He did not calculate a numerical value for G , however 12
!
Since time is not a parameter in equation (1.1-1), time does
not enter into Newton’s force law of gravity. We then have the additional implication that the action-at-a-distance of gravity occurs instantaneously. Therefore gravitational force appears to propagate with infinite speed according to Newton’s force law of gravity. !
On the basis of our everyday experience, this physical
interpretation of gravity does not appear plausible. As Dolbear (1897) stated "every body moves because it is pushed, and the mechanical antecedent of every kind of phenomena is to be looked for in some adjacent body possessing energy; that is, the ability to push or Figure 1-1!
1.1.3! !
Gravitational force F on a material body of mass m due to a material body of mass M .
PHYSICAL INTERPRETATION
produce pressure." It is hard to imagine how a force could act over some large distance without an intervening medium. It is even harder to believe that this action could be instantaneous. We must conclude, therefore, that Newton’s force law of gravity expressed in the form of equation (1.1-1) does not lend
Newton’s force law of gravity given in equation (1.1-1) can
itself to a direct physical interpretation as to the cause of
be interpreted physically to imply that some sort of action-at-a-
gravity. This prompts us to ask what Newton thought about the
distance is occurring as each of the two material bodies exerts
relation he had discovered for gravitational forces.
an attractive force on the other; that is, the two material bodies
!
will tend to accelerate towards each other. Action-at-a-distance
purposed “only to give a mathematical representation of those forces,
is defined here to mean that two bodies separated in space and
without considering their physical causes and seats.” Newton also
having no intervening medium exert a physical effect on each
noted that he used “the words attraction, impulse or propensity of
other across the empty space between them.
any sort towards a center, interchangeably, one for another;
Newton stated very clearly in his Principia that he
13
considering those forces not physically, but mathematically.” He was not attempting “to define the kind or the manner of any action, the
1.1.4!
PHYSICAL CAUSE OF GRAVITY
causes or the physical reason thereof,” nor was he attributing
!
“forces, in a true and physical sense, to certain centers (which are
He was certain that gravity “must proceed from a cause that
only mathematical points)” when he referred to ”centers as
penetrates to the very centers of the Sun and planets, without
attracting, or as endued with attractive powers.” Moreover, Newton
suffering the least diminution of its force; that operates not according
(1693b) wrote that the very thought that “one body may act upon
to the quantity of the surfaces of the particles upon which it acts (as
another at a distance through a vacuum without the mediation of any
mechanical causes do), but according to the quantity of solid matter
thing else by and through which their action or force may be conveyed
which they contain, and that propagates in all directions to immense
from one to another is to me so great an absurdity that I believe no
distances, decreasing always by the square of the distances.” He
man who has in philosophical matters any competent faculty of
suspected that the ultimate cause of gravity was to be found in
thinking can ever fall into it.” Newton clearly did not believe in
“an ethereal medium”, a physical nonmaterial medium thought
action-at-a-distance. He felt the need for a medium that serves
to permeate all space (Newton, 1730; and Jammer, 1957). He
to transmit force (Jourdain, 1915c). Newton’s theory of gravity,
thought that gravity was the result of this ethereal medium
therefore, clearly does not postulate instantaneous action-at-a-
flowing with accelerated motion into material bodies and
distance.
then collecting within these bodies (Aiton, 1969). Newton
!
Newton realized that he was not defining the “physical
thought the ethereal medium to be composed of distinct
qualities of forces, but investigating the quantities and mathematical
particles and to be mechanical in nature (see Hall and Hall,
proportions of them” in his Principia. We see, therefore, that
1960).
Newton considered his force law of gravity to be useful for
!
mathematically calculating gravitational forces, but not for
gravity, and he was not prepared to hypothesize on its physical
determining their physical causes.
cause in his Principia. He concluded that the “cause of gravity is
Newton did, however, attempt to find the cause of gravity.
Nevertheless, Newton could not verify the cause of
what I do not pretend to know, and therefore would take more time to consider of it“ (Newton, 1693a). 14
Faraday employed the term magnetic field to denote the
1.2!
POISSON’S FIELD EQUATION OF GRAVITY
distribution of magnetic forces in a region (see Gooding, 1980). !
A magnetic line of force is a line having the direction of the
In 1813, more than one hundred years after Newton’s
magnetic force at each point along the line. The concentration
Principia was first published, Siméon Denis Poisson derived an
of such lines in a given region indicates the intensity of the
equation that can be used to describe the acceleration due to the
force in the region. Faraday surmised that these lines of force
gravitational force of a material body. The equation describing
exert an influence upon each other by transferring their action
gravitational force developed by Poisson is of a type know as a
from particle to contiguous particle of a physical medium.
field equation. Using vector analysis methods, we will now
!
show that Poisson’s field equation of gravity can be obtained
"we cannot help thinking that in every place where we find these lines
from Newton’s force law of gravity, and so the two laws are
of force, some physical state or action must exist in sufficient energy
compatible. Subsequently we will consider the physical
to produce the actual phenomena." Maxwell proceeded to "examine
interpretation of Poisson’s field equation. We will begin with a
magnetic phenomena from a mechanical point of view" based upon
brief discussion of the concept of a physical field.
an interpretation of Faraday's concept of a physical medium of
!
Considering these lines of force, Maxwell (1861a) stated
contiguous particles (Maxwell, 1861a, b).
1.2.1! !
PHYSICAL FIELDS
The concept of a physical field was developed first by
1.2.1.1!
CONTIGUOUS PARTICLES
Euler in the 1750s during his studies of the kinematics of fluids
!
Contiguous particles are simply particles packed so
(see Truesdell, 1954). It was James Clerk Maxwell, however,
closely together that there is essentially no space between them.
who more generally introduced the field concept into physics in
Faraday (1844) envisaged contiguous particles as being next to
the 1860s. Maxwell derived his field concept from the magnetic
each other, but not touching each other. He did not consider the
lines of force that Michael Faraday had proposed to explain
contiguous particles of his physical medium to be matter. He
magnetic force patterns in space (Faraday, 1844, 1847, 1855).
noted that: “matter is not essential to the physical lines of magnetic force any more than to a ray of light or heat.” Faraday’s ideas were 15
used by Maxwell to formulate the concept of a continuous
region of space with no gaps between the points. The result is a
physical field in which all actions occur only between
one-to-one correspondence between point particles of the
contiguous particles. We will designate these contiguous
continuum and geometrical points assigned to the field
particles as field particles.
particles in the region of space occupied by the field medium.
!
A disturbance resulting from a change in the motion of a
The mathematical continuum of abstracted physical particles
field particle can then propagate to a distance only by a
provides justification for using the limiting process of the
succession of actions between contiguous particles. Action-at-a-
calculus on physical problems involving fields, and for using
distance is thereby reduced to actions between particles that are
differential equations to describe changes in the fields.
contiguous. Some time duration is required for all these actions
Each point particle of the continuum always retains its
to occur, and so field disturbances can only propagate with a
individual identity as well as all the physical properties
velocity that is finite.
(including the kinematical and dynamical properties) of its
!
We see then that the existence of a continuous physical
corresponding field particle except for volume and extension (a
field in a region requires the existence of a real physical
point particle has neither). Nevertheless the same density and
medium consisting of contiguous particles. We will designate
pressure are assumed to exist in the continuum as in the field
this real physical medium as the field medium. The particles of
medium.
the field medium must be small enough so that a large number
!
of these field particles are contained within a volume element
capable of being abstracted to form a mathematical continuum,
that is considered infinitesimal relative to the volume of the
this is not what defines a physical field. Rather, a physical field
entire region.
is some specific attribute associated with each of the field
While a physical field requires contiguous field particles
These contiguous field particles can then be used to form a
particles being abstracted to form the mathematical continuum.
continuous mathematical entity known as a continuum. This
A physical field is not a physical medium then, but is some
continuum is obtained by abstracting the field particles of the
attribute of the contiguous field particles that form the real
field medium to point particles, thereby allowing the field
physical medium. A physical field can never have an existence
medium to be treated as a mathematical continuum filling a
independent of the field particles that constitute the field 16
medium. A physical field is more, therefore, than simply a mathematical device used to solve a physical problem. Physical
1.2.1.3!
fields are always real.
!
TYPES OF VECTORS
Since vector fields can, by definition, vary from point to
point, the vectors associated with a vector field are known as 1.2.1.2! !
TYPES OF PHYSICAL FIELDS
point vectors. They are defined for and occupy only a single
Depending upon properties of the field particle attribute
point. Point vectors and line vectors are not the same then since
that is the physical field, several different types of physical
all nonzero line vectors occupy more than a single point and
fields can be identified. When the field particle attribute is a
can be slid over many points. The magnitude of a point vector,
scalar quantity so that a scalar quantity is associated with each
therefore, cannot be equated with the length of the vector,
point of the mathematical continuum, these quantities together
unlike the magnitude of a line vector. Finally, since the
form a scalar field. Similarly, when the field particle attribute is
addition of two or more point vectors at any given point is
a vector quantity or a tensor quantity so that a vector quantity
always valid, the superposition of different vector fields
or a tensor quantity is associated with each point of the
representing the same type of physical entity is always possible.
mathematical continuum, these quantities form a vector field or a tensor field, respectively (in this book we will use the term
1.2.2!
GRAVITATIONAL FIELD
tensor to refer to tensors of order two or higher). The
!
temperature in the Earth’s atmosphere is an example of a
masses M and m as shown in Figure 1-1. We will define a
physical scalar field, and the wind velocity in the atmosphere is
material point particle to be a material body that has been
an example of a physical vector field. The real medium that is
abstracted to an infinitesimal size so that: “the distances between
abstracted to a mathematical continuum in both these cases is
its different parts may be neglected” as Maxwell (1877) suggested.
air, and the field particles are extremely small air parcels.
A material point particle can then be considered to have no
!
A field that does not vary with time (is independent of
volume or extension, and so it will occupy a single point in
time) is termed a stationary field or a steady-state field. A field
space. We will assume, nevertheless, that a material point
that does vary with time is termed non-stationary.
particle has density and that this density is constant. We will let
We will now once again consider two material bodies of
17
the material body of mass m be a material point particle.
we can conclude that the vector field g surrounding the body
Newton formulated his gravitational theory in terms of such
of mass M is a real physical vector field.
material point particles. We will specify the location of this particle in space by the line position vector r that has its coordinate system origin in the center of the body of mass M . !
In order to avoid the action-at-a-distance concept of force,
we will now consider that at all points in space surrounding the material body of mass M a gravitational acceleration field due to the matter in this body exists. This gravitational acceleration field of the material body of mass M exists in space whether or not any material point particle is present at a particular point in space to experience the acceleration. The vector quantity F m in equation (1.1-4) has the dimensions of acceleration and represents the gravitational acceleration existing around the material body of mass M . We will define F m to be the gravitational field intensity or gravitational field g . A vector quantity g can be associated, therefore, with each point of space surrounding the material
Figure 1-2!
Line position vector r from the material body of mass M to a body of mass m .
body of mass M .
!
particle of mass m will experience due to the gravitational pull
space about the body of mass M : ! F ! M ! ≡ g = − G 2 rˆ ! m r
The vector g is the acceleration that a material point
of the matter of mass M if the point particle is placed at a point in space specified by the line position vector r (see Figure 1-2). Since g is a real physical vector quantity that can be measured,
Using Newton’s law of gravity as given in equation (1.1-4), we can then determine F m (and so g ) for all points in
(1.2-1)
where the unit vector rˆ is given by: 18
! ! r r rˆ = ! = ! r r
!
(1.2-2)
Equation (1.2-1) shows that the acceleration g of a body of mass
is quantity of matter, then matter density must be Qm per unit volume, and so for a material body: !
Qm =
∫∫∫ ρ dV !
(1.2-3)
V
m due to the gravitational pull of a material body of mass M is a function of M , but is independent of m .
where V is the volume of the material body. It is important to
!
If the material body of mass M is stationary, its gravitational field g will also be stationary. A material point
notice that matter density ρ is not defined in terms of mass.
particle of mass m placed within such a stationary field will instantaneously experience the gravitational acceleration F m
of the quantity of matter, we have:
of the field. We see now that the reason gravitational force appears to propagate with infinite speed according to Newton’s force law of gravity is that this law effectively assumes the gravitational field is stationary, and so gravitational force already exists at all points within the field. The gravitational force that is associated with a stationary gravitational field will always be constant (not a function of time) at any given point within the field. It is then not correct to state that Newton’s law of gravity assumes that gravitational force propagates with an infinite speed.
1.2.3! !
MATHEMATICAL EXPRESSION
The density of a substance is defined as the quantity of
the substance per unit volume. If ρ is matter density and Qm
Since, for a material body, mass has been found to be a measure
!
Qm =
∫∫∫ ρ dV ≈ M !
(1.2-4)
V
where M is the mass of the material body. In fact, mass “serves for measuring a portion of matter so well that matter and mass appear to be synonyms” as noted by Rougier (1921). However, mass and matter are not the same type of thing (see Section 2.2). For example, some particles do not contain matter but still have mass (see Section 5.4.6). Only for a free neutron does the mass of a material body exactly equal the quantity of matter of the body (see Section 5.11). !
Instead of considering the entire material body of mass
M , we will now consider only a single very small volume element ΔV of this body. The matter density ρ of this volume element can be taken as constant since the dimensions of ΔV
19
are taken to be very small. The mass ΔM of the matter contained within ΔV is then given by: ! !
ΔM =
∫∫∫
ρ dV = ρ
ΔV
∫∫∫
dV = ρ ΔV !
ΔV
(1.2-5)
To obtain the gravitational field g at a point r in space
!
! ! ∇• g = − G ρ
∫∫∫ 4 π δ (r ) dV ΔV
!
We then have finally: ! ∇ • g = − 4π G ρ !
(1.2-9)
(1.2-10)
due to the gravitational pull of only the matter of mass ΔM
This equation, which is known as Gauss's law of gravity or as
contained within the volume element ΔV , we can use equations (1.2-1) and (1.2-5) to express the gravitational field g
Gauss's flux theorem of gravity, describes the gravitational field g due to matter of mass ΔM and density ρ contained
as: !
! g = −Gρ
∫∫∫
ΔV
rˆ ! 2 dV r
(1.2-6)
The factor rˆ r 2 is under the integral sign since the coordinate origin of the position vector r is taken to be in the volume element ΔV . The divergence of the gravitational field g is then determined by the relation:
! ! ! ! ∇• g = − G ρ∇•
∫∫∫
ΔV
rˆ dV = − G ρ r2
! rˆ ∇ • 2 dV ! (1.2-7) r ΔV
∫∫∫
within a very small volume element ΔV . !
The law of gravitation defined by Newton’s force law of gravity is linear. Therefore, the gravitational field g due to each volume element of a material body can be calculated individually using Newton’s force law of gravity, and then a vector summation can be performed to determine the total gravitational field of the entire material body. In other words, the gravitational field at a point r due to a body of mass M can be obtained by summation of g at the point r for all the
as is shown in Appendix A (the matter density ρ is constant).
volume elements ΔV in the body of mass M . If this body is located far from the point r , it is possible to use equation
From vector analysis we have:
(1.2-10) directly for the entire mass M with ρ representing the
!
! rˆ ∇ • 2 = 4 π δ (r ) ! r
(1.2-8)
where δ ( r ) is the Dirac delta function (see Appendix A), and so
average density of the body. !
For a stationary gravitational field resulting from matter of
mass ΔM contained in a volume element ΔV , we can define
we can rewrite equation (1.2-7) as: 20
the Newtonian stationary gravitational potential ϕ at a point r by: ΔM ! ϕ ≡G ! (1.2-11) r
Mathematically we must obtain the same results from Poisson’s
We then have:
since Newton’s force law of gravity is not a function of time,
! ΔM ∇ϕ = − G 2 rˆ ! r
!
field equation of gravity (1.2-15) as from Newton’s force law of gravity (1.1-1) since, as just shown, Newton’s force law can be used to derive Poisson’s gravitational field equation. Moreover, Poisson’s field equation of gravity (1.2-15) only applies to
(1.2-12)
gravitational fields that are stationary. Therefore, Poisson’s field
and so, for matter of mass ΔM contained in a volume element
equation of gravity and Newton’s force law of gravity must be
ΔV , we can use equations (1.2-1) and (1.2-12) to write the stationary gravitational field g in the form:
mathematically equivalent for a stationary gravitational field.
! ΔM ! ˆ g = − G 2 r = ∇ϕ ! r
!
(1.2-13)
!
We see, therefore, that it is possible to express a stationary gravitational field g in terms of the gradient of a scalar potential function ϕ . From equations (1.2-10) and (1.2-13) we have: ! or
! ! ! ! ∇ • g = ∇ • ∇ϕ = − 4 π G ρ !
1.2.4!
PHYSICAL INTERPRETATION
The physical interpretation of Poisson’s field equation of
gravity (1.2-15) is, however, very different from the physical interpretation of Newton’s force law of gravity (1.1-1). Rather than the action-at-a-distance interpretation of gravity obtained from Newton’s force law of gravity, Poisson’s field equation is based upon and must be interpreted in terms of the existence of
(1.2-14)
a physical medium consisting of contiguous particles about any
which is Poisson’s field equation of gravity for matter of mass
material body. ! Since g is a physical field, g must be an attribute of real field particles of a real field medium. Since g is a vector
ΔM and density ρ contained within a volume element ΔV .
acceleration field, this attribute of the particles must be
!
!
! ! ∇ • g = ∇ 2ϕ = − 4 π G ρ !
(1.2-15)
Poisson’s field equation describes the gravitational field
acceleration. We can conclude then that, at each point of a
ρ.
gravitational field, real particles must be in accelerated motion.
existing around a material body of matter density
21
We can also conclude, therefore, that a real physical medium must exist in space and that some flowing motion must be
1.2.5!
occurring within this medium to produce the observed
!
gravitational acceleration.
associated with the field particles of a real field medium as was
!
This is analogous to observing the effects of wind velocity,
noted in Section 1.2.1. If within the region of the vector field
which is a vector velocity field, and concluding that a real
there exists an entity that produces a discontinuity in the
physical medium (air) must exist and must be in motion. For
particular particle attribute that is the vector field, then such an
example, when we observe through a closed window the
entity is known as a source or a sink for the field (see Kellogg,
autumn leaves blowing about, we know that air is in motion
1929). Only vector fields have field sources and sinks. From
although we cannot directly see the air.
vector analysis, we know that a vector equation in the form of ! equation (1.2-10) can always be written for a vector field ϒ
!
The real physical medium whose flowing motion
produces the gravitational field must be consistent with Poisson’s field equation (1.2-15). We will refer to this real field medium as aether for historical reasons to be discussed in Section 1.2.7. Aether particles are then the field particles of gravitational fields. The mathematical continuum for the gravitational acceleration field is abstracted from aether particles with each aether particle corresponding to a point in the gravitational field. The acceleration of aether particles is the attribute of the field particles that is the gravitational field. Since the gravitational field is a real physical acceleration field, we can conclude that the field medium, aether, is real.
VECTOR FIELD SOURCES AND SINKS
A physical vector field consists of some vector attribute
having a source or a sink. We have then: ! ! ! ∇ • ϒ = ± 4π K ρ !
(1.2-16)
where the left side of this equation is the divergence of a vector ! field ϒ (vector attribute of field particles), and where the right side of this equation has a positive sign for a source and a negative sign for a sink. The parameter K is a constant required to make the units consistent. The density ρ is a scalar that is the strength of the source or sink per unit volume, and so is a measure of the effect the source or the sink has on the vector ! field ϒ . !
For the special case where the attribute that constitutes the
vector field is a kinematic flow property (flow velocity or flow 22
acceleration) of the field medium, sources and sinks of the
Sections 4.10 and 5.4.4. We can summarize by noting that there
vector field can be interpreted as places where this real physical
exist two types of vector field sources and, correspondingly,
field medium is being continuously created and continuously
two types of vector field sinks:
destroyed, respectively (Lamb, 1879; and Granger, 1985). For
1.! Field flow sources and field flow sinks.
such cases, we will designate any source as a field flow source
2.! Field non-flow sources and field non-flow sinks (field positive sources and field negative sources).
and any sink as a field flow sink. When the rate at which a source continuously creates a field medium (or a sink continuously destroys a field medium) is constant, the source (or sink) is referred to as steady. !
If the vector field does not consist of a kinematic flow
1.2.6! !
MATTER AS A FIELD FLOW SINK FOR AETHER
From equations (1.2-15) and (1.2-16), we see that since the
attribute of the field medium, then the source and sink will not
density of a material body is ρ > 0 and since the sign is
represent places where the field medium is being, respectively,
negative on the right side of equation (1.2-15), ρ must represent
created and destroyed. That is, they will not represent a field
the density of a sink for the gravitational field. Since ρ is the
flow source or a field flow sink. Rather, both source and sink
density of matter within the volume element ΔV of a material
will produce the attribute constituting the vector field. The only
body, the sink for the gravitational field is matter. The field
constraint is: when a source and a sink of this type and of equal
medium for a gravitational field is aether, and the attribute of
strength come together, they can nullify each other. We will
aether that is the gravitational field is the acceleration of aether
designate this type of source as a field non-flow source and the
particles. Gravitational acceleration is then a kinematic flow
corresponding sink as a field non-flow sink. Since a field non-
property of aether: flow acceleration. Therefore, matter is a
flow sink is really just another kind of source, we will also
field flow sink for aether. Poisson’s field equation (1.2-15) for
designate a field non-flow source to be a field positive source
the gravitational field is an equation describing the accelerating
and a field non-flow sink to be a field negative source
flow of aether into matter.
(recognizing that non-flow sources and sinks can nullify each
!
other). Such non-flow sources and sinks will be discussed in
there is no known entity that is a field flow source for a
While matter is a field flow sink for a gravitational field, 23
gravitational field. Other fields such as electromagnetic fields
that particles of aether form a continuum, everywhere pressing
do have both source and sink entities (as will be discussed in
upon other particles of aether except in the very few places
Chapters 4 and 5). It has sometimes been thought that material
where ordinary matter exists. He conceived of aether as a
particles are inconsistent with field theory (e.g., Einstein, 1950).
medium that can move very rapidly towards Earth. He thought
We now see, however, that material particles simply represent
that aether is capable of transmitting force so that all action-at-
discontinuities (sinks) in the field medium (aether), and so are
a-distance can be explained as the result of forces acting
consistent with field theory.
between contiguous aether particles. He also considered that aether provided the means by which light is propagated.
1.2.7! !
AETHER
Following Descartes, Huygens (1690a, b) postulated aether as
The concept of a real physical medium called aether (or
the nonmaterial physical medium in which light waves
ether) is not new. The origins of aether as a metaphysical
propagate. Newton (1693b) thought that: “It is inconceivable, that
concept can be traced back to antiquity. The word ‘aether’ is
inanimate brute matter should, without the mediation of something
from a Greek word meaning ‘the upper purer air’. Aether was
else, which is not material, operate upon and affect other matter
considered by Aristotle (c. 384 BCE to 322 BCE) to be the
without mutual contact, . . .” He considered aether to be the cause
“primary” substance of the Universe and the substance that fills
of gravity and of certain wavelike properties evident in the
up all empty parts of space. He made aether his quintessence
reflection and refraction of light.
(fifth essence) of the Universe, completing the essences of air,
!
water, fire and earth.
physical nonmaterial medium called aether. The basis for this
!
The scientific concept of aether has existed for almost four
support was often one of the following: to avoid the action-at-a-
hundred years, beginning in the seventeenth century with
distance explanation of physical forces (such as gravitational,
Kepler (1620) and Descartes (1637, 1638, 1644). Kepler thought
electrical, and magnetic forces), or to provide a real medium for
that the space between planets is filled with aether, whereas
light waves to propagate in. These are the historical reasons we
Descartes thought that aether permeates the entire Universe,
are calling the physical medium introduced in the previous
filling all voids where matter does not exist. Descartes believed
section ‘aether’.
Since then many scientists have supported the concept of a
24
!
After the existence of aether as a physical medium was
first proposed, numerous aether theories were developed to
1.2.7.1!
AETHER IS AN INVISCID FLUID
explain various physical phenomena. Nevertheless the physical
!
properties and the functions of aether remained unclear and,
gravity as describing the accelerating flow of aether into matter,
for most of the twentieth century, the very existence of aether
we are now able to determine some of the physical properties
was doubted for reasons which will be discussed later in this
of aether. For aether to be the physical medium whose motion
book.
results in the observed gravitational field, aether must be able
!
Before considering the nature of aether, we need to
to flow rapidly. Aether is therefore a fluid. The flow of aether
provide a definition of a fluid. A fluid can be defined as a
associated with gravity appears to be inviscid since there is no
physical medium that can flow (a fluid cannot sustain shear
evidence of internal friction in aether (see Section 3.2). No
stress without moving). A solid can be defined as a physical
internal cohesion then exists within aether. Sir Oliver Lodge
medium that maintains a definite shape and volume due to
(1923) noted that: “ether has nothing of the nature of viscosity.”
Using our interpretation of Poisson’s field equation of
large cohesive internal forces. A solid is the frozen state of a fluid. All material media can flow at some temperature, and so
1.2.7.2!
all material media have a fluid state. If a fluid has no internal
!
friction, it is considered to be capable of ideal flow.
so the field medium of gravity (aether) must exist wherever
!
Viscosity of a fluid is a measure of its internal resistance
there is matter. Moreover, aether must be continuous down to
to gradual deformation by shear stress (resistance to flow).
some very small dimension since the gravitational field appears
This internal resistance to gradual deformation is due to
to be continuous. We can conclude, therefore, that aether must
internal cohesion. A fluid having no viscosity is known as an
be ubiquitous and continuous in our Universe. Only at
inviscid fluid, and the flow of such a fluid is known as inviscid
dimensions approaching the size of the aether particles is the
flow.
aether discontinuous. We will designate an aether particle as an
AETHER IS UBIQUITOUS AND CONTINUOUS
A gravitational field is always associated with matter, and
aetheron. 25
!
As J. J. Thomson (1909) noted, matter occupies “but an
insignificant fraction of the universe, it forms but minute islands in
1.2.7.4!
NUCLEONS ARE STEADY SINKS FOR AETHER
the great ocean of the ether, the substance with which the whole
!
universe is filled.” Lodge (1925a) also noted: “The first thing to
steady sinks for the field medium, aether. Therefore aether
realise about the ether is its absolute continuity.” While some
must flow continuously and uniformly into matter.
Since nucleons do not vary with time, they must constitute
scientists around the beginning of the twentieth century thought that aether was not discontinuous at any dimension
1.2.7.5!
(see Doran, 1975), aether actually consists of extremely small
!
physical particles.
aetherons, we can conclude that the density of aether must be
THE FLUID AETHER HAS HIGH DENSITY
Given the continuity of aether and the very small size of
extremely high (see Section 5.3.1). In comparison, the density of
1.2.7.3! !
AETHER IS HOMOGENEOUS
any material body is very low. The distances between the nuclei
Since matter, which is a field flow sink for aether, exists in
of atoms in a material body are very great relative to the
the form of nucleons (protons and neutrons), aether must
dimensions of aetherons. Moreover, wherever there are
appear continuous at the dimensions of nucleons. This requires
nucleons, the aether flows into these nucleons. This explains
particles of aether to have dimensions that are orders of
how aether can be very dense and yet flow through material
magnitude smaller than nucleons. Moreover, since all protons
bodies without any apparent resistance. We recall Newton's
are identical and all neutrons are identical, aether must also be
words for the cause of gravity: it “penetrates to the very centers of
homogeneous in composition. As Dalton (1808) noted, “the
the Sun and planets, without suffering the least diminution of its
ultimate particles of all homogeneous bodies are perfectly alike.” If
force“ (see Section 1.1.4).
aether were not homogeneous, we would expect variations to be observed in both protons and neutrons since they are field
1.2.7.6!
flow sinks for aether.
!
AETHER IS NEARLY INCOMPRESSIBLE
Since aether consists of extremely small contiguous
particles and so is an extremely dense medium, aether is nearly incompressible. The compressibility of a physical substance is 26
a measure of the relative change in volume that occurs in the
rigidity (which is defined as resistance to relative motion
substance as a result of an applied force. The physical
between particles of the medium). Therefore, aether must
properties of aether causing it to be nearly incompressible
have rigidity.
generally lead to the flow of aether being incompressible. In
!
fluid dynamics, incompressibility describes fluid flow in which
electromagnetism binding atoms together (while their nuclei
no change in fluid volume occurs. When the flow of a fluid is
remain widely separated by a vacuum). This binding force
incompressible, the fluid’s density remains constant.
generally prevents material solid bodies from flowing (it
The rigidity of material solid bodies is due to the force of
provides the mechanism whereby material bodies can exist in
1.2.7.7! !
AETHER IS NONMATERIAL
We can also conclude that aether must be entirely different
the frozen state known as solid). !
