Physical and Epistemological Foundations of Einstein's General Theory
of Relativity
-- A Critique and Alternative View
Ilan Kroo
10-Mar-1980
Not submitted for publication.
Introduction
Although a great deal of experimental evidence for Einstein's General Theory
of Relativity now exists, the theory was supported for nearly 45 years by
data that would now be considered very meager. Unlike, Quantum Theory,
the acceptance of General Relativity was based on the simplicity and universality
of its underlying postulates rather than on its ability to resolve apparent
contradictions of experiments with existing theories. It was not the failure
of the predictive capabilities of Newton's laws which led Einstein to his
general theory but philosophical difficulties with their basic tenets. Newton's
theories formed the basis for most interesting physics for hundreds of years
and yet, as Einstein found, the most interesting physics was not that which
resulted from the application of the theory but rather the physics that
formed the basis for Newton's theories. The same is, no doubt, true of Einstein's
theory. The philosophically interesting physics is not only that resulting
from General Relativity but also the epistemological and physical foundations
of the theory itself.
In this paper some of the concepts and empirical bases underlying General
Relativity are examined. Einstein's motivation for adopting these views
is discussed along with an alternative theory which seems to resolve some
of the philosophical difficulties associated with the general theory.
The Need for a General Theory -- Problems with Newton's View
Berkeley and Mach
Bishop Berkeley and Ernst Math criticized Newton's mechanics from an epistemological
viewpoint, noting that the concept of acceleration or rotation as employed
by Newton, was not well-defined without a specified system of reference.
Newton did specify a reference system which he described as absolute a priori
, "in its own nature, without relation to anything external, remains
always similar and immovable."(1)
It appears that Newton did consider the idea that accelerations should be
measured with respect to something more substantive than empty space, but
his reported experiments with rotating vessels convinced him that the acceleration
giving rise to inertial 'forces' was an acceleration independent of the
experimental environment.
Newton's error seems to stem from an over simplified Gedanken experiment
in which the only relevant element is taken to be a vessel filled with liquid.
When the vessel is rotated no inertial effects are observed in the liquid,
although a relative rotation is present. As the liquid's rotation approaches
that of the vessel, inertial effects are observed, although no relative
rotation exists. Newton's conclusion was that the frame of reference with
respect to which this rotation must be measured is not determined by the
circumstances of the experimental arrangement but is an a priori entity.
Although Berkeley's objections to Newton's absolute space stemmed from
religious attitudes, Berkeley did recognize the problem with Newton's reasoning.
He noticed that Newton's experiments with inertial forces involved accelerations
with respect to the "fixed stars" and that the idea of absolute
space was not necessary:
" ... it would be enough to bring in, instead of absolute
space, relative space as confined to the heavens of the fixed stars, considered
at rest. But motion at rest marked out by such relative space, can conveniently
be substituted in the place of the absolutes which cannot be distinguished
from them by any mark.
-from De Motu Published <20 years after the Principia.
(2)
Mach
Some one hundred fifty years later the discussion was taken up again by
Ernst Mach. Mach's conception of space and time required that the frame
with respect to which rotation was measured be defined only by the distribution
of surrounding matter. Mach realized, as Berkeley had, that perhaps it was
rotation with respect to the 'distant stars' that resulted in the inertial
forces. Mach wont on to postulate that it was the distant stars which caused
inertial forces.
The idea that inertial forces arise from accelerations with respect to other
matter, not only fit the experimental evidence but avoided the epistemological
problems with Newton's absolute space. But Mach was bothered by the mysterious
nature of inertial "forces". Having asserted that forces produce
accelerations with respect to distant stars, the following question immediately
suggests itself: What have the distant stars to do with these motions?
Mach considered various explanations in which the stars played both fundamental
and collateral roles in the determination of the motion of a body. He suggested,
at first, that the stars' role might not be fundamental "in the determination
of the motion of a body K and that this motion is determined by a medium
in which K exists <and with respect to which the stars are at rest>...
It is known from recent hydrodynamic investigations that a rigid body experiences
resistance in a frictionless fluid only when its velocity changes..."
