The Equivalence Principle, the Covariance Principle
and
the Question of Self-Consistency in General Relativity
C. Y. Lo
Applied and Pure Research Institute
17 Newcastle Drive, Nashua, NH 03060, USA
September 2001
Abstract
The equivalence principle, which states the local equivalence between acceleration and gravity, requires that a free falling observer must result in a co-moving local Minkowski space. On the other hand, covariance principle assumes any Gaussian system to be valid as a space-time coordinate system. Given the mathematical existence of the co-moving local Minkowski space along a time-like geodesic in a Lorentz manifold, a crucial question for a satisfaction of the equivalence principle is whether the geodesic represents a physical free fall. For instance, a geodesic of a non-constant metric is unphysical if the acceleration on a resting observer does not exist. This analysis is modeled after Einstein illustration of the equivalence principle with the calculation of light bending. To justify his calculation rigorously, it is necessary to derive the Maxwell-Newton Approximation with physical principles that lead to general relativity. It is shown, as expected, that the Galilean transformation is incompatible with the equivalence principle. Thus, general mathematical covariance must be restricted by physical requirements. Moreover, it is shown through an example that a Lorentz manifold may not necessarily be diffeomorphic to a physical space-time. Also observation supports that a spacetime coordinate system has meaning in physics. On the other hand, Pauli version leads to the incorrect speculation that in general relativity space-time coordinates have no physical meaning
1. Introduction.
Currently, a major problem in general relativity is that any Riemannian geometry with the proper metric signature would be accepted as a valid solution of Einstein equation of 1915, and many unphysical solutions were accepted [1]. This is, in part, due to the fact that the nature of the source term has been obscure since the beginning [2,3]. Moreover, the mathematical existence of a solution is often not accompanied with understanding in terms of physics [1,4,5]. Consequently, the adequacy of a source term, for a given physical situation, is often not clear [6-9]. Pauli [10] considered that he theory of relativity to be an example showing how a fundamental scientific discovery, sometimes even against the resistance of its creator, gives birth to further fruitful developments, following its own autonomous course." Thus, in spite of observational confirmations of Einstein predictions, one should examine whether theoretical self-consistency is satisfied. To this end, one may first examine the consistency among physical rinciples" which lead to general relativity.
The foundation of general relativity consists of a) the covariance principle, b) the equivalence principle, and c) the field equation whose source term is subjected to modification [3,7,8]. Einstein equivalence principle is the most crucial for general relativity [10-13]. In this paper, the consistency between the equivalence principle and the covariance principle will be examined theoretically, in particular through examples. Moreover, the consistency between the equivalence principle and Einstein field equation of 1915 is also discussed.
The principle of covariance [2] states that he 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 (generally covariant)." The covariance principle can be considered as consisting of two features: 1) the mathematical formulation in terms of Riemannian geometry and 2) the general validity of any Gaussian coordinate system as a space-time coordinate system in physics. Feature 1) was eloquently established by Einstein, but feature 2) remains an unverified conjecture. In disagreement with Einstein [2], Eddington [11] pointed out that pace is not a lot of points close together; it is a lot of distances interlocked." Einstein accepted Eddington criticism and no longer advocated the invalid arguments in his book, he Meaning of Relativity" of 1921. Einstein also praised Eddington book of 1923 to be the finest presentation of the subject ever written
Moreover, in contrast to the belief of some theorists [14,15], it has never been established that the equivalence of all frames of reference requires the equivalence of all coordinate systems [9]. On the other hand, it has been pointed out that, because of the equivalence principle, the mathematical covariance must be restricted [8,9,16].
Moreover, Kretschmann [17] pointed out that the postulate of general covariance does not make any assertions about the physical content of the physical laws, but only about their mathematical formulation, and Einstein entirely concurred with his view. Pauli [10] pointed out further, he generally covariant formulation of the physical laws acquires a physical content only through the principle of equivalence...." Nevertheless, Einstein [2] argued that "... there is no immediate reason for preferring certain systems of coordinates to others, that is to say, we arrive at the requirement of general co-variance."
Thus, Einstein covariance principle is only an interim conjecture. Apparently, he could mean only to a mathematical coordinate system for calculation since his equivalence principle, among others, is an immediate reason for preferring certain systems of coordinates in physics (壯 5 & 6). Note that a mathematical general covariance requires, as Hawking declared [18], the indistinguishability between the time-coordinate and a space-coordinate. On the other hand, the equivalence principle is related to the Minkowski space, which requires a distinction between the time-coordinate and a space-coordinate. Hence, the mathematical general covariance is inherently inconsistent with the equivalence principle.
Although the equivalence principle does not determine the space-time coordinates, it does reject physically unrealizable coordinate systems [9]. Whereas in special relativity the Minkowski metric limits the coordinate transformations, among inertial frames of reference, to the Lorentz-Poincaré transformations; in general relativity the equivalence principle limits the physical coordinate transformations to be among valid space-time coordinate systems, which are in principle physically realizable. Thus, the role of the Minkowski metric is extended by the equivalence principle even to where gravity is present.
Mathematically, however, the equivalence principle can be incompatible with a solution of Einstein equation, even if it is a Lorentz manifold (whose space-time metric has the same signature as that of the Minkowski space). It has been proven that coordinate relativistic causality can be violated for some Lorentz manifolds [9,16]. Unfortunately, due to inadequate physical understanding, some relativists [19-23] believe that a proper metric signature would imply a satisfaction of the equivalence principle. The misconception that, in a Lorentz manifold, a ree fall" would automatically result in a local Minkowski space [20,23], has deep-rooted physical misunderstandings from believing in the general mathematical covariance in physics.
Although the equivalence principle for a physical space-time1) is clearly stated, the conditions for its satisfaction in a Lorentz manifold have been misleadingly over simplified. Thus, it is necessary to clarify first, in terms of physics, the meaning of the equivalence principle and its satisfaction (§2 & §3). The crucial condition for a satisfaction of the equivalence principle is that the geodesic represents a physical free fall. The mathematical existence of local Minkowski spaces means only mathematical compatibility of the theory of general relativity to Riemannian geometry. Then, it becomes possible to demonstrate meaningfully through detailed examples that diffeomorphic coordinate systems may not be equivalent in physics (§5 & § 6). Moreover, to avoid prejudice due to theoretical preferences, these demonstrations are based on theoretical inconsistency.