For aether to possess a frozen state would require that a
from matter since matter is a field flow sink for aether. The
binding force exist between aether particles. Since aether is
existence of aether must then be independent of matter. We see,
perfectly elastic (see Chapter 3) and flows easily, no binding
therefore, that aether is not a material medium; aether is
forces can exist between aether particles. Without such binding
nonmaterial. Nevertheless, aether is a real physical medium.
forces aether cannot have a frozen state, and so aether does not
Because aether is nonmaterial, it has no material particles that
exist as a solid.
can be in motion, and so it cannot possess temperature; aether
!
is perfectly cold.
in a frozen state. Rather, the rigidity of aether must be due to its
The rigidity of aether, therefore, is not due to aether being
extremely high density whereby all aether particles are
1.2.7.8!
AETHER HAS RIGIDITY
contiguous with other aether particles. Any given aether
While aether has the flow properties of a perfectly
particle has almost no room to move relative to other aether
frictionless fluid, it also has the elastic properties of a perfectly
particles since very little void space exists between aether
elastic solid. This is evident since light waves, which are
particles.
transverse elastic waves, propagate in aether (see Chapter 3).
electromagnetic forces, but by neighboring aetherons that are
Transverse waves can only propagate in a medium that has
contiguous.
!
Aetherons
are
held
in
place
then,
not
by
27
!
We see, therefore, that the mere possession of rigidity by a
is also incompressible, then aether will act as a perfect fluid. No
medium is not a sufficient criterion to indicate whether the
deformation will occur, and so no stresses resulting from elastic
medium is a fluid or a solid. The physical cause of the rigidity
restoring forces will be present.
must also be considered. When this is done we see that, although aether has rigidity (a property previously thought to
1.2.7.10! AETHER AND MATTER
characterize only solids), aether is a fluid and not a solid.
!
Material fluids do not possess rigidity, but the nonmaterial
aware of a couple of perplexing problems that existed in
fluid, aether, does. This explains why it is that when relative
physics. The British physicist Lord Kelvin (1901) referred to
displacements of contiguous aether particles occur, the fluid
these problems as two “nineteenth century clouds” over physics.
aether acts as a perfectly elastic solid. The rigidity of aether
!
causes elastic restoring forces to result when any displacement
was how matter can move through aether if aether is an elastic
of the aether from an equilibrium position occurs.
solid. We see now that this cloud disappears once we realize
As the nineteenth century was ending, scientists were
The first “nineteenth century cloud” that concerned Kelvin
that aether is a fluid.
1.2.7.9! !
AETHER FLOW IS FRICTIONLESS
!
The second “nineteenth century cloud” that concerned
Given the cause of aether rigidity, it is clear that no
Kelvin was how aether can act as a fluid for a solid body
internal resistance exists for the flowing motion of aether (since
moving through it, but act as a solid for light propagating in it.
there is then no relative motion between contiguous aetherons).
We see now that this cloud also disappears once we realize that
Aether, therefore, has zero viscosity and so aether is inviscid.
aether is a fluid that has rigidity.
Moreover, aether has no internal friction, and so the flow of aether is frictionless or ideal. When there exists a void into
1.2.8!
PHYSICAL CAUSE OF GRAVITY
which aether particles can flow without any relative motion
!
occurring between contiguous aether particles, aether will flow
flowing into matter. Matter is required for the generation of a
freely (uniformly) from a region of high pressure to a region of
gravitational field. Since matter is a gravitational field flow
low pressure. Because the flow of aether is always inviscid, if it
sink, matter is a sink into which the gravitational field medium
Gravitational acceleration is the acceleration of aether
28
(aether) is actually flowing. For a spherical material body, the direction of aether flow into the body is vertical across the entire surface of the body. Aether does not flow out of matter because matter is not a field flow source for aether. Aether, therefore, must be consumed at a steady rate by any nucleon. !
Since aether has extremely high density and is nearly
incompressible, the density of aether can be expected to remain constant during the flow of aether into matter, except in proximity to nucleons where the density of aether will increase (see discussion of optical density in Section 3.6.2). The flow of aether will then be incompressible when not in proximity to nucleons, and nearly incompressible when in proximity to nucleons. As the flow of aether converges into a nucleon, the near incompressibility of aether causes an increase in the flow velocity of aether so that the quantity of aether flowing per unit time can remain constant. It is just this increasing flow rate of aether as it approaches matter that creates the observed gravitational acceleration field (see the discussion in Section 1.5.3 on the continuity of aether flow). We can conclude then that the gravitational acceleration field of a material body is caused by three factors: 1.! Aether being ubiquitous in our Universe. 2.! Matter being a field flow sink for aether.
!
If we once again consider the material body of mass M
shown in Figure 1-1, we can conclude that aether must flow continuously into the body from all directions. This aether flow will draw towards the body of mass M the aether that exists in the surrounding region. A negative pressure gradient in the aether will result (see Section 1.6), and this negative pressure gradient will cause a continuous flow of aether into the region surrounding the body of mass M
to replace the aether
constantly disappearing into the body. !
If a second material body of mass m is within this region
as shown in Figure 1-1, the aether flow into both the body of mass M and the body of mass m will draw towards the bodies the aether that exists between them; as a result a negative pressure gradient will be created in the aether. Since no aether flows out of the material bodies, the flow of aether into the two bodies will cause the bodies to be pushed closer together. Moreover, the converging flow of aether into each body will result in an increase in velocity of the aether, and so also of the two material bodies. In other words, the acceleration of aether towards the material bodies results in the acceleration of the two material bodies towards each other. This is gravitational attraction. The force of gravity, therefore, is the action of contiguous particles of aether, and is not some action-at-a-
3.! Aether being nearly incompressible. 29
distance. Lodge (1908) was correct when he stated: “Matter acts
dependent on the mass M , but not on the mass m . The
on matter only through the ether.”
acceleration of the body of mass m is due entirely to (and is
We also see that two material bodies do not directly
equal to) the acceleration of the aether flowing towards the
attract each other; the apparent attraction is due to the flow of
body of mass M .
aether into matter creating a pressure differential. As Lodge
!
(1908) noted, “when the mechanism of attraction is understood, it
perhaps the most fundamental property of gravity: why the
will be found that a body really only moves because it is pushed by
acceleration experienced by a body in a gravitational field is not
something from behind.” Without the physical medium aether,
a function of the quantity of matter in the body. In other words,
gravitational force would not exist. Flowing aether carries with
the flow of aether explains the results of Galileo Galilei’s
it any material body within the flow. This answers the age-old
famous experiments with free falling bodies from which he
question of how matter can appear to act where it is not. Matter
determined that the gravitational acceleration of a body is
acts through the aether it disturbs.
independent of the body’s mass and composition. From his
!
From Poisson’s field equation (1.2-15), we see that the flow
experiments, Galileo came to the conclusion “that in a medium
of aether into a material body of mass M is proportional to the
totally devoid of resistance all bodies would fall with the same
matter density ρ of the body. Matter density is the field flow
speed” (Galilei, 1638).
sink strength for the field medium aether. Obviously the greater
!
the matter density of a body, the greater will be the acceleration in the flow of aether into the body.
bodies is independent of their mass, we see from equation (1.2-1) that gravitational force F will always be proportional to
!
Since mass is a measure of the quantity of matter in a
the mass m of the body on which it acts. Since matter is a field
material body, mass is a measure of field flow sink strength.
flow sink and not a field flow source for aether, the
The acceleration of aether flowing into a body of mass M can
gravitational acceleration field associated with a material body
then be expressed as a function of M . We also see that the
is always directed towards and never away from the body. Therefore gravitational force F will always be attractive.
acceleration of a body of mass m resulting from the body being
Aether flow, therefore, very simply explains what is
Because the gravitational acceleration g experienced by all
drawn along by aether flowing into a body of mass M will be 30
!
The aether flow into a material body of mass M can be
matter. Without aether no material bodies could exist in our
visualized as a liquid flowing towards a drain. The motion of a
Universe.
material body of mass m towards the body of mass M is in
!
some ways similar to a cork floating in the liquid, and being
aether has previously been suggested in a number of studies
drawn towards the drain. Obviously the motion of the cork will
[for example, Riemann, 1853; Thomson, 1870 (as discussed by
be independent of the cork’s mass. The increase in fluid speed
Ball, 1892); Pearson, 1891; and Ellis, 1973, 1974]. Riemann was
as it approaches the drain will cause the floating cork to
aware that gravity could be explained by the continuous flow
accelerate towards the drain. This analogy is not perfect since
of aether into every material particle, and that this requires
the body of mass M is entirely surrounded in all directions by
material particles to be aether flow sinks. Pearson showed that
aether flowing into it. Moreover, a material body of mass m
an inverse distance-squared law of attraction could be obtained
remains a field flow sink itself as it moves toward the field flow
by simply considering ‘negative’ matter to be a field flow sink
sink of mass M .
for aether, and matter to be a field flow source of aether (see
A physical cause of gravity involving sinks or sources of
The aether that flows continually into matter is consumed
Kragh, 2002). Also Kirkwood (1953, 1954) noted that the
and does not flow out. Aether flowing into matter must then
acceleration field of gravity is “strongly suggestive of the flow of a
replace lost energy since matter is in equilibrium with the flow
fluid medium,” and he considered this fluid medium to be
of aether into it (we know from observation that the quantity of
aether.
matter in a material body does not increase nor decrease with
!
time if the body is left undisturbed).
have determined that matter is a field flow sink for aether and
!
expends energy continuously. For matter to exist in the form of
that the acceleration of flowing aether particles constitutes the gravitational acceleration field g . This provides a very simple
nucleons, a continuous and steady flow of aether into matter
physical cause for gravity. Since gravitational acceleration is
must be required. This can also be inferred from the fact that
real, the field medium, aether, and the field flow sink for aether,
matter can be defined as a field flow sink for aether. We see
matter, must both also be real. We see too that the laws of fluid
We can conclude, therefore, that matter by simply existing
In summary, from Poisson’s field equation of gravity we
then that gravitation reveals some aspects of the essence of 31
dynamics as applied to an inviscid fluid are appropriate for
of the resulting equations for the given field. For example, the
describing the mechanism of gravitation.
spherical coordinate system is a logical choice for the case of the two material bodies shown in Figure 1-1.
1.2.9!
GRAVITATIONAL FIELD EXPRESSED IN GENERAL CURVILINEAR COORDINATES
1.2.9.1!
REFERENCE FRAMES
! We will now briefly consider some results that can be obtained by using a general curvilinear coordinate system x1, x 2 , x 3 to represent the gravitational field g of a material body of mass M . This review will be useful as we proceed to the following sections.
!
!
First we note that point vectors and tensors are invariant
entities. Coordinate systems are not physical entities in
under coordinate system transformation for all coordinate
themselves, however. A coordinate system cannot then be a
systems not in motion relative to each other. This invariance is
reference frame. Moreover, a coordinate system cannot interact
assured since point vectors and tensors are defined so as to
with physical fields and forces. A coordinate system can, of
remain unchanged under a rotation of coordinate axes. When a
course, be placed within a frame of reference for use in
change in coordinate system is made, the components of the
describing parameters of the physical entities interacting with
vectors and tensors change in such a way as to leave the vectors
the reference frame.
(
)
Spatial position and motion can only be defined relative to
some real physical entity. Such a physical entity is known as a reference frame or a frame of reference. Coordinate systems are geometrical definitions or concepts that are extremely useful mathematical constructions for describing physical
and tensors invariant. As a result, any point vector or tensor equation valid in one coordinate system will also be valid after
1.2.9.2!
THE METRIC TENSOR
transformation to any other coordinate system not moving
!
relative to the first.
properties of any general curvilinear coordinate system being
!
Although the choice of a general curvilinear coordinate system to represent the vector gravitational field g is arbitrary,
used. The metric tensor also provides the necessary information
in practice this choice is usually made so as to simplify the form
another. It does this by relating an invariant differential
The metric tensor gi j provides information concerning
for translating measurements from one coordinate system to
32
distance ds (which describes an invariant physical quantity) to
gi j = δ i j !
!
i
differentials dx along the coordinate curves: ! ( ds )2 = dr! • dr! = gi j dx i dx j !
(1.2-17)
where dr is the differential of the position vector r , and where
where δ i j is the Kronecker delta defined as:
⎧⎪ 0 for i ≠ j δi j = ⎨ ! 1 for i = j ⎪⎩
!
the Roman indices i and j assume values of 1, 2, and 3. We are using the Einstein summation convention (Einstein, 1916a) which specifies that any index appearing twice in a term is summed over its entire range unless otherwise noted. Since a scalar product is invariant to coordinate transformation, we see from equation (1.2-17) that ( ds ) is invariant as expected. The differential dr is a point vector given by: ! ! ! ! ∂r ! ∂r i ∂r 1 ∂r 2 ! dr = i dx = 1 dx + 2 dx + 3 dx 3 ! (1.2-18) ∂x ∂x ∂x ∂x 2
!
The metric tensor can be used to define coordinate space.
For example, Riemannian space ( R )
N
is a coordinate space
specified by N real coordinates and by the fundamental metric form given by equation (1.2-17). Euclidean coordinate space is that part of Riemannian space containing the Cartesian coordinate system. For orthogonal coordinate systems, we have: !
gi j = 0 !
(i ≠ j ) !
(1.2-19)
For the rectangular Cartesian (Euclidean) coordinate system, we have:
(1.2-20)
(1.2-21)
Equation (1.2-17) then becomes: !
( ds )2 = δ i j dx i dx j = dx i dx i = ( dx1 )
2
( ) ( )
+ dx 2
2
2
+ dx 3 ! (1.2-22)
From equations (1.2-17) and (1.2-18) we can write: ! ! ! ! ∂r ∂r ! ! ∂r i ∂r j ! dr • dr = i dx • j dx = i • j dx i dx j = gi j dx i dx j ∂x ∂x ∂x ∂x ! ! (1.2-23) The metric tensor gi j can be written, therefore, as a function of the position vector r : ! ! ! ! ∂r ∂r ∂r ∂r ! (1.2-24) gi j = i • j = • = gj i ! ∂x ∂x ∂x j ∂x i and so the metric tensor is symmetrical. 1.2.9.3!
THE METRIC TENSOR IN TERMS OF THE GRAVITATIONAL FIELD ! The gravitational field g is, of course, also a function of the position vector r . From equations (1.2-13) and (1.2-2), we can write for a material body of mass M : 33
M M ! ! ! g = ∇ϕ = − G 2 rˆ = − G 3 r ! r r
!
(1.2-25)
and so !
r3 ! r3 ! ! r =− g=− ∇ϕ ! GM GM
(1.2-26)
From equations (1.2-26) and (1.2-24) we therefore obtain: ! !
gi j =
1
[G M ]
2
! ! ∂ ⎡⎣ r 3 g ⎤⎦ ∂ ⎡⎣ r 3 g ⎤⎦ ! • ∂x i ∂x j
(1.2-27)
Information concerning the gravitational field g has been
introduced into equation (1.2-24) resulting in equation (1.2-27). Equations (1.2-24) and (1.2-27) must then be equivalent. This shows that, for any coordinate system being used to describe a gravitational field g , it is possible to express the metric tensor gi j at any point in terms of the gravitational field g at the same point. The field g is determined using Newton’s force law of gravity in the form of equation (1.2-1). !
Obviously, it is also possible to express the metric tensor
gi j in terms of any other vector field (such as an electromagnetic field). The gravitational field, therefore, is not unique in its ability to specify components of the metric tensor. Any physical vector field can be represented in terms of geometry. !
gi j =
!
Using equation (1.2-25), we can write equation (1.2-27) in
1
[G M ]
2
! ! 3 3 ⎡ ⎤ ⎡ ∂ ⎣ r ∇ϕ ⎦ ∂ ⎣ r ∇ϕ ⎤⎦ • ! ∂x i ∂x j
(1.2-28)
In general curvilinear coordinates, ∇ϕ is given by: ! ∂ϕ ! ∇ϕ = k e k ! ∂x
!
(1.2-29)
! where e k is a reciprocal base vector. From equations (1.2-28) and (1.2-29), we see that the metric tensor gi j can be expressed as a function of the first and second derivatives of the gravitational potential ϕ .
1.3! !
EINSTEIN’S GENERAL THEORY OF RELATIVITY In 1915, more than one hundred years after Poisson
derived his field equation, Einstein published his field equation of gravity (Einstein, 1915a, b, c, d). He obtained his field equation of gravity as a result of his ingenious efforts to extend his special theory of relativity so that it would also encompass gravity. Einstein referred to his work on gravity as the general theory of relativity. !
The special theory of relativity (which will be discussed in
Chapter 4) is founded on two postulates (assumptions):
terms of the gravitational potential ϕ : 34
1.! There is no preferred non-accelerating reference frame, and so physical laws described with respect to one non-accelerating reference frame will have the same form when described using any other non-accelerating reference frame.
(essentially at each point) by a uniformly accelerating frame of reference. Each different point r in the gravitational field g will generally require a differently accelerating reference frame in order to eliminate gravity. The acceleration of these reference frames is such that the reference frames are freely falling and so
2.! The speed of light in vacuum is a constant independent of the state of motion of its source.
will experience no gravitational force (since a freely falling
The general theory of relativity is based upon the
reference frames (which are essentially infinitesimal parts of the
assumption that a uniform acceleration field and a spatially
gravitational field itself), the gravitational force at each point is
uniform gravitational field have an equivalent effect on a non-
eliminated. A theory of gravitation can then be developed
rotating material body (Einstein, 1907a). This assumption is
based upon the geometric structure of reference frames as
known as the weak equivalence principle. By making this
revealed in the transformation properties of the coordinate
assumption, Einstein was in effect assuming that the inertial
systems used to represent these reference frames.
mass and the passive gravitational mass of material bodies
!
must be identical (see Section 2.3.5).
field is not defined so as to be a true physical field; that is, it
!
does not represent some attribute of real particles of a physical
!
Another form of the equivalence principle known as the
body is weightless). By transforming to such freely falling
In the theory of general relativity then, the gravitational
strong equivalence principle assumes that locally (in an
medium.
infinitesimal region of space in which the gravitational field is
!
uniform), all laws of physics will follow the postulates
transformed away in general relativity, the special theory of
(assumptions) of special relativity in some non-rotating freely-
relativity should be valid within each very small freely falling
falling reference frame.
reference frame. According to the theory of general relativity
!
From these initial assumptions, Einstein constructed his
then, the speed of light is constant only within vanishingly
general theory of relativity. Based upon his principle of
small non-accelerating reference frames. For more realistic
equivalence, he found that he could replace gravity locally
physical reference frames, the speed of light is no longer a
Because the gravitational field at each point is effectively
35
constant (Einstein, 1916b; and Cook, 1994), and light no longer
!
must travel in a straight line. In a very real sense, therefore, the
Minkowski (and then Einstein) made the replacement:
theories of special relativity and general relativity are not
!
compatible. As Pauli (1921) noted, “the special theory of relativity can only be correct in the absence of gravitational fields.” Dingle
To obtain equation (1.3-1) in quasi-Euclidean form,
c dt → i c dt !
(1.3-4)
where i 2 = −1 so that equation (1.3-1) becomes:
ds 2 = dx 2 + dy 2 + dz 2 + c 2 dt 2 !
(1972) observed that: “Einstein’s general theory does not include the
!
second postulate of the special theory but depends only on a
Equation (1.3-1) can then be written in the form of a four-
generalization of the first postulate.”
dimensional quasi-Euclidean coordinate system:
1.3.1!
!
!
MATHEMATICAL EXPRESSION
In an effort to incorporate the assumptions of special
relativity into his general theory of relativity, Einstein (1916b) followed Minkowski (1908, 1909) in expressing the space-time differential ds using four-dimensional coordinates in the form:
ds 2 = dx 2 + dy 2 + dz 2 − c 2 dt 2 !
!
(1.3-1)
( ds )2 = ( dx1 )
2
(1.3-5)
( ) + ( dx ) + ( dx ) !
+ dx 2
2
3 2
4 2
(1.3-6)
where !
dx 4 = i c dt !
(1.3-7)
The term dx 4 now acts like a spatial coordinate (although it remains a different type of entity). Equation (1.3-6) can be compared to the three-dimensional rectangular Cartesian
where c is the speed of light in a vacuum and dt is the time
coordinate system equation (1.2-22).
differential. The constraint that the speed of light in a vacuum
!
is a constant is included in equation (1.3-1) in the form of a
(associated with a freely falling reference frame) from a general
fourth dimension. For a light ray we must have:
curvilinear coordinate system (associated with a reference
ds = 0 !
!
(1.3-2)
To transform to a quasi-Euclidean coordinate system
frame in which gravitational forces are acting), ( ds ) can be 2
since
written in the form:
! ! 2 2 2 2 2 dr dr ⎛ dr ⎞ ⎛ dx ⎞ ⎛ dy ⎞ ⎛ dz ⎞ ⎛ ds ⎞ ! c =⎜ ⎟ +⎜ ⎟ +⎜ ⎟ = • = ⎜ ⎟ = ⎜ ⎟ ! (1.3-3) ⎝ dt ⎠ ⎝ dt ⎠ ⎝ dt ⎠ ⎝ dt ⎠ dt dt ⎝ dt ⎠
!
2
( ds )2 = gα β dxα dx β !
(1.3-8) 36
where the Greek indices α and β assume values of 1, 2, 3, and
of the metric tensor. For the sink/source term of Poisson’s
4. As a result of this transformation, space and time dimensions
equation, Einstein chose the stress-energy tensor Tα β which
become intertwined as space-time and are no longer separate
includes not only gravitational energy, but also electromagnetic
for many curvilinear coordinate systems. Coordinates therefore
energy. Based partly upon symmetry considerations, Einstein
lose their expected geometric and temporal interpretation.
then decided that the stress-energy tensor Tα β must be related
!
To obtain the mathematical equations that reflect a change
from a reference frame in which gravity is acting to a freely falling reference frame in which gravity is not acting, an appropriate coordinate system must first be selected and associated with the gravitational field. Then the corresponding transformation relations can be developed. Einstein (1916a) noted that the metric tensor components gα β are “to be regarded from the physical standpoint as the quantities which describe the gravitational field in relation to the chosen system of reference.” The equivalent statement for Poisson’s field equation of gravity is given by equation (1.2-27). !
Einstein knew that if an analog of Poisson’s equation
exists for the general theory of relativity, then this equation could be written in terms of both the metric tensor gα β and the gravitational potential. He assumed that the metric tensor
to Rαβ , a contracted form of the Riemann tensor R αµ β γ , by the equation: !
1 Rα β − gα β R = −κ Tα β ! 2
(1.3-9)
where R is the Ricci curvature and κ is a constant. This equation states that variations in the geometry of space-time are proportional to the stress-energy of the source of the variations. By constraining equation (1.3-9) to be identical to Poisson’s equation in the limit of a weak gravitational field, κ is found to be given by: !
κ=
8π G ! c4
(1.3-10)
and so equation (1.3-9) becomes: !
would, in fact, play the role of the gravitational potential in his
1 8π G Rαβ − gα β R = − 4 Tα β ! 2 c
(1.3-11)
general theory of relativity. This assumption led him to select
!
Equation (1.3-11) is Einstein’s field equation of gravity
the Riemann tensor Rαµ β γ as the differential part of Poisson’s
from the general theory of relativity (see Einstein, 1916b; de
equation since the Riemann tensor contains second derivatives
Sitter, 1921; Eddington, 1921b; McVittie, 1956; Synge, 1960; 37
Bergmann, 1962b; Adler et al., 1965; Rindler, 1969; Ehlers, 1973;
transformations will ( ds ) be invariant. We will see that the
Kilmister, 1973; Atwater, 1974; Dirac, 1975; Geroch, 1978;
Lorentz transformations modify both displacement and time so
Clarke, 1979; Bose, 1980; Jones, 1981; Wald, 1984; Schutz, 1985;
that they become functions of velocity (Section 4.23.1). Both
Martin, 1988b; Hughston and Tod, 1990; Kenyon, 1990; Foster
displacement and time are then dependent upon the velocity of
and Nightingale, 1995; Kriele, 1999; Ludvigsen, 1999; Callahan,
the frame of reference being used.
2
2000; Hartle, 2003; Carroll, 2004; Capria, 2005; Dalarsson and Dalarsson, 2005; Khriplovich, 2005; Hobson et al., 2006;
1.3.2!
Plebanski and Krasinski, 2006; Ferraro, 2007; Grøn and Hervik,
!
2007; Walecka, 2007; Woodhouse, 2007; Chow, 2008; Lieber,
field equation (1.3-11) is that the existing distribution of matter
2008; Choquet-Bruhat, 2009; Ryder, 2009; and Ni, 2015).
and energy in our Universe influences the geometry of space-
!
equation (1.3-11) can be interpreted, it is necessary to transform
time. In other words, the existence of a relation between the gravitational field g and the metric tensor gαβ is taken to
the differential i c dt back using:
indicate that space and time are being modified (warped) by
Finally, we note that before any results obtained using
i c dt → c dt !
!
We also note that
(1.3-12)
( ds )2
given by equation (1.3-1) in four-
dimensional coordinates is not the result of a scalar product as is
( ds )2
given by equations (1.2-22) and (1.2-17) in three-
dimensional coordinates. This can be seen by writing equation (1.3-1) in the form: !
( ds )
2
! ! = dr • dr − c 2 dt 2 !
PHYSICAL INTERPRETATION
The physical interpretation proposed by Einstein for his
the presence of matter and energy. The nature of space-time in a given region is then determined by the presence of matter and the type of physical processes occurring in the vicinity.
1.3.3! !
PHYSICAL CAUSE OF GRAVITY
In the general theory of relativity, gravitational force is
eliminated and gravitational effects are explained in terms of (1.3-13)
Therefore, ( ds ) in four-dimensional coordinates will generally 2
not be invariant to coordinate transformation. In fact, only for coordinate transformations of the type known as Lorentz
the curvature of space-time. Gravity is then not really a force at all, but the result of the space-time geometry through which a material body moves. Gravitational acceleration results then not from force, but by space-time changing in a nonlinear 38
manner. A force-free material point particle responds to the
!
curvature of space-time by moving along a geodesic of space-
case must have spherical symmetry, and so only time and
time as defined by:
radial distance will be modified by gravity. Moreover, he knew
∫
δ ds = 0 !
!
that the solution of Einstein’s field equation of gravity must (1.3-14)
where δ indicates variation. The shape of space-time, therefore, affects the motion of material point particles. Or, as Misner et al. (1970) state, “space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.” A material point particle that moves along a geodesic is free of any force. Force is only required to cause a particle to deviate from a
!
provide the line element of space-time ds which, in turn, provides the space-time curvature that is gravity. At great distances from all matter, a material point particle is thought to move in a straight line; at infinity the metric must then become Euclidean. Schwarzschild assumed, therefore, that the solution in spherical coordinates to Einstein’s field equation of gravity has the form: !
geodesic.
1.3.4!
THE SCHWARZSCHILD SOLUTION
No general solution to Einstein’s field equation of gravity
has ever been found, and such a solution may, in fact, not even exist (Torretti, 1983). Immediately after Einstein published his equations of general relativity in 1915, Schwarzschild, from the battlefield of World War I, was able to determine an exact solution to the field equation for the special case of a stationary gravitational field surrounding a large spherical material body in a vacuum. Einstein was surprised that such an exact solution of his field equation could be formulated.
Schwarzschild (1916a, b) knew that the solution for this
ds 2 = A ( r ) dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 − B ( r ) c 2 dt 2 !
(1.3-15)
where the functions A ( r ) and B ( r ) depend only upon the radial coordinate r . He used the field equation of general relativity to obtain the two radial functions A ( r ) and B ( r ) . The solution Schwarzschild (1916a, b) derived can be written in spherical coordinates as: ! ds 2 = !
⎡ GM ⎤ dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 − ⎢1− 2 2 ⎥ c 2 dt 2 c r⎦ ⎡ GM ⎤ ⎣ 1− 2 ⎢ ⎥ c2 r ⎦ ⎣ ! (1.3-16) 1
where M is the mass of the large spherical material body that is the source of the gravitational field. The differential ds given by 39
equation (1.3-16) is known as the Schwarzschild solution or
be considered to have no volume or extension. Unlike a
the Schwarzschild metric.
material point particle, however, the density of a fluid point
!
particle is not necessarily constant.
Birkhoff (1923) showed that the Schwarzschild solution
We will now consider matter of mass ΔM and density ρ
(1.3-16) of the general relativity field equation (1.3-11) is the
!
most general solution having spherical symmetry. Adler et al.
contained within a very small spherical volume element ΔV .
(1965) noted that the Schwarzschild solution “must be considered celestial mechanics; it is an exact solution, which corresponds
We will take the matter in ΔV to be the reference frame. Aether at a point r outside of ΔV is flowing continuously towards the matter within ΔV with a flow velocity υ that is a function of r ,
historically to Newton’s treatment of the 1 r 2 force law of classical
the position vector whose origin is centered in the matter
gravitational theory.”
within ΔV . We will focus on a very small parcel of the aether flowing towards the matter within ΔV . We will abstract this
1.4!
very small parcel of aether to an infinitesimal size, making it a
to be the main achievement of general relativity theory in the field of
AETHER FIELD EQUATION OF GRAVITY
fluid point particle. Note that a fluid point particle of aether is
With the insight into the physical cause of gravity
not an aetheron, but is an abstraction of many aetherons. Since
obtained from Poisson’s field equation of gravity, we will now
in this section we will be describing fluid flow by tracing the
develop a new gravitational field equation. This field equation,
dynamical history of a single fluid point particle of fixed
based upon the concept of flowing aether, provides more
identity in its trajectory, we will be using a Lagrangian
detailed information concerning the physical processes that
specification of the flow.
produce gravitational fields.