Mach did not take this idea too seriously but was evidently looking for
a mechanism to explain how the distant stars could act to produce acceleration-dependent
inertial forces. Later Mach attributes a more fundamental role to the stars
in influencing a body's motion and suggests that perhaps it is the large
mass of the stars that makes them important in these considerations. He
goes on to suggest that perhaps even all masses contribute some to the inertial
effects:
"Newton's experiment with the rotating vessel of water simply
informs us that the relative rotation of the water with respect to the sides
of the vessel produces no noticeable centrifugal forces, but that such forces
are produced by its relative motion with respect to the mass of the earth
and the other celestial bodies. No one is competent to say how the experiment
would turn out if the sides of the vessel increased in thickness and mass
until they were several leagues thick." (3)
Mach did not explain further how it is that such masses may produce acceleration-dependent
forces, and it is to this problem that more recent work, starting with Einstein,
addresses itself.
Einstein
Einstein's interest in the epistemological difficulty with Newton's laws
resulted from his attempts to generalize special relativity:
"When by the special theory of relativity I had arrived at
the equivalence of all so-called inertial systems... the question whether
there was not a further equivalence of coordinate systems followed naturally,
to say the least of it... if only a relative meaning can be attached to
the concept of velocity, ought we nevertheless to preserve in treating acceleration
as an absolute concept?" (4)
But he admits:
"...physically speaking, the inertial system seemed to occupy a
privileged position, which made the use of coordinate systems moving in
other ways appear artificial."
Mach's views were just those needed if special relativity were to be generalized,
and Einstein adopted Mach's epistemological objection:
" Of all imaginable spaces... in any kind of motion relatively to one
another, there is none which we may look upon as privileged a priori..."
(5)
Einstein then states a form of covariance principle which he seems to take
as following in some unspecified wag from the above view:
"The laws of physics must be of such a nature that they apply to systems
of reference in any kind of motion."
This principle is extended to the following which Einstein refers to as
the General Principle of Covariance:
"The general laws of nature are to be expressed by equations which
hold good for all systems of coordinates, that is, are covariant with respect
to any substitutions whatever." (5)
This principle is not merely a reformulation of Mach's principle. Mach
objected to the concept of an a priori privileged frame but did not believe
that the actual laws of nature should not have some frame in which they
might be written most simply -- in fact he suggested what frame that might
be -- that fixed with respect to the distant stars. The difference in these
views comes from the fact that Mach was not looking for 'general laws of
nature' that would apply to isolated bodies in free space. Mach objected,
in fact, to programs which attempted to formulate laws based on such notions
of empty space. He did not object to local laws of nature which could have
some externally-defined frame, so long as that frame were recognized as
an a posteriori entity.
It is clear that the 'general laws of nature' to which Einstein believes
the general principle of covariance must apply are laws that relate the
motion of two bodies relative to another in the absence of other bodies.
That is, GPOC is a global principle which we expect not to apply to systems
for which an external reference frame may be defined.
Einstein's objections to Newton's theories must then be distinguished from
Mach's criticism as follows: Mach objected to Newton's a priori conception
of a preferred frame. Einstein objected to the fact that Newton's laws were
not generally covariant and hence were not general laws of nature
(they could not be said to hold in free space).
Additional Objections to Newton's Theories
Mach's postulate that accelerations in Newton's dynamics be measured with
respect to distant stars removes the epistemological problem with Newton's
view but there remains Einstein's objection that it cannot be considered
a general law. A further difficulty with Newton's third law arises from
the concept, not of acceleration but of mass. The relation between mass,force,
and acceleration (w.r.t. distant stars) may be interpreted in two ways:
1) It may be taken as the defining relation for a concept called inertial
mass; or 2) It may be a relation between force and the concept of gravitational
mass which occurs elsewhere in physics.
In the first case, it remains a complete mystery as to why this new concept
should be so intimately related to the concept of gravitational mass. In
the second interpretation Newton's third law appears as an ad-hoc law with
no physical motivation other than the empirical evidence in its favor.
It is the resolution of this problem that forms the physical foundation
of general relativity and it is on this point that Einstein and Mach come
to very different conclusions.
The Physical Axioms of the General Theory
The Search for a Physical Basis
Having decided that the correct general laws of nature ought to satisfy
the general principal of covariance, Einstein's task became that of producing
a physical law consistent with this requirement. He apparently attempted
to do this first by extending the ideas of Mach upon which he had relied
so heavily in the development of the ideas of covariance. Einstein comments
on his approach:
"... it appeared conceivable that what inertial resistance
counteracts is not acceleration as such but acceleration with respect to
the masses of other bodies existing in the world. There was something fascinating
about this idea to me, but it provided no workable basis for a new theory."