To this end, Einstein illustration of the equivalence principle in his calculation of the light bending is used as a model for this analysis. However, in his calculation, there are related theoretical problems that must be addressed. First, the notion of gauge used in his calculation is actually not generally valid [9] as will be shown in this paper. Also, it is known that validity of the 1915 Einstein equation is questionable [7,8,24-26]. For a complete theoretical analysis, these issues should, of course, be addressed thoroughly. Nevertheless, for the validity of Einstein calculation on the light bending [2], it is sufficient to justify the linear field equation as a valid approximation. For this purpose, the Maxwell-Newton Approximation (i.e., the linear field equation) is derived directly from the physical principles that lead to general relativity (§4).
Moreover, there are intrinsically unphysical Lorentz manifolds none of which is diffeomorphic [21] to a physical space-time (§7). Thus, to accept a Lorentz manifold as valid in physics, it is necessary to verify the equivalence principle with a space-time coordinate system for physical interpretations. Then, for the purpose of calculation only, any diffeomorphism can be used to obtain new coordinates. It is only in this sense that a coordinate system for a physical space-time can be arbitrary.
In this paper, the requirement of a general covariance among all conceivable mathematical coordinate systems [2] will be further confirmed to be an over-extended demand [9]. (Note that Eddington [11] did not accept the gauge related to general mathematical covariance.) Analysis shows that a satisfaction of the equivalence principle restricted covariance (壯 3-5). After this necessary rectification, some currently accepted well-known Lorentz manifolds would be exposed as unphysical (§7). But, general relativity as a physical theory is unaffected [9]. It is hoped that this clarification would help urther fruitful developments, following its own autonomous course [10]".
2. Einstein Equivalence Principle, Free Fall, and Physical Space-Time Coordinates
Initially based on the observation that the (passive) gravitational mass and inertial mass are equivalent, Einstein proposed the equivalence of uniform acceleration and gravity. In 1916, this proposal is extended to the local equivalence of acceleration and gravity [2] because gravity is in general not uniform. Thus, if gravity is represented by the space-time metric, the geodesic is the motion of a particle under the influence of gravity. Then, for an observer in a free fall, the local metric is locally constant. To be consistent with special relativity, such a local metric is required to be locally a Minkowski space [2].
Thus, a central problem in general relativity is whether the geodesic represents a physical free fall. However, validity of this global property is realized locally through a satisfaction of the equivalence principle. Moreover, Eddington [11] observed that special relativity should apply only to phenomena unrelated to the second order derivatives of the metric. Thus, Einstein [27] added a crucial phrase, t least to a first approximation" on the indistinguishability between gravity and acceleration.
The equivalence principle requires that a free fall physically result in a co-moving local Minkowski space2) [3]. However, in a Lorentz manifold, although a local Minkowski space exists in a ree fall" along a geodesic, the formation of such co-moving local Minkowski spaces may not be valid in physics since the geodesic may not represent a physical free fall [9,16]. In other words, given the mathematical existence of local Minkowski space co-moving along a time-like geodesic, the crucial physical question for the satisfaction of the equivalence principle is whether the geodesic represents a physical free fall.
Einstein [28] pointed out, s far as the prepositions of mathematics refers to reality, they are not certain; and as far as they are certain, they do not refer to reality." Thus, an application of a mathematical theorem should be carefully examined although ne cannot really argue with a mathematical theorem [18]". If, at the earlier stage, Einstein arguments are not so perfect, he seldom allowed such defects be used in his calculations. This is evident in his book, he Meaning of Relativity' which he edited in 1954. According to his book and related papers, Einstein viewpoints on space-time coordinates are:
1) A physical (space-time) coordinate system must be physically realizable (see also 2) & 3) below).
Einstein [29] made clear in hat is the Theory of Relativity? (1919)' that n physics, the body to which events are spatially referred is called the coordinate system." Furthermore, Einstein wrote f it is necessary for the purpose of describing nature, to make use of a coordinate system arbitrarily introduced by us, then the choice of its state of motion ought to be subject to no restriction; the laws ought to be entirely independent of this choice (general principle of relativity)". Thus, Einstein coordinate system has a state of motion and is usually referred to a physical body. Since the time coordinate is accordingly fixed, choosing a space-time system is not only a mathematical but also a physical step.
2) A physical coordinate system is a Gaussian system such that the equivalence principle is satisfied.
One might attempt to justify the viewpoint of accepting any Gaussian system as a space-time coordinate system by pointing out that Einstein [3] also wrote in his book that n an analogous way (to Gaussian curvilinear coordinates) we shall introduce in the general theory of relativity arbitrary co-ordinates, x1, x2, x3, x4, which shall number uniquely the space-time points, so that neighboring events are associated with neighboring values of the coordinates; otherwise, the choice of co-ordinate is arbitrary." But, Einstein [3] qualified this with a physical statement that n the immediate neighbor of an observer, falling freely in a gravitational field, there exists no gravitational field." This statement will be clarified later with a demonstration of the equivalence principle (see eqs. [6] & [7]).
3) The equivalence principle requires not only, at each point, the existence of a local Minkowski space2)
ds2 = c2dT2 - dX2 - dY2 - dZ2, (1)
but a free fall must result in a co-moving local Minkowskian space (see also [10-13]). Note that the equivalence principle requires that such a local coordinate transformation be due to a specific physical action, acceleration in the free fall alone. Einstein [2] wrote, " For this purpose we must choose the acceleration of the infinitely small (ocal") system of co-ordinates so that no gravitational field occurs; this is possible for an infinitely small region."
Also, for a Lorentz manifold, if a ree fall" results in a local constant metric, which is different from Minkowski metric, then the equivalence principle is not satisfied in terms of physics. Einstein [2] wrote, "...in order to be able to carry through the postulate of general relativity, if the special theory of relativity applies to the special case of the absence of a gravitational field."
According to Einstein, the body to which events are spatially referred is called the coordinate system. To be more precise, a spatial coordinate system attached to a body (i.e., no relative motion nor acceleration) is its rame of reference" [2,3]. These coordinates together with the time-coordinate form the space-time coordinate system. A frame of reference can be chosen physically and, due to the equivalence principle, the time-coordinate is determined accordingly (壯 5 & 6). Thus, one may call loosely the frame of reference as a coordinate system. In this paper, for the purpose of considering a satisfaction of the equivalence principle, a frame of reference and a related space-time coordinate system, are distinguished as above.
To clarify the theory, Einstein [3] wrote, ccording to the principle of equivalence, the metrical relation of the Euclidean geometry are valid relative to a Cartesian system of reference of infinitely small dimensions, and in a suitable state of motion (free falling, and without rotation)." Thus, at any point (x, y, z, t) of space-time, a ree falling" observer P must be in a co-moving local Minkowski space L as (1), whose spatial coordinates are attached to P, whose motion is governed by the geodesic,
= 0, where , (2)
ds2 = g((dx(dx( and g(( is the space-time metric. The attachment means that, between P and L, there is no relative motion or acceleration. Thus, when a spaceship is under the influence of gravity only, the local space-time is automatically Minkowski. Note that the free fall implies but is beyond just the existence of rthogonal tetrad of arbitrarily accelerated observer" [4].