!
!
We will define aether density ζ as the quantity of aether
per unit volume. Aether density ζ is then analogous to matter We will define a fluid point particle to be a very small
density ρ (see Section 1.2.3). Therefore, aether momentum per unit volume or aether momentum density ζ υ is analogous to
fluid parcel that has been abstracted to an infinitesimal size so
the momentum density of matter. The change with time in
1.4.1!
MATHEMATICAL EXPRESSION
that its dimensions can be neglected. A fluid particle can then 40
aether momentum density for a fluid point particle of aether as it flows along is ∂(ζ υ ) ∂t . !
Since the gravitational force due to matter within ΔV is
proportional to ΔM , the time derivative of the momentum density of aether flowing through the surface area ΔS of ΔV must also be proportional to ΔM . This is consistent with the following facts: 1.! Matter within ΔV is a field flow sink for aether. 2.! Nearly incompressible inviscid flow of aether towards this sink causes fluid point particles of aether to accelerate as they approach the sink. Their momentum will change with time in proportion to the strength of the sink. 3.! Mass ΔM is a measure of the quantity of matter (strength of the sink) within ΔV .
and k is an undetermined constant. Using equation (1.2-5), we can also write equation (1.4-1) as: !
!
−
∫∫
∫∫
ΔS
∂ ! (ζ υ ) • nˆ dS = 4 π G k ∂t
∫∫∫
ΔV
ρ dV !
(1.4-2)
From Gauss’s theorem given in equation (C-59) of Appendix C, we can rewrite the left side of equation (1.4-2) as: !
−
∫∫
ΔS
∂ ! ζ υ ) • nˆ dS = − ( ∂t
! ∂ ! ∇ • (ζ υ ) dV ! ∂t ΔV
∫∫∫
(1.4-3)
and so equation (1.4-2) becomes: !
! ∂ ! ∇ • (ζ υ ) dV = − 4 π G k ∂t ΔV
∫∫∫
∫∫∫
ΔV
ρ dV !
(1.4-4)
Since ΔV is arbitrary, we must have: !
We can therefore write:
∂ ! (ζ υ ) • nˆ dS = 4 π G k ΔM ! ∂t
−
∂ ∇ • (ζ υ ) = − 4 π G k ρ ! ∂t
(1.4-5)
Taking the time derivative, we obtain: (1.4-1)
spherical volume element ΔV which has surface area ΔS . The
! ⎡ ! ∂ζ ! ⎤ (1.4-6) ∇ • ⎢ζ α + υ ⎥ = − 4 π G k ρ ! ∂t ⎦ ⎣ where α is the time variation ∂υ ∂t of the aether flow velocity.
constant of proportionality in this equation is taken to be
Using the vector identity given in equation (C-16) of Appendix
4 π G k where G is the gravitational constant of proportionality
C, this equation can also be written as:
ΔS
where nˆ is the outward directed unit normal vector to the
!
from Newton’s force law of gravity given in equation (1.1-1), 41
!
! ! ! ! ! ⎡ ∂ζ ! ⎤ ζ ∇ • α + α • ∇ζ + ∇ • ⎢ υ ⎥ = − 4 π G k ρ ! ⎣ ∂t ⎦
flow acceleration as noted in Section 1.2-8, we must therefore (1.4-7)
!
To determine the constant k , we will now assume that the gravitational field is stationary. We will consider a point r that is not in proximity to nucleons and that is on the flow trajectory (pathline or particle path) of a given fluid point particle of aether. As was noted in Section 1.2.8, the flow of aether in a gravitational field is incompressible when not in proximity to
have: !
!
!
(1.4-8)
∇ζ = 0 !
(1.4-9)
Equation (1.4-7) can then be written as: ! ! ! ζ ∇ •α = − 4π G k ρ ! Using equation (1.2-10), equation (1.4-10) becomes: ! ! ! ! ! ζ ∇ •α = k ∇ • g ! !
k =ζ !
(1.4-13)
Equations (1.4-5) and (1.4-6) can then be written as: !
1 ! ∂ ! ∇ • (ζ υ ) = − 4 π G ρ ! ζ ∂t
(1.4-14)
!
1 ! ⎡ ! ∂ζ ! ⎤ ∇ • ⎢ζ α + υ ⎥ = − 4 π G ρ ! ζ ∂t ⎦ ⎣
(1.4-15)
constant: !
(1.4-12)
and so the constant k is:
nucleons. Therefore, the density of aether in such a flow will be
∂ζ =0! ∂t
α = g!
Using equation (1.2-10), we can also write equations (1.4-14) and (1.4-15) as: !
! ! 1 ! ∂ ! ∇ • (ζ υ ) = ∇ • g ! ζ ∂t
(1.4-16)
!
1 ! ⎡ ! ∂ζ ! ⎤ ! ! ∇ • ⎢ζ α + υ ⎥ = ∇ • g ! ζ ∂t ⎦ ⎣
(1.4-17)
(1.4-10)
(1.4-11)
At any point r not in proximity to nucleons, the aether
flow acceleration toward the matter of mass ΔM will simply equal α because of the relations given in equations (1.4-8) and (1.4-9). Since the gravitational acceleration g is just the aether
Equations (1.4-16) and (1.4-17) describe the gravitational field g at a point r completely in terms of properties of the aether at this point. Since equations (1.4-14) through (1.4-17) are simply different forms of the same equation, each of these equations can be considered to be the aether field equation of gravity. 42
! !
! area. The gravitational flux vector Λ G is then the quantity of
Equation (1.4-4) can now be written as:
∫∫∫
ΔV
∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV = − 4 π G ζ ∂t
∫∫∫
ΔV
ρ dV !
aether flowing per unit time through a unit area and is given (1.4-18)
where we have interchanged the time and space differential operations and have used the equality k = ζ . We note that the
by:
! ! ΛG = ζ υ !
!
and equations (1.4-19) and (1.4-20) can be written as:
volume flux of an entity is defined to be a scalar that specifies the quantity of the entity flowing per unit time per unit volume. In equation (1.4-18), the term ∇ • (ζ υ ) is the aether flux (quantity of aether flowing per unit time per unit volume) into
ΔV . The aether field equations of gravity (1.4-14) and (1.4-16) can be written in terms of aether flux: !
!
1 ∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = − 4 π G ρ ! ζ ∂t ! ! 1 ∂ ! ! ⎡∇ • (ζ υ ) ⎤⎦ = ∇ • g ! ζ ∂t ⎣
(1.4-21)
!
1 ∂ ! ! ⎡∇ • Λ G ⎤⎦ = − 4 π G ρ ! ζ ∂t ⎣
(1.4-22)
!
! ! 1 ∂ ! ! ⎡⎣∇ • Λ G ⎤⎦ = ∇ • g ! ζ ∂t
(1.4-23)
Bridgman (1941) stated that the need in physics for a (1.4-19)
gravitational flux vector is imperative.
1.4.2! (1.4-20)
!
PHYSICAL INTERPRETATION
The aether field equation of gravity can be interpreted
physically as describing the flow acceleration of a real physical
The units of G are cm 3 sec-2 gm -1 as given in equation (1.1-2).
medium, aether, into matter. The same physical interpretation
These units can be seen to be correct and perhaps more
pertains to Poisson’s field equation of gravity. This agreement
intuitive from equation (1.4-19) than from equation (1.1-1).
in interpretation is to be expected since the aether field
!
The flux vector of an entity is defined to be a vector
equation of gravity is based upon knowledge obtained from
representing the momentum density of the entity (the entity density multiplied by the entity velocity), and so is the
Poisson’s field equation of gravity. ! The gravitational field g at a point r is completely
quantity of the entity flowing per unit time through a unit
determined by properties of the aether at this point as can be 43
seen from equations (1.4-16) and (1.4-20). The energy of
!
If aether flow is accelerating without the presence of
gravitational fields must then reside entirely within the aether.
matter, an acceleration field will still exist. We will denote such an acceleration field as an aether acceleration field, and we will
1.4.3! !
PHYSICAL CAUSE OF GRAVITY
Gravitational acceleration is caused by aether flow that is
reserve use of the term gravitational acceleration field for accelerations resulting from gravitational aether flow.
accelerating towards matter. As we have seen, this accelerating flow of aether results from matter being a field flow sink for the
1.5!
nearly incompressible aether. We will designate all aether flow
CONTINUITY OF AETHER DENSITY IN A VACUUM
(incompressible and nearly incompressible) resulting from
!
aether flowing into matter as gravitational aether flow.
space having no field flow sinks for aether. Such a region of
!
Matter has an associated gravitational acceleration field
space contains a vacuum since matter is the only field flow sink
only because there is an accelerated flow of aether into matter
for aether, and absence of matter defines a vacuum. We will
resulting from matter being a field flow sink for aether. The
describe the aether flow in the vacuum of this region of space
field particles for gravitational fields are aetherons, and the
by considering a very small volume element ΔV of vacuum.
flow acceleration of aetherons is the attribute of the field
We will let the volume element ΔV itself remain constant with
particles that is the gravitational field. Finally, we note that the
time, although aether can flow freely through the surface of the
aether field equation of gravity is completely consistent with
element. Such a volume element is known as a control volume
both Newton’s force law of gravity and Poisson’s field equation
in fluid mechanics (see Granger, 1985), and the open surface of
of gravity. Moreover the aether field equation of gravity shows
the control volume is known as the control surface.
us that the gravitational field is linear. !
For a large material body such as the Earth, the flow of
The quantity of aether Q in the volume element ΔV of vacuum is:
aether from outside the body into the body will always be vertical. There is then no aether flow tangential to the surface of the Earth.
We will now consider the flow of aether in a region of
!
Q=
∫∫∫
ζ dV !
ΔV
(1.5-1)
44
where ζ is the aether density. This equation can be compared with equation (1.2-3). Because the volume element ΔV itself is not a function of time, the rate of any increase of the total quantity of aether Q within the volume element ΔV is given by: !
∂Q = ∂t
∫∫∫
ΔV
∂ζ dV ! ∂t
(1.5-2)
Since we are considering a region of space that has no field flow sinks for aether, and since no field flow sources of aether are known to exist in our Universe, there will be no sinks or sources for aether flow within ΔV . Therefore ∂Q ∂t must equal the rate of influx of aether through the surface area ΔS bounding ΔV : !
∂Q =− ∂t
∫∫
ΔS
! ζ υ • nˆ dS = −
∫∫
ΔS
! ! ζ υ • dS !
∫∫∫
!
ΔV
! ⎤ ⎡ ∂ζ ! + ∇ • ζ υ ( )⎥ dV = 0 ! ⎢⎣ ∂t ⎦
(1.5-5)
Since the volume element ΔV is arbitrary within the vacuum, we obtain the equation of continuity for aether density:
∂ζ + ∇ • (ζ υ ) = 0 ! ∂t
! !
(1.5-6)
From this equation of continuity for aether density, we see
that the rate at which the density of aether ζ increases within a very small volume element ΔV will equal the convergence of flux of aether into ΔV (assuming no sinks or sources for aether are present within ΔV ). In fact, this equation can be interpreted as stating that no sinks or sources for aether exist within ΔV . Similar continuity equations apply for any physical density
(1.5-3)
entity that has no sinks or sources within ΔV (and so the physical density entity cannot be destroyed or created within
where nˆ is the outward directed unit normal vector to the surface area ΔS , and υ is the velocity of aether flowing into
ΔV ).
ΔV . Invoking Gauss’s theorem given in equation (C-59) of
1.5.1!
Appendix C, we can rewrite the relation for ∂Q ∂t given in
!
equation (1.5-3) as:
in equation (C-16) of Appendix C, we have for any point within a small constant volume element ΔV moving with a velocity υ :
!
∂Q =− ∂t
∫∫∫
ΔV
! ! ∇ • (ζ υ ) dV !
From equations (1.5-2) and (1.5-4) we then have:
(1.5-4)
!
THE SUBSTANTIVE DERIVATIVE
Expanding equation (1.5-6) using the vector identity given
∂ζ + υ • ∇ζ + ζ ∇ • υ = 0 ! ∂t
(1.5-7) 45
The first two terms of equation (1.5-7) can be defined to be: !
Dζ ∂ζ ! ! ≡ + υ • ∇ζ ! Dt ∂t
(1.5-8)
where Dζ Dt is the total rate of change or total derivative of aether density ζ within the constant volume element ΔV . ! !
The derivative D Dt is defined as:
D ∂ ! ! ≡ +υ •∇ ! Dt ∂t
spatial rate of change results from the convection of the constant volume element ΔV from one position within the medium (where the physical quantity has one value) to a different position within the medium (where the physical quantity may have another value). For a steady-state medium, the local time derivative will always be zero. If no convection is occurring or if the medium
(1.5-9)
When applied to some physical quantity, the derivative D Dt is
is homogeneous, the convective derivative will always be zero.
1.5.2!
known as the substantive derivative, substantial derivative, total derivative, or material derivative of the physical quantity. This derivative is calculated with respect to a coordinate system attached to a constant volume element ΔV moving with velocity υ . Because the derivative D Dt is a scalar operator, it
! !
INCOMPRESSIBLE FLOW OF AETHER IN A VACUUM
From equations (1.5-7) and (1.5-8) we obtain:
Dζ + ζ ∇ •υ = 0 ! Dt
(1.5-10)
When flowing aether is not in proximity to a nucleon, no
can be applied to scalars, vectors, and tensors.
compression of the aether will occur, and so aether density will
!
be constant. From equations (1.4-8), (1.4-9), and (1.5-8), we
The substantive derivative consists of two operator terms:
the term ∂ ∂t is the temporal rate of change and is known as the local time derivative or the Eulerian derivative; the term υ • ∇ is the spatial rate of change and is known as the convective derivative. The local time derivative represents the change with respect to time of the physical quantity, and the convective derivative represents the change of the physical quantity with respect to spatial position within the flow. The
obtain the equation for the incompressible flow of aether: !
Dζ =0! Dt
(1.5-11)
This equation can be considered to be the definition of incompressible flow for a fluid of density ζ and flow velocity υ (Kellogg, 1929). For such a flow, we have from equations (1.5-11) and (1.5-10): 46
∇ •υ = 0 !
!
(1.5-12)
and so the incompressible flow of aether is solenoidal. Equation (1.5-12) will always be true for any fluid whose flow is incompressible. While aether is not completely incompressible, the flow of aether is incompressible except in proximity to nucleons. !
When the flow of aether is incompressible, the aether
density ζ remains constant by definition. Equation (1.5-2) can then be written as:
∂Q = ∂t
!
∂ζ dV = 0 ! ΔV ∂t
(1.5-13)
∫∫
! ! υ • dS = − ζ
ΔS
∫∫∫
! ! ∇ • υ dV = 0 !
(1.5-14)
ΔV
quantity of aether within any constant volume element ΔV (not containing field flow sinks or sources) remains constant for the incompressible flow of aether.
!
streamline of the flow. Streamlines will then always indicate instantaneous fluid flow direction. By definition, therefore, no flow can cross a streamline, and streamlines can never cross one another. The collection of all streamlines in a region at a given instant constitutes the instantaneous flow pattern. Note that a of time while a fluid point particle trajectory describes the flow !
where we once again obtain equation (1.5-12). Therefore, the
1.5.3!
each of the points of the line, then the line is defined as a
pattern over some finite period of time.
and so from equations (1.5-3) and (1.5-4), we have: !
dynamics. If at any given instant in a flowing fluid, the tangent to a line is in the direction of the velocity vector υ of the fluid at
streamline describes the velocity field pattern at a given instant
∫∫∫
∂Q = −ζ ∂t
along a line in the flow known as a streamline in fluid
STREAMLINES OF AETHER FLOW IN A VACUUM
Aether flow is described in terms of a velocity field. We
will now consider incompressible aether flow in a vacuum
If the flow is steady (does not vary with time), the flow velocity υ of fluid point particles will be constant so that at ! each point ∂υ ∂t = 0 . Streamlines will then coincide with fluid point particle trajectories. If the flow is unsteady (varies with time), the flow velocity υ of fluid point particles will not be constant at each point. Streamlines will then vary with time, and will not coincide with fluid point particle trajectories.
1.5.4! !
CONTINUITY OF AETHER FLOW IN A VACUUM
A surface enclosing a portion of a fluid’s streamlines will
have the shape of a tube and is known as a stream tube. 47
Maxwell (1855) noted that for stream tubes “Since this surface is
!
υ1 ΔS1 = υ 2 ΔS2 !
(1.5-17)
generated by lines in the direction of fluid motion no part of the fluid can flow across it, so that this imaginary surface is as impermeable to the fluid as a real tube.” The boundary of a stream tube consists of streamlines. !
If a volume element ΔV is taken to be a section of a stream
tube of incompressible aether flow in a vacuum, then from equation (1.5-14) we obtain: !
∫∫
! ! υ • dS = 0 !
(1.5-15)
ΔS
where ΔS is the surface area of this section of stream tube. From equations (1.5-14) and (1.5-12) we see that fluid flow described by equation (1.5-15) is solenoidal. !
If ΔS1 and ΔS2 are the cross-sectional surface areas of the
stream tube section as shown in Figure 1-3, equation (1.5-15) can be written as: !
∫∫
! ! υ • dS = 0 =
ΔS
∫∫
! ! υ • dS +
ΔS1
∫∫
! ! υ • dS !
Figure 1-3! Incompressible fluid flow through a section of a ! stream tube. (1.5-16)
ΔS2
!
An integral over the sides of the tube is not included on the right side of equation (1.5-16) since υ is orthogonal to dS for the sides of the tube (by definition of a streamline), and so no flow occurs through the sides. We then have:
where υ1 is the speed of aether flowing through ΔS1 , and υ 2 is the speed of aether flowing through ΔS2 . Note υ and dS are in opposite directions for ΔS1 , but in the same direction for ΔS2 . !
Equation (1.5-17) is the equation of continuity of
incompressible flow, and is valid for any stream tube of aether flow in a vacuum not in proximity to a nucleon. From this 48
equation we see that the volume of flowing aether through area
where the area element vector is taken as positive in the
ΔS1 per unit time is the same as the volume of aether flowing
outward direction. Using the gradient theorem given in
through area ΔS2 per unit time. As Maxwell (1855) stated “The
equation (C-62) of Appendix C, we can rewrite equation (1.6-1)
quantity of fluid which in unit time crosses any fixed section of the
as:
tube is the same at whatever part of the tube the section be taken.” If the stream tube contracts so that ΔS1 > ΔS2 , we must then have
υ 2 > υ1 . Aether must accelerate, therefore, within the stream tube section between the surface areas ΔS1 and ΔS2 . This is exactly the cause of gravitational acceleration since aether converges into a nucleon.
1.6!
steady flowing aether is acting. When the aether in a region is flowing in a steady manner, no shear stresses will be acting. Therefore, the force density f must result from normal forces acting upon the surface area ΔS of the volume element ΔV . Such normal forces can be expressed in terms of the scalar
ΔV
∫∫
ΔS
(1.6-2)
Since ΔV is arbitrary, the force density f of flowing aether in terms of the hydrodynamic pressure P of aether is given by: ! (1.6-3) f = − ∇P !
Since gravitational force results from the steady flow of aether into matter, gravitational force density f can be expressed in terms of the gradient of the hydrodynamic pressure P of aether using equation (1.6-3). Gravitational force density f is, therefore, the negative pressure gradient resulting from the steady flow of aether into matter.
1.7!
hydrodynamic pressure P :
∫∫∫
ΔV
!
We will now consider a small volume element ΔV of aether upon which a force density f (force per unit volume) of
!
∫∫∫
that, if unbalanced, will cause aether acceleration.
!
! P dS !
ΔV
! ∇P dV !
A pressure differential in aether is, therefore, a force density
FORCE DENSITY OF FLOWING AETHER
! f dV = −
∫∫∫
!
! f dV = −
(1.6-1)
!
EQUATION OF MOTION FOR FLOWING AETHER We can obtain the general equation of motion for flowing
aether (where both time and position can vary) by calculating the aether flow acceleration using the substantive derivative of 49
aether flow velocity υ . The substantive derivative is given in equation (1.5-9). We obtain: Dυ ∂υ ! = + υ •∇ υ ! Dt ∂t
(
as:
)
(1.7-1)
Using the vector identity given in equation (C-46) of Appendix C, equation (1.7-1) becomes: Dυ ∂υ ∂υ ⎛ υ • υ ⎞ ! = + υ •∇ υ = + ∇⎜ −υ × ∇ ×υ ! ⎝ 2 ⎟⎠ Dt ∂t ∂t
(
)
(
)
!
!
! ! ! Dυ f =ζ = − ∇P ! Dt
(1.7-6)
or
! ! ! ! ! !⎤ ⎡ ∂υ f = − ∇P = ζ ⎢ + υ • ∇ υ ⎥ ! ⎣ ∂t ⎦
(
! (1.7-2)
)
(1.7-7)
This equation can be considered Euler’s equation of motion for
or !
aether acceleration, and so can be written using equation (1.6-3)
the flow of aether.
! ! ! Dυ ∂υ ! ! ! ∂υ ! ⎛ υ 2 ⎞ ! ! ! = + υ •∇ υ = + ∇⎜ ⎟ −υ × ∇ ×υ ! Dt ∂t ∂t ⎝ 2⎠
(
)
(
)
(1.7-3)
In a region of space in which a steady aether flow exists,
we will have at any given point: ∂υ ! =0! ∂t
For the steady flow of aether, the force density f of the
!
steady flow of aether at a given point on a streamline can be expressed using equation (1.7-5) as:
! ! ⎡ ! ⎛ υ2 ⎞ ! ! ! ! ! ! f = − ∇P = ζ υ • ∇ υ = ζ ⎢∇ ⎜ ⎟ − υ × ∇ × υ ⎣ ⎝ 2⎠
(
! (1.7-4)
)
(
)
⎤ ⎥ ! (1.7-8) ⎦
This equation can be considered Euler’s equation of motion for
and so equation (1.7-3) becomes for the steady flow of aether: ! Dυ ! ! ! ! ⎛ υ2 ⎞ ! ! ! ! (1.7-5) = υ • ∇ υ = ∇⎜ ⎟ −υ × ∇ ×υ ! Dt ⎝ 2⎠
the steady flow of aether. This equation is nonlinear. Note that in equations (1.7-7) and (1.7-8) f is not necessarily gravitational
Equation (1.7-5) is the general equation of motion for the
1.8!
(
)
(
)
steady flow of aether. !
The force density f of the flowing aether at a given point
on a streamline is given by the product of aether density and
force density.
!
GRAVITATIONAL ENERGY
Since the gravitational acceleration g is the aether flow
acceleration, we have: 50
!
Dυ g= ! Dt
(1.8-1)
!
where υ is the aether flow velocity. The gravitational force density f can be defined as: Dυ ! (1.8-2) f =ζ g =ζ = − ∇P ! Dt
!
For a stationary gravitational field, the gravitational aether flow
!
is steady, and so we can rewrite equation (1.8-2) using equation (1.7-8): !
)
(1.8-3)
If dr is an element of length along a streamline, we have: !
! ! ⎡ ! ⎛ υ2 ⎞ ! ! ! ! !⎤ f • dr = ζ ⎢∇ ⎜ ⎟ • dr − υ × ∇ × υ • dr ⎥ ! ⎣ ⎝ 2⎠ ⎦
(
)
(1.8-4)
Using the vector identity given in equation (C-3) of Appendix C, we can write: !
! ! ⎡ ! ⎛ υ2 ⎞ ! ! ! ! ! ⎤ f • dr = ζ ⎢∇ ⎜ ⎟ • dr + υ × dr • ∇ × υ ⎥ ! ⎣ ⎝ 2⎠ ⎦
)
(1.8-5)
Since υ is parallel to dr by definition of a streamline, the equation of a streamline is given by: ! υ × dr = 0 !
(1.8-7)
where ΩK is flow kinetic energy density for the incompressible flow of aether and is given by:
1 ΩK = ζ υ 2 ! 2
(1.8-8)
Because gravitational acceleration in a stationary gravitational as given in equation (1.2-13), we will now let: ! ! ! ! ! f = ζ g = ζ ∇ϕ = − ∇ΩV !
(1.8-9)
where ΩV is flow potential energy density of aether. Equation (1.8-7) can then be rewritten as: ! ! ! − ∇ΩV • dr = − dΩV = dΩK !
(1.8-10)
or !
(
! ⎛ υ2 ⎞ ! ! ! ⎛1 ⎞ f • dr = ζ ∇ ⎜ ⎟ • dr = d ⎜ ζ υ 2 ⎟ = dΩK ! ⎝2 ⎠ ⎝ 2⎠
field can be written in terms of the gradient of a scalar potential
! ⎡ ! ⎛ υ2 ⎞ ! ! ! ⎤ ! f = ζ g = ζ ⎢∇ ⎜ ⎟ − υ × ∇ × υ ⎥ ! ⎣ ⎝ 2⎠ ⎦
(
For incompressible flow ζ is constant, and so we have:
!
d ( ΩK + ΩV ) = 0 !
(1.8-11)
We now can see that, for the incompressible flow of aether
due to gravity, any change in the aether flow kinetic energy density between two points of a streamline is equal to the negative of the change in the gravitational potential energy
(1.8-6)
density between the same two points. Gravitational energy 51
density is then conserved for the incompressible flow of aether
for steady flow along a streamline. Equation (1.8-16) is a
along a streamline. From equation (1.8-11) we have:
Bernoulli equation for the steady frictionless incompressible
d ( ΩK + ΩV ) = 0 ! dt
!
(1.8-12)
which agrees with the empirical observation that energy is conserved in a stationary gravitational field. Gravitational energy can be considered as negative energy that resides in the aether. !
will remain the same for any given streamline, but can be different for different streamlines.
1.9! !
! ⎛ υ2 ⎞ ! f = ζ ∇⎜ ⎟ ! ⎝ 2⎠
! ! ⎛ υ2 ⎞ ! ! ∇P • dr + ζ ∇ ⎜ ⎟ • dr = 0 ! ⎝ 2⎠
(1.8-13)
!
!
1 P + ζ υ 2 = P + ΩK = constant ! 2
ΩG = ΩK + ΩV !
(1.9-1)
entity. Since within a vacuum there can be no sinks or sources (1.8-14)
of ΩG , we know that within a vacuum the gravitational energy density ΩG must satisfy the equation of continuity: !
(1.8-15)
where P is hydrodynamic pressure. Integrating, we obtain: !
If we denote gravitational energy density as ΩG , we then
The gravitational energy density ΩG is a physical density
or for incompressible flow of aether:
⎛1 ⎞ dP + d ⎜ ζ υ 2 ⎟ = 0 ! ⎝2 ⎠
GRAVITATIONAL ENERGY DENSITY FLUX WITHIN A VACUUM
can write:
Using equation (1.8-2), we can write equation (1.8-7) as: !
proximity to a nucleon. For such a flow the integration constant
From equation (1.8-7) we also have for gravitational force
density: !
flow of aether in a stationary gravitational field when not in
(1.8-16)
∂ΩG ! ! + ∇ • ( ΩG υ ) = 0 ! ∂t
(1.9-2)
where υ is the flow velocity of aether due to gravity. The ! gravitational energy flux vector SG (momentum density of gravitational energy) is given by: ! ! ! SG = ΩG υ !
(1.9-3) 52
The gravitational energy flux (quantity of gravitational energy flowing per unit time per unit volume) is then: ! ! ! ! ! ∇ • SG = ∇ • ( ΩG υ ) !
1.10! (1.9-4)
From equations (1.9-4) and (1.9-2) we obtain for a vacuum:
∂ΩG ! ! (1.9-5) + ∇ • SG = 0 ! ∂t ! We will designate SG as the gravitational Poynting vector. !
!
For a stationary gravitational field in a vacuum, the
gravitational energy density ΩG must not change with time: !
∂ΩG = 0! ∂t
From equation (1.9-5) we must then also have: ! ! ! ∇ • SG = 0 !
(1.9-6)
(1.9-7)
Heaviside (1912) postulated the existence of this equation.
IRROTATIONAL FLOW OF AETHER IN A GRAVITATIONAL FIELD
!
We will now consider the gravitational aether flow associated with a stationary gravitational field. At any point r of the gravitational field g , the aether flow is characterized by a ! flow velocity υ and a flow acceleration Dυ Dt . From equation (1.8-1) we have: Dυ ! = g! Dt
(1.10-1)
When not in proximity to nucleons, the aether flow will be incompressible and so we will have from equation (1.5-12): ! (1.10-2) ∇ •υ = 0 ! Using equation (1.2-13), we can express the stationary gravitational field g in terms of the gradient of a scalar
possess energy density without containing matter. In Einstein’s
potential function ϕ : Dυ ! = g = ∇ϕ ! Dt
general theory of relativity, the gravitational Poynting vector
Since the gravitational field is stationary, we can use equation
does not exist (Narlikar, 2010).
(1.7-5) to write:
!