(4)
Whether Mach's suggestion that other MASSES affected a body's inertia led
Einstein to next consider the role of gravitation in the general theory
is not clear, but the next step was to try to represent the law of gravitation
within the framework of special relativity. The project proceeded as follows:
1) "Like most writers at the time I tried to form a field
law for gravitation. " (4)
"In classical mechanics, interpreted in terms of fields,
the potential of gravitation appears as a scalar field. ... the following
program appears natural therefore: the total physical field consists of
scalar field (gravitation) and a vector field (electromagnetic field)."
(7)
2) "The simplest thing was... to retain the Laplacian scalar
potential of gravity, and to complete the equation of Poisson in an obvious
way by a term, differentiated with respect to time in such a way that the
special theory of relativity was satisfied." (4)
Einstein was unable to successfully pursue this idea as the theory needed
to combine two things:
i) "From general considerations of special relativity theory
it was clear that the inert <inertial> mass of a physical system increases
with the total energy ..."(7)
ii) "From very accurate experiments (especially from the
torsion balance experiments of Eotvos) it was empirically known that the
gravitational mass of a body is exactly equal to its inert mass.
It followed from (i) and (ii) that the weight of a system depends in a precisely
known manner on its total energy."
Einstein could find no way to incorporate such an effect in the manner described
above and thus "abandoned as inadequate the attempt to treat the problem
of gravitation in the manner outlined above..."
The Equivalence Principle
The difficulty in deriving this relation between inert and heavy mass from
special relativity indicated to Einstein that this relation was a significant
principle which had to be incorporated in the general theory. It is not
clear why he was willing to accept this "empirical fact" so readily
as a basic tenet of his analysis but he was "in the highest degree
amazed at its existence and guessed that in it must lie the key to a deeper
understanding of inertia and gravitation. I had no serious doubts about
its strict validity even without knowing the results of the admirable experiments
of Eotvos ...which I only came to know later." (4)
Einstein observed that if the gravitational mass were taken to be exactly
equal to the inertial mass one could 'explain' inertial forces by transforming
to a frame at rest with respect to the mass and and postulating the existence
of a gravitational field proportional to the acceleration. If inertial forces
did arise in such a manner it would indeed follow that inertial mass and
gravitational mass were equal. But the framing of this idea as a postulate
for a general theory (that is, a "general law of nature" in the
strict sense that Einstein would like) is a large inductive leap from the
properly stated Physical fact that the acceleration produced by gravity
on a mass m is independent of its composition. In order that this principle
(for which the experimental evidence of Eotvos, and later of Dicke and Braginsky
(8,9) lends support) have anything to do with Einstein's principle of equivalence,
it is necessary to make clear the definitions of inertial and gravitational
mass. Such definitions are not made explicit in Einstein's discussion but
these ideas seem to refer to the terms in Newton's laws of gravitation and
dynamics. In this way the proportionality of the two terms may be inferred
from the experiments:
If m_g is such that m_g (m_e / G r^2) = W and m_i is such that F = m_i
a, then:
m_g / m_i = a G r^2 / m_e
where m_e is the mass of the earth, G is the Gravitational constant, and
r is the distance from earth center to the object of interest.
However, even, if these 'definitions' are accepted it may be seen that
they apply only for a small region of space-time whereas Einstein apparently
asserts that their ratio is constant in space and time.
Einstein's postulation of this form of the equivalence principle as
"one of the most universal which the observation of nature has yielded"
(6) is not only incompletely supported by empirical evidence but is also
subject to the same criticism as that levelled against Newton, namely that
there is no more physical motivation for the appearance of a gravitational
field in an accelerated frame than there is for the idea that F = m a.
Mach's idea that inertial forces might in some way be related to the presence
of distant masses has been incorporated not as a result ( as Mach and Einstein
had at first hoped) but as a postulate in the general theory.
Correspondence
The next assumption that must be made in the development of the field equations
of general relativity is the requirement that they reduce to those of special
relativity in matter-free space. The assumption is not a trivial one, as
without it many solutions to the field equations are possible. The requirement
is empirically motivated and seems to suffer from the same epistemological
problems encountered in the discussion of Newton's theory. This problem
is presented in a thought experiment devised by R.H. Dicke which seems to
show that this requirement on the boundary conditions of general relativity
contradicts Mach's Principle. (10)
An Alternative Set of Physical Axioms which Satisfy the Epistemological
Requirements
Reexamining Einstein's attempts at identifying a good law of nature:
In his argument for the requirement of general covariance Einstein assumes
that the theory for which the principle must hold is a global one -- that
is, the law of nature must be a general law of nature, applicable to bodies
in free space. In the formulation of general relativity this postulate is
added to a second one: the principle of equivalence. Yet this principle
seems to be a local postulate. We have no reason to expect that in the absence
of all other matter, two bodies will obey the principle of equivalence.