Einstein equivalence principle is very different from the version formulated by Pauli [10, p.145], or every infinitely small world region (i.e. a world region which is so small that the space- and time-variation of gravity can be neglected in it) there always exists a coordinate system K0 (X1, X2, X3, X4) in which gravitation has no influence either in the motion of particles or any physical process." Note that in Pauli misinterpretation, gravitational acceleration as a physical cause is not mentioned, and thus Pauli version3), which is now commonly but mistakenly regarded as Einstein version of the principle [30], actually is not a physical principle. Based on Pauli version, it was believed that in general relativity space-time coordinates have no physical meaning. In turn, diffeomorphic coordinate systems are considered as equivalent in physics [21] not just in certain mathematical calculations. However, according to Einstein calculations [2,3], this is simply not true (see section 3).
The initial form of the equivalence principle is a relation between acceleration and gravity. However, in the above clarification, the role of acceleration is not explicitly shown. One may ask if acceleration does not exist for a static object, would the equivalence principle be satisfied? One must be careful because a geodesic may not represent a physical free fall.
There are three physical aspects in Einstein equivalence principle as follows [3]:
1) In a physical space, the motion of a free falling observer is a geodesic.
2) The co-moving local space-time of an observer is Minkowski, when 1) is true.
3) A physical transformation transforms the metric to the co-moving local Minkowski space.
Point 3) must indicate that this physical local coordinate transformation is due to the free fall alone. In other words, the physical validity of the geodesic 1) is a prerequisite for the satisfaction of the equivalence principle, and validity of 3) is an indication of such a satisfaction. Thus, a satisfaction of the equivalence principle is beyond the mathematical tangent space (壯 5-7).
Perhaps, this inadequate understanding is, in part, due to the fact that it is often difficult to see the physical validity of point 2) directly, i.e., how the metric transformed automatically to a local Minkowski space. To this end, examining point 1) and/or point 3) would be useful. Point 1) is a prerequisite of the equivalence principle point 2). For Point 1) to be valid, i.e., the geodesic representing a physical free fall, it is required that the metric of such a manifold should satisfy all physical principles. Needless to say, such a metric must be physically realizable. If point 1) is valid in physics, point 3) should produce valid physical results. Thus, one can check point 3) to determine the validity of point 1) or vice versa.
The mathematical existence of a co-moving Local Minkowski space along a ree fall" geodesic implies only that Riemannian geometry is compatible with the equivalence principle. The physics is whether the existence of a physical local transformation which transforms the metric to the co-moving local Minkowski space. This is possible only if the geodesic represents a physical free fall, i.e., the equivalence principle ensures the existence of such a physical transformation. Thus, one must carefully distinguish mathematical properties of a Lorentz metric from physical requirements. Apparently, a discussion on the possible failure of satisfying the equivalence principle was over-looked by Einstein and others (see 壯6 & 7).
3. Einstein Illustration of the Equivalence principle
Einstein [3] illustrated his equivalence principle in his calculation of the light bending around the sun. (Note, the other method does not have such a benefit.) His 1915 equation for the space-time metric g(( is
G(( ( R(( -R g(( = -KT(m)(( (3)
where K is the coupling constant, T(m)(( is the energy-stress tensor for massive matter, R(( is the Ricci curvature tensor, and R = R((g((, where g((, is the inverse metric of g((. Now, he considered a coordinate system S (x, y, z, t) with the sun attached to the spatial origin. Based on eq. (3), and the notion of weak gravity, Einstein ustified" the linear equation,
= 2K(T(( - g((T), where ( (( = g(( - ((( , (4a)
((( is the flat metric, and T = T((g((. Then, from eq. (4a), and Ttt = (, otherwise T(( is zero, by using the asymptotically flat of the metric, Einstein obtained, to a sufficiently close approximation, the metric for coordinate system S
ds2 = c2(1 - )dt2 - (1 + )(dx2 + dy2 + dz2), (4b)
where ( is the mass density and r2 = x2 + y2 + z2.
However, since eq. (3) itself is questionable for dynamic problems [7,8,24-26], it is necessary to justify eq. (4a) again. Also, the notion of weak gravity may not be compatible with the principle of general covariance [3]. In the next section, eq. (4a) will be justified directly and is independent of the details of higher order terms of an exact Einstein equation. For this reason and the dynamical incompatibility with eq. (3), eq. (4a) is called the Maxwell-Newton Approximation [7]. In other words, eq. (4a) should be valid for dynamic problems. Also, an implicit assumption of Einstein calculation is that the gravitational effects due to the light itself, is negligible. To address this issue theoretically, would be complicated and is beyond the scope of this paper [9]. Here, this negligibility is justified from the viewpoint of practical observations only.
Now, according to the geodesic eq. (2), one has d2x /ds2 = 0 for x( (= x, y, z) since (gtt/(x( ( 0. Thus, the gravitational force is non-zero, and the equivalence principle would be applicable. (For the non-applicable cases, please see 壯5-7.) Consider an observer P at (x0, y0, z0, t0) in a ree falling" state,
dx/ds = dy/ds = dz/ds = 0. (5)
According to the equivalence principle and eq. (1), state (5) implies the time dt and dT are related by
c2(1 - )dt2 = ds2 = c2dT2 (6)
since the local coordinate system is attached to the observer P (i.e., dX = dY = dZ = 0 in eq. [1]). This is the time dilation of metric (4b). Eq. (6) shows that the gravitational red shifts are related to gtt, and is compatible with his 1911 derivation [2]. Moreover, since the space coordinates are orthogonal to dt, at (x0, y0, z0, t0), for the same ds2, eq. (6) implies [3]
(1 + )(dx2 + dy2 + dz2) = dX2 + dY2 + dZ2 . (7)
On the other hand, the law of the propagation of light is characterized by the light-cone condition,
ds2 = 0. (8)
Then, to the first order approximation, the velocity of light is expressed in our selected coordinates S by
= c(1 - ). (9)
It is crucial to note that the light speed (9), for an observer P1 attached to the system S at (x0, y0, z0), is smaller than c; and this condition is required by the coordinate relativistic causality for a physically realizable space-time coordinate system (see §6). Observer P1 shares the same frame of reference with the sun, and the velocity of light is clearly frame-dependent, but restricted.
