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 r²) = W, and m_i is such that F = m_i a, then: m_g / m_i = a G r² / 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:

Lorentz covariance requires that we write:

Now we abandon Einstein's idea that gravitation might be represented as a scalar field. r_g is introduced as the fourth component of a four-vector . This identification requires that the scalar potential, f_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, ir] and [A_x, A_y, A_z, if] preserving Lorentz invariance in the process. The real components of the four-vector associated with r_g may be thought of as currents with j_x = r_g v_x / c .

In the event that j and r are not time dependent we can write:

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:

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:

So,

Now we may write the force law as:

The quantity q_g may be related to the gravitational mass as follows. From our assumption that:

in the static case, together with the force law and the definition:

in the case of two particles:

We thus have, q_g = m G^1/2 and write the force exerted by a homogeneous, spherically symmetric mass distribution as:

In this way a force proportional to the gravitational mass, m G^1/2 times the acceleration is derived. In order for this to be identified with Newton's inertia forces we would require that:

with:
r_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 = 10²⁶ m and r_mass = 2 x 10^-26 kg/m³, (the critical density according to GTR for a closed universe and close to the value accepted today), we find 2 p r_mass R² G / c² = 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 R² ~ 10⁵² m²

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 r_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 = G^-1/2 = f(R,r,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 r 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:

If one treats q_g as the good quantum number and assumes it is conserved:

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:

while:

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:

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 m G^1/2 that 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 m G^1/2 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

1. Newton, Mathematical Principles of Natural Philosophy, in Problems of 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", 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)