So even if this principle is "the most fundamental property of gravitation",
Einstein's objection to Newton's laws must apply as well to this one : It
cannot be a general law of nature. Mach's view of the problems with Newton's
reasoning can be applied to this view also: Just as "no one is competent
to say how <Newton's> experiment would turn out if the sides of the
vessel were increased in thickness and mass..." or if the distant stars
were removed, so too no one is competent to say whether the equivalence
principle would hold were the stars to be removed.
An alternative principle which is not subject to these sorts of criticism
may be derived by returning to Einstein's program of formulating a field
law for gravitation compatible with special relativity. It is first necessary,
however, to demonstrate why it is not the case, as Einstein suggested, that
"the possibility of the realization of such a program <is>, however,
dubious from the very first..."(7)
The dilemma that Einstein encountered in such a formulation has been mentioned
previously. The first two premises, which are apparently well established
empirical facts, force the conclusion that weight increases with energy.
The argument is:
1) The inert mass of a system increases with energy (by S.R.)
2)Weight and inert mass are proportional (by Eotvos)
=> Weight increases with energy.
The error in this argument has been discussed with reference to the motivation
for the equivalence principle and rests in the fact that the second premise
is stronger than it need be. A more careful wording of this premise is:
2') The acceleration produced by gravity on a body is independent
of its composition.
Then, in order to carry out the argument, an additional promise is required:
3) The weight of a body is equal to the product of its inert
mass and the acceleration produced by gravity.
But postulate 3) is just the statement that F = ma, which ought not be
taken as an a priori postulate. (It seems very conceivable that the correct
relation between inertial reaction and mass, itself contains terms that
are functions of the system's energy.) The consistency of any theory of
gravitation with special relativity must be checked but the particular concepts
associated with Newton's mechanics and assumed in special relativity for
lack of a more general view need not be assumed first. The discussion will
be extended later but because of fundamental ambiguities in the definitions
of inertial and heavy mass in the absence of a general theory we should
not as yet be convinced that within the framework of special relativity
there is no room for a satisfactory theory of gravitation."
The idea here will be to start as Einstein suggested, "by completing
the equation of Poisson in on obvious way by a term, differentiated with
respect to time in such a way that the special theory of relativity is satisfied."
Whereas Newton would write:
del^2 phi = - 4 pi rho
Lorentz covariance requires that we write:
DeLambertian phi = del^2 phi_g - 1/c^2 \par^2 phi_g / \par t^2 = - 4 pi
rho_g
Now we abandon Einstein's idea that gravitation might be represented as
a scalar field. rho_g is introduced as the fourth component of a four-vector
. This identification requires that the scalar potential, phi_g also be
a component of a four-vector potential.
As in electromagnetism, we obtain the relation between the four-vectors
[j_x, j_y, j_z, i rho]
and [A_x, A_y, A_z, i phi] preserving Lorentz invariance in the process.
The real components of the four-vector associated with rho_g may
be thought of as currents with j_x = v_x / c rho_g
In the event that j and rho are not time dependent we can write:
phi_g = \int_{all space} rho(r') / |r -r'| dV'
A_g = \int_{all space} j(r') / |r -r'| dV'
By analogy with electromagnetism we can find the components of the gravitational
field as elements of a second rank tensor (the math is omitted here but
follows essentially the discussion in Ref.10)
We find that the gravitational field may be written:
G = - del phi_g - 1/c \par A / \par t
At this point we speculate that the inertial forces discussed by Mach may
come about as the result of the addition of this second term. Such speculation
is motivated by the results from electrodynamics which predict a force (arising
from the analogous term which varies as 1/r x a between two charged particles
with relative acceleration, a . To investigate this idea we consider a test
mass near the center of a homogeneous, spherically symmetric mass distribution
and calculate the magnitude of the force produced by the gravitational vector
potential term. (This is, admittedly a very simplified picture, but serves
to demonstrate the theory 's basic results. )
We calculate the vector potential, A, in the instantaneous rest frame of
a particle initially moving with velocity v:
A \approx \int_{all space} rho_g(r') v/c / |r -r'| dV'
= rho_g v / c \int_{extent of all massive bodies} 1/r r^2 dr sin\theta
d\theta d\psi
= 2 \pi \rho v / c R^2 / 2 (2)
= 2 pi rho v / c R^2
=> G = -grad phi - 2 pi rho R^2 a / c^2
\approx -2 pi rho R^2 a / c^2
Now we may write the force law as:
F = -q_g G = 2 pi rho R^2 q)g a / c^2
The quantity q_g may be related to the gravitational mass as follows:
From our assumption that deLambertian phi = - 4 pi rho_g in the static
case, together with the force law and the definition: G = -grad phi,
in the case of two particles:
q G = -q^2/ r^2 == -Gm^2/r^2.