Finally, we note that since gravitational energy density can
exist in the aether in a vacuum, it is obvious that an entity can
!
(1.10-3)
! Dυ ! ! ! ! ! ! ⎛ υ2 ⎞ ! ! ! = g = ∇ϕ = υ • ∇ υ = ∇ ⎜ ⎟ − υ × ∇ × υ ! Dt ⎝ 2⎠
(
)
(
)
(1.10-4) 53
Taking the curl of this equation, we have: !
! ! ! ! ! ! ⎛ υ2 ⎞ ! ! ! ! ⎤ ⎡ ∇ × g = ∇ × ∇ϕ = ∇ × ∇ ⎜ ⎟ − ∇ × ⎣υ × ∇ × υ ⎦ ! (1.10-5) ⎝ 2⎠
(
)
!
C, we obtain: ! ! ! ! ! ! ! ! (1.10-6) ∇ × g = 0 = − ∇ × ⎡⎣υ × ∇ × υ ⎤⎦ ! We see that the gravitational field g is an irrotational vector field. For equation (1.10-6) to be true for any υ , we must have: ! (1.10-7) ∇ ×υ = 0 !
!
This can be verified by expanding equation (1.10-6) using the
!
)
vector identity given in equation (C-41) of Appendix C and using equation (1.10-2). Gravitational aether flow, therefore, is irrotational. !
That gravitational aether flow is now determined to be
(1.10-8)
From equations (1.10-8) and (1.2-11) we have:
Using the vector identity given in equation (C-29) of Appendix
(
! ⎛ υ2 ⎞ ! ! g = ∇ϕ = ∇ ⎜ ⎟ ! ⎝ 2⎠
υ2 ΔM ! ϕ= =G 2 r
(1.10-9)
for the Newtonian stationary gravitational potential ϕ at a point r from a material body having mass ΔM . The speed υ of aether flowing toward the material body of mass ΔM is therefore given by:
!
υ 2 = 2G
ΔM ! r
(1.10-10)
! From equation (1.10-7) we see that aether flow velocity υ
can be expressed as the gradient of a scalar velocity potential
φ:
! ! υ = ∇φ !
irrotational could be expected since the gravitational field
!
resulting from this flow is irrotational. More fundamentally, it
and so the irrotational flow of frictionless aether associated
could be expected since aether flow is frictionless, and the
with a gravitational field can be termed potential flow. A
aether flow of a stationary gravitational field has no curvature.
velocity potential, therefore, exists for gravitational aether flow.
In fact, it is for these reasons that the Newtonian gravitational
!
potential ϕ exists for a stationary gravitational field. !
This can be seen by using equation (1.10-7) to rewrite
!
From equation (1.10-2) we then have: ! ! ! ! ∇ • υ = ∇ • ∇φ = ∇ 2φ = 0 !
(1.10-11)
(1.10-12)
equation (1.10-4): 54
when not in proximity to nucleons. This Laplace’s equation can
!
be
equation (C-37) of Appendix C, we see that the vorticity field is
considered
to
be
the
equation
of
continuity
for
incompressible potential flow (see Malvern, 1969). !
When not in proximity to nucleons, gravitational aether
flow is incompressible and is both solenoidal and irrotational, as can be seen from equations (1.5-12) and (1.10-7): ! ∇ •υ = 0 ! (1.10-13) !
∇ ×υ = 0 !
(1.10-14)
These equations generally characterize gravitational fields.
1.11!
VORTICITY
!
A vortex is a circular or near-circular flow about a common axis. The vorticity ψ of a fluid flow having a flow velocity field υ is defined as: ! (1.11-1) ψ = ∇ ×υ ! and represents the rotation of the individual fluid point particles as they flow with velocity υ about a common axis. If a fluid flow has vorticity, each point of the fluid will have a vorticity vector associated with it, thereby creating a field of vorticity vectors known as a vorticity field. Any line in the fluid that is everywhere tangent to a vorticity vector at a given time
From the definition in equation (1.11-1) and using
solenoidal: ! ∇ •ψ = ∇ • ∇ × υ = 0 !
(
)
(1.11-2)
If the fluid particles have a constant angular velocity ω , we will have: !
1 ω = ∇ ×υ ! 2
(1.11-3)
ψ = 2ω !
(1.11-4)
and so: ! !
At any point along a vortex line, the axis of the rotation vector ω will be parallel to the vortex line. From equations (1.10-7) and (1.11-1) we see that for irrotational flow (such as gravitational aether flow) the vorticity ψ is: ! (1.11-5) ψ =0! and so no vorticity exists for gravitational aether flow. !
For the general flow of aether, the force density equation
can be written using equations (1.7-6) and (1.7-3): !
! ! ! ⎡ ∂υ! ! ⎛ υ 2 ⎞ ! ! ! ⎤ Dυ f =ζ = ζ ⎢ + ∇ ⎜ ⎟ − υ × ∇ × υ ⎥ = − ∇P ! (1.11-6) Dt ⎝ 2⎠ ⎣ ∂t ⎦
(
)
is known as a vortex line. 55
Taking the curl of this equation and using the definition of vorticity given in equation (1.11-1), we have: ! ! ! ! ! 2 ! ⎛ ⎞ ∂ψ υ ∇ × ∇P ! ! ! + ∇ × ∇ ⎜ ⎟ − ∇ × (υ × ψ ) = − ! ∂t 2 ζ ⎝ ⎠
(1.11-7)
With the vector identity given in equation (C-29) of Appendix C, we obtain: ∂ψ ! − ∇ × (υ × ψ ) = 0 ! ∂t
(1.11-8)
We can rewrite this equation using the vector identity given in equation (C-41) of Appendix C: ∂ψ ! − ∇ • ψ υ + ∇ • υ ψ − ψ • ∇ υ + υ • ∇ ψ = 0 ! (1.11-9) ∂t
(
) (
) (
) (
)
Using equation (1.11-2) we then obtain: ∂ψ ! + ∇ •υ ψ − ψ • ∇ υ + υ • ∇ ψ = 0 ! ∂t
(
) (
) (
)
(1.11-10)
From the definition of the substantive derivative given in equation (1.5-9) we have the Helmholtz vorticity equation: Dψ ∂ψ ! (1.11-11) = + υ •∇ ψ ! Dt ∂t
(
)
Comparing equations (1.11-10) and (1.11-11), we can rewrite equation (1.11-10) in the form:
! !
Dψ + ∇ •υ ψ − ψ • ∇ υ = 0 ! (1.11-12) Dt For incompressible flow ∇ • υ = 0 as given by equation
(
) (
)
(1.5-12) and so equation (1.11-12) becomes the Helmholtz vorticity equation for an incompressible fluid: Dψ ! = ψ •∇ υ ! Dt
(
)
(1.11-13)
This is the equation of motion for the incompressible flow of aether in terms of vorticity. This equation shows how vorticity is transported in the flowing aether, and so it can be considered the vorticity transport equation for the incompressible flow of aether. We also have using equation (C-43) of Appendix C: ! ! ! ! ! ! ! ! ! ! (1.11-14) ∇ × ψ = ∇ × ∇ × υ = ∇ ∇ • υ − ∇ 2υ !
(
)
(
)
For incompressible flow ∇ • υ = 0 and so this equation becomes: ! ! ! ! (1.11-15) ∇ × ψ = − ∇ 2υ ! !
In cases where the flow of aether can be considered to be essentially two-dimensional, the vorticity vector ψ will be orthogonal to and independent of the velocity vector υ . From equation (1.11-8) we then have: ! ∂ψ ! ! = 0! ∂t
(1.11-16)
56
We also have: ! ! ! ! ! υ •∇ ψ = 0 !
(
)
!
Γ=
∫∫ (
! ! ! ∇ × υ • dS !
S
(1.11-17)
)
(1.12-2)
since ψ is orthogonal to υ (this can be seen by choosing a coordinate system such that υ is in the x-y plain and ψ has
where S is the surface area enclosed by the curve C . From the definition of vorticity ψ given in equation (1.11-1) we have:
only a component in the z direction). From equation (1.11-11)
!
we therefore obtain: ! Dψ ! ! =0! Dt
∫∫
! ! ψ • dS =
S
(1.11-18)
when the flow of aether is two-dimensional.
1.12!
Γ=
∫∫
! ψ • nˆ dS !
(1.12-3)
S
and so the circulation Γ represents the normal component of the vorticity ψ over the surface area S . If υ is irrotational, the vorticity is ψ = 0 and so we have Γ = 0 . !
The vortex lines passing through the closed simply-
connected curve C that bounds the surface area S form another
CIRCULATION
surface known as a vorticity tube. The circulation Γ provides a
!
If we consider aether flowing around a closed simply connected curve C with a flow velocity υ , the circulation Γ of
measure of the strength of the vorticity (fluid rotation) within
the aether flow is defined by:
that is irrotational is known as a vortex tube. The circular flow
!
Γ=
"∫
! ! υ • dr !
(1.12-1)
C
the vorticity tube. A vorticity tube surrounded by a fluid flow bounded by a vortex tube is then a vortex. The vortex that exists within a vortex tube is known as a vortex filament.
where dr is an infinitesimal distance element tangent to the
!
curve C . The circulation Γ , therefore, is the line integral of the
using equation (1.12-1):
tangential (linear) velocity component to the curve C . Invoking Stokes theorem given in equation (C-63) of Appendix C, we can rewrite equation (1.12-1) as:
!
The rate of change of the circulation can be calculated
DΓ = Dt
!∫
C
D ! ! (υ • dr ) ! Dt
(1.12-4)
or 57
"∫
! Dυ ! • dr + Dt
!∫
! D " ! υ• (1.12-5) ( dr ) ! Dt C C ! Letting the flow velocity at a point r of the closed curve C be ! ! υ , the flow velocity at a point r + dr will be: ! Dr D ! ! ! D ! ! ! D ! υ + dυ = ( r + dr ) = + ( dr ) = υ + ( dr! ) ! (1.12-6) Dt Dt Dt Dt DΓ = Dt
Three important vortex theorems were established by Helmholtz (1858): 1.! The circulation on any closed curve C around a vortex tube is constant all along the tube. 2.! A vortex tube cannot come to an end within a fluid. This means all vortex tubes must close on themselves becoming vortex rings, or they must terminate at a fluid boundary.
where we used equation (C-24) of Appendix C. We then have: !
! D ! dυ = ( dr ) ! Dt
Using equation (1.12-7), equation (1.12-5) can now be written: ! DΓ Dυ ! ! ! ! (1.12-8) = • dr + υ • dυ ! Dt C Dt C
"∫
!∫
!
"∫
C
! Dυ ! • dr + Dt
!∫
C
⎛ υ2 ⎞ d⎜ ⎟ ! ⎝ 2⎠
(1.12-9)
exact differential, the rate of change of circulation around a
"∫
These vortex laws are all derived in Appendix D. From
Helmholtz’s third theorem we have for a vortex tube in a !
Since the second term on the right side of this equation is an closed curve C is given by: ! DΓ Dυ ! ! = • dr ! Dt C Dt
!
frictionless fluid:
or
DΓ = Dt
3.! For an inviscid fluid the circulation or strength of a vortex tube is invariant, and the vortex tube can exist forever.
(1.12-7)
Γ = constant !
independent of the cross-sectional area S of the vortex tube. Using equations (1.12-2) and (1.12-3), we then have: !
Γ=
∫∫ ( S
! (1.12-10)
This equation is known as Kelvin’s circulation theorem.
(1.12-11)
! ! ! ∇ × υ • dS =
)
∫∫
! ! ψ • dS = constant !
(1.12-12)
S
This means that, for a frictionless fluid, the vorticity is
constant within a vortex tube, and the vortex tube always contains the same fluid particles. Once a vortex ring of frictionless fluid is created, therefore, the motion of the fluid 58
within the ring cannot be changed by any external force as long
!
Any pushing action that occurs directly through physical
as the ring exists.
contact of contiguous real particles and causes a body to accelerate is known as mechanical force. Mechanical force is,
1.13!
GENERATION OF VORTICITY IN AETHER
therefore, contact force involving a push action between contiguous real particles. Material particles are, however, never
Generation of vorticity in aether flow can occur whenever
in physical contact with each other, but are always surrounded
there is an abrupt change in pressure or density in the aether.
by aether (see Chapter 5). “All pieces of matter and all particles are
Such changes are most likely to occur at physical boundaries of
connected together by the ether and by nothing else,” as Lodge
the aether flow, but can also occur as a result of sudden
(1925a) noted. Therefore, mechanical force defined as contact
insertion of an object into the flow.
push action between real particles exists only because aether
!
particles can be in physical contact both with other aether
1.14!
NATURE OF MECHANICAL FORCE
particles and with material particles. Material particles act upon
In his Principia, Newton defines an impressed force as “an
each other only through (by means of) the aether. When
action exerted upon a body, in order to change its state, either of rest,
contiguous aether particles that are accelerating cause other
or of uniform motion in a straight line.” Generalizing from this
aether particles or a material particle to accelerate, this action is
definition, we will define (unbalanced) force to be any action
mechanical force.
on a body that causes the body to accelerate.
!
!
Galileo Galilei (1638) first discovered that a force exerted
generally regarded as merely an artificial invention since force
upon a body causes the body to accelerate rather than to move
often appears to serve only as an “intermediate term” between
with constant velocity. Application of a nonzero resultant force
two motions (Jammer, 1957). Nevertheless, the changing
upon a body will always cause a sudden discontinuous change
momentum of aether is an action that can cause the acceleration
in the acceleration of the body. Any resulting changes in the
of a body. Defined as an action that causes acceleration, force
velocity and position of the body will not be discontinuous, but
certainly can be considered to be as real as is any motion.
!
In twentieth century physics, the concept of force was
will represent the integration over time of the acceleration. 59
If a force appears to be acting-at-a-distance, the force is
physical contact of contiguous aetherons. We can conclude that
really acting through contiguous aether particles as Descartes
gravitational force is a mechanical force.
proposed. Action-at-a-distance force does not exist. The concept
!
of action-at-a-distance is counterintuitive and has no physical
individual particles of aether do not act over any measurable
validity.
distance. The long-range force known as gravity is produced by
Since aether particles act only upon neighboring particles,
Finally, we note that the nonexistence of any force that
a flow motion involving a series of contiguous aether particles.
acts-at-a-distance has a very important implication: there can be
The flow of the fluid aether into matter is evident to us through
no such thing as a true attractive force. In physics this term
the resulting gravitational pull. Gravitational force is an
should not then be taken to mean force acting-at-a-distance. In
attractive force. Finally, we note that gravitational force is not
this book we will follow Newton (1687) by taking attractive
action-at-a-distance.
force as indicating only the relative direction in which forces
We will designate any force produced by the flowing
are acting. When we say that an attractive force exists between
motion of a fluid as a flow force. All other forces will be
two bodies, we will mean, therefore, only that some mechanical
designated as non-flow forces. Gravitational force is then a
forces are pushing the two bodies closer together.
flow force. Flow forces are distinguished from non-flow forces by the following:
1.15! !
NATURE OF GRAVITATIONAL FORCE
Gravitational force is the direct result of material particles
(nucleons) being field flow sinks for the nearly incompressible aether. The accelerating flow of aether into a nucleon produces a negative pressure gradient in the aether surrounding the nucleon. This negative pressure gradient is gravitational force. We see then that gravitational force both results from and causes the acceleration of aether. Gravitational force can be characterized, therefore, as an action occurring through the
1.! If a non-flow force is accelerating you, it is possible for you to detect this acceleration without reference to any other material bodies. We will reserve the designation impressed force for non-flow forces. 2.! If a flow force is uniformly accelerating you, it is impossible for you to detect this acceleration without reference to other material bodies that are not being so accelerated. You would seem to 60
!
be weightless. Examples of flow forces acting on a body are:
aether particles since they are field flow sinks and not field flow
a.! A material body in free fall in a uniform gravitational field.
be aether particles. Gravitons have not been discovered nor do
b.! A material body floating beneath the surface of a river whose flow is uniformly accelerating.
only particles involved in the transmission of gravitational
Because gravitational force results from the action of
sources for aether. The hypothetical gravitons, therefore, cannot we expect they ever will be; gravitons simply do not exist. The force are nonmaterial aether particles.
1.17!
ANTIGRAVITY
aether particles, such force will penetrate through the surface of
!
Since matter is a field flow sink for aether, but is not a field
any body composed of atoms and will affect the entirety of the
flow source for aether, the gravitational flow of aether must
body. A force such as gravity that can act directly upon not only
always be towards and never away from matter. Therefore, the
the surface but also the interior parts of a material body is
traditional concept of antigravity as a gravitational force
known as a body force.
directed away from matter is not possible. Aether flow in space far from nucleons, however, may not be gravitational aether
1.16! !
NONEXISTENT GRAVITONS
Gravitons are hypothetical particles proposed under a
theory that assumes the force of gravity is transmitted by some sort of particle that mediates or transmits the force over a measurable distance through an exchange of these hypothetical
flow. If the aether flow is not gravitational aether flow, the flow can be in any direction. Any acceleration field in space resulting from such non-gravitational aether flow can then appear to be a type of antigravity (see Glanz, 1998).
particles between material bodies. Since the transmission of
1.18!
SUMMARY
gravitational force is by mechanical contact of contiguous
!
aether particles, however, there is no particle that is exchanged
were derived by Newton and published in 1687 in his Principia.
between material bodies to transmit in some unknown manner
Using Newton’s force law of gravity, Poisson’s field equation of
the gravitational force. Moreover, material bodies do not emit
gravity can be derived. From Poisson’s equation we find that a
The first mathematical expressions for gravitational force
61
real physical field medium, aether, must exist and be ubiquitous in space, and that some motion must be occurring within this medium to produce the observed gravitational fields. With vector analysis we then determine that matter is a field flow sink for aether. The accelerating flow of aether into matter is manifest as gravitational acceleration. The flow of aether into matter completely explains Galileo’s observation that the gravitational acceleration of a body is independent of the body’s mass and composition. !
Using the concept of matter as a field flow sink for aether,
an aether field equation of gravity can be derived. This field equation expresses the gravitational field g at a point r in space completely in terms of properties of the aether at this point. Gravity is clearly a fluid dynamic phenomenon of aether. Gravitational fields result from mechanical processes occurring within the aether as it flows into matter. Finally, we note that Newton’s force law of gravity, Poisson’s field equation of gravity, and the aether field equation of gravity are all mutually consistent, and are all valid for the gravitational field surrounding any quantity of matter. It is particularly interesting to note that Newton’s force law of gravity is valid for a stationary gravitational field of any strength, weak or strong (contrary to an opinion formed after the publication of Einstein’s general theory of relativity). 62
Chapter 2 The Nature of Mass and of Inertia
“One has to admit that in spite of the concerted effort of physicists and philosophers, mathematicians and logicians, no final clarification of the concept of mass has been reached.” Max Jammer Concepts of Mass in Classical and Modern Physics
63
!
The scientific concept of mass has existed for over four
gravitational mass and inertial mass. We will derive Newton’s
hundred years. During that time there have been countless
laws of motion from the aether field equation of gravity. We
studies of mass and numerous revisions in how mass is viewed
will also determine the source of inertia and the nature of
in physics. Nevertheless, in all that time no clear definition of
inertial forces. Finally, we will determine if dark matter really
mass has emerged. This prompted Brown (1960) to state:
exists.
“Nothing in the history of science is perhaps so extraordinary as the doubt and confusion surrounding the definition of mass.” The
2.1!
concept of mass has only become more complex through the
!
years: for bodies in motion, mass has appeared to depend on
quantity of matter, a metaphysical idea developed in the Middle
certain kinematic properties of the bodies; for all material
Ages (Jammer, 1961). In the seventeenth century, Kepler
bodies, mass has been considered equivalent to energy. The
introduced the concept of force to provide a cause for planetary
physical reality of mass has even been questioned (Roche,
motions and the concept of inertia of matter to describe the
2005).
resistance of matter to impressed force (see Barbour, 1989).
!
Another question that has puzzled scientists since the time
Newton then proposed that inertia is a material body’s
of Kepler is the source of inertia. Because the source of inertia
resistance to a change in motion and that mass is a measure of
could not be found, some scientists have even appealed to the
inertia. He also found that the gravitational force or weight of a
far distant stars for a mechanism to explain inertia.
body is proportional to its quantity of matter and that mass is a
!
measure of the quantity of matter. Kepler and Newton thereby
One of the outstanding mysteries of modern cosmology is
CLASSICAL CONCEPTS OF MASS The concept of mass originated from quantitas materiae or
the nature of dark matter. This postulated matter is thought to
began the process of creating the scientific concept of mass.
be the predominant form of matter in our Universe, and yet it
!
has never been detected; its presence has only been inferred.
mass was generally viewed as real and substantial. All objects
!
In this chapter, we will briefly review classical concepts of
were thought to consist of mass, and conservation of mass
mass. We will then determine the definition and nature of mass.
became a principle of physics. By the twentieth century,
We will show how this new definition of mass explains both
however, the concept of mass in physics had changed greatly.
During most of the eighteenth and nineteenth centuries,
64
Mass was no longer regarded as substantial, but only as a
aether. If the time variation of aether flux within ΔV is zero,
proportionality factor, inertial factor, or a form of energy. Mass
then the mass within the volume element ΔV is zero.
was often defined then as the quotient of force and acceleration
!
(Huntington, 1918). Of course, this definition was not without
measured; mass is a measure. The fact that mass is not an entity
its own problems since force as a concept in physics was also
itself, but is a measure, may have contributed to the difficulty
being questioned in the twentieth century (Jammer, 1957).
of determining the nature of mass. Jackson (1959) stated, “there
As Brown (1960) noted, mass is not an entity that can be
is no experiment in which mass reveals itself directly.”
2.2!
NATURE OF MASS
!
According to equation (2.2-2), mass exists within a given
To derive a definition of mass, we will consider matter
volume element whenever there is a time variation in the flux
having mass ΔM and density ρ contained within a very small
of aether flowing in the volume element, whether matter is
volume element ΔV of a material body. From equation (1.4-18)
present or not. Equation (2.2-2) is also valid for a volume
we have:
element ΔV that contains only aether. Therefore, mass can exist
!
!
1 − 4 π Gζ
∫∫∫
ΔV
∂ ! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV = ∂t
∫∫∫
ΔV
ρ dV ! (2.2-1)
1 ΔM = − 4 π Gζ
∫∫∫
ΔV
∂ ∂t
! ! ⎡⎣∇ • (ζ υ ) ⎤⎦ dV !
thing as matter. This can also be seen from the fact that matter is a field flow sink for aether whereas mass is not. It is matter,
Using equation (1.2-5), we then obtain: !
without matter. It is evident then that mass is not the same
not mass, that is the cause of gravitational fields. Gravitation is (2.2-2)
not a property of mass, and so mass is not a source of gravitational force. Finally, it is not correct to state that matter
Equation (2.2-2) provides us with a definition of mass from
alone is endowed with mass.
which we can determine the nature of mass. From this equation
!
we see that the mass within a given volume element ΔV is a
of the aether flux (mass) is proportional to the strength of the
measure of the variation with time of the flux of aether in the
sink. The strength of the sink is, in turn, proportional to the
volume element. Therefore, mass is a measure of the variation
quantity of matter. It is for this reason that mass is a measure of
with time of the flux of aether. Mass cannot exist without
the quantity of matter in a body as Newton (1687) proposed.
For aether flowing into a sink (matter), the time variation
65
Newton’s definition of mass as a measure has puzzled scientists for a long time (e.g., see Roche, 1988), but we see now that he was correct. Newton never said that mass is the same thing as matter. !
If ΔS is the surface area of the volume element ΔV , we
can use Gauss’s theorem given in equation (C-59) of Appendix C to rewrite equation (2.2-2) as: !
∫∫
1 ΔM = − 4 π Gζ
ΔS
! ∂ ! ζ υ ) • dS ! ( ∂t
(2.2-3)
This equation shows that the mass within a given volume
1 ΔM = − 4 π Gζ
ΔV
Since aether flow is incompressible when not in proximity to nucleons, the density of aether is then constant: !
∂ζ =0! ∂t
(2.2-6)
!
! ! ∇ζ = 0 !
(2.2-7)
and so the definition of mass given in equations (2.2-3) and (2.2-5) can be rewritten as:
element is a measure of the variation with time of the momentum density of aether flowing through the surface of the
∫∫∫
! ⎛ ∂ζ ! ⎞ ⎤ ⎡ ! ! ! ! ⎢ζ ∇ • α + α • ∇ζ + ∇ • ⎜⎝ ∂t υ ⎟⎠ ⎥ dV ! (2.2-5) ⎣ ⎦
!
ΔM = −
1 4π G
ΔM = −
1 4π G
volume element. Note that this is just equation (1.4-1) with
k = ζ . From equation (2.2-3) we see that mass is a scalar which can be uniquely specified since it is defined relative to a unique reference frame, aether (see Section 2.7). We also see that mass ! ! is always positive (since υ and dS are oppositely directed). !
The definition of mass given in equation (2.2-2) can be
written in the form: !
1 ΔM = − 4 π Gζ
!
∫∫
ΔS
! ! α • dS !
∫∫∫
ΔV
! ! ∇ • α dV !
(2.2-8)
(2.2-9)
!
Because of the relations given in equations (2.2-6) and ! (2.2-7), α represents the acceleration of gravitational aether ! ! flow when not in proximity to nucleons. Therefore, α = g as given in equation (1.4-12), and we can write equation (2.2-9) as:
∫∫∫
ΔV
! ! ⎛ ∂ζ ! ⎞ ⎤ ⎡! ⎢∇ • (ζ α ) + ∇ • ⎜⎝ ∂t υ ⎟⎠ ⎥ dV ! (2.2-4) ⎣ ⎦
! ! where α is ∂υ ∂t (the time variation of the aether flow velocity ! ! υ at a point r ). We then have:
!
1 ΔM = − 4π G
∫∫∫
ΔV
! ! ∇ • g dV !
(2.2-10)
Using equation (1.2-10) we then have: 66
!
ΔM = −
1 4π G
∫∫∫
ΔV
[ − 4 π G ρ ] dV = ∫∫∫
ΔV
ρ dV ! (2.2-11)
5). We will begin by specifying the location in space of the body ! of mass m by the line position vector r that has its coordinate
which is equation (1.2-5).
system origin at the center of the stationary body of mass M .
2.3!
2.3.1!
!
MASS TYPES The nature of mass is completely specified by the
definition of mass given in equation (2.2-2), and so all mass
!
ACTIVE GRAVITATIONAL MASS
! If g is the gravitational field resulting from aether flowing
into the material body of mass M shown in Figure 1-1,
must be consistent with this definition. Therefore, only one
equation (1.2-1) applies and we have:
kind of mass exists. While different mass types have been
!
previously identified, we will now show that the designation of a mass type is based, not upon differences in the nature of mass
M ! g = − G 2 rˆ ! r
(2.3-1)
The mass M
itself, but upon differences in the nature or situation of entities
of the material body responsible for the ! gravitational field g is designated as the active gravitational
possessing the mass. We will also see that not all entities are
mass for this field. Active gravitational mass is then a measure
capable of possessing all types of mass; some entities having
of the quantity of matter (the strength of the sink) that is
mass are not such that the mass they possess can ever be
producing the gravitational field. The strength of a sink within
designated active gravitational mass.
a given volume element ΔV is, in turn, proportional to the
!
We will examine three different types of mass that have
variation with time of the flux of aether in ΔV as can be seen
been identified in the scientific literature: active gravitational
from equation (2.2-2). Active gravitational mass, therefore, is
mass, passive gravitational mass, and inertial mass. To do this,
consistent with the definition of mass given in equations (2.2-2)
we will again consider two bodies of mass M and m ,
and (2.2-3).
respectively (see Figure 1-1). In this case, however, we will
!
require only the body of mass M to be material, while the body
gravitational mass, it must contain matter since only matter can
of mass m can be either material or nonmaterial. An example of
create a gravitational field. Gravity is a property of matter, not
a nonmaterial body is the electron (as will be shown in Chapter
of mass. The mass of electrons, for example, cannot be
For a body to have mass that is designated as active
67
designated as active gravitational mass since electrons do not
!
possess matter. Electrons then do not possess a gravitational
flux of aether in the body of mass m . If the body of mass m is
field. The designation of mass as active gravitational mass
material, for example, the actual source of the mass m is the
arises simply from a consideration of the source of the variation
quantity of matter in the body of mass m . Therefore its mass m
with time of the flux of aether that is mass.
can be designated as both active and passive gravitational
2.3.2!
PASSIVE GRAVITATIONAL MASS
!
The body of mass m shown in Figure 1-1 will experience ! acceleration due to the gravitational field g of the material body of mass M . From equation (1.2-1) we have: ! M ! F ! g ≡ = − G 2 rˆ ! (2.3-2) m r ! where F is the gravitational force on the body of mass m . Regardless of whether the body of mass m is a material body or a nonmaterial body, it will experience the same gravitational ! acceleration g since the gravitational field of the body of mass
M is independent of both the composition and mass of the body of mass m . When the body of mass m is experiencing a ! gravitational force due to the gravitational field g of the material body of mass M , the mass m is designated as passive gravitational mass. The designation of passive gravitational mass for mass m does not arise then from a consideration of the source of the mass m but only from the fact that the body of mass m lies within a gravitational field of some other body.