This difference from c is due to gravity (or the curved space) together with the equivalence principle. The observer P is in a free falling frame of reference and thus would not experience the gravitational force as P1. Note that eq. (9) is consistent with eqs. (6) and (7) which are due to the equivalence principle. A reason for deriving eq. (6) and eq. (7) is that if the metric of a manifold does not satisfy the equivalence principle, ds2 = 0 would lead to an incorrect light velocity (see §5-7). Thus, not only eq. (6), which leads to gravitational red shifts, but also eq. (9) is a test of the equivalence principle.
Einstein [3] wrote, e can therefore draw the conclusion from this, that a ray of light passing near a large mass is deflected." Thus, Einstein has demonstrated that the equivalence principle requires that a space-time coordinates system must have a physical meaning; and a space-time coordinate system cannot be just any Gaussian coordinate system. It seems, Einstein [2] chose this calculation method to clarify his statements on the equivalence principle. In many textbooks [12,13,21-23], derivation of the coordinate light speed is circumvented, and the deflection angle is obtained directly. But, such a manipulation has not really achieved a derivation independent of the coordinate system since a particular type is needed to define the angle.
However, although Einstein emphasized the importance of satisfying the equivalence principle, he did not discuss what could go wrong. For instance, if the requirement of asymptotically flat were not used, one could obtain a solution, which does not satisfy the equivalence principle. Another interesting question is whether the equivalence principle is satisfied if ((tt = 0 (( = x, y, z). What has been missing is a discussion on the validity of the geodesic representing a physical free fall. Understandably, such a discussion was not provided since the validity of (4b) can be decided only through observations. This illustrates also that to see whether the equivalence principle is satisfied, one must consider beyond the Einstein equation (see §5).
4. Derivation of the Maxwell-Newton Approximation for Massive Matter
For massive matter, it has been proven [7] that eq. (4a) is dynamically incompatible with eq. (3). The binary pulsar experiments [31] make it necessary to modify eq. (1) to a 1995 update version,
Gab ( Rab -R gab = -K[T(m)ab - t(g)ab]. (10a)
and
(cT(m)cb = (ct(g)cb = 0, (10b)
where t(g)ab is the gravitational energy-stress tensor. The first order approximation of eq. (10a) is
(c(cab = -KT(m)ab (10c)
Eq. (10c) is called the Maxwell-Newton Approximation [7] and is equivalent to eq. (4a).
The above modification is based on the facts that, as a first order approximation, eq. (10c) is supported by experiments [7,19] and that it is the natural extension from Newtonian theory. However, one may argue that this is not yet entirely satisfactory since it has not been shown rigorously that eq. (10c) is compatible with general relativity. In particular, one might still argue [32] that the wave component in gat (for a = x, y, z, t) as artificially induced by the harmonic gauge.
It will be shown that the Maxwell-Newton Approximation (10c) can be rigorously derived from the equivalence principle and related physical principles that lead to general relativity. Since linear eq. (10c) is supported by experiments, to reaffirm the validity of general relativity, one must show clearly that eq. (10c) is compatible with the theoretical framework of relativity. Thus, such a proof of eq. (10c) not only provides a theoretical foundation for eq. (10) but also reaffirms general relativity.
In general relativity [2] there are three basic assumptions namely: 1) the principle of equivalence; 2) the principle of covariance (as will be shown necessarily be restricted to space-time coordinate systems which are compatible with the equivalence principle.) and 3) the field equation whose source can be modified. Note that eq. (10c) is invariant with respect to the Lorentz transformations. Moreover, eq. (10c) is compatible with the notion of weak gravity. Thus, eq. (10c) as an approximation for a specified coordinate system, is compatible with the requirement of covariance and compatibility with weak gravity. It remains to show that eq. (10c) is derivable from the equivalence principle.
The equivalence principle and the principle of general relativity imply that the geodesic equation (2) is the equation of motion for a neutral particle [2,3]. In comparison with Newton theory, Einstein [2] obtains the gravitational potential,
( " c2g00/2. (11)
Since ( satisfies the Poisson equation (( = 4(((, according to the correspondence principle, one has the field equation, (g00/2 = 4((c-2T00, where T00 "(, the mass density and ( is the coupling constant.
Then, according to special relativity and the Lorentz invariance, one has
(c( cgab = (c( c (ab = -4((c-2((T(m)ab + ((m)(ab(, (12a)
where
( + ( = 1, (m) = (cd T(m)cd , (12b)
T(m)ab is the tensor for massive matter, (ab is the Minkowski metric, and ( and ( are constants. Eq. (12) is a field equation for the first order approximation (as assumed) for weak gravity of moving particles. An implicit gauge condition is that the flat metric (ab is the asymptotic limit at infinity. To have the exact equation, since the left hand side of eq. (12a) does not satisfy the covariance principle, one must search for a tensor whose difference from (c( c (ab/2 is of second order in (c-2.
In Riemannian geometry, it has been proven [12] that the curvature tensor (((( is the only tensor that can be constructed from the metric tensor and its first and second derivatives, and is linear in the second derivatives." Einstein identified the Ricci curvature tensor Rab (( R(a(b) as the required tensor. If Rab includes no other first order sum, the exact equation would be
Rab = X(2)ab - 4((c-2((T(m)ab + (T(m)gab, (13)
where T(m) (= gcdT(m)cd) is the trace, X(2)ab is a second order unknown tensor chosen by Einstein to be zero. However, a non-zero X(2)ab may be needed to ensure eq. (12) as an approximation of eq. (13) [7].
Now, let us examine Rab further whether the above physical requirement can be valid. Let us decompose
Rab = R(1)ab + R(2)ab , (14a)
where
R(1)ab = (c( c (ab - (( c((b(ac ( (a (bc( ( (a(b ( , (14b)
and R(2)ab consists of higher order terms. If eq. (12) provides the first order approximation, the sum of other linear terms must be of second order. To this end, let us consider eq. (12a), and obtain K = 8((c-2 and
(c( c(( a(ab) = (K((( aT(m)ab + ((b(m)( . (15a)
From (cT(m)cb = 0, it is clear that K ( cT(m)cb is of second order but K(b(m) is not. However, one may obtain a second order term by a suitable linear combination of ( c(cb and (b (. From (15a), one has
(c( c(( a(ab ( C (b() = (K ((( aT(m)ab + (( + 4C( + C()(b(m)( . (15b)
Thus, simply choosing the harmonic coordinates (i.e., (( a(ab ( (b(/2 " 0), can lead to inconsistency. It follows eq. (14b) and eq. (12b) that, for the other terms to be of second order, one must have
( ( 4C( + C( = 0, 2C + 1 = 0, and ( + ( = 1. (15c)
The solution of eq. (15c) is C = -1/2, ( = 2, and ( = -1. Thus, for the first order approximation,
(c( c (ab = (K (T(m)ab + (m) (ab( , (16)
which is equivalent to eq. (10c), has been determined to be the field equation of massive matter.