we thus have, q_g = m G and write the force exerted by a homogeneous, spherically
symmetric mass distribution as:
F = 2 pi rho_mass R^2 m G a / c^2
In this way a force proportional to the gravitational mass, mG times the
acceleration is derived. In order for this to be identified with Newton's
inertia forces we would require that
2 pi rho_mass R^2 G / c^2 \approx 1.
with
rho_mass = average mass density of matter in the universe
R = radius of matter distribution in the universe
G = universal gravitational constant of Newton
c = speed of light.
Using R \approx 10^26 m and rho_mass = 2 x 10^-26 kg/m^3, [The critical
density according to GTR for a closed universe and close to the value accepted
today]
We find 2 pi rho_mass R^2 G / c^2 = 0.9
It is quite remarkable that this number should be so close to 1.0 when we
are
dealing with numbers as large as R^2 ~ 10^52 m^2
Several simplifications have been made here and we expect that careful attention
to the requirements of covariance and full account of the time-dependent
nature of the interaction is necessary. However, the basic program seems
promising as it shows quantitatively Mach's idea that inertial forces come
about as a result of accelerations with respect to distant masses. The weak
equivalence principle is a derived result of this theory. The idea is based
on the postulates that the proper field theory meet the requirements of
special relativity (Lorentz invariance) and that the gravitational field
is represented by a vector potential.
Differences between this theory and GTR
The primary difference between this theory and the general theory of relativity
(in fact, nearly all other theories of gravity) involves the fundamental
concept of mass. By identifying a vector potential with the gravitational
field, and deriving the force-acceleration relation as a consequence, the
inertial mass of a body is seen to be an emergent property --
the interaction of a gravitating body with other masses in the universe.
By introducing rho_g as one component in a four-vector we have asserted
that there exists a fundamental property (which may be called gravitational
charge to make clear the strong analogy with electromagnetic theory) which
is invariant under Lorentz transformations. It is suggested by this argument
that inertial mass may be written m_i = k q_g where the constant of proportionality
is k = 1 / G = f(R,rho,c).
It may be seen then that on this view only the weak equivalence principle
is expected to hold, for although m_i and q_g are proportional for sufficiently
small regions of space-time. k may change with time and/or position if the
values of R or rho change.The view that the "good quantum number"
is gravitational charge, not mass, is a radical departure from the standard
view and the implications are far-reaching. Among these are the conservation
laws for energy and momentum which are based on the force-mass relation
with classical concepts of inertial mass. If the F = m a relation is replaced
by F = M(r,t) a, energy and momentum conservation seem to hold only in small
regions of space-time (at least with the current expressions for energy
and momentum.)
Furthermore the energy-mass relation is seen to be a consequence of the
interaction with distant masses and it is not clear that the global theory
(including field equations for G ) ought to include the non-linear terms
which Einstein believes must be included with his view of mass as a fundamental
entity of a global theory.
The implications of the view that the concept of inert mass should not be
part of a global (generally covariant) theory are beyond the scope of this
paper and I am not aware of other discussions related to this idea. One
important implication should be discussed, however, as it distinguishes
this theory from others and seems to be empirically verifiable.