The mass m is a measure of the variation with time of the
mass. Both the mass M and the mass m are then defined by equations (2.2-2) and (2.2-3). The designation of passive gravitational mass arises simply from a consideration of the situation within which the body possessing the mass m exists.
2.3.3! !
EQUALITY OF ACTIVE AND PASSIVE GRAVITATIONAL MASS
Since active gravitational mass and passive gravitational
mass are both defined by equations (2.2-2) and (2.2-3), the active and passive gravitational mass of any given material body must not only be equal but identical. Experiments by Kreuzer (1968) and by Bartlett and Van Buren (1986) have indeed found the active and passive gravitational mass of a material body to be equal (see Section 2.4.3). !
For a nonmaterial body, passive gravitational mass is also
defined by equation (2.2-2). A nonmaterial body can never have mass that is designated active gravitational mass, however, since such a body does not contain matter. If a nonmaterial body has mass that is designated passive gravitational mass, 68
the source of this mass must be other than matter (see Section
accelerate as given by equation (2.3-3). The accelerating aether
5.4.6).
will in turn cause the body of mass m to accelerate. !
2.3.4! !
INERTIAL MASS
Since the aether is being accelerated by an impressed force
rather than by flowing into a nucleon, aether density will
We will now consider a material body with gravitational
remain constant. Both equations (2.2-6) and (2.2-7) will then
mass m . Using equation (1.2-1), we can rewrite the aether field
always hold. Therefore, the term containing the aether flow ! ! ! velocity υ in equation (2.3-4) will always be zero (even if υ ≠ 0 )
equation of gravity (1.4-20) for this mass in the form: ! ! 1 ∂ ! F ⎡ ⎤ ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • ⎢ ⎥ ! ! (2.3-3) ζ ∂t ⎣m⎦ This is basically a field equation describing the relation between ! the motion of aether and the acceleration F m . Whenever
because ∂ζ ∂t = 0 , and we will have: ! ! ! ! ⎡F⎤ ! ∇ •α = ∇ • ⎢ ⎥ ! ⎣m⎦
(2.3-5)
aether is being accelerated, equation (2.3-3) must apply.
and so: !
!
Taking the time derivative in equation (2.3-3), we have: ! 1 ! ⎡ ! ∂ζ ! ⎤ ! ⎡ F ⎤ ! ∇ • ⎢ζ α + υ ⎥ = ∇ • ⎢ ⎥ ! (2.3-4) ζ ∂t ⎦ ⎣ ⎣m⎦ ! There is nothing in equation (2.3-3) restricting F to be only a ! flow force, such as gravitational force. The force F can also be a
!
non-flow force, which is impressed force, and equation (2.3-3)
!
must remain valid. This is evident since impressed force acting
constant since motion of the body will not change the quantity
on a body involves the acceleration of aether. ! If F is an unbalanced impressed force acting on the body
of matter in the body. Because the body is being accelerated by
of gravitational mass m , then this force will cause aether to
designated as inertial mass.
! ! mα = F !
(2.3-6)
! Due to the impressed force F , the aether will have an ! acceleration α . The accelerating aether will cause the material ! body of gravitational mass m to acquire an acceleration a ! ! where a = α . Equation (4.3-6) can then be written: ! ! ! F = ma! (2.3-7) The body still has its gravitational mass m . This mass is
an impressed force, however, the gravitational mass m is now
69
pendulum experiments and torsion balance experiments for
2.3.5! !
EQUALITY OF INERTIAL MASS AND PASSIVE GRAVITATIONAL MASS
bodies composed of different materials. These are classic tests in physics.
Clearly gravitational mass and inertial mass are not only
equal, but identical. The equality of gravitational mass and
2.3.6.1!
inertial mass is not some incredible coincidence, but follows
!
directly from the aether field equation of gravity. It is then not
mass for material bodies was first determined experimentally
necessary to assume the equality of passive gravitational mass
by Newton (1687), and later by Bessel (1832), Southerns (1910),
and inertial mass as has been done in the weak equivalence
and Potter (1923) using observations of the period (swing time)
principle of general relativity. We can conclude that, for a
of pendulums made of different materials.
material body, active gravitational mass, passive gravitational
!
mass, and inertial mass are all equal since they are all simply
end of a string of length L is shown in Figure 2-1. The angle the
different designations for the same mass. For a nonmaterial
pendulum string makes with the vertical is given by θ , and the
body, passive gravitational mass and inertial mass are equal. A
arc length from the vertical equilibrium position is given by s .
nonmaterial body cannot have mass that is designated as active
!
gravitational mass.
we can write an equation for forces acting upon the pendulum
PENDULUM EXPERIMENTS
The equality of passive gravitational mass and inertial
A simple pendulum consisting of a bob of mass m at the
Using Newton’s second law of motion (see Section 2.4.2),
bob:
2.3.6!
TESTS OF THE EQUALITY OF PASSIVE GRAVITATIONAL MASS AND INERTIAL MASS
!
d 2s ˆ mI 2 T = − mG gsin θ Tˆ ! dt
(2.3-8)
of inertial mass to passive gravitational mass is dependent
where mI is the inertial mass of the pendulum bob, mG is the passive gravitational mass of the pendulum bob, g is the magnitude of gravitational acceleration, and Tˆ is a unit vector tangent to the swing arc. We also have:
upon the composition of the body. We will now consider both
!
!
Observations of bodies experiencing both a gravitational
force and an inertial force can be used to determine if the ratio
s = Lθ !
(2.3-9) 70
d 2θ g mG + θ = 0! dt 2 L mI
!
(2.3-12)
and so:
g L
θ = A cos
!
mG g t + Asin mI L
mG t! mI
(2.3-13)
The period T of the pendulum is then given by:
T = 2π
!
L g
mI ! mG
(2.3-14)
The acceleration of gravity g experienced by a body is independent of the body’s weight and composition as was first determined by Galileo. !
In this experiment two equal-length pendulums are found
to always have the same period even when their pendulum Figure 2-1!
bobs are made of different substances. Since the observed
Simple pendulum of length L .
period of a pendulum does not depend upon the mass of the and so equation (2.3-8) can be written as:
pendulum bob, mI and mG must then be equal (within the
d 2θ mI L 2 = − mG gsin θ ! dt
measurement error). It was determined, therefore, that passive
!
(2.3-10)
five parts in 10 6 .
or !
gravitational mass and inertial mass must be equal to within
d 2θ g mG + sin θ = 0 ! dt 2 L mI
For small oscillations this becomes:
(2.3-11)
2.3.6.2! !
TORSION BALANCE EXPERIMENTS
The equality of passive gravitational mass and inertial
mass has been tested to greater precision by torsion balance 71
experiments. All these experiments are patterned after the original torsion balance experiment conducted by von Eötvös (1890). In this type of experiment, two material bodies A and B are suspended from the ends of a beam that is, in turn, suspended at its center by a fine wire (see Figure 2-2). !
The compositions and weights of bodies A and B are
different (although their weights are usually nearly equal). We will denote the passive gravitational mass and inertial mass of body A by mG A and mI A , respectively, and the passive gravitational mass and inertial mass of body B by mG B and mI B , respectively. !
Acting on each body are two forces: gravitational force
and centrifugal force. The gravitational force is directed vertically down into the Earth. The centrifugal force is due to the rotation of the Earth and is not vertical at the location of
Figure 2-2!
Experiment of von Eötvös type.
Budapest where von Eötvös conducted his experiments (see Figure 2-3). !
For body A, we will denote the gravitational force as ! ! mG A g and the centrifugal force as mI A a with vertically and ! ! horizontally resolved forces of mI A aZ and mI A aH , respectively. For body B, similar notation applies. We can then write the
!
LA ⎡⎣ mG A g − mI A aZ ⎤⎦ = LB ⎡⎣ mG B g − mI B aZ ⎤⎦ !
where LA and LB are the beam lengths from the center of the beam to where the bodies A and B are suspended, respectively. From equation (2.3-15) we have:
equation for equilibrium of the beam under vertical torque about the beam center as:
(2.3-15)
!
LB = LA
mG A g − mI A aZ mG B g − mI B aZ
!
(2.3-16)
72
or
⎡ mG A g − mI A aZ mI B ⎤ ! horizontal torque = LA mI A aH ⎢1− ⎥ ! (2.3-19) m g − m a m ⎢⎣ GB IB Z IA ⎥ ⎦ This can be rewritten as:
m I A m I B aZ ⎡ m − IB ⎢m mG A g I A horizontal torque = LA mG A aH ⎢ − aZ ⎢ mG A m − m G B I B ⎢ g ⎣
⎤ ⎥ ⎥ !(2.3-20) ⎥ ⎥ ⎦
Since g >> aZ , we then have: !
Figure 2-3!
!
! Gravitational acceleration g and centrifugal ! acceleration a of a body on the surface of the rotating Earth.
The horizontal torque about the wire suspending the beam
horizontal torque = LA mI A aH − LB mI B aH !
(2.3-21)
Therefore, any inequality in the ratios of inertial mass to passive gravitational mass for the bodies A and B will produce a horizontal torque. !
Since no horizontal torque is observed when bodies A and
B have different compositions and weights, the ratio of inertial mass to passive gravitational mass remains the same to within
at its center is given by: !
⎡ mI A mI B ⎤ horizontal torque = LA mG A aH ⎢ − ⎥! m m ⎢⎣ G A GB ⎥ ⎦
(2.3-17)
Using equation (2.3-16), we have:
⎡ ⎤ mG A g − mI A aZ horizontal torque = LA ⎢ mI A aH − mI B aH ⎥ ! (2.3-18) m g − m a ⎢⎣ ⎥⎦ GB IB Z
one part in 10 9 . This ratio must then equal a constant (which can be chosen to be one) since it is the same for bodies of different compositions and weights. That is, within the limits of the resolution of this experiment, passive gravitational mass must always equal inertial mass. Torsion balance experiments 73
by von Eötvös (1890) and von Eötvös et al. (1922) together with the more recent torsion balance experiments of Dicke (1961a),
2.4.1!
Roll et al. (1964), Braginsky and Panov (1972), Heckel et al.
!
(1989), Adelberger et al. (1990), and Touboul, P., et al. (2017)
inertia is:
show that the passive gravitational mass and inertial mass are
!
DERIVATION OF NEWTON’S LAWS OF MOTION Newton’s three laws of motion describe how the motion of
a body changes when the body is acted upon by a force. Newton’s laws of motion, therefore, are laws pertaining to acceleration. !
Newton’s laws of motion appeared for the first time in
definitive form in his Principia. These laws were developed by Newton from empirical observations. They are the foundation for the entire field of mechanics. Newton formulated his laws of
Newton’s first law of motion known as the principle of Every body remains in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.
equal to within several parts in 1015 (see Cook, 1988).
2.4!
NEWTON’S FIRST LAW OF MOTION
!
Newton’s first law of motion can be obtained directly from
equation (2.3-7) which was derived using the aether field equation of gravity. We will now consider a material point ! particle of mass m located at a point r . From equation (2.3-7) we then have: ! ! ! F = ma!
(2.4-1)
! We will let the force F in equation (2.4-1) be an impressed (non! flow) force acting on the material point particle at point r . ! ! ! If F = 0 so that no force is acting on the material point
based, therefore, upon the theoretical concept of material
particle of mass m , we will have from equation (2.4-1): ! ! ! a = 0! (2.4-2)
particles abstracted to material point particles (which have no
Therefore, if no force is impressed on the material point
volume but do have constant density). Newton’s laws of
particle, the particle will remain in a state of rest or of uniform
motion can all be derived directly from the aether field
rectilinear motion.
equation of gravity.
!
motion for material point particles. Newtonian mechanics is
! ! If an impressed force F ≠ 0 is acting on the point particle
of mass m , we will have from equation (2.4-1): 74
!
! ! a ≠ 0!
(2.4-3)
If a force is impressed on a material point particle, therefore, the particle will be compelled to change its state of motion (the particle will accelerate). !
2.4.2! !
Newton’s second law of motion, which is known as the
fundamental principle of dynamics, is:
! ! By considering the two cases of impressed force F = 0 and
The change in momentum of a body is proportional to the force impressed upon it and is in the direction of the straight line in which that force is impressed.
! ! F ≠ 0 , we have obtained Newton’s first law of motion from
equation (2.3-7). We have shown, therefore, that Newton’s first law of motion can be derived from the aether field equation of gravity. Newton’s first law of motion provides the qualitative definition of force (Jammer, 1957). !
If there are several forces acting on the material point
particle at the same time, the above discussions will remain ! valid providing that F is taken to be the resultant force (the ! ! vector sum of all forces acting). If the resultant force F = 0 , the forces acting on the material point particle are balanced and are
NEWTON’S SECOND LAW OF MOTION
!
Newton considered the momentum of a body to change in
a series of velocity jumps resulting from force impulses. Today, however, his second law of motion is generally written in terms of either acceleration or the time derivative of momentum. !
Newton’s second law of motion can be obtained directly
from the aether field equation of gravity. To show this, we will again consider a material point particle of mass m (as Newton ! did) located at a point r . We can then use equation (2.4-2) in the
then in equilibrium. The particle, therefore, will remain in its
form:
state of rest or of uniform motion in a straight line. The concept
!
of balanced forces is fundamental to the field of statics in ! ! mechanics. If the resultant force F ≠ 0 , the forces acting on the
! where v is the velocity of the material point particle. Since the
material point particle are unbalanced, and the particle will be compelled to change its state of motion (accelerate).
! ! dv ! F = ma = m ! dt
(2.4-4)
density of a material point particle is constant, its mass will be constant. We can therefore rewrite equation (2.4-4) as: !
! d ! F = [m v ]! dt
(2.4-5)
Equation (2.4-5) is Newton’s second law of motion. 75
!
We see, therefore, that Newton’s second law of motion can
be derived from the aether field equation of gravity. The derivation of equation (2.4-5) given above is consistent with Newton’s concept of “the accelerative action of force as a series of successive actions that imparts to the moving object successive increments of velocity,” as stated by Jammer (1957). In his second law of motion, therefore, Newton gave a relation for impulsive forces, not for continuous forces (Cohen, 2002). From this derivation follows the definition: The total impulse of external forces on a body in a given small time interval is equal to the change in linear momentum of the body in the same time interval: ! ! ! F Δt = Δ ( m v ) ! (2.4-6)
2.4.3! !
NEWTON’S THIRD LAW OF MOTION
Newton’s third law of motion is: To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal and oppositely directed.
!
! d ! ! 0 = [ m1 v1 + m2 v2 ] ! dt
(2.4-7)
d d ! ! m1 v1 ] = − [ m2 v2 ] ! [ dt dt
(2.4-8)
or !
Using equation (2.4-5) again, we see from equation (2.4-8) that the force one material point particle of the system exerts on the other material point particle of the system is given by: ! ! ! F1 = − F2 ! (2.4-9) which is Newton’s third law of motion for two material point particles interacting with each other. Therefore, Newton’s third law of motion follows from the aether field equation of gravity. ! ! Note that F1 and F2 do not act on the same point particle but on two different point particles. Also note that both forces must be acting simultaneously for this law to be valid. !
The proportionality of active and passive gravitational
mass is easily demonstrated from Newton’s third law of
Newton’s third law of motion can be obtained directly
motion. We will consider two material bodies that are
from Newton’s second law of motion. If we consider a physical
stationary relative to each other. We will let mA1 and mP1 be the
system consisting of two material point particles having masses ! ! m1 and m2 and velocities v1 and v2 , respectively, and if no
active and passive gravitational masses, respectively, of body 1
!
and mA2 and mP2 be the active and passive gravitational
resultant external force is acting upon this system, we can use
masses, respectively, of body 2. From Newton’s force law of
equation (2.4-5) to write for the system:
gravity as given in equation (1.1-4) we then have: 76
! m m F12 = − G A1 2 P2 rˆ ! r
!
(2.4-10)
for the gravitational force on body 2 exerted by body 1 and
! m m F21 = G A2 2 P1 rˆ ! r
!
(2.4-11)
classical mechanics is to be found in aether field theory. Newton’s laws of motion are valid only because of the existence and properties of aether. Moreover, Newton’s laws of motion are valid only with respect to inertial frames of reference (see Section 2.7).
for the gravitational force on body 1 exerted by body 2 (where we are using the same unit vector rˆ in both equations). From
2.5!
Newton’s third law of motion we can write: ! ! m m m m ! F12 = − G A1 2 P2 rˆ = − F21 = − G A2 2 P1 rˆ ! r r
! (2.4-12)
We can now derive a number of important conservation
laws. A physical entity is considered to be conserved if it is invariant (does not change) with time.
and so:
mA1 mA2 ! = mP1 mP2
!
DERIVATION OF CONSERVATION LAWS
(2.4-13)
2.5.1!
CONSERVATION OF LINEAR MOMENTUM
which must hold for any two material bodies regardless of
If the resultant impressed force on a body of mass m is F = 0 , we have from Newton’s second law of motion given in
composition. These ratios must then equal a constant (which
equation (2.4-5):
can be chosen to be one) since they are the same for bodies of
d ! ! mv] = 0! [ dt
different compositions and weights.
!
2.4.4!
! and so the momentum p is given by: ! ! ! p = m v = constant vector !
!
ORIGIN OF NEWTON’S LAWS OF MOTION
To summarize, Newton’s laws of motion are all contained
(2.5-1)
(2.5-2)
in and can be derived from the aether field equation of gravity.
Therefore, if no resultant force is acting on a body of mass m ,
The origin of Newton’s three laws of motion can all be found
its momentum will be a constant vector. This is known as the
then in the motion of aether. This means that the foundation of
law of conservation of linear momentum or simply as the law 77
of conservation of momentum. Because this law was derived using Newton’s second law of motion, the conservation of momentum is limited to inertial frames of reference (see Section 2.7).
2.5.2! !
CONSERVATION OF ANGULAR MOMENTUM
! ! ! When m is a constant and since v × v = 0 , we can use
equation (2.5-1) to write:
! d ! d ! ! ! r × [ m v ] = ⎡⎣ m ( r × v ) ⎤⎦ = 0 ! dt dt
!
(2.5-3)
! where r is the position vector for the body of mass m . We then have:
! ! ! ! ! ! m ( r × v ) = r × ( m v ) = r × p = constant vector !
!
(2.5-4)
Therefore if no resultant torque is acting on a body of constant ! ! mass m , its angular momentum r × p will be a constant vector. This is known as the law of conservation of angular ! momentum. In a central force field, the linear momentum p is ! ! ! ! in the same direction as r and so we have r × p = 0 . Angular momentum is always conserved then in a central force field.
2.5.3! !
CONSERVATION OF ENERGY
! ! When there exists a force field F ≠ 0 that is derivable from
a scalar potential V so that:
! ! F = − ∇V ! (2.5-5) ! ! the force F is known as a conservative force if F does not ! depend on time. If F is acting along some path C , we have: !
!
! ! F • dr = −
∫
C
∫
! ! ∇V • dr = −
C
∫ dV !
(2.5-6)
C
Using Newton’s second law of motion for a material point particle of mass m , we can rewrite equation (2.5-6) as: ! d ! dr ! ( m v ) • dt dt = − dV ! (2.5-7) dt C C
∫
∫
or !
m
∫
C
! dv ! • v dt = − dt
∫ dV !
(2.5-8)
∫ dV !
(2.5-9)
C
Therefore we have: !
1 m 2
∫
K=
1 m v2 ! 2
C
! ! d (v • v ) = −
C
Letting !
(2.5-10)
We can rewrite equation (2.5-9) as: !
∫ dK = − ∫ dV ! C
C
(2.5-11)
78
where K is defined as the kinetic energy and V is defined as the potential energy of the point particle. ! !
Letting E = K + V , from equation (2.5-11) we have:
dE d ( K + V ) = = 0! dt dt
(2.5-12)
dE ! ! ! ! ∂V ∂V = F • v − F• v + = ≠ 0! dt ∂t ∂t
(2.5-17)
and so energy is not conserved. !
The potential energy V in equation (2.5-5) should not be
confused with the gravitational potential ϕ , which was defined in equation (1.2-11); potential energy and gravitational potential
or !
!
K + V = constant !
have different physical dimensions and refer to different (2.5-13)
entities. Gravitational potential characterizes a gravitational
which is known as the law of conservation of energy.
field while gravitational potential energy refers to the energy a
!
If the force is a function of time so that the force is not
given body has by being in a certain position within a
conservative, then energy will not be conserved. We can show
gravitational field. From equations (1.2-1) and (1.2-13) we have
this using E = K + V :
for a material point particle of mass m : ! ! ! ! F = m g = m ∇ϕ !
(2.5-18)
Using equation (2.5-5), we can write: ! ! ! − ∇V = m ∇ϕ !
(2.5-19)
!
dE dK dV ! = + dt dt dt
(2.5-14)
! where V is a function of position and time: V = V ( r, t ) . We can rewrite this equation using equation (2.5-10): ! ! dE ! d v ∂V d r ∂V ! ! = mv • + ! + dt dt ∂r dt ∂t
or (2.5-15)
V = − mϕ !
(2.5-20)
where ϕ is the gravitational potential of a gravitational field
or !
!
dE ! ! ! ! ∂V ! = F • v + ∇V • v + dt ∂t
(2.5-16)
and V is the gravitational potential energy of a material point particle having mass m .
Using equation (2.5-5): 79
4.5.4! !
FUNDAMENTAL JUSTIFICATION FOR CONSERVATION LAWS
∫
!
! ! F • dr = −
C
The three conservation laws: conservation of linear
∫
! ! ∇V • dr = −
C
∫ dV = ∫ dW ! C
C
(2.5-21)
conservation of energy can all be obtained from Newton’s laws
where dW is defined as the differential work done by the force ! ! F acting over a distance dr in the force field. From equation ! (2.5-21) we see that the net work done by F on a particle in
of motion. Therefore, these conservation laws all derive
moving it around any closed path is zero.
indirectly from the aether field equation of gravity. These
!
conservation laws all have their origin then in the motion of
(2.5-9) and (2.5-7) to write:
momentum,
conservation
of
angular
momentum,
and
aether, and properties of aether provide the fundamental justification for the conservation laws. Or as J. J. Thomson (1904a) said, “all mass is mass of the ether, all momentum, momentum of the ether, and all kinetic energy, kinetic energy of the ether.” For this reason the three conservation laws: 1.! Conservation of momentum. 2.! Conservation of angular momentum. 3.! Conservation of energy. have not only an empirical basis, but also a theoretical basis (applying equally at macroscopic and subatomic levels).
For a material particle of mass m , we can use equations
1 m 2
∫
C
! ! d (v • v ) =
∫
C
! d ! dr ( m v ) • dt dt = dt
∫
! ! F • v dt =
C
∫ dW ! (2.5-22) C
or using equation (2.5-10): !
∫
C
d ⎡1 ⎤ m v 2 ⎥ dt = ⎢ dt ⎣ 2 ⎦
∫
C
dK dt = dt
∫
C
! ! F • v dt =
∫
dW dt ! (2.5-23) C dt
and so we have: !
dK ! ! dW ! = F•v = dt dt
(2.5-24)
Therefore, the rate at which a conservative force field moving a material particle does work is equal to the time variation of the
2.5.5! !
WORK
! If F is a conservative force field, then for a material
particle’s kinetic energy.
particle of mass m moving along some path C in the field we can rewrite equation (2.5-6) as: 80
involved for inertia to be present, the force must be a non-flow
2.6!
NATURE OF INERTIA
force. Such a force will then be an impressed force. Inertia can
From his study of spherical bodies moving on inclined
be defined, therefore, as a measure of the resistance of a body
planes, Galileo Galilei (1638) determined that: “any velocity once
to any change in its state of motion due to an impressed force.
imparted to a moving body will be rigidly maintained as long as the
The concept of inertia is obviously associated with Newton’s
external causes of acceleration or retardation are removed.” Galileo
laws of motion and, in particular, with Newton’s first law of
thought, however, that the velocity would remain constant only
motion (which is also referred to as Newton’s principle of
for horizontal motion along the Earth’s surface (where gravity
inertia).
could not provide horizontal acceleration). Descartes (1644)
!
stated that: “each thing, as far as is in its power, always remains in
the source of inertia has remained a mystery ever since Kepler
the same state; and that consequently, when it is once moved, it
(1620) first associated inertia with matter almost four centuries
always continues to move,” and that: “all movement is, of itself,
ago. As Feynman (1965) noted, “The reason why things coast for
along straight lines.” Descartes was the first to realize the more
ever has never been found out. The law of inertia has no known
general law: a body will remain in a state of rest or of uniform
origin.” Or as Pais (1982) stated, “It must also be said that the
rectilinear motion unless acted upon by an external force. This
origin of inertia is and remains the most obscure subject in the theory
law is known today as Galileo’s law of inertia.
of particles and fields.”
!
!
Newton defined inertia to be a measure of the resistance of
Inertia is a fundamental concept in physics. Nevertheless
a body to any change being made in its state of motion. A
2.6.1!
change in the state of motion is, of course, acceleration. A body
!
in a state of rest or of uniform rectilinear motion will then not
aether field equation of gravity. From the definition of inertia ! we see that a body of mass m located at a point r will possess
have inertia since the body is not changing its state of motion (accelerating). Because a body cannot be accelerated without a force acting on the body, a body will possess inertia only if a force is acting on the body. Moreover, since resistance is
DERIVATION OF GALILEO’S LAW OF INERTIA
It is possible to derive Galileo’s law of inertia from the
inertia only if a nonzero impressed force: ! ! ! F ≠ 0!
(2.6-1) 81
is acting to accelerate this body by causing the aether to
!
accelerate so that: ! F ! ! ≠ 0! m
inertia only if at least one of the two terms of this equation is
!
nonzero. Since the density of aether ζ is constant for aether (2.6-2)
For equation (2.6-2) to be true in general, we see from
equation (2.3-3) that we must have: ! ! 1 ∂ ! F ⎡ ⎤ ! ⎡⎣∇ • (ζ υ ) ⎤⎦ = ∇ • ⎢ ⎥ ≠ 0 ! ! ζ ∂t ⎣m⎦
(2.6-3)
Equation (2.6-3) is the necessary condition for a body to possess inertia. Taking the time derivative we have: !
1 ! ⎡ ! ∂ζ ! ⎤ ∇ • ⎢ζ α + υ ⎥ ≠ 0 ! ζ ∂t ⎦ ⎣
!
flow not in proximity to nucleons, we have ∂ζ ∂t = 0 , and so, for aether motion resulting from an impressed force, the second ! term will always be zero regardless of the value of υ : !
∂ζ ! ! υ = 0! ∂t
(2.6-6)
! ! For inertia to exist, therefore, we must have α ≠ 0 . Any ! ! body at rest or moving at a constant velocity (so that α = 0 ) will !
not have inertia since both terms of equation (2.6-5) will then be (2.6-4)
zero. An impressed force must be acting to accelerate a body for the body to have inertia. !
or
! ∂ζ ! ζ α + υ ≠ 0! ∂t
From equation (2.6-5) we also see that a body can have
We can now understand why a body moving at a constant
velocity does not have inertia, but an accelerating body does. (2.6-5)
!
From equation (2.6-5) we see that the necessary condition ! for a body at any point r to possess inertia can be expressed completely in terms of properties of the aether at this same ! point r . The origin of inertia must then be found in the physical properties of aether. This explains how it is that inertia can be present instantaneously wherever an impressed force
We can also see why uniform motion through space does not produce observable effects, while accelerated motion does. !
We have found that Galileo’s law of inertia can be derived
from the aether field equation of gravity. For the first time we have an explanation why it is that, for inertia to be present in a body, a fundamental distinction must exist between accelerated and unaccelerated motion.
acts on a body. Aether is ubiquitous in our Universe. 82
2.6.2! !
MACH’S PRINCIPLE
Mach (1883) thought that a material body has inertia only
2.7!
because it interacts physically in some manner with all the
ABSOLUTE MOTIONS AND GALILEAN RELATIVITY
other material bodies in the Universe. Mach regarded “all
!
masses as related to each other.” Mach’s principle (as named by
and the reference frame must be a physical entity (see Section
Einstein, 1918b) proposes that all matter is coupled together so
1.2.9.1). Galileo Galilei (1632) investigated the relativity of
that inertia has its physical origin in the totality of the matter of
bodies in motion in an attempt to explain why the rotation of
the Universe. Since most matter is located in far distant ‘fixed
the Earth had not been detected by observations on Earth (in
stars’, Mach concluded that the accelerated motion of a body
his day). He determined that “motion, in so far as it is and acts as
relative to the far distant stars must be the primary cause of the
motion, to that extent exists relatively to things that lack it; and
inertia of the body. Why this might be so has never been
among things which all share equally in any motion, it does not act,
explained.
and is as if it did not exist.” Galileo found that the velocity of a
!
Mach’s principle has the serious problem of requiring
material body is always relative to the velocity of the reference
instantaneous action-at-a-distance between a material body and
frame chosen. The velocity of a material body is not uniquely
the far distant stars. Furthermore as Bondi (1952) noted, it has
specified until the frame of reference is also specified. This
not been possible to express Mach’s principle in mathematical
frame of reference can be any other physical entity.
form, nor has it been possible to verify it experimentally.