This derivation is independent of the exact form of an Einstein equation. An implicit gauge condition is that the flat metric (ab is the asymptotic limit. Eq. (16) is compatible with the equivalence principle as demonstrated by Einstein [2] in his calculation of the bending of light. Thus, the derivation is self-consistent.
One might argue that Einstein equation (3) could be erived" from a linear equation more general than eq. (12a), if one regards the gravitational field as a spin-2 field coupled to the energy tensor [19,33]. However, such a ure" theoretical approach is not really consistent with Newton theory and related observations because the notion of gauge is used. Moreover, in such a roof", the existence of bounded dynamic2) solutions for eq. (3) must be invalidly assumed.
Note that Einstein obtained the same values for ( and ( by considering eq. (13) after assuming X(2)ab = 0 [34]. The present approach makes it possible to obtain from eq. (13) an equation with an additional second order term, i.e.,
Gab ( Rab ( gabR = - K(T(m)ab ( Y(1)ab(, (17)
where
KY(1)ab = X(2)ab - g ab(X(2)cd gcd(
is of second order. The conservation law (cT(m)cb = 0 and (cGcb ( 0 implies also (a Y(1)ab = 0. If Y(1)ab is identified as the gravitational energy tensor t(g)ab , eq. (10) is reaffirmed.
The anti-gravity coupling of t(g)ab that explains the dynamical failure of eq. (1), is due to Einstein radiation formula [7]. However, Pauli [10] was the first to point out explicitly the possibility of such an antigravity coupling. Moreover, the existence of such a coupling is, in a way, implicitly suggested by the singularity theorems, which show that if all the couplings are of the same sign [21], the existence of unrealistic spacetime singularities would b inevitable. The need of an anti-gravity coupling was first discovered in calculating the gravity of an electromagnetic wave [6].
Moreover, it was Einstein and Rosen [35] who first discover that the 1915 equation may not have a propagating wave solution. In 1953 Hogarth [36] conjectured that this equation does not have a dynamic solution. A definitive indication of this is the non-existence of the plane-wave solution [8]. Note that the lane waves" proposed by Bondi, Pirani, and Robinson [37], are actually unbounded although they believe that a plane-wave is an idealization of a weak wave from a distant isolated source. These unbounded solutions satisfy the condition of planeness but are not related to any weak wave. Also, although Misner, Thorne & Wheeler [13] conclude correctly that the plane-waves are bounded, their equation for plane-waves actually has no bounded wave solution [8]. This illustrates that over confidence may lead to careless, and result in inconsistency.
In short, the theoretical framework of general relativity permit an additional term Y(1)ab ( 0 whose existence is required by the dynamic cases. The 1915 equation is only an over simplified special choice of Einstein. Note, however, such a choice is consistent with the equivalence principle is known only for the static case.
5. Validity of a Space-Time Metric and the Equivalence Principle
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). In fact, the equivalence principle is satisfied, if and only if the space-time manifold is physically realizable, since a satisfaction of the equivalence principle requires that the geodesic represent a physical free fall. Thus, although defining a coordinate system for the purpose of calculation is only a mathematical step, choosing a space-time coordinate system requires physical considerations.
It will be shown that not all mathematical coordinate systems are equivalent in physics as claimed by Bergmann [14] and Liu [15]. For clarity, this will be illustrated with a few simple Lorentz metrics without gravity.
Example 1. To see the need of considering beyond the metric signature, consider the artificial metric,
ds2 = (2dt2 - dx2 - dy2 - dz2, (18a)
the time unit of t is second, the space unit is cm, and ( (( 2c, c = 3x1010 cm/sec). If the equivalence principle were valid, ds2 = 0 would imply the light speed to be (. Immediately, there is a contradiction, and thus the equivalence principle cannot be valid.
Nevertheless, one might argue that metric (18a) can be transformed to
ds2 = c2dt'2 - dx'2 - dy'2 - dz'2, (18b)
by the following diffeomorphism,
x' = x, y' = y, z' = z, and t' = t(/c. (19a)
Eq. (19a) implies, however, that the unit of t' is c/( (sec). The light speed in the x'-direction is
= c() = (((cm/sec). (19b)
Thus, the light speed remains (. If ( = 2c, Metric (19a) implies that the light speed would be 6 x 1010 cm/sec; and metric (19b) implies that the light speed is 3 x 1010 cm/half-sec.
In the literature, the units of the coordinates are usually not specified. Then, the distinct metric (18a) would be confused with a rescaling of the Minkowski metric. This creates a illusion that all constant metrics were equivalence in physics. This example illustrates also that it is invalid to efine" light speed in terms of local Minkowski spaces in a manifold.
Moreover, if the equivalence principle were valid, according to Einstein approach, one would obtain
c2dT2 = (2dt2 , and (dX2 + dY2 + dZ2) = (dx2 + dy2 + dz2), (20a)
for a resting observer at a point (x0, y0, z0, t0). Eq. (20a) and ds2 = 0 imply that the light speed is
= = 1 (20b)
Eq. (20b) implies, however, that the light speed is c in the local Minkowski coordinate, but is ( ((2c) in the (x, y, z, t) space. On the other hand, since there is no gravitational force for this case, we can have also
x = X, y = Y, and z = Z (21)
Eq. (20b) and eq. (21) absurdly mean that for the same frame of reference, we have different light speeds. This is in disagreement with the principle of uniqueness for a physical measurement.
Example 2, consider the Minkowski flat metric and the transformation, which is a diffeomorphism,
t = C[exp(T/C) - exp(-T/C)]/2, where C = constant. (22a)
Then
ds2 = [exp(T/C) + exp(-T/C)]2dT2 - dx2 - dy2 - dz2, (22b)
is the metric transformed from the Minkowski metric. If metric (22b) is realizable, according to ds2 = 0, the measured light speed would be [exp(T/C) + exp(-T/C)]/2.
From (22b), the Christoffel symbols ((,(( are zeros except (t,tt = (tgtt/2. Then, according to the geodesic equation, the equation of motion for a particle at (x, y, z, T) is
+ (t,tt= 0, and === 0 (23a)
where
(t,tt = {ln[exp(T/C) + exp(-T/C)]}.