An Experimental Test of this Idea
If one takes the astronomical observations of red-shift to be an indication
of the expansion of the extent of matter distributed in the universe then
it might be expected that the relation between q_g and m_i will change as:
m_i = q_g / G ~ c^2 / rho_mass R^2 ~ c^2 G / rho_g R^2
If one treats q_g, as the good quantum number and assumes it is conserved:
c^2 G / (rho_g R^2) = constant = c^2 R G / q_universe = constant
=> 1/G dG/dt = 1/R dR/dt = Ho (Hubble's constant)
The effects of a change in G on geophysics, planetary motion, and stellar
evolution are discussed in refs. 9,12-14. Measurements based on radar observations
of Mercury and Venus, laser ranging of the Earth-Moon system, and ancient
eclipse records (ref. 12) have not been able to determine the value of this
parameter other than to indicate that:
|1/G dG/dt| < 4 x 10^-10, while Ho ~ 10^-10
The analysis of such data must be carried out carefully in that the usual
constants of motion may vary over large time scales. The standard practice
has been to write:
G m1 m2 / r^2 = a m1 m2 / (m1+m2)
Using data on the observed secular acceleration (and assuming that m is
constant, the equations are solved for G(t). According to the view presented
here, however, it is the quantity mG which must be taken as constant in
time and m is found to vary. A derivation of the secular acceleration expected
from this theory has been made using perturbation theory. Results lie within
the bounds of present experimental accuracy.
The difference between this theory and other theories in which G is allowed
to vary is apparent in the interpretation of a recently-proposed experiment
designed to determine the time dependence of G. (14) This laboratory experiment
attempts to compare the ratio of electrostatic and static gravitational
forces over a long period of time using a sort of Cavendish balance arrangement.
It is claimed that a change in G with time will result in an observed change
in the ratio of these forces. Yet on the interpretation given here, there
is no reason to expect that the fundamental interaction is getting weaker
or stronger. It is taken to be constant and only the inertial reaction varies
with time. Since the proposed experiment involves static interactions the
theory predicts (along with GTR and contrary to Brans-Dicke) that a null
result will be obtained. The experiment actually tests for a change in the
quantity mG which, in this account, is constant. The view taken here is,
therefore, in agreement with general relativity and may be distinguished
from other theories of gravitation in which G is taken as time-dependent
while mass is retained as a fundamental concept.
Conclusions and Further Work
It has been argued that a generally covariant theory of gravitation
may be developed from physical postulates, conceptually distinct from those
of general relativity. These postulates are seen to be less problematic
than those of relativity in that epistemological difficulties associated
with the boundary conditions of Einstein's theory are avoided. It is shown
that assumptions regarding the form of the equations of the gravitational
field lead naturally to the empirical result that the gravitational and
inertial interactions are proportional.
The discussion presented here is intended as a motivation for the conceptual
view, it is expected that a complete development might include the full
time-dependent tensor nature of the gravitational interaction and a more
complete analysis of the need for non-linear
terms associated with self-gravitation of the gravitational field. Such
a program may be undertaken only after a reexamination of the mass-energy
relation and conservation laws on large scales is completed with the proper
view of the nature of mass.
The implications of the concept of mass as an emergent, time-dependent quantity
are far-reaching and provide the basis for further work Including a search
for empirical evidence, the development of a cosmological model consistent
with these ideas, and a reexamination of related conceptual primitives in
physics.
References
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Space and Time", ed. J.J.C. Smart, MacMillan Co., 1964
2.Berkeley, 'De Motu', excerpts in Sciama, D. 'Physical Foundations of General
Relativity', Doubleday & Co.,Inc., 1969
3.Mach, E., 'The Science of Mechanics' in Smart collection.
4.Einstein, A., 'Notes on the Origin of the General Theory of Relativity'
in 'Ideas and Opinions', Dell Publishing Co. 1973
5.Einstein, A. , 'The Foundation of the General Theory of Relativity', in
'The Principle of Relativity', Dover Publications 1953
6.Einstein, A. , 'On the Influence of Gravitation on the Propagation of
Light' in same collection.
7.Einstein, A. , 'Autobiographical Notes' in 'Albert Einstein: Philosopher-Scientist'
ed. P.A. Schlipp, Open Court Publishing Co., 1951
8.Dicke, R.H., 'The Theoretical Significance of Experimental Relativity',
Gordan and Breach, 1964
9.Will, C., The Confrontation Between Gravitation Theory and Experiment',
to be published in Einstein Centenary Volume Cambridge University Press,
1979
10.Brans, C., and Dicke, R.H., "Mach's Principle and a Relativistic
Theory of Gravitation", Phys. Rev. 124 (1961)
11.Schwartz, M.,'Principles of Electrodynamics', McGraw-Hill Book Co.,1972
12. Weinberg, S.,'Gravitation and Cosmology', Wiley & Sons Inc. 1972
13. Van Flandern,T. C. 'A Determination of the Rate of Change of G', Mon.
Not.Roy. Ast. Soc.,170 (1975)
14. Van Flandern,T. C., 'Is Gravity Getting Weaker?', Scientific American,234
(1976)