!
Mach’s principle predicts that the inertia of any given material
a reference frame since acceleration is motion. The acceleration
body will increase as the amount of matter in proximity to the
of a material body is found to be absolute, however. A
body increases. No such increase in inertia has ever been
kinematic absolute designates a kinematic entity that is the
observed, however (see Hughes et al., 1960; and Drever, 1960).
same for all observers everywhere, and so a kinematic absolute
!
must always be relative to a unique reference frame. A single
From equation (2.6-3) we see that Mach’s principle cannot
be correct. The inertia of a body is not directly dependent upon
Motion can only be defined relative to a reference frame,
The acceleration of a material body must also be relative to
reference frame must then exist for all accelerated motion.
any other body. 83
!
A unique reference frame for any given motion is
demonstrate Galilean relativity. We will now consider two such
considered an absolute reference frame for that motion. We see
material reference frames S and S’ indicated by the xi and xj′
then that the designation of absolute or relative for motion is
coordinate systems, respectively, in Figure 2-4.
based upon the existence or nonexistence of a unique frame of reference. !
A material reference frame that is at rest or is moving with
a uniform rectilinear motion is referred to as an inertial reference frame (Lange, 1886). In other words, a material reference frame that is not accelerating is an inertial reference frame. Therefore, an accelerating or rotating material body cannot be an inertial reference frame. !
Newton’s laws of motion contain only accelerations, not
velocities. In fact, “Newton’s laws of motion are statements about acceleration,” as Clotfelter (1970) observed. Newton’s laws of motion, therefore, are true relative to the absolute reference frame for accelerations, or to any inertial frame. This is the reason all inertial reference frames are equivalent in Newtonian mechanics. The fact that acceleration of a material body is absolute, while velocity of a material body is relative is known
Figure 2-4!
as Galilean relativity or Newtonian relativity. !
By considering the transformation laws that allow us to
!
A reference frame S’ is moving with a constant ! velocity U relative to an inertial reference frame S.
We will assume that the two reference frames S and S’
pass from a coordinate system within one material reference
were co-located and had identical coordinate system origins
frame to a coordinate system within a second material reference
and orientations at time t = 0 . The reference frame S’ is now ! moving with a constant velocity U with respect to the reference
frame that is moving relative to the first reference frame, we can
84
frame S, which is at rest relative to the absolute reference frame
!
for accelerations. Both reference frames S and S’ are therefore
respect to the inertial reference frames S and S’ are equal, and
inertial. A material particle of mass m is referred to both
so acceleration is absolute. If acceleration were not absolute,
reference frames. The position of the material particle of mass ! m is given by line position vector r with respect to the ! reference frame S, and by line position vector r ′ with respect to
then a body could accelerate without the application of a force
the reference frame S’, where: ! ! ! ! r′ = r − U t !
acceleration without any consideration of the body’s motion (2.7-1)
The three component equations represented by the vector equation (2.7-1) are known as Galilean transformations. In Galilean transformations, time intervals are taken to be absolute (which in fact they are as shown in Section 8.6). Therefore, time is the same in reference frames S and S’. Galilean transformations are valid between inertial reference frames.
!
From equation (2.7-1), the velocity of the particle is: ! ! dr ′ dr ! = −U ! (2.7-2) dt dt
We see from equation (2.7-3) that accelerations with
(by simply changing the reference frame). If a non-flow force is accelerating a material body, it is possible to detect this relative to other material bodies. !
We also see from equation (2.7-2) that velocities of material
particles with respect to the inertial reference frames S and S’ are not equal, and so velocity is not absolute, but relative to the velocity of the reference frame. We then cannot determine an absolute velocity for a material particle. As Davies and Gribbin (1992) noted, “The contrast between uniform and nonuniform motion is deep.” !
Similarly, we see from equation (2.7-1) that position is not
absolute, but relative to the position of the reference frame. We then cannot determine an absolute position for a material particle.
This equation is known as the Galileo addition theorem for
Since the acceleration of a material body is absolute,
velocities. From equation (2.7-2), the acceleration of the
implicit in Newton’s laws of motion is the assumption that an
material particle is:
absolute reference frame exists relative to which these laws are
!
! ! d 2r ′ d 2r = 2! dt 2 dt
(2.7-3)
valid. In fact, Newton’s laws of motion are only valid relative to the absolute reference frame or to inertial reference frames. 85
Newton’s laws of motion are invariant with respect to
frame. Empty space has no physical properties, however, and
Galilean transformations, therefore. This means that a law in
so cannot be a reference frame (see Section 8.5).
mechanics that is valid for one inertial reference frame is also
!
valid for any other reference frame moving with constant
has to be a physical entity with physical properties so that the
velocity relative to the first frame. Relative to inertial reference
motion of material bodies relative to this reference frame can be
frames, laws of mechanics take the form of Newton's laws of
determined. Aether is a real physical entity that is nonmaterial
motion.
but, nevertheless, has real physical properties. Moreover aether
The nonmaterial absolute reference frame for accelerations
Newton’s laws of motion also require the concept of
is ubiquitous and permeates all space. We will now show that
absolute time as was noted by Hawking (1987). There certainly
aether does indeed serve as the absolute reference frame for
are no kinematic reasons for assuming that time is in any way
accelerations.
dependent on motion. Moreover the fact that acceleration is
!
absolute for all inertial reference frames argues against any
equation (2.3-4) is:
dependence of time on motion.
! 1 ! ⎡ ! ∂ζ ! ⎤ ! ⎡ F ⎤ ! ∇ • ⎢ζ α + υ ⎥ = ∇ • ⎢ ⎥ ! (2.7-4) ζ ∂t ⎦ ⎣ ⎣m⎦ ! where we will now consider F to be a non-flow (impressed)
!
Newton struggled with the problem of absolute versus
relative motion, and he tried to find the absolute reference frame for accelerations. He knew that: “True motion is neither
The aether field equation of gravity written in the form of
generated nor altered, but by some force impressed upon the body
force. We then have ∂ζ ∂t = 0 since the aether not in proximity
moved; but relative motion may be generated or altered without any force impressed upon the body.” Newton realized that no material
to nucleons is incompressible, and so equation (2.7-4) will hold ! ! true for any value of υ but for only one value of α . We now see
body may really be at rest and that: “the true and absolute motion
how a reference frame can exist that is absolute for acceleration,
of a body cannot be determined by the translation of it from those
but is not absolute for velocity. The absolute reference frame for
which only seem to rest.” Therefore, he sought an absolute
acceleration must be aether. This is confirmed by the fact that
reference frame for acceleration that was nonmaterial. He
Newton’s three laws of motion (which pertain to the
thought that empty space might provide this absolute reference
acceleration of bodies) can all be derived from a consideration 86
of the motion of aether. Since there is no absolute reference
reference frames, and they depend on the inertial reference
frame for velocity, the velocity of a body will be relative to any
frame.
given inertial reference frame (i.e., any reference frame that is not accelerating relative to aether). We can now understand why inertial reference frames are
2.8! !
INERTIAL FORCES Aether is the unique reference frame for the acceleration of
distinguishable from accelerating material reference frames.
a material body. Any non-inertial frame will be accelerating
Aether is the special, preferential, or privileged reference frame
with respect to the absolute reference frame for accelerations.
for accelerating bodies. Aether is the absolute reference frame
!
relative to which Newton’s laws of motion are valid. If aether
of a material body is a non-inertial frame of reference, a force
did not exist, no absolute reference frame for accelerations
known as an inertial force will act on the body. This inertial
would exist.
force will be in a direction opposite to the force causing the
When the reference frame used to describe the acceleration
Finally, the fact that someone in free fall in a uniform
acceleration of the body. Inertial forces are often referred to as
gravitational field does not sense acceleration can now be
fictitious forces or pseudo forces because no physical source
explained. Gravitational force is a flow force (see Section 1.15).
capable of producing these forces is apparent when referred to
The aether surrounding a person in free fall is also being
the non-inertial reference frame.
accelerated (causing the gravitational field), and so there exists
!
no acceleration of the person relative to the absolute reference
g-force, centrifugal force, Coriolis force, and Euler force.
frame that immediately surrounds the person. Note that in
Material bodies that are speeding up or slowing down
general relativity the seeming elimination of gravitational force
represent a non-inertial reference frame and experience g-
in free fall is taken as an assumption in the form of a postulate
forces. Centrifugal forces, Coriolis forces, and Euler forces are
known as the equivalence principle.
all inertial forces that can be experienced by rotating material
!
bodies (which are also non-inertial reference frames).
The conservation laws of linear and angular momentum
There are four inertial forces that have been identified: the
and of energy are all functions of velocity (see Section 2.5).
!
Therefore, these conservation laws are valid only for inertial
passenger of a car that is accelerating forward. The passenger’s
As an example of a g-force, we will consider the force on a
87
seat will become compressed against the passenger’s back as the car accelerates. !
With respect to an inertial reference frame, the seat is
2.8.1! !
NEWTON’S LAWS OF MOTION FOR NON-INERTIAL REFERENCE FRAMES
compressed because the seat must accelerate with the car, and
!
Newton’s laws of motion pertain to accelerations and so
so must exert force on the passenger so that the passenger
are based upon inertial frames of reference. With respect to
accelerates with it. Since the passenger will then be accelerating
non-inertial reference frames, Newton’s laws of motion are not
relative to an inertial reference frame, the total force on the
valid without some modification since such non-inertial frames
passenger will be unbalanced. The passenger (and the car) will
are not the absolute reference frame for acceleration.
move forward.
!
With respect to the non-inertial reference frame of the car,
For example, if the reference frame S’ is non-inertial, and if ! S’ is being accelerated with a constant acceleration a with
the seat will appear to be compressed because a fictitious g-
respect to the inertial reference frame S, then in place of
force is pushing the passenger back against the seat. The force
equation (2.7-3) we have:
!
of the compressed seat will balance the g-force and so the force on the passenger will be balanced. The passenger will therefore be stationary with respect to the non-inertial reference frame of the accelerating car. The source of the g-force will not be apparent to the passenger. !
Since
aether
is
the
unique
reference
frame
for
accelerations, only with respect to an inertial reference frame
!
! ! d 2r ′ d 2r ! = 2 −a ! dt 2 dt
(2.8-1)
Newton’s second law of motion in the non-inertial reference frame S’ must then be modified to include all the forces acting: !
! d 2r′ ! ! m 2 = F − ma ! dt
(2.8-2)
can an accurate description of the true force producing
! ! where F is reduced by the force m a , which is a fictitious
accelerated motion be realized. When a non-inertial reference
inertial force. The acceleration due to any inertial force is
frame is used to describe accelerated motion, fictitious forces
proportional to the mass of the material body being accelerated.
must be postulated to explain the observations.
88
2.8.2! !
INERTIAL FORCES AND MACH’S PRINCIPLE
Inertial forces are obviously independent of the local
2.8.3! !
NEWTON’S BUCKET EXPERIMENT
Centuries ago Newton determined empirically that
distribution of material bodies since these forces always follow
centrifugal force is not caused by rotation relative to other
the same laws for all observed distributions of matter. Mach’s
material bodies, but is due to an absolute rotation. Newton’s
principle is sometimes invoked to explain the origin of inertial
famous experiment on rotational motion can be visualized by
forces as due to the accelerating motion of a body relative to the
considering a bucket filled with water and suspended by a long
far distant ‘fixed’ stars. In other words, Mach’s principle is
cord that has been strongly twisted. The experiment consists of
equivalent to the assumption that far distant ‘fixed’ stars
four stages.
provide the absolute reference frame for the accelerations associated with inertial forces. For example, Mach (1883) thought that centrifugal forces are produced only when a material body rotates relative to the fixed stars; when a material body rotates relative to some other material body, but not relative to the fixed stars, centrifugal forces are not produced. Mach’s principle is not valid, however, as we have seen in Section 2.6.2. !
Aether, not distant matter, provides the absolute reference
frame for the acceleration of a material body. Einstein attempted to include Mach’s principle in his general theory of relativity. As Whitehead (1922) noted, however, “The Einstein theory in explaining gravitation has made rotation an entire mystery.”
1.! The bucket is held at rest. There is no relative motion between the bucket and the water. During this time period the surface of the water will remain flat. 2.! The bucket is then suddenly whirled about in the direction that will unwind the cord. At first the bucket will rotate but the water will not. Relative motion exists between the bucket and the water. During this time period the surface of the water will continue to remain flat although the bucket is rotating. 3.! Soon the motion of the bucket will be communicated to the water, and the water will begin to rotate. Centrifugal force on the water will then cause the water to begin ascending the sides of the bucket, forming a concave water surface as bucket and water rotate together. There is still relative motion between the bucket and the water. 89
4.! When the bucket and the water are finally rotating at the same speed so that no relative motion exists between them, the surface of the water will show its greatest deviation from a flat surface. !
Relative motion between the water and the bucket clearly
velocities are relative, not absolute. Mach attempted to explain Newton’s bucket experiment with his principle (see Earman, 1989), but as we have shown Mach’s principle is not correct.
2.9!
DARK MATTER
does not produce the same effect on the water’s surface as
!
Ever since 1932 when Oort made the first measurements of
rotation of the water. The effect on the water’s surface is
the rotational velocity of our spiral galaxy, the Milky Way, a
greatest when the relative motion between the bucket and the
problem with the amount of matter in our galaxy has been
water is essentially zero. The shape of the water’s surface then
recognized. There appears to be too little matter in our galaxy.
indicates absolute (not relative) motion.
The outward directed centrifugal force on the stars and clouds
!
Newton’s bucket experiment shows that the centrifugal
of gas in our galaxy due to galaxy rotation has always been
force on the water depends only upon the absolute rotation of
assumed to be balanced by the inward directed gravitational
the water and not upon the relative rotation between the bucket
force resulting from the matter in our galaxy. From Newton’s
and the water. In other words, centrifugal force depends only
law of gravity we know that this gravitational force is
upon absolute rotation, and not upon rotation relative to any
proportional to the quantity of matter in our galaxy, and
material body.
inversely proportional to the distance squared from this matter.
!
Newton concluded, therefore, that absolute motion exists
Since the luminous matter of a spiral galaxy decreases rapidly
in the form of "the true and absolute circular motion of the water."
with distance from the bright center region of the galaxy, the
A unique reference frame for accelerations must then exist. The
gravitational force is also expected to decrease with radial
unique reference frame for accelerations is aether as we have
distance. In order for the balance between gravitational force
seen. This explains why it is possible to determine if a material
and centrifugal force to be maintained at greater radial
body is rotating (or not) without reference to any other material
distances, the centrifugal force (and rotational velocity) of our
body. The same does not apply for determining if a material
galaxy must then also decrease.
body is moving with a constant velocity or is at rest, since 90
Studies by Oort (1932, 1960), Bahcall (1984a, b, c, 1987), and Carney and Latham (1987) have found, however, that the rotational (tangential) velocity of our galaxy does not decrease with radial distance. Moreover, Freeman (1970), Rubin and Ford (1970), Rogstad (1971), Rogstad and Shostak (1972), Ostriker and Peebles (1973), Roberts and Rots (1973), Roberts (1976), Faber and Gallagher (1979), Rubin (1979, 1983a, b, 1987), and Rubin et al. (1980, 1985) have discovered that it is not only our spiral galaxy that exhibits this rotational velocity problem, but also all spiral galaxies examined regardless of their size (see Sanders, 2010). It has become obvious that a serious problem exists in our understanding of galaxy structure and perhaps of physics. !
A typical rotational velocity curve measured for a spiral
Figure 2-5! A typical rotational (tangential) velocity curve for a spiral galaxy.
galaxy is shown in Figure 2-5. In the bright center region of a
!
galaxy, the rotational velocity increases linearly with radial
constant (independent of radial distance) and remains constant
distance from the galactic center. This type of rotational motion
to great distances. In fact, the rotational velocity of galaxies is
is generally associated with a rotating solid body. The
almost never observed to decrease with radial distance in the
explanation usually given for this observed rotational velocity
galaxy. Moreover this rotational velocity is simply too large in
increase with radial distance is that, because matter is so tightly
the outer regions of spiral galaxies for the centrifugal force to be
packed in the bright center region of any galaxy, all this matter
balanced by gravitational force (calculated on the basis of the
behaves as one solid body.
amount of luminous matter in the galaxy). Stars in the outer
At greater radial distances, the rotational velocity becomes
regions of the galaxy should fly off into deep space, and yet they don’t. The conclusion usually made then is that the 91
gravitational force in the galaxy must be stronger than estimated, suggesting that there may exist more matter in spiral
2.9.1.1!
DARK MATTER
galaxies than the radiating matter that we can detect.
!
!
Studies by Zwicky (1933, 1937c), Smith (1936), Aaronson
the form of undetected dark matter is required to provide the
(1983), Fabricant and Gorensten (1983), Faber and Lin (1983),
necessary additional gravitational force. The designation dark
Aaronson and Olszewski (1987), and Kormendy (1987) show
matter indicates that this undetected matter is non-luminous
that not only spiral galaxies but also all types of galaxies and
and does not emit or reflect detectable electromagnetic
even certain clusters of galaxies (that appear not to be
radiation at any wavelength. Also this dark matter must be
dispersing when it seems they should be) have a similar
transparent and electrically neutral. Otherwise the presence of
undetected matter problem. It is no exaggeration to say that
this matter would have been detected by now.
astronomy has reached a crisis because of the undetected
!
matter problem.
must possess mass that is designated as active gravitational
For the first proposed solution, at least 90% more matter in
To provide the necessary gravitational force, dark matter
mass. Moreover dark matter must be distributed in a huge
2.9.1!
PREVIOUSLY PROPOSED SOLUTIONS
cloud that pervades the galaxy and extends well beyond the
Two solutions for the unexplained rotational velocities
visible disk of luminous matter (a factor of four or five times
observed in spiral galaxies have been considered previously.
farther). The dark matter must then be distributed quite
They are:
differently from the luminous matter. As van Albade and
!
1.! The matter contained in galaxies is much greater than is apparent from luminous matter radiating at any wavelength. 2.! Some modification of Newton’s force law of gravity is required.
Sancisi (1986) noted, “Luminous matter and dark matter seem ‘to conspire’ to produce the flat observed rotation curves in the outer region.” !
To date the search for possible dark matter candidates has
been unsuccessful, as all known types of matter have been eliminated, and no new type of matter has been discovered (Cline, 2003). Dark matter, if it exists, must consist of a type of 92
elementary particle as yet unknown. The failure to discover
matter problem. As noted in Section 1.2.8, flowing aether can
dark matter has occurred despite the fact that dark matter, if it
carry with it any material body within the flow. The aether flow
exists, must also be present right here on Earth. Moreover, if
in a region of space influences the motion of all matter within
dark matter exists and is by far the predominant form of matter
the flow. Matter is carried along by the flow of aether just as
in our Universe, it is really incredible that we have no direct
floating material is carried along by a current (Very, 1919).
laboratory evidence for this dark matter. In any case, as Krauss
Freeman and McNamara (2006) noted that stars and galaxies
(2000) noted: “Requiring an entirely new form of matter in the
behave as if they are “mere flotsam on a cosmic sea.” Descartes
universe is a radical concept that must not be taken lightly.”
(1644) observed: “when an entire fluid body moves simultaneously
2.9.1.2! !
MODIFIED LAW OF GRAVITY
For the second proposed solution, modifications of the
distance dependence in Newton’s force law of gravity have been proposed by Finzi (1963), Tohline (1983), and Sanders (1984). Other modifications to Newton’s force law of gravity have been proposed for the case of small accelerations by Milgrom (1983a, b, c, 1989, 2002a, b), Bekenstein and Milgrom (1984), Milgrom and Bekenstein (1987), Bekenstein (1992, 2004, 2010), and Sanders and McGaugh (2002). All of these modifications to Newton’s force law of gravity are, however, inconsistent with the physical cause of gravity (see Sections 1.2.8 and 1.4.3).
2.9.2! !
NONEXISTENT DARK MATTER
Despite the difficulties encountered with previously
in some direction, it must necessarily carry along with it any solid body which is immersed in it.” For a region of space defined by the boundaries of a galaxy, a large circular flow or vortex of aether can produce all the observed motions of luminous matter as we will now demonstrate. The idea that some type of celestial vortices of aether may exist was originally proposed by Descartes in 1644, and was an important concept in European physics for several centuries thereafter (see Burtt, 1924; and Aiton, 1972). !
We will begin by noting that the whirlpool appearance of
many galaxies (see Figure 2-6) certainly suggests that some real physical medium is in circular motion about the central axis of each galaxy. Based upon the typical rotational velocity distribution for galaxies shown in Figure 2-5, we can conclude that for any given galaxy the circular flow of aether around the
proposed solutions, we are now able to resolve the undetected 93
galactic axis must consist of two distinct vortices, with each
within the flows will be carried along at the observed rotational
vortex having a different rotational (tangential) velocity.
velocity of each flow. !
! Writing the aether flow velocity υ around a galaxy in
cylindrical coordinates, we have: ! ! υ = υr eˆr + υθ eˆθ + υz eˆz !
(2.9-1) ! From Figure 2-5, we see that the aether flow velocity υ1 of the inner vortex of the galaxy is a tangential (linear) velocity given by: !
! υ1 = K1 r eˆθ = υθ 1 eˆθ !
(2.9-2)
υθ 1 = K1 r !
(2.9-3)
and so: !
! where K1 is a constant. The aether flow velocity υ 2 of the outer vortex of the galaxy is a tangential (linear) velocity given by: ! ! υ 2 = K2 eˆθ = υθ 2 eˆθ ! (2.9-4) Figure 2-6! !
Galaxy M81 (Photo credit: NASA/JPL-Caltech)
The inner vortex occupies the bright central region of the
and so:
υθ 2 = K 2 !
!
(2.9-5)
galaxy and has a rotational velocity proportional to radial
where K2 is a constant.
distance. The outer vortex occupies the galactic region outside
!
the bright central region and has a rotational velocity that is
between the two concentric flows is located by rB , then for
constant. These two aether flows circle the galactic axis much as
continuity of the aether flow in the vortices we must have at
hurricane winds circle the central axis of a hurricane. Matter
r = rB :
Denoting the radial distance at which the boundary
94
! ! υ1 = υ 2 = K1 rB eˆθ = K 2 eˆθ !
!
(2.9-6)
body rotation. Obviously tightly packed matter need not be
and so: ! !
hurricane is a vortex with a central core that has a similar rigid-
K2 = K1 rB !
(2.9-7)
present to cause this rigid-body rotation (see Figure 2-7).
The inner vortex flow has the characteristics known as
wheel flow in fluid mechanics. The circulating aether flow in this bright central region of the galaxy has the same motion as a ! rigid body rotating with a constant angular velocity ω 1 as can easily be shown: ! ! ω 1 = ω 1 eˆz ! !
(2.9-8)
! ! ! υ1 = ω 1 × r = ω 1 eˆz × r eˆr = ω 1 r eˆθ = K1 r eˆθ = υθ 1 eˆθ !
(2.9-9)
where we have used equation (2.9-2). We then have: !
ω 1 = K1 =
υθ 1 r
!
(2.9-10)
and so the angular speed of circulation ω 1 of the inner vortex flow is constant (time independent). Such rotary motion is known as rigid-body rotation. !
Since flowing aether carries along any matter within the
flow, no assumptions concerning dark matter are necessary in order to obtain the observed rotation of luminous matter within the bright central region of the galaxy. The type of vortex
Figure 2-7! !
Hurricane Isabel 2003 (Photo Credit NASA).
Using the aether flow velocity given by equation (2.9-4),
we can write for the outer vortex flow: ! ! ω 2 = ω 2 eˆz ! !
(2.9-11)
! ! ! υ 2 = ω 2 × r = ω 2 eˆz × r eˆr = ω 2 r eˆθ = K 2 eˆθ = υθ 2 eˆθ ! (2.9-12)
defined by equation (2.9-9) is known as a forced vortex. A 95
! where ω 2 is the angular velocity of circulating aether in the outer vortex. We then have: !
ω 2 r = K2 !
!
! ! 1 ∂( K1 r ) ∇ • υ1 = =0! r ∂θ
(2.9-16)
!
! ! 1 ∂( K 2 ) ∇ • υ2 = = 0! r ∂θ
(2.9-17)
(2.9-13)
or
K 2 υθ 2 ω2 = = ! r r
!
(2.9-14)
and so both flows are incompressible as expected. !
Assuming that the vortex flows in a galaxy are steady, the
and so the angular speed of circulation ω 2 of the outer vortex
accelerations of the inner and outer vortex aether flows are
flow is inversely proportional to the distance from the axis of
given by equations (1.7-5), (2.9-2), and (2.9-4): ! Dυ1 ! ⎡ K12 r 2 ⎤ ! =∇⎢ ⎥ − K1 r eˆθ × 2 K1 eˆz ! Dt ⎣ 2 ⎦
rotation. Since flowing aether carries along any matter within the flow, no assumptions concerning dark matter are necessary in order to obtain the observed rotation of luminous matter within the outer vortex of the galaxy. !
We will now examine the aether flow velocities of the two
types of vortices found in galaxies to determine if they are consistent with known properties of aether. Since aether flow is incompressible when not in proximity to nucleons, we expect the two aether flows in a galaxy to be incompressible. We then should have: ! ! ! ∇ •υ = 0 !
(2.9-15)
as given by equation (1.5-12) for the two aether flows. We actually have:
!
! Dυ 2 ! ⎡ K 22 ⎤ K2 ˆ =∇⎢ − K e × eˆz ! 2 θ ⎥ Dt 2 r ⎣ ⎦
(2.9-18)
(2.9-19)
Using equations (2.9-3) and (2.9-5), the aether flow accelerations of the inner and outer vortices of a galaxy are: !
! 2 υ Dυ1 θ = K12 r eˆr − 2 K12 r eˆr = − K12 r eˆr = − 1 eˆr ! Dt r
(2.9-20)
! υθ22 Dυ 2 K 22 ! =− eˆr = − eˆr ! (2.9-21) Dt r r ! ! The centrifugal accelerations a1 and a2 of matter within the inner and outer vortices of a galaxy are given by: 96
!
Dυ1 ! υθ a1 = 1 eˆr = − ! r Dt
(2.9-22)
!
Dυ 2 ! υθ 2 ˆ ! a2 = er = − r Dt
(2.9-23)
2
2
where we have used equations (2.9-20) and (2.9-21). The aether ! ! flow accelerations Dυ1 Dt and Dυ 2 Dt of the inner and outer vortices, respectively, of a galaxy are then exactly equal and oppositely
directed to the corresponding centrifugal ! ! accelerations a1 and a2 of matter in the galaxy. Therefore, no unbalanced forces exist in galaxies, and so no exotic dark matter is required to balance these forces. Galaxies primarily consist of two large concentric aether vortices. ! ! ! For any galaxy, the vorticities ψ 1 and ψ 2 of the inner and
Figure 2-8!
Side view of vortex structure of a galaxy.
! ! The total rate of change in the vorticities ψ 1 and ψ 2 can be
outer vortices, respectively, can be calculated to determine if the
!
vortices are rotational or irrotational. We have: ! ! ! ! ˆ ˆ ! ψ 1 = ∇ × υ1 = 2 K1 ez = 2 ω 1 ez ≠ 0 !
determined using equation (1.11-12): ! ! ! ! Dψ 1 ! ! ! ! = − ∇ • υ1 ψ 1 + ψ 1 • ∇ υ1 ! Dt
!
! ! ! K2 ! ˆ ˆ ψ 2 = ∇ × υ2 = ez = ω 2 ez ≠ 0 ! r
(2.9-24) (2.9-25)
Therefore, both the inner vortex and the outer vortex have vorticities and are rotational. This type of composite vortex structure consisting of concentric cylinders (see Figure 2-8) is similar to a Rankine vortex (see Robertson, 1965).
(
!
)
(
)
! ! ! ! Dψ 2 ! ! ! = − ∇ •υ2 ψ2 + ψ2 • ∇ υ2 ! Dt
(
)
(
)
From equations (2.9-16) and (2.9-17) we have: ! Dψ 1 ! ! ! ! = ψ 1 • ∇ υ1 ! Dt
(
)
(2.9-26)
(2.9-27)
(2.9-28) 97
!
! Dψ 2 ! ! ! = ψ 2 • ∇ υ2 ! Dt
(
)
Using equations (2.9-24) and (2.9-25) we obtain: ! ! ∂( r eˆθ ) ! Dψ 1 ∂υ1 ! = 2ω1 = 2 ω 1 K1 =0! Dt ∂z ∂z ! !
(2.9-29)
of aether may be one reason the rotation of the inner vortex is of the rigid body type.
2.10!
APPLICABILITY OF NEWTON’S LAW OF GRAVITY
(2.9-30)
! ! ∂( eˆθ ) ! Dψ 2 ∂υ 2 = ω2 = ω 2 K2 = 0! (2.9-31) Dt ∂z ∂z ! ! Therefore, the vorticities ψ 1 and ψ 2 are constant. From
!