It follows eq. (23a) that one obtains, for some constant k
dT/ds = k[exp(T/C) + exp(-T/C)]-1 and dx(/ds = Constant (23b)
Now, consider the case dx/dT = dy/dT = dz/dT = 0. For this case, one has dx/ds = dy/ds = dz/ds = 0 and dx2/dT2 = dy2/dT2 = dz2/dT2 = 0. Thus, in such a ree fall", there is no change in the spatial position or acceleration. Physically, this means that such an observer would have the same frame of reference, whether ree fall" or not. Thus, he would absurdly have two different light speeds from the same frame of reference. Accordingly, the equivalence principle is not satisfied and metric (22) is not realizable. Note that, from metric (22), there is no acceleration for a static particle.
Nevertheless, some theorists would disregard all these inconsistency because they believe that space-time coordinates have no physical meaning. Therefore, they also disagree with Einstein [2,3] and regard that coordinate light speeds as meaningless.
6. Incompatibility of the Galilean Transformation to the Equivalence Principle
It will be shown that a Galilean transformation, which is unrealizable, is incompatible with the equivalence principle. Some theorists, however, considered incorrectly that the Galilean transformation would lead to a space-time coordinate system. The root of their problem is that they mistaken the existence of the tetrad as equivalent to a satisfaction of the equivalence principle. They do not understand that a satisfaction of this principle requires the geodesic of ree fall" must be valid in physics.
Consider the Galilean transformation from (x, y, z, t) to the K' coordinates,
t = t', x = x', y = y', and z = z' - vt', (24a)
where v is a constant. Eq. (24a) transforms metric (10a) to another constant Lorentz metric
ds2 = ((c - v)dt' + dz' ( ((c + v)dt' - dz'( - dx'2 - dy'2, (24b)
Metric (24b) is a special case of a space with an indefinite metric. Then, for light rays in the z'-direction, ds2 = 0 would imply at any point the light speeds were
dz'/dt' = c + v, or dz'/dt' = -c + v. (25)
Clearly, ight speed" (25) violates coordinate relativistic causality (i.e. no cause event can propagate faster than the velocity of light in a vacuum). Thus, metric (24b) is not physically realizable, and those in (25) are not coordinate light velocities.
Moreover, according to the geodesic equation (2), metric (24b) implies d2x'(/ds2 = 0, and thus
dx'(/ds = constant. where x'( (= x', y', z', or t') (26a)
at any point. Now, according to metric (24), consider the case of a ree fall" at (x'0, y'0, z'0, t'0)
dx'/ds = dy'/ds = dz'/ds = 0, and dt'/ds = (c2 - v2)-1/2 (26b)
Since there is no acceleration or motion, such a ree falling" observer P' carries with him the frame of reference K'. Since a ree fall" does not automatically obtain a local Minkowski space, point 2) of the equivalence principle is violated. Also, for observer P', according eq. (1) the measured light speed is c, but according (24b) the light speed in the x-direction is (c2 - v2)1/2. This inconsistency also implies that point 3) of the equivalence principle is not satisfied in K'.
Nevertheless, mathematics ensures the existence of a local Minkowski space, which can be obtained by choosing first the path of a particle to be the time coordinate and then the other three space coordinates by using orthogonality. According to condition (26b), the time coordinate would remain the same dt'. But, the coordinate dz' is not orthogonal to dt'. Now, let us work out the local orthogonal tetrad of P', whose direction vP' is (0, 0, 0, dt'). Then, the orthonormal vectors of the tetrad are
a1 = (1, 0, 0, 0), a2 = (0, 1, 0, 0), a3 = (0, 0, (, (), and bp' = (0, 0, 0, () (27a)
where
( = ( -1, ( = -( v/c2, and ( = (c2 - v2)-1/2.
The corresponding transformations is as follows:
dt' = ( (dT - v/c2 dZ) , dz' = ( -1dZ, dx' = dX, and dy' = dY. (27b)
Thus, (dx', dy', dz') and (dX, dY, dZ) share the same frame of reference since there is no acceleration. But, there is a space measurement change in the z-direction. Metric (24b) does not satisfy point 2) of the equivalence principle since there is no physical cause for transformation (27b). In relativity, such a physical transformation happens only when there is relative motion or acceleration. But, P' is rest at K'. Thus, (27b) illustrates also that geodesic (26) does not represent a physical free fall.
A misunderstanding of the equivalence principle, as Yu (p. 42 of [23]) believed, is that at any space-time point, it is always possible to establish a local Minkowski space, which is related to a ree fall". However, this is necessary but insufficient. For instance, at any space-time point of manifold (18a), (22b) or (24b), there is a local Minkowski space, which is co-moving with a ree falling" observer in the manifold. But, the geodesic does not represent a physical free fall. Note that Yu interpretation is essentially rephrasing Pauli misinterpretation [3, p.145].
The Galilean transformation is an unphysical transformation, and it simply takes another unphysical transformation to cancel out the unphysical properties so introduced. In fact, (24a), and (27b) imply
dt = ( (dT - v/c2dZ) , and dZ = (dz' = ( (dz + v dt). (27c)
Transformation (27c) is just a Lorentz-Poincaré transformation. (27b) completes the transformation (27c) starting form (24a).
It has been shown in different approaches that metric (24b) is incompatible with physics and in particular the equivalence principle. Since (24a) is a Galilean transformation, the Galilean transformation is also not physically valid in general relativity. The failure of satisfying the equivalence principle should be expected since the Galilean transformation is experimentally not realizable. This analysis shows also that the Minkowski metric is only valid constant metric in physics. In fact, a general result is that if ((tt = 0 for ( = x, y, or z, then the equivalence principle is satisfied only if the metric is Minkowski.
Another consequence is the reaffirmation of coordinate relativistic causality in vacuum. That the speed of light could be larger than c through a coordinate transformation is inconsistent with the notion that the light speed c is the maximum possible speed. The equivalence principle rules out such a possibility. It thus follows that physically the speed of light cannot be larger than c at the presence of gravity. In fact, observation confirms that gravity only leads to a reduction of the light speed.
It has been illustrated that the Galilean transformation is incompatible with the equivalence principle in the absence of gravity. In fact, the incompatibility is also true even when gravity is present. To illustrates this, let us consider physical metric (4b) and the physical situation that a particle at (0, 0, z0, t0) moving with velocity v at the z-direction. The Galilean transformation (24a) transforms metric (4b) to
ds2 = c2(1 - )dt'2 - (1 + )(dx'2 + dy'2 + [dz' - v dt 2 (28a)
If metric (28a) had a physical realizable coordinate system S', the particle would be at (0, 0, z'0, t'0) in the state (0, 0, 0, dt') and the local spatial coordinates dx', dy', and dz' would be attached to the particle at the instance t'0. The problem can be reduced to previous case by considering the limits (? 0.
