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The Equivalence Principle, the Covariance Principle and the Question of Self-Consistency in General           ★★★ 【字体:
The Equivalence Principle, the Covariance Principle and the Question of Self-Consistency in General
作者:佚名    论文来源:本站原创    点击数:    更新时间:2008-12-20    

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 byc2(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-2.33. R. P. Feynman, The Feynman Lectures on Gravitation (Addison-Wesley, New York, 1995).
34. A Pais, Subtle is the Lord ... (Oxford University Press, New York, 1996), pp 255-261.
35. A. Einstein & N. Rosen, J. Franklin Inst. 223, 43 (1937).
36. J. E. Hogarth, articles, Fields, and Rigid Bodies in the Formulation of Relativity Theories", Ph. D. thesis 1953, Dept. of Math., Royal Holloway College, University of London (1953), p. 6.
37. H. Bondi, F. A. E. Pirani, & I. Robinson, Proc. R. Soc. London A 251, 519-533 (1959).
38. R. Penrose, Rev. Mod. Phys. 37 (1), 215-220 (1965).
39. V. A. Fock, The Theory of Space Time and Gravitation, trans. N. Kemmer (Pergamon Press, 1964), pp 6, 119, & 231.
40. S. W. Hawking & G. F. R. Ellis, The large Scale Structure of Space-Time (Cambridge: Cambridge Univ. Press, 1979).
41. V. F. Weisskopf, The Privilege of Being a Physicist (Freeman, San Francisco, 1988), p. 129.
42. O. Klein, Z. F. Physik 37, 895 (1926).
43. V. I. Denisov, & A. A. Logunov, in: Current Problems in Mathematics, Vol. 24: 3, 219 (Moscow: Vsesoyuz. Inst. Nauchn. Tekhn. Informatsii, 1982).
44. A. A. Vlasov, & V. I. Denisov, Teoret. Mat. Fiz. 53, 406 (1982).
45. H. Yilmaz, Nu. Cim. 107B, 941 (1992).
46. C. Y. Lo, Phys. Essays, 12 (3), 508-526 (September, 1999).
47. J. Weber and J. A. Wheeler, Revs. Modern Phys. 29 (3) 509 (1957).
48. I. Robinson and A. Trautman, Physical Review Letters 4 (8), 431 (April 1960).
49. A. Einstein, L. Infeld, and B. Hoffmann, Annals of Math. 39 (1), 65-100 (Jan. 1938).
50. H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Proc. R. Soc. Lond. A 269, 21 (1962).
51. L. Blanchet, & T. Damour, Phil. Trans. R. Soc. Lond. A 340, 379-430 (1986).
52. T. Damour in 300 Years of Gravitation, ed. S. W. Hawking & W. Israel (Cambridge: Cambridge Univ. Press, 1987), 128.
53. J. B. Griffiths, olliding Plane Waves in General Relativity" (Oxford Univ. Press, 1991).
54. F. E. Low, Dept. of Physics, M.I.T., Mass., private communications, 1997.
55. D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space (Princeton University Press, 1993).
56. C. Y. Lo, Phys. Essays, 13 (1) 109-120 (March, 2000).
57. Volker Perlick, Zentralbl. f. Math. (827) (1996) 323, entry Nr. 53055.
58. Volker Perlick (republished with an editorial note), Gen. Relat. Grav. 32 (2000).6

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