Newton (1730) posed the problem: “what hinders the fix’d
stars from falling upon one another?” as a direct result of their gravitational pull. Another problem was noted by Seeliger (1895, 1896, 1909): if our Universe is infinite and has uniform
equation (1.11-18) we see that this is consistent both with the
matter distribution, the gravitational force exerted on any
flow of aether in galaxies being essentially two-dimensional
material point particle by all the other matter in our Universe
and with the known stability of galaxies.
would appear to be infinite (see Norton, 1999). This can be seen
!
by using equations (1.2-11) and (1.2-5) to write the gravitational
Since the requirement for dark matter within galaxies or
within clusters of galaxies does not exist, we can conclude that
potential ϕ for all the matter in our Universe:
dark matter does not exist. This explains why dark matter does not appear to be present in our solar system (no effects are observed), and why dark matter remains unidentified in our
!
ϕ=
∫∫∫
Gρ dV = r
Universe
2π
∫ ∫ dθ
0
π
0
sin φ dφ
∫
∞ R
Gρ 2 r dr ! r
(2.10-1)
Universe. We can also conclude that gravitational forces are not
where ρ is the mean density of matter in our Universe. We then
acting alone to hold galaxies together.
have:
Within the inner vortex, in the very center or eye of a spiral galaxy, there is a region into which aether flows from
!
ϕ = 4π G ρ
∫
∞ R
r dr !
(2.10-2)
both above and below the galaxy. It is possible that this region
Therefore, as r → ∞ , the gravitational potential also becomes
consists of a black hole. In fact, a black hole may exist at the
infinite.
very center of all galaxies (see Section 3.13). The vertical inflow 98
!
The resolution of these two problems in gravitational
larger region of space encompassed by the entire Milky Way
physics is obtained by recognizing that Newton’s law of gravity
galaxy. Since the non-gravitational acceleration field is not
does not control the acceleration fields existing throughout
observed within our solar system, it must be approximately
much of space. Wherever aether is accelerating, a force field
constant in this region. This is consistent with Corollary VI of
will exist. If an accelerating flow of aether is due to the presence
Newton’s laws of motion: “If bodies, moved in any manner among
of matter, which is a field flow sink for aether, the resulting
themselves, are urged in the direction of parallel lines by equal
force field will be a gravitational field and will always follow
accelerative forces, they will all continue to move among themselves,
Newton’s law of gravity. If an accelerating flow of aether is not
after the manner as if they had not been urged by those forces.” It is
due to the presence of matter, the resulting force field generally
for this reason the presence of dark matter has been postulated
will not follow Newton’s law of gravity. Since accelerating
to balance forces for our galaxy, but not for our solar system.
flows of aether exist that are not gravitational aether flows (as
The flow of aether in galaxies is not controlled just by the
evidenced by the structure of spiral galaxies), and since matter
presence of matter.
does not directly attract matter, the resultant gravitational force
!
on any given single material point particle is not simply due to
that is not consistent with Newton’s law of gravity has an
all matter in our Universe.
aether flow that is not totally controlled by the presence of
!
A local gravitational field can exist within an extended
matter. From our knowledge of spiral galaxy rotational
non-gravitational acceleration field. In other words, there can
velocities and the velocities of galaxies within clusters, we can
be some local variation in aether flow within a large regional
conclude that aether flow in our Universe is not totally
flow. In terms of vector flow velocities, this can be represented
controlled by the presence of matter. Therefore, gravity does
as the sum of two vector fields. For example, while Newton’s
not totally control the structure of our Universe. On an even
law of gravity represents the gravitational acceleration field
larger scale it is possible that immense currents of aether exist
that we observe within the relatively small region of space
in our Universe.
encompassed by our solar system, it does not represent the
!
non-gravitational acceleration field that extends over the much
of gravity always correctly describes the acceleration of aether
Any region of space having an observed acceleration field
Nevertheless, it should be mentioned that Newton’s law
99
into matter, irrespective of where in our Universe that matter
filtering is performed before any search for decay patterns even
exists. Therefore, Newton can be said to have provided the first
begins. Statistical methods are then used to detect Higgs
unification of physical forces, showing that the terrestrial and
particles based on a theoretical model known as the Standard
the celestial laws of gravity are one and the same.
Model of particle physics. If more decay patterns are found having the expected Higgs decay pattern than predicted by the
2.11! !
NONEXISTENT HIGGS PARTICLES
Higgs particles are hypothetical particles that were
Standard Model assuming Higgs particles do not exist, then the existence of Higgs particles is inferred.
proposed as part of a theory advanced by Higgs (1964) and
!
others. This theory postulates the existence throughout our
upon such indirect evidence. Clearly the validity of this
Universe of a special scalar field known as a Higgs field from
inference depends not only on the functioning of the extreme
which some particles acquire mass. The mass acquired by these
amount of computer filtering necessary to obtain a meaningful
particles would depend upon how well the particles couple
signal from the immense collision-created noise background,
with the Higgs field. Particles such as neutrons and protons
but also on the validity of the Standard Model (see Unzicker,
were postulated by the theory not to acquire their mass from
2013).
the Higgs field. The Higgs field, therefore, cannot account for
!
gravitational mass.
almost certainly wrong (see Section 6.15). Ironically, the
!
inferred existence of the Higgs particle is now being used to
A Higgs particle can never be directly detected. It is
In 2013 the existence of Higgs particles was inferred based
The Standard Model has many problems, however, and is
thought to exist for only about 10 − 22 second. The existence of
argue for the validity of the Standard Model.
Higgs particles can only be inferred from indirect evidence.
!
This inferential process involves examining the decay products
(2.2-2), we can now see that there is no Higgs field, and Higgs
of high energy particle collisions that occur within a particle
particles do not exist. Moreover, the mass that all entities have
accelerator in a search for a certain decay pattern that has been
originates from aether processes described by equation (2.2-2).
From the nature of mass as determined using equation
predicted for Higgs particles. Because of the massive amount of data produced in these high energy collisions, extensive data 100
!
2.12! !
SUMMARY
Mass can be defined as a measure of the time variation of
Aether is the absolute reference frame for all accelerating
bodies. Fictitious forces in the form of inertial forces must be postulated to explain observations only when a non-inertial
the flux of aether. All mass is consistent with this definition.
reference frame is used to describe accelerated motion.
There is only one kind of mass. Active gravitational mass,
!
passive gravitational mass, and inertial mass are designations
huge vortices of aether, and not by gravity. Centrifugal
based, not upon differences in the nature of mass itself, but
acceleration within these galaxies is exactly balanced by aether
upon differences pertaining to the source of the mass or to the
flow acceleration. Dark matter does not exist.
The motion of matter in galaxies is largely controlled by
situation of an entity possessing the mass. !
A material body can have mass that is designated as active
gravitational mass, passive gravitational mass, or inertial mass. A nonmaterial body can only have mass that is designated as passive gravitational mass or inertial mass. !
Newton’s three laws of motion can all be derived from the
aether field equation of gravity. The theorems for conservation of momentum and conservation of energy can, in turn, be obtained from Newton’s laws of motion. Therefore, Newton’s laws of motion and the laws of conservation of momentum and conservation of energy all have their origin in the motion of aether. The foundation of classical mechanics is to be found in aether field theory. !
The inertia a material body is independent of any other
material body. Mach’s principle is not correct.
101
Chapter 3 “But without a medium how can the propagation of lightwaves be explained?”
The Nature of Light
A. A. Michelson Studies in Optics
102
!
The study of light has fascinated philosophers and
but as a substance with ascertainable physical properties,” as Lodge
scientists for many centuries and has led to the development of
(1925a) noted. The only exceptions will be where the spelling
some of the most important theories in physics. For example,
‘ethereal’ has been used in a quoted reference or book title.
the theories of special relativity and quantum mechanics have their roots in attempts to explain phenomena associated with
3.1!
LUMINIFEROUS AETHER AND LIGHT
the generation and propagation of light. Despite all this study
!
and theorizing, however, light remains a mysterious entity
vibrations of a nonmaterial elastic medium that permeates
propagating as electromagnetic energy in an unknown manner
space. As a result of their research in optics, many scientists
through space. As Feynman says (in Feynman et al., 1964):
from the seventeenth to the early twentieth century concluded
“When I talk about the fields swishing through space, I have a terrible
that an aether must exist and must serve as the propagation
confusion between the symbols I use to describe the objects and the
medium for light. For example, Descartes (1637) proposed that
objects themselves. I cannot really make a picture that is even nearly
aether permeates all of space and that light propagates
like the true waves.”
instantaneously through aether in the form of a pressure pulse.
!
In this chapter we will determine the nature of light, and
Hooke (1665) postulated that light is a rapid vibratory motion
we will examine the nature of the physical medium in which
of very small amplitude that propagates very fast in all
light propagates. The effects of material body motion and of
directions as spherical waves in aether, so that light reaches “the
gravity on the propagation of light will be described. We will
greatest imaginable distance in the least imaginable time.” In 1676
provide a new interpretation of the classic Michelson-Morley
Rømer discovered that light does not propagate with infinite
experiment. Finally, we will consider gravitational lenses and
speed, but with a finite speed.
black holes.
!
!
We will use the term ‘light’ in this book to mean not just
with finite speed through aether in the form of corpuscles.
visible radiation but any electromagnetic radiation. When
Under the corpuscular theory, light is assumed to consist of
discussing aether we will use the spelling ‘aetherial’ rather
material particles emitted as tiny projectiles by the luminous
than ‘ethereal’ since aether “is dealt with not as a rarefied essence
source.
Under the wave theory, light is assumed to consist of
Then Newton (1687, 1730) suggested that light propagates
103
!
1.! A physical medium exists in which a disturbance occurs when some of the physical particles of the medium are displaced from their equilibrium positions.
If light consists of corpuscles, then a sharp shadow must
result from any obstacle in the path of light. It was already known in Newton's time, however, that shadows are not perfectly sharp, but diffuse (an effect known as diffraction). It
2.! Displacement of physical particles of the medium from their equilibrium positions must result in restoring forces that cause the displaced particles to attempt to return to their equilibrium positions.
was also known that two crossed beams of light can pass through each other and continue on without any effect. Huygens (1690a, b) reasoned, therefore, that light cannot consist of corpuscles since corpuscles in crossed beams would
3.! Inertia of the displaced particles causes them to overshoot their equilibrium positions and so to oscillate about their equilibrium positions.
collide with one another, resulting in scattering of the light beams. From his work on the propagation, reflection, and refraction of light, he came to the conclusion that light
4.! A disturbance at any point in the physical medium must produce a similar disturbance at a neighboring point at a slightly later time, resulting in propagation of the disturbance.
propagates in aether with a finite speed as waves.
3.1.1! !
DEFINITION OF A WAVE
To understand the nature of light, it is first necessary to
!
Therefore only in a physical medium having restoring
understand what a wave is. A wave is really a descriptive term
forces can vibratory motions exist and propagate. The physical
for a particular type of disturbed state that can occur within
disturbance that constitutes a wave is clearly a mechanical
physical media. As Leighton (1959) noted, “all wave motion with
phenomenon. All physical waves are mechanical waves. As
which we are familiar possesses a medium for its propagation.” In
Lodge (1909) noted, “Waves we cannot have, unless they be waves
fact, a wave is a propagating disturbance in and of a physical
in something.” Waves consist then of the oscillatory motion
medium; therefore a wave can only exist in a physical medium.
about their equilibrium positions of real particles of a physical
The type of propagating disturbance known as a wave is
medium. If there is nothing to oscillate, then waves cannot
formed by physical medium particles moving in an organized
exist. This is why “one cannot really think of waves without some
way. A wave requires that:
substantial medium for their conveyance,” as Lodge (1922) said. 104
!
Waves involve the transportation over distance of energy,
however, can propagate through a vacuum within a glass
but not the transportation over distance of the physical
vessel while sound cannot. This was demonstrated by von
medium in which the waves are propagating. Therefore, it is
Guericke in 1654 by showing that a ringing bell inside an
only a change in the state of the medium that is propagated.
evacuated glass jar can be seen but not heard. Huygens
Particles of the medium simply oscillate in place about their
concluded that, since a vacuum is devoid of matter, the elastic
equilibrium positions. This oscillation is often visualized as a
medium in which light waves propagate must be nonmaterial,
cork on water: the cork moves up and down as a wave passes,
and so the propagation medium for light waves must be an
then returns to its former position. “The essential characteristic of
elastic aether. In fact light is nothing more than undulations in
wave motion is that a periodic disturbance is handed on successively
a ubiquitous elastic aether.
from one portion of a medium to another,” noted Preston (1890). In this manner energy flows through the medium during wave
3.1.3!
propagation.
!
CORPUSCLES
Huygen’s conclusions were not generally endorsed during
the 1700s. Instead the theory that light propagates through
3.1.2! !
LIGHT WAVES
aether as corpuscles was accepted by most scientists based
For all real waves, elasticity is the restoring force causing
primarily on their interpretation of Newton’s work (including
particles of a physical medium to oscillate about their
his criticism of wave theory). An exception was Euler (1746,
equilibrium positions, resulting in wave motion. Light waves
1768, 1772) who thought that all space was filled with aether
must then be associated with the elastic oscillations of particles
and that, “rays of light are nothing else but the shakings or
of a physical medium. Huygens (1690a, b) thought that the
vibrations transmitted by the ether.”
propagation of light waves requires a medium that possesses
!
"springiness." He knew, therefore, that some type of elastic
interference of light in the early 1800s, Young concluded that
medium must exist in which particles vibrate as light
interference cannot be explained by the corpuscular theory of
propagates. He thought these vibrations to be analogous to the
light, and that light waves must be “undulations of the
From his work on the constructive and destructive
longitudinal vibrations of air that produce sound. Light, 105
luminiferous ether.” Subsequently, Young (1802a) came to the
of light is, however, easily explained by the wave theory of
conclusion that:
light.
1.! A rare and highly elastic aether pervades the Universe. 2.! A luminous body excites undulations in this aether. 3.! The different color sensations depend upon the frequency of vibrations excited by light in the retina. !
Young also concluded that a superposition principle
applies for light waves so that, for two intersecting beams of light, “the joint motion may be the sum or difference of the separate motions, accordingly as similar or dissimilar parts of the undulations are coincident.” At any point the vibrations due to all passing waves add as vectors in a linear sum. This is verified by the fact that two intersecting beams of light can interfere and yet continue on unchanged. !
Fresnel (1815, 1816a) found that it is impossible to explain
the observed expansion of a beam of light upon passing through a narrow opening if light consists of corpuscles. As Ditchburn (1953) noted, “the spreading of light by diffraction shows that the energy of a photon cannot be permanently concentrated in a small volume like the energy of a material particle.” Such diffraction
3.1.4!
TRANSVERSE WAVES
Fresnel (1816b) discovered that two beams of light polarized at right angles do not interfere with each other. This confirmed a suspicion that Young had voiced earlier: light waves consist of transverse, not longitudinal, vibrations of aether. The aether particles oscillate about their equilibrium positions with the vibrations occurring in a plane orthogonal to the direction of propagation of the light beam. !
Transverse waves are distortional waves. Only in a
physical medium possessing rigidity can transverse waves exist and propagate. Young and Fresnel determined, therefore, that aether must have rigidity so that it responds to distortion elastically as a material solid does, rather than as a material fluid does. Of course we now know that, unlike any material substance, aether is a fluid having flow properties of a liquid and elastic properties (including rigidity) of a solid. This emphasizes a point made by Lodge (1925a): “Ether is not to be explained in terms of matter.” Fresnel (1821a, b) determined that the transverse vibrations of aether can be resolved into two linearly independent directions of polarization and, using this
106
discovery, he derived the reflection and refraction laws for light
3.1.6!
waves.
!
3.1.5! !
LUMINIFEROUS AETHER
PHYSICAL MEDIUM FOR LIGHT WAVES
There is no question that light exhibits definite wave
characteristics such as those of constructive and destructive
Following the work of Young and Fresnel, the scientific
interference. To produce these observed physical effects, light
community almost universally accepted the wave theory of
must propagate as waves, not corpuscles. Light waves must
light, and many significant scientific efforts were undertaken
then consist of motions occurring in a physical medium, and so
(beginning in the 1830s and continuing throughout the
the existence of a real physical medium is required in order to
remainder of the 1800s) to design mathematical models for an
provide something real that can oscillate as waves. This
aether that flows easily, but has the mechanical properties of a
physical medium must be pervasive in our Universe, filling all
solid body. This aether was thought to be ubiquitous in space
empty space, in order to account for the ability of stellar light to
and to constitute the real physical medium that provides the
travel across the vastness of space allowing us to see stars. This
necessary elastic restoring force for light waves to occur. It was
is why, for over two hundred and fifty years, the existence of a
referred to as luminiferous aether in order to distinguish it
ubiquitous luminiferous aether in which light propagates as
from any other type of aether that might possibly exist.
waves was considered necessary.
!
!
Based primarily on interpretations of results obtained
As we have seen in Section 1.2.8, a ubiquitous aether must
from experiments conducted by Michelson and Morley and on
also exist to provide the physical cause of gravity. It is very
assumptions contained in Einstein’s special and general
unlikely, if not impossible, for two different types of aether to
theories of relativity, the concept of a luminiferous aether was
be ubiquitous simultaneously in our Universe. It is reasonable
abandoned by most scientists in the 1900s. As will be discussed
to conclude that the gravitational aether and the luminiferous
in the following sections, however, a luminiferous aether is
aether are then one and the same aether. Therefore, light waves
completely consistent with the results obtained by Michelson
must consist of vibrations of the aether of gravity, and the
and Morley in their famous experiments.
nature of light is transverse elastic waves propagating in this aether. Since the aether of gravity and the luminiferous aether 107
are one and the same aether, we will now refer to this aetherial
aether. We will then review the concepts of stress and strain
medium simply as aether.
that are employed to develop the light wave equation.
!
!
Having discovered that the existence of a ubiquitous
The aether that exists in free space is homogeneous,
aether is necessary to explain gravity, mass, and inertia, it is not
isotropic, and has no sinks nor sources. For light waves to be
surprising to find that this same aether profoundly influences
able to propagate in this aether, the aether must be elastic.
other phenomena of physics. We can use some properties of the
Maxwell (1665) thought that the medium for light waves “must
aether to explain certain observations of light waves, and we
be capable of a certain kind of elastic yielding, since the
can use other observations of light waves to obtain additional
communication of motion is not instantaneous, but occupies time.”
information about the aether.
Since observations of spectral lines of stellar light exhibit no
!
As a preliminary, we will define a vacuum as a region of
increase in blurring with distance to the star, we know that light
space in which there is no matter, but in which aether is
waves can propagate across the vast distances of space without
ubiquitous and homogeneous. A vacuum is then a region of
any attenuation except that due to geometrical spreading.
space having no sinks for aether. However, a vacuum is not a
Therefore, aether must be perfectly elastic. As Lodge (1925a)
void (empty space); a vacuum is a region of space entirely filled
noted, “No law of dissipation applies to the ether, . . . Ether fritters
with aether (and nothing else). Nevertheless a vacuum can
away no energy, it preserves all: it is perfectly transparent; it
contain sources and sinks of electromagnetic fields since such
transmits light from the most distant stars without waste or loss of
sources and sinks are nonmaterial (see Sections 5.4.2 and 5.4.4).
any kind.”
We will define free space for electromagnetic fields as a
!
vacuum with no sources or sinks of electromagnetic fields.
that the particles of aether constitute a physical medium of
3.2!
extremely high density ζ . The particles of aether within this medium are contiguous and in equilibrium subject to mutual
ELASTIC PROPERTIES OF AETHER WITHIN FREE SPACE
From our study of gravity in Chapter 1 we determined
forces that are exhibited as the rigidity of the aether. Any
We will now examine the physical properties of aether
motion in and of the perfectly elastic aether that causes a
that make it possible for elastic waves to exist and propagate in
displacement of one particle of aether relative to the contiguous
!
108
particles of aether will then be resisted by the contiguous
accompany elastic wave propagation in aether. Since ΔV in free
particles of aether. This will result in:
space contains only aether, it is the aether that is actually being
1.! A restoring force that is oppositely directed to the displacement of the aether particle.
!
deformed during the straining of ΔV .
3.2.1!
STRAIN IN AETHER
2.! A force per unit area that is proportional to the relative displacement of the aether particle.
!
The fundamental cause of the elasticity of aether is the
within ΔV
Deformation of the aether within a very small volume
element ΔV of free space will result when particles of aether are in motion relative to each other. This
extremely high density of aether whereby all aether particles
deformation is known as strain and consists of the relative
are contiguous with other aether particles. A propagating
displacement of aether within ΔV . This deformation can take
elastic wave will exist when a disturbance of one aether particle
the form of both a change of volume and a change of shape of
produces a similar disturbance of a neighboring aether particle
ΔV .
an infinitesimally short time later.
!
To describe the deformation of aether within
ΔV
A very small volume element ΔV of free space will
associated with the vibrational motion of elastic waves
contain only a high density of aether particles. Motion of this
propagating in the aether, we will use a rectangular Cartesian
!
volume element can result when volume elements contiguous
coordinate system ( x1, x2 , x3 ) having its origin at a point P0
to ΔV are in motion. As was shown by Helmholtz (1858), any
within ΔV (see Figure 3-1).
displacement of a volume element ΔV can be represented as
We will denote the displacement of aether particles from ! ! ! their equilibrium positions by e ( r, t ) , where e is a function of ! position r and time t . We will let P and Q be two neighboring ! points in the aether within ΔV . These points are located at r ! ! and r + δ r , respectively, when the aether within ΔV is in ! equilibrium. The magnitude of δ r is very small compared to ! that of r .
the sum of three motions: a translation, a rotation, and a deformation. The translation and rotation of ΔV are rigid body motions, and so do not result in any restoring forces within ΔV . Deformation of ΔV , however, will result in restoring forces within ΔV causing elastic vibrations to occur. Therefore, only motion that deforms (strains) the volume element ΔV can
109
! ! ! e ( r + δ r, t ) from its equilibrium position to point Q’. The ! ! separation δ r between points P and Q then changes by δ e : ! ! ! ! ! ! ! δ e = e ( r + δ r, t ) − e ( r, t ) ! (3.2-1) ! ! where the magnitude of δ e is very small compared to that of e . ! The components of δ e are given by the linear relations: !
δ ei =
∂ei δ xj ! ∂xj
(3.2-2)
The linearity expressed by equation (3.2-2) is required since aether is perfectly elastic, and so recovery from all deformations must be single-valued and complete. Any index appearing twice in an equation term, as does j in equation (3.2-2), is summed over its range. ! Figure 3-1!
Solid dots show the equilibrium positions of two aether particles located at points P and Q within a very small volume element ΔV . Open dots show the strained positions of the two particles after deformation of the aether within ΔV .
!
When the aether within the volume element
ΔV
undergoes deformation from its equilibrium state, the aether ! ! particle at P is displaced by e ( r, t ) from its equilibrium position to point P’, and the aether particle at Q is displaced by
∂ei 1 ⎡ ∂ei ∂ej ⎤ 1 = ⎢ − ⎥+ ∂xj 2 ⎣⎢ ∂xj ∂xi ⎦⎥ 2
⎡ ∂ei ∂ej ⎤ + ⎢ ⎥ = ω i j + ε i j ! (3.2-3) ∂x ∂x ⎢⎣ j ⎥ i ⎦
where ω i j is the rank two rotation tensor given by: !
!
We can write ∂ei ∂xj in the form:
1 ωi j = 2
⎡ ∂ei ∂ej ⎤ − ⎢ ⎥! ⎢⎣ ∂xj ∂xi ⎥⎦
(3.2-4)
and where ε i j is the rank two strain tensor given by: !
1 εi j = 2
⎡ ∂ei ∂ej ⎤ + ⎢ ⎥! ⎢⎣ ∂xj ∂xi ⎥⎦
(3.2-5) 110
!
The quantities ω i j are components of infinitesimal rigid
deformation is expressed in terms of the strain tensor ε i j
rotation. The quantities ε i j are components of infinitesimal
which represents relative displacement per unit length. At a
strain. Note that the components of the rank two tensor ω i j ! ! given in equation (3.2-4) are equal to components of 12 ∇ × e and ! ! that ∇ × e is a measure of rotation of the aether. ! ! The components of δ e given in equation (3.2-2) can now
given point P in a continuous medium, the strain tensor ε i j
be rewritten as: !
δ ei = ω i j δ xj + ε i j δ xj !
(3.2-6)
! Using these components, the vector δ e can be rewritten in terms of two vector displacements: ! ! ! ! δe = δw +δq ! ! !
expected for motion of the aether within ΔV due entirely to ! elastic wave propagation. The components of δ e are then: !
δ ei = ε i j δ xj !
(3.2-10)
From equation (3.2-4) we see that the rotation tensor is antisymmetric:
(3.2-7)
! The components of δ w are:
δ wi = ω i j δ xj !
completely specifies the state of strain. ! ! ! ! ! Since δ w = 0 we now have δ e = δ q , which is to be
!
ωi j = −ω ji !
(3.2-11)
and from equation (3.2-5) we see that the strain tensor is (3.2-8)
symmetric:
ε i j = εj i !
! and so δ w represents rigid body rotation of the aether within
!
ΔV . For motion of ΔV due entirely to elastic wave propagation ! ! through the aether, we must have δ w = 0 since rigid body
!
rotation is not elastic motion. ! ! The components of δ q are:
components of strain. The strain tensor components ε11 , ε 22 ,
!
δ qi = ε i j δ xj !
(3.2-12)
Because of these symmetries, there are then only three
independent components of rotation and six independent and ε 33 correspond to extensional deformation of ΔV along
(3.2-9)
! and so δ q represents relative displacement (deformation) of the aether within ΔV due to elastic wave propagation. This
the 1, 2, and 3 coordinate directions, respectively. The other strain tensor components correspond to shear deformation of
ΔV . The strain tensor ε i j is a measure of deformation and is a rank two tensor. 111
!
After straining, the volume element ΔV becomes ΔV ′ ,
3.2.2!
where ΔV ′ is given by:
⎡ ∂e ⎤ ⎡ ∂e ⎤ ⎡ ∂e ⎤ ΔV ′ = Δx1 ⎢1+ 1 ⎥ Δx2 ⎢1+ 2 ⎥ Δx3 ⎢1+ 3 ⎥ ! (3.2-13) ⎣ ∂x1 ⎦ ⎣ ∂x 2 ⎦ ⎣ ∂x3 ⎦
!
or to first order:
⎡ ∂e ∂e ∂e ⎤ ΔV ′ = ΔV ⎢1+ 1 + 2 + 3 ⎥ ! ⎣ ∂x1 ∂x2 ∂x3 ⎦
!
(3.2-14)
some volume V of vacuum, the forces associated with the wave will produce displacements in the aether in V . These forces are force acting per unit area in known as the stress. For a very small volume element ΔV of V , displacement of the aether will cause deformation of the aether within ΔV . The forces that
ΔV ′ − ΔV ∂e1 ∂e2 ∂e3 ! = + + ΔV ∂x1 ∂x2 ∂x3
produce these displacements must be proportional to the (3.2-15)
Θ=
∂e1 ∂e2 ∂e3 ! + + ∂x1 ∂x2 ∂x3
displacements since aether is perfectly elastic. !
Letting !
When an elastic wave is propagating in the aether within
exerted on each other by contiguous particles of aether. The
Therefore !
!
STRESS IN AETHER
! ! For example, within ΔV the displacement e ( r, t ) of the
aether particle at point P shown in Figure 3-1 is caused by a (3.2-16)
force proportional to and in the same direction as the ! ! displacement e ( r, t ) . This force can be specified as acting upon
then Θ represents the fractional volume change (change in
a very small area element ΔA containing the point P.
volume per unit volume) due to strain. For this reason Θ is
!
known as the cubical dilatation or simply dilation. We also see
one side, and having three equal orthogonal area elements ΔA1 ,
using the Einstein summation convention that:
ΔA2 , and ΔA3 as the other sides (see Figure 3-2). When we let
!
! ! ∂e ∂e ∂e Θ = 1 + 2 + 3 = ∇ • e = ε11 + ε 22 + ε 33 = ε i i ! ∂x1 ∂x2 ∂x3
We can construct a very small tetrahedron having ΔA as
all four area elements approach zero in size, all the area (3.2-17)
and so Θ is an invariant of the strain tensor. A relative decrease in volume (cubical compression) yields a negative value for Θ .
elements will then contain only the point P. The force per unit area acting on ΔA at point P is defined as the stress vector or ! traction t on ΔA at point P.
112
components depend on the directions of two vectors: the stress ! vector t and the outward directed normal vector nˆ to area element ΔA . !
Since the stress tensor represents force per unit area, for a
complete specification of the stress tensor acting at a point P, it is necessary to know the stress vector acting upon all area elements containing P. Fortunately, however, the stress tensor at point P can be completely specified if the stress vector is known for three mutually perpendicular area elements that contain P (see Love, 1892; and Malvern, 1969). The nine stress tensor components will then be determined for a Cartesian coordinate system. At a given point P in a continuous medium, the stress tensor σ k l completely specifies the state of stress. Figure 3-2!
! The stress vector t acts on area element ΔA at point P.