Moreover, according to Einstein [3], the equivalence principle is valid only if ds2 = 0 produces the correct light speeds. Thus, if S' were realizable, the light speeds in the z-direction would be
= c(1- () + v, or = -c(1- () + v, where ( = (28b)
according to metric (28a). Thus, coordinate relativistic causality is violated for sufficiently large r. In other words, point 1) of the equivalence principle cannot be satisfied and metric (28a) is not realizable.
This illustrates that the equivalence principle is a requirement for a valid physical space-time coordinate system.
7. Restriction of Covariance, and Intrinsically Unphysical Lorentz Manifolds
Einstein proposed that the equivalence principle is satisfied in a physical space-time1). Moreover, the equivalence principle is satisfied only in a physical space-time since the existence of a local Minkowski space has been proven by mathematics and a satisfaction of the equivalence principle requires sufficient satisfactions of all physical conditions. For example, when coordinate relativistic causality is not satisfied, the equivalence principle is proven directly to be not valid for this manifold. The current confusion was due to that the equivalence principle has not been understood correctly from the viewpoint of physics.
However, one may still wonder whether a Lorentz manifold is always diffeomorphic to a physical space. If this were true, then the metric signature would be essentially equivalent to the equivalence principle. But, there are Lorentz manifolds any of which cannot be diffeomorphic to a physical space. In view of this, such misunderstanding of relativity must be rectified. Since the belief that a Lorentz manifold were diffeomorphic to a physical space, has never been proven; the burden of proof is on such believers. Nevertheless, it is desirable to give an example of an intrinsically unphysical Lorentz manifold. This can even be a solution of Einstein equation if it fails a physical requirement, which is independent of a coordinate system [8,9,16].
For instance, an accepted solution of metric for an electromagnetic plane wave [38] is
ds2 = du dv + H du2 - dxi dxj , where H = hij(u)xixj, hii(u) 0, hij = hji , (29)
u = t - z, v = t + z. This is a Lorentz manifold since its eigen values are H ( (H2 + 1)1/2, -1, and -1. However, since the condition 1 ( (1 + H)/(1 - H) may not be valid, metric (29) does not satisfy coordinate relativistic causality and therefore the equivalence principle. Moreover, since H can be arbitrarily large, metric (29) is incompatible with Einstein notion of weak gravity4) and the correspondence principle. Also, in the light bending experiment, the gravitational effect of the light is implicitly assumed to be negligible. Thus, metric (29) cannot be valid in physics.
Nevertheless, to show that metric (29) cannot be diffeomorphic to a physical space, needs more work. The gravitational force (related to (ztt = (1/2)((hijxixj)/(t has arbitrary parameters (the coordinate origin). This arbitrariness in the metric violates the principle of causality (i.e., the causes of phenomena are identifiable) [8,11]. Thus, the manifold (29) cannot be diffeomorphic to a physical space since a diffeomorphism cannot eliminate the parameters, which violate the principle of causality.
8. Conclusions and Discussions
Einstein [2,3] proposed the equivalence principle for the reality, which he models as a Riemannian physical space-time. However, Pauli [10, p.145] version implies that the equivalence principle would be satisfied even though the coordinate system is not physically realizable. Now it is clarified that Einstein correctly objected Pauli version as a misinterpretation [30]. Also, it is proven that the equivalence principle is satisfied if and only if a manifold is physically realizable.
In general relativity, the Minkowski metric in special relativity is obviously a special case. However, it was not clear that the rinciples" which lead to general relativity are compatible with each other even in this special case. Some theorists believe incorrectly that the Galilean transformation were valid for general relativity, although Einstein [2] has made clear, pecial theory of relativity applies to the special case of the absence of a gravitational field". To rectify this, it is shown directly that, due to the equivalence principle, the Minkowski metric is the only valid constant space-time metric (§6).
To establish special relativity, the Galilean transformation is proven to be unrealizable by experiments. Thus, theoretically a Galilean transformation should be incompatible with the equivalence principle, which is applicable to only a physical space. This means, in contrast of the belief of some theorists [14,15], that the equivalence of all frames of reference is not the same as the physical equivalence of all mathematical coordinate systems. In fact, it is invalid in physics to extend the space-time physical coordinate system to an arbitrary Gaussian system [9]. For instance, the time coordinate is not arbitrary [2,3].
The Galilean transformation implied that there is no limit on the velocity of light. This, in principle, disagrees with the notion of invariant light speed. However, due to entrenched misconceptions on covariance, this problem was not even recognized for further investigations [20,23]. Moreover, some supported such a misconceptions with other errors and misunderstandings. In other words, such current heories" are characterized and maintained with a system of errors. Thus, it is necessary to calculate examples that directly demonstrate a violation of the equivalence principle.
Some theorists incorrectly claimed that the equivalence principle is equivalent to the mathematical existence of the tetrad. They over simplified Einstein principle merely as the mathematical existence of a co-moving local Minkowski space along a time-like geodesic. However, the physics is not only just such an existence, but also the formation of such local space by the free fall alone. For instance, the local space-time of a spaceship under the influence of only gravity is a local Minkowski space. Thus, the real question for the equivalence principle is whether the geodesic represents a physical free fall.
The fact that there is a distinction between the equivalence principle and the proper metric signature would imply also that the covariance principle must be restricted. An important function of the equivalence principle test is to eliminate unphysical Lorentz manifolds (see §7). For example, the fact that metric (29) is intrinsically unphysical resolves its seemingly paradox with the light bending calculation in which the gravity due to the light is implicitly assumed to be negligible [2,3]. This is another example that a misunderstanding of the equivalence principle can leads to disagreements with experiments.
Perhaps, due to confusing mathematical theorems with Einstein equivalence principle as Pauli did, this principle is often not explained adequately in some books [21-23]. To deal with all the theoretical inconsistence superficially, some theorists claimed that the space-time coordinates have no physical meaning in general relativity. Such a speculation disagrees with the fact that there are non-scalars in physics. The deflection of light is related to the light ray being observed as an almost straight line away from the sun, and gravitational red shifts are related to gtt - the time-time component of space-time metric.
Nevertheless, based on such an absurd claim, Hawking [18] declares, n general relativity, there is no real distinction between the space and time coordinates, just as there is no real difference between any two space coordinates." On the other hand, Hawking [18] also believed, n arrow of time, something that distinguished past from future, giving a direction of time". Apparently, he did not see that there is an inconsistency between these two statements. Moreover, like others, Hawking accepted the deflection of light. He probably did not realize that the deflection angle could be defined only in a certain type of physical coordinate systems, where the trajectory of a light ray, when far away from the sun, is approximately a straight line. Note that such logical deficiency is a common problem among those so-called tandard" relativists.