! The components of the stress vector t along coordinate directions k can be written in terms of the components of the stress tensor σ k l at point P: !
t k = σ k l nl !
!
The stress tensor associated with elastic wave propagation
is symmetric: !
σ kl = σ l k !
(3.2-19)
This equation follows from the law of conservation of angular momentum (see Malvern, 1969; and Mase et al., 1992). If the stress tensor were not symmetric, a force associated with elastic
(3.2-18)
where nl are the components of the unit normal vector nˆ to the ! area element ΔA upon which the stress vector t is acting (see Figure 3-2). The stress tensor σ k l is a rank two tensor whose
wave propagation would produce a rigid body rotation of ΔV ! ! so that we would have δ w ≠ 0 in equation (3.2-7). The motion of ΔV when an elastic wave propagates in the aether would then no longer consist entirely of elastic vibrations. 113
!
3.2.3! !
HOOKE’S LAW FOR AETHER
The proportionality of force to the resulting displacement
from equilibrium of an elastic body is a law discovered by
We know that the superposition principle applies to light
Hooke (1678): “the rule or law of nature in every springing body is,
waves, and that light waves propagate unattenuated through
that the force or power thereof to restore it self to its natural position
an elastic aether. These facts provide confirmation that aether is
is always proportionate to the distance or space it is removed
perfectly elastic. Therefore, aether strained by propagating light
therefrom,” (see Todhunter and Pearson, 1886). Equation (3.2-20)
waves will recover completely after the light waves pass.
can be considered to represent a generalized form of Hooke’s
Energy expended on deforming the aether must then be
law for aether.
transformed into potential energy that is stored in the strained
!
aether. This energy is known as strain energy.
expect the elastic properties of aether to be identical at all
!
Since aether is perfectly elastic, any displacement of an
points in free space. The elastic moduli of aether in free space
aether particle relative to the contiguous aether particles must
are then simple constants, independent of both position and
be proportional to the force producing the displacement. A
time. Moreover, the elastic properties of aether in free space do
linear relation between the force per unit area (stress) and the
not have a preferred direction in space, but are the same in all
relative displacement per unit length (strain) is then dictated
directions. Light travels in stationary aether in free space with
(see Cannon, 1967). This means that the stress tensor σ k l in a
the same speed in any direction. Therefore, aether is elastically
very small volume element ΔV in free space will be related
isotropic.
linearly to the strain tensor ε i j in ΔV :
!
!
σ kl = ckl i j ε i j !
(3.2-20)
Since aether is homogeneous in composition, we can
We can now show that of the 81 elastic moduli of aether,
only two elastic constants are independent. We begin by noting that, since aether is elastically isotropic, the rank four Cartesian
where the 81 coefficients ckl i j are the elastic moduli of aether
tensor of elastic moduli must be an isotropic tensor. This simply
and are components of a rank four Cartesian tensor (since
means that we expect the elastic moduli ckl i j to be the same in
they relate components of two rank two tensors).
all directions, and so have components that do not change upon transformation (through an arbitrary rotation about the origin) 114
to any other rectangular Cartesian coordinate system. From
!
tensor analysis we know that all isotropic Cartesian tensors ckl i j
λ and η are Lamé constants or Lamé moduli of aether, and
of rank four must have the form:
that η is the shear modulus or rigidity of aether. These
!
ckli j = λ δ kl δ i j + α δ ki δ l j + β δ kj δ l i !
(3.2-21)
where λ , α , and β are some scalars, and where δ i j is the Kronecker delta defined in equation (1.2-21). !
Letting α = η + γ and β = η − γ where η and γ are scalars,
) (
constants represent physical properties of aether. !
From equation (3.2-26) with l = k we also have:
! or !
we can rewrite equation (3.2-21) as:
(
From the form of equation (3.2-26), we see that the scalars
)
σ k k = 3λ Θ + 2η ε k k = 3λ Θ + 2η Θ !
(3.2-27)
σ k k = ( 3λ + 2η ) Θ !
(3.2-28)
! ckl i j = λ δ kl δ i j + η δ ki δ l j + δ kj δ l i + γ δ ki δ l j − δ kj δ l i ! (3.2-22)
Hydrostatic pressure P is given by:
From the symmetry of the strain and stress tensors we have:
!
!
ckl i j = cl ki j = ckl ji = cl k ji !
(3.2-23)
and so equation (3.2-22) becomes: !
ckl i j = λ δ kl δ i j + 2 η δ ki δ l j !
pressure will result in a fractional volume change: (3.2-24)
!
1 1 − P = σ k k = ( 3λ + 2η ) Θ = K Θ ! 3 3
(3.2-30)
and so:
law:
σ kl = λ δ kl δ i j ε i j + 2 η δ ki δ l j ε i j !
(3.2-25)
Hooke’s law for aether is therefore: !
(3.2-29)
and so P is an invariant of the stress tensor. Hydrostatic
From equations (3.2-20) and (3.2-24) we then have for Hooke’s !
1 − P = σ11 = σ 22 = σ 33 = σ k k ! 3
σ kl = λ δ kl Θ + 2 η ε kl !
!
K=−
P 2 =λ+ η! Θ 3
(3.2-31)
where K is the bulk modulus of elasticity or modulus of (3.2-26)
where Θ is the cubical dilatation given in equation (3.2-17).
compression, and represents the ratio of applied hydrostatic pressure to resulting fractional decrease in volume. Therefore,
115
K is the ratio of invariants of the stress and strain tensors for a
within ΔV . Using equation (3.2-18), we have for the k th
given medium.
component of Newton’s second law of motion:
The Lamé constant λ of aether can be expressed in terms
!
of K :
2 λ = K − η! 3
! !
(3.2-32)
Finally, we note that although the relation between stress
and strain in material fluids can be expressed in terms of one elastic constant, two elastic constants are required to express such a relation for the fluid, aether. This is because aether has rigidity similar to material solids.
!
∫∫
t k dS =
ΔS
∫∫
σ kl nl dS =
ΔS
D Dt
∫∫∫
ζ vk dV !
ΔV
(3.3-1)
where vk is the k th component of the aether vibrational ! velocity v : ! ! de ! v= ! (3.3-2) dt !
The substantive derivative D Dt accounts for changes in
ζ vk within ΔV not only with time, but also with position. Using Reynold’s transport theorem for aether in free space (see
3.3! !
ELASTIC WAVE EQUATIONS FOR AETHER WITHIN FREE SPACE An elastic wave propagating in aether will act as a stress
wave causing aether particles to change momentum. We will consider a very small volume element ΔV of this aether in free
Appendix F), we can rewrite equation (3.3-1) in the form: !
ΔS
flow freely through the surface ΔS of ΔV . The total force of the ! stress vector t produced by the propagating elastic wave acting
∫∫∫
ζ
ΔV
Dvk dV ! Dt
(3.3-3)
From Gauss’s theorem given in equation (C-59) of Appendix C and using equation (1.5-9), we can then write:
space. We will also take this volume element to be a control volume; it will remain constant with time although aether can
∫∫
σ kl nl dS =
!
∫∫∫
ΔV
∂σ kl ∂xl
dV =
∂v ⎤ ⎡ ∂v ζ ⎢ k + υl k ⎥ dV ! ∂xl ⎦ ΔV ⎣ ∂t
∫∫∫
(3.3-4)
where ∂σ kl ∂xl is the k th component of force per unit volume
of the momentum of the aether within ΔV at any instant. This
due to elastic wave vibration in aether, and where υl is the l th ! component of aether flow velocity υ . The term on the left of
is just Newton’s second law of motion as applied to aether
this equation represents the force associated with the elastic
upon the surface ΔS of ΔV must equal the time rate of change
116
wave acting upon the aether within ΔV . The first term on the right of this equation represents the resulting momentum change with time of the aether within ΔV . The second term on the right of this equation represents the resulting momentum change with position of the aether within ΔV
due to
convection. This second term will always be equal to zero for elastic waves since they do not result in the flow of the physical ! ! medium in which they are propagating, and so υ = 0 . Only for ! ! gravitational waves (which are non-elastic waves) will υ ≠ 0 (see Section 4.25). ! !
Equation (3.3-4) then becomes:
∫∫∫
ΔV
∂σ kl ∂xl
dV =
∫∫∫
ΔV
ζ
∂vk dV ! ∂t
(3.3-5)
and since ΔV is arbitrary we obtain: !
∂σ kl
∂vk ∂ 2 ek =ζ =ζ 2 ! ∂xl ∂t ∂t
or
∂ε kl ∂ 2 ek ∂Θ λ + 2η =ζ 2 ! ∂xk ∂xl ∂t
!
From the definition of the strain tensor given in equation (3.2-5) we have:
⎡ ∂ 2 ek ∂ 2 el ⎤ ∂ 2 ek ∂Θ λ +η ⎢ + ⎥ =ζ 2 ! ∂xk ∂x ∂x ∂x ∂x ∂t ⎢⎣ l l l k ⎥ ⎦
!
In vector form equation (3.3-9) can be written: 2! ! ! ∂ e ! 2 ! λ ∇Θ + η ∇ e + η ∇Θ = ζ 2 ! ∂t
(3.3-9)
(3.3-10)
or from equation (3.2-17):
! ! ! ! ∂ 2e 2! (λ + η ) ∇ ∇ • e + η ∇ e = ζ 2 ! ∂t
(
! (3.3-6)
(3.3-8)
)
(3.3-11)
This can be considered to be the Cauchy equation for elastic waves propagating in aether. From this equation the elastic
where we have used equation (3.3-2) to change from vibrational
wave equations for compressional and transverse waves in
velocity components vk to displacement components ek .
aether can be obtained (see Sections 3.3.1, 3.3.2, and 3.3.3).
!
Using Hooke’s law for aether given in equation (3.2-26),
3.3.1!
we can write equation (3.3-6) in the form: !
∂ε kl ∂ 2 ek ∂Θ λ δ kl + 2η =ζ 2 ! ∂xl ∂xl ∂t
(3.3-7)
!
DECOMPOSITION OF ELASTIC DISPLACEMENT FIELD
From Helmholtz’s decomposition theorem (Helmholtz,
1858) we know that any vector field can be written as the sum 117
of an irrotational vector field and a solenoidal vector field provided that these two fields vanish sufficiently rapidly at infinity (which will generally be true for any physical field). ! Therefore, we can write the displacement field e as: ! ! ! ! e = eI + eS ! (3.3-12) ! ! where eI is the irrotational component vector, and eS is the ! solenoidal component vector. The vector displacement field e ! ! is uniquely determined by eI and eS . ! ! For the irrotational component vector eI we must have: ! ! ! ! ∇ × eI = 0 ! (3.3-13) ! and for the solenoidal component vector eS we must have: ! ! ! ∇ • eS = 0 ! (3.3-14) ! ! The two vector components eI and eS are independent. !
Both displacement components must then separately
satisfy Cauchy’s equation for elastic waves given in equation (3.3-11): !
!
! ! ! ! ∂ 2 eI 2! ( λ + η ) ∇ ∇ • eI + η ∇ eI = ζ 2 ! ∂t
(
)
! ! ! ( λ + η ) ∇ ∇ • eS + η ∇ 2 e!S = ζ
(
)
! ∂ 2 eS ∂t 2
!
From equations (3.3-12), (3.3-13), and (3.2-14) we also have: ! ! ! ! ! ! ! ! ! ∇ × e = ∇ × eI + ∇ × eS = ∇ × eS ! (3.3-17) !
(3.3-16)
(3.3-18)
! and so the irrotational component vector eI does not contribute ! ! ! to ∇ × e , and the solenoidal component vector eS does not ! ! contribute to ∇ • e = Θ .
3.3.2!
COMPRESSIONAL ELASTIC WAVES IN AETHER WITHIN FREE SPACE
! Considering the irrotational component vector eI of the ! displacement vector field e , we can use the vector identity !
given in equation (C-43) of Appendix C and equation (3.3-13) to write: !
! ! ! ! ! ! ! 2! ∇ × ∇ × eI = ∇ ∇ • eI − ∇ eI = 0 !
(3.3-19)
! ! ! ! ∇ ∇ • eI = ∇ 2 eI !
(3.3-20)
(
)
(
)
and so: !
(3.3-15)
! ! ! ! ! ! ! ! ∇ • e = Θ = ∇ • eI + ∇ • eS = ∇ • eI !
(
)
Equation (3.3-15) can then be written in the form: !
2! ∂ eI ! 2 ( λ + 2 η ) ∇ eI = ζ 2 ! ∂t
(3.3-21)
or 118
!
wave propagation with a corresponding periodic increase and
2! 1 ∂ eI 2! ∇ eI = 2 2 ! C ∂t
(3.3-22)
where, from equations (3.3-21), (3.3-22), and (3.2-32), the wave speed C is given by:
!
C=
decrease of volume and, therefore, of the density of aether. Compressional waves are also known as dilatational waves or longitudinal waves. Compressional waves in aether have not been detected for reasons which will be explained in Section
λ + 2η = ζ
4 K+ η 3 ! ζ
3.12. (3.3-23)
The wave speed C is a function of the bulk modulus of
3.3.3!
TRANSVERSE ELASTIC WAVES IN AETHER WITHIN FREE SPACE
the density ζ of aether. Taking the divergence of equation
! Considering the solenoidal component vector eS of the ! displacement vector field e , we can use equations (3.3-16) and
(3.3-22), we have:
(3.3-14) to write:
elasticity K , the shear modulus of elasticity (rigidity) η , and
!
! ! 2 ∂ ∇ • eI ! ! 1 2 ! ∇ ∇ • eI = 2 C ∂t 2
(
(
)
)
(3.3-24)
and using equation (3.3-18), we obtain the scalar wave equation:
∇ 2Θ =
1 ∂ Θ ! C 2 ∂t 2
(3.3-25)
We see that equations (3.3-25) and (3.3-22) are wave equations for compressional elastic waves propagating in aether with a speed C . !
2! ∂ eS ! 2 η ∇ eS = ζ 2 ! ∂t
!
The speed C is with respect to the propagation medium,
aether. Compressional waves vibrate only along the direction of
(3.3-26)
or 2! ∂ eS 1 ! 2 ! ∇ eS = 2 2 c ∂t
!
2
!
!
(3.3-27)
where, from equations (3.3-26) and (3.3-27), the wave speed c is given by: !
c=
η ! ζ
(3.3-28)
Taking the curl of equation (3.3-27) we have: 119
!
! ! 2 ∂ ∇ × eS ! 1 ! 2 ∇ ∇ × eS = 2 ! c ∂t 2
(
)
(
)
! (3.3-29)
From equation (3.2-26) with k ≠ l so that δ k l = 0 , we have:
η=
!
We see that equations (3.3-29) and (3.3-27) are vector wave equations
for
transverse
elastic
waves
(shear
waves)
Transverse elastic waves in aether are light waves. The
( k ≠ l and no sum)!
(3.3-30)
We can then rewrite equation (3.3-28) in the form:
propagating in aether with a speed c . !
1 σ kl ! 2 εkl
c=
!
speed c is with respect to the propagation medium, aether.
1 σ kl ! 2ζ ε k l
( k ≠ l and no sum)!
(3.3-31)
Transverse waves vibrate in a plane that is orthogonal to the
Therefore, the propagation speed c of transverse waves in
direction of wave propagation. The speed c is a function of the
aether is proportional to the square root of the ratio of stress to
rigidity η and the density ζ of aether. Transverse elastic waves
resulting strain. The constant speed of light, therefore, is a
are also known as equivoluminal waves since no change in
consequence of the linearity of stress and strain in aether
volume occurs as a result of their propagation. Therefore, no
(Whittaker, 1910). As Huygens (1690a, b) noted, “by supposing
variation in aether density occurs during the propagation of
springiness in the ethereal matter, its particles will have the property
transverse elastic waves in aether. Deformation occurring
of equally rapid restitution whether they are pushed strongly or feebly;
within a medium without any change in medium volume is
and thus the propagation of light will always go on with an equal
known as isochoric deformation.
velocity.”
!
!
We conclude then that both compressional and transverse
elastic waves can exist in aether as was determined by Green (1838a, b). Aether is similar in this regard to elastic material solids for which Poisson (1829a, b) showed that both compressional and transverse waves can exist. We can also conclude that mechanical waves in aether result from the deformation of an elastic aether.
Since transverse waves vibrate in a plane orthogonal to
the direction of wave propagation, the vibrational displacement ! vector eS within this plane can be resolved into two orthogonal ! ! component displacement vectors eE and eH which will, of course, be linearly independent as Fresnel (1821b) determined: ! ! ! ! eS = eE + eH ! (3.3-32)
120
! The subscripts of the orthogonal displacement vectors eE and ! eH are designated as E and H in anticipation of discussions of Maxwell’s field equations to follow in Chapter 4. Equation ! ! (3.3-27) can then be written in terms of eE and eH as two independent vector wave equations: 2! 1 ∂ eE 2! ! ∇ eE = 2 ! c ∂t 2 !
! ∇ 2 eH =
2! 1 ∂ eH 2 2
c
∂t
(3.3-33)
2! 1 ∂ vE = 2 ! 2 c ∂t
!
! ∇ 2 vH
2! 1 ∂ vH ! = 2 c ∂t 2
3.4! !
!
(3.3-34)
! Since eS is a solenoidal displacement field as given by equation ! ! (3.3-14) and since eE and eH are independent, we also must have:
!
! ∇ 2 vE
(3.3-38)
(3.3-39)
LIGHT WAVES Two theories were initially proposed to explain the
propagation of light: the corpuscular theory and the undulatory or wave theory. Experiments on the refraction, interference, and diffraction of light eventually showed conclusively that light propagates as waves. We now know that light waves are transverse elastic waves. They consist of transverse vibrations
!
! ! ∇ • eE = 0 !
(3.3-35)
!
! ! ∇ • eH = 0 !
(3.3-36)
From equations (3.3-2) and (3.3-32) we can write: ! ! ! ! vS = vE + vH !
propagating as spherical waves in the perfectly elastic aether.
3.4.1! !
SPEED OF LIGHT
The physical medium in which the transverse waves
known as light waves propagate is aether. Relative to the aether (3.3-37)
in which the light waves are propagating, the speed of light is
! where vS is the transverse vibrational velocity vector, and ! ! where vE and vH are orthogonal component velocity vectors of ! ! ! vS . Therefore, the vectors vE and vH are linearly independent.
completely determined by physical properties of the aether as
Taking the derivative with respect to time of the two vector
waves since, relative to the material medium in which sound
wave equations (3.3-33) and (3.3-34), we have:
waves are propagating, the speed of sound is completely
can be seen from equation (3.3-28). The speed of light, therefore, is a characteristic property of aether. This is analogous to sound
121
determined by physical properties of the material medium.
placed several kilometers apart across a valley. Upon seeing
Lodge (1919a) observed that aether is needed “for any reasonable
light from the first lantern, the shutter was opened on the
understanding of what is meant by the velocity of light.”
second lantern, and a light signal was sent back. The resulting
Any medium in which waves propagate will always form
travel time interval was, however, too small to be detected with
a unique reference frame for these waves. Therefore wave
the available instrumentation (including the reaction time of the
speed is always absolute relative to its propagation medium.
observers) given the very high speed of light and the relatively
The aether in which light waves propagate then serves as the
short travel path. Galileo could only conclude that the speed of
absolute (unique) reference frame for their speed. This means
light must be “at least extremely fast.”
that Galilean relativity cannot apply to the speed of light, just
The first successful determination of the speed of light was
as it does not apply to the speed of sound.
made by Rømer in 1676 using observations of the eclipse
!
Given the linearity of stress and strain in aether, and given
periods of four of Jupiter’s moons (which had been discovered
the expected homogeneity of the elastic properties of aether in
in 1610 by Galileo). These observations showed that when the
free space throughout our Universe, we can conclude that the
Earth was farthest from Jupiter the eclipses were delayed by
speed of light waves propagating within this aether should be
about 1320 seconds relative to when the Earth was closest to
constant. The fact that light waves have a characteristic
Jupiter. Rømer correctly concluded that this delay was just the
constant speed in vacuum is thereby explained. The speed of
time required for light to travel the diameter of the Earth’s
light c relative to the aether of free space in which the light
orbit. Using the information available to Rømer on the diameter
waves are propagating is one of the universal constants of
of the Earth’s orbit, he obtained an approximate value of
nature. As Young (1807) noted, “the undulations of every
2.20 × 1010 cm /sec for the speed of light. Rømer was the first to
homogeneous elastic medium are always propagated, like those of
determine that the speed of light is finite rather than infinite.
sound, with the same velocity, as long as the medium remains
!
unaltered.”
made by Fizeau (1849) using a rapidly rotating toothed wheel
!
The first attempt to measure the speed of light was made
having 720 teeth to modulate a light beam. The beam was sent
by Galileo Galilei (1638) using lanterns equipped with shutters
through the gap between two teeth along a path about 9
The first terrestrial determination of the speed of light was
122
kilometers long to a mirror, and then the beam was reflected back along the same path. The uniform speed of rotation of the
3.4.2!
toothed wheel could be adjusted so that the returning light
WAVE EQUATIONS FOR LIGHT WAVES WITHIN FREE SPACE
beam was not eclipsed by any of the following teeth, but was
!
seen through the following gaps. From a knowledge of the total
light waves propagating within free space can be obtained from
distance the light travels and the speed of rotation of the wheel,
Maxwell’s field equations (see Section 4.7.1). Since the
the speed of light could be determined. Using this method
derivation of these equations will be discussed in Chapter 4, we
Fizeau determined the propagation time for the light beam and
will only present the electromagnetic wave equations here: ! ! 1 ∂2E 2 ! ∇ E= 2 2 ! (3.4-1) c ∂t
obtained a value of about 3.153 × 1010 cm /sec for the speed of light. !
Similar methods were used by Cornu (1876) and by Young
and Forbes (1881). A rotating mirror system developed by Arago (1850) and Foucault (1850) was used by Foucault (1862a, b) to obtain a light speed of 2.980 × 1010 cm /sec . This same method was used by Michelson (1879a, b, c, d) to obtain a light
Using electromagnetic parameters, the wave equations for
! ! 1 ∂2H ! ! ∇ H= 2 (3.4-2) c ∂t 2 ! where E is the electrical field intensity or electric field ! strength, H is the magnetic field intensity or magnetic field 2
been determined in many different experiments and is now
strength, and c is the speed of electromagnetic waves ! ! propagating through free space. Both E and H are point
estimated to be 2.99792 × 1010 cm /sec (see Evenson et al., 1972).
vectors.
In fact, the length of a meter is now defined in terms of the
!
distance light travels in a specified time.
light waves propagating through free space. Since light waves
speed of 2.9991× 1010 cm/s . Since then the speed of light has
Equations (3.4-1) and (3.4-2) are vector wave equations for
are propagating transverse vibrations of an elastic aether, it is not surprising that the electromagnetic wave equations (3.4-1) and (3.4-2) can be shown to be equivalent to the two elastic wave equations (3.3-38) and (3.3-39). This will be done in 123
Chapter 4 by deriving Maxwell’s electromagnetic field equations from classical elasticity theory for transverse elastic
3.5!
MOTION OF AETHER AND OF THE LIGHT SOURCE OR LIGHT RECEIVER
waves in aether. ! !
From equation (3.3-28) we have for the speed of light:
η c= ! ζ
(3.4-3)
root of the rigidity of aether η , and inversely proportional to the square root of the density of aether ζ . The rigidity controls how fast an aether particle will snap back after being moved from its equilibrium position. The restoring forces of aether must be very strong since both the density of aether and the speed of light in aether are so great (see Sections 5.3.1 and 5.3.2). Using equation (3.4-3), the electromagnetic wave
!
2 ! ! ζ ∂ H ! ∇2 H = η ∂t 2
Relative to the medium in which waves are propagating,
wave speed depends only upon intrinsic physical properties of
and so the speed of light is directly proportional to the square
equations for free space can also be written as: ! ! ζ ∂2E 2 ! ! ∇ E= η ∂t 2
!
(3.4-4)
the propagation medium. Therefore, the velocity of the wave source or of the wave receiver can have no effect on the wave speed. For example, the speed of sound waves does not change according to whether the sound source is approaching or receding relative to the material propagation medium. !
The speed of light waves in aether is likewise determined
entirely by physical properties of the nonmaterial aether. Therefore, the velocity of the light source or of the light receiver can have no effect on the speed of light. The speed of light waves must then always be independent of source velocity and ! receiver velocity. Consequently, the velocity of light c cannot be combined with the velocity of a source or receiver of light to obtain their relative motion. In other words, Galilean relativity does not apply to the velocity of light. This has been verified
(3.4-5)
experimentally in a number of studies, including studies by de Sitter (1913a, b), Majorana (1917, 1918a, b, 1919a), Kennedy and Thorndike (1932), Alväger et al. (1963, 1964), James and Sternberg (1963), Babcock and Bergman (1964), and Brecher (1977). 124
!
The fact that the speed of light is independent of source
change in frequency is just the well-known Doppler effect
velocity provides additional evidence that light propagates as
discovered by Christian Doppler in the 1840s. He determined
waves, not corpuscles. The speed of waves is always
that for sound waves the detected frequency varies according
independent of the source velocity; the speed of corpuscles is
to whether the sound source is approaching or receding relative
always dependent upon source velocity.
to the propagation medium. Doppler (1843) then speculated
!
that the same frequency shift effect must exist for light waves as
Wave speed is affected, however, by any motion of the
medium in which the wave is propagating. The velocity of light
for sound waves.
through free space, therefore, is a function of both the velocity ! of light c with respect to the aether and the velocity of the
!
aether itself. If aether in free space is in motion with a velocity ! υ , the true velocity of light waves propagating through free ! ! space will then be c + υ . Therefore, the speed of light waves
equations for the Doppler effect are, in fact, similar for sound
propagating through free space can actually be greater than or
effect is a purely kinematic phenomenon (it is true for sound as well as
less than the speed of light c . For light waves traveling through
for light),” as was noted by Sommerfeld (1942d).
a vacuum, it is only with respect to the aether in which the light
!
waves are propagating that the speed of light is always
emitted by a moving light source and detected by a stationary
constant with a value of c . The speed c corresponds then to a
light receiver. The designation ‘moving’ and ‘stationary’ are
reference frame relative to which aether is at rest such as a
relative to the aether in which the light waves are propagating.
reference frame of the aether itself.
Light waves emitted by the source will have a definite period
3.5.1! !
DOPPLER EFFECT
While motion of the light source or light receiver will not
affect the speed of light, any such motion relative to the aether
Since sound and light both propagate as waves in a
physical medium: material and aetherial, respectively, the and light. Only the value of the wave speed is different. This similarity in the equations is not surprising since “the Doppler
We will now consider the Doppler effect for light waves
τ S , frequency νS , and wavelength λ S such that: !
c=
λS = λ S νS ! τS
(3.5-1)
will affect the received frequency of the light waves. This 125
where c is the speed of light relative to the aether in which the
For a light source moving directly towards a stationary receiver
light waves are propagating. Light waves detected by the
with a speed US relative to the propagation medium (aether),
receiver will have a period τ R , frequency νR , and wavelength
we will similarly have:
λ R such that: !
!
λ c = R = λ R νR ! τR
(3.5-2)
⎡ U λ R′ = γ L λ S ⎢1− S c ⎣
⎤ ⎥⎦ !
(3.5-7)
The parameter γ L in equations (3.5-6) and (3.5-7) is a
Only if the motions of the light source and of the light receiver
limiting parameter. This parameter is required because there
relative to the aether are identical will we have:
exists a maximum value of US such that US < c which cannot be
! !
λ S = λR !
τS = τ R !
νS = νR !
(3.5-3)
For a light source moving directly away from a stationary
receiver with a speed US relative to the propagation medium (aether), after the time τ S the source will move a distance: !
U US τ S = S ! νS
(3.5-4)
For each wavelength λ S emitted by the light source, the receiver will detect a wavelength λ R such that:
U ⎡ ! λ R = γ L ( λ S + US τ S ) = γ L ⎢ λ S + S νS ⎣
US ⎤ ⎡ = γ λ 1+ ⎥ L S⎢ λ ν S S ⎦ ⎣
Doppler equations is given in Section 3.5.2. !
⎤ ⎥ ! (3.5-5) ⎦
US → 0 the limiting parameter is no longer required in equations (3.5-6) and (3.5-7). Since γ L
⎤! ⎥⎦
must satisfy both
Doppler equations (3.5-6) and (3.5-7), we can use these equations to write:
⎡ U ⎤⎡ U ⎤ λ R λ R′ = γ L2 λ S2 ⎢1+ S ⎥ ⎢1− S ⎥ ! c ⎦⎣ c ⎦ ⎣
(3.5-8)
or !
⎡ U λ R = γ L λ S ⎢1+ S c ⎣
The limiting parameter γ L for US can be expected to be a
function of US and c , but not a function of wavelength since as
!
or !
exceeded. The physical cause of this limiting parameter in the
λ R λ ′R
1
γ L2
=
λ 2S
⎡ US2 ⎤ ⎢1− 2 ⎥ ! c ⎦ ⎣
(3.5-9)
(3.5-6) 126
Since γ L is independent of wavelength and since we know that
λ R λ ′R =
λ 2S
when US