Theorists such as Synge [19], Fock [39] and more recently, Hawking [18,40], Ohanian, Ruffini, and Wheeler [22], who do not understand Einstein equivalence principle for various reasons including inadequate understanding of physics or mathematics at the fundamental level or deficiency in logic, advocated essentially that the basis of general relativity should be the Einstein field equation alone. However, experimentally the unrestricted validity of Einstein equation has not yet been established beyond reasonable doubt [7,25,26,41].
Theoretically there is no satisfactory proof of rigorous validity of Einstein field equation [42-44] (e.g., the inadequate source term mentioned in §1, is the cause of the unphysical solution (29) [6,8]). In fact, in 1953 Hogarth [36] conjectured that the 1915 Einstein equation is invalid for a dynamic two-body problem; and Einstein himself had pointed out that his equation might not be valid for matter of very high density [3]. Moreover, it has been proven by the binary pulsar experiment that Einstein equation must be modified [7] and Yilmaz [45] pointed out that Einstein equation of 1915 is only a test particle theory. Moreover, in terms of physics, a static solution is only an approximation for some dynamical problems. This means that, to support Einstein illustration of the equivalence principle with calculations on the light bending, it is necessary to show that his linear equation is justifiable for dynamical problems. Thus, it is necessary to derive the Maxwell-Newton Approximation independently from physical principles (§4) since the 1915 Einstein equation is valid for static problems only.
A simple dynamical problem would be the gravity due to the interaction of two massive particles. Then, Einstein condition of weak gravity, which is also due to the equivalence principle [46], requires such a solution of gravity must be bounded. Thus, satisfying boundedness of gravity due to a weak source is independent of the field equation. But, no such solution has ever been proven to be in existence. Being aware of the unboundedness of cylindrical and spherical aves" [47,48], after proving the existence of Cauchy solutions, Bruhat [4] remarked that the physical validity of any Cauchy solution is up to the experiments to decide. While her clarification is reasonable for a mathematician, physicists should have known physics better.
The equivalence principle remains indispensable because of its solid experimental foundation such as gravitational red shifts and the blending of light [7,20]. Theoretically, as illustrated, its failure is always accompanied with a violation of another physical requirement. Thus, as Weinberg [5] points out, t is much more useful to regard general relativity above all as a theory of gravitation, whose connection with geometry arises from the peculiar empirical properties of gravitation, properties summarized by Einstein Principle of the Equivalence of Gravitation and Inertia."
The long-standing errors in general relativity have profound historical reasons. Most physicists are used to linear equations, and unavoidably they would apply techniques, which are valid for linear equations. But, in nonlinear field equation, a second order term from the viewpoint of physics, may be crucial for the existence of a bounded physical solution. In other words, one may not take it for granted (i.e., without a proof) that a physical requirement is compatible with a field equation. Even well known physicists such as Einstein [49] and more recently Feymann [33] made such mistakes in general relativity.
In general relativity, assuming the existence of a bounded dynamics physical solution, has never been proven, but it has been used with blind faith [50-53]. This is essentially where many of the mathematical errors come from. A most telling evidence is that the lane-waves" proposed by Bondi et al. [37] are not bounded although they believe that they are. Furthermore, it has been proven that there are no bounded plane-waves [8] for the 1915 Einstein equation. Then, the non-existence of a dynamic solution for massive matter is proven [7] because experiments support the Maxwell-Newton Approximation. This approach of proof has been completed since the Maxwell-Newton Approximation can be derived from physical principles (§4).
Currently, relativists often ignore physical requirements [1,5] because they misunderstood the equivalence principle and accepted the covariance principle, which was rejected by Eddington. Historically, after early observational confirmations of Einstein predictions, Einstein declared logical completeness of his theory [34] although such confirmations verify only the theoretical framework of general relativity. Subsequently, a blind faith on the theoretical self-consistency of general relativity was developed. Many physicists working on general relativity, in spite of warnings from Gullstrand [25,26], Bohr & Klein [40], and other physicists [10,37], tend to have over confidence on Einstein equation (except a few such as N. Rosen [35]).
A dynamic physical solution, as pointed out by Low [54], is not just a time-dependent solution, which can be obtained from the Minkowski metric by making a coordinate transformation. In physics, such a dynamic solution must be related to the dynamics of source matter and gravitational radiation. Nevertheless, Christodoulou and Klainerman [55] claimed the existence of dynamical solutions by their construction although such olutions" are unrelated to dynamical sources or radiation. It is also surprising that their main mathematical mistakes are actually at the fundamental level [56-58]. As pointed out by Kramer et al. [1], many relativists have a problem in distinguishing a physically valid solution from mathematical solutions. Bonnor et al [5] further confirm this problem by pointing out that it is not possible to have a consistent physical interpretation.
9. Acknowledgments
This paper is dedicated to my grandfather Lu Zhu Qiu. The author gratefully acknowledges stimulating discussions with Professors C. Au, C. L. Cao, S.-J. Chang, A. J. Coleman, Li-Zhi Fang, L. Ford, R. Geroch, J. E. Hogarth, Liu Liao, F. E. Low, P. Morrison, A. Napier, H. C. Ohanian, R. M. Wald, Erick J. Weinberg, J. A. Wheeler, Chuen Wong, H. Yilmaz, Yu Yun-qiang, and Y. Z. Zhang. This work is supported in part by Innotec Design, Inc., U.S.A.
ENDNOTES
1) In general relativity, Einstein [2,3] considers the four-dimensional space-time reality as a physical space-time modeled as a Riemannian space-time (M, g). The Riemannian space M is characterized by a space-time metric gik that can be determined by physical considerations such as the distribution of matter. In elativity and the problem of space", Einstein [27] wrote,
or the functions gik describe not only the field, but at the same time also the topological and metrical structural properties of the manifold. ... There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field."
Moreover, since such a Riemannian space-time models reality, all the physical requirements must be sufficiently satisfied.
2) A local Minkowskian space is a short hand to express that special relativity is locally valid, except for phenomena involving the space-time curvature.
3) For example, the Wheeler-Hawking School [13,18,40] follows Pauli misinterpretation, and thus, their theories are different from general relativity. They, different from Einstein [2,3], believe that space-time coordinates have no physical meaning. Hawking [18] makes no secret of his disagreements with Einstein [2,3]. More recently, based on misinterpretations of Fock [39], Ohanian, Ruffini, and Wheeler [22] openly criticized Einstein theory as confusing and his principles invalid.
4) Some theorists believe that the solution of gravity for a weak source need not be bounded [38]. However, it has been shown that the equivalent principle implies compatibility with Einstein notion of weak gravity [46].
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