Applied and Pure Research Institute

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Physics Essays, vol. 13, no. 4, 2000

Abstract

It is shown that the 1915 Einstein equation is incompatible with the physical notion that a wave carries away energy-momentum. This proof is compatible with that Maxwell-Newton Approximation (the linear field equation for weak gravity), and is supported by the binary pulsar experiments. For dynamic problems, the linear field equation is independent of, and furthermore incompatible with the Einstein equation. The linear equation, as a first-order approximation, requires the existence of the weak gravitational wave such that it must be bounded in amplitude and be related to the dynamics of the source of radiation. Due to neglecting these crucial physical associations, in addition to inadequate understanding of the equivalence principle, unphysical solutions were mistaken as gravitational waves. It is concluded theoretically that, as Einstein and Rosen suggested, a physical gravitational wave solution for the 1915 equation does not exist. This conclusion is given further supports by analyzing the issue of plane-waves versus exact wave solutions. Moreover, the approaches of Damour and Taylor for the radiation of binary pulsars would be valid only if they are as an approximation of the equation of 1995 update. In addition, the update equation shows that the singularity theorems prove only the breaking down of Wheeler-Hawking theories, but not general relativity. It is pointed out that some Lorentz manifolds are among those that actually disagree with known experimental facts.

Key Words: compatibility, dynamic solution, gravitational radiation, principle of causality, plane-wave, Wheeler-Hawking theories

1. Introduction

In physics, the existence of a wave is due to the fact, as required by special relativity, that a physical cause must propagate with a finite speed [1]. This implies also that a wave carries energy-momentum. Thus, the field equation for gravity must be able to accommodate the gravitational wave, which carries away gravitational energy-momentum. In this paper, it will be shown that the Einstein equation of 1915 fails this.

In general relativity, the Einstein equation of 1915 [2] for gravity of space-time metric g(( is

G(( ( R(( - g((R = - KT (m)(( , (1)

where G(( is the Einstein tensor, R(( is the Ricci curvature tensor, T(m)(( is the energy-stress tensor for massive matter, and K (= 8((c-2, and ( is the Newtonian coupling constant) is the coupling constant1). Thus,

G(( ( R(( - g((R = 0, or R(( = 0, (1‘)

at vacuum. However, (1‘) also implies no gravitational wave to carry away energy-momentum.

An incompatibility with radiation was first discovered by Einstein & Rosen [3,4] in 1936. However, due to conceptual and mathematical errors then, their discovery was not accepted. These errors form the basis of the so-called geometric viewpoint of the Wheeler-Hawking school [5,6] (see also Section 4). An obvious problem of their viewpoint is that one cannot distinguish a physical solution among mathematical solutions [7].

Conceptually, one would argue incorrectly that (1‘) carries energy-momentum because

G(( ( G(1)(( + G(2)(( (2a)

where G(1)(( consists of the linear terms (of the deviation ((( = g(( - ((( from the flat metric ((() in G(( , and G(2)(( consists of the others. Since G(2)(( has been identified as equivalent to the gravitational energy-stress of Einstein‘s notion [8], it seemed obvious that G(2)(( carries the energy-momentum. However, unless (1) can accommodate a physical gravitational wave, such an argument has no meaning. Moreover, no wave solution has ever been obtained for equation (1). In fact, this is impossible (see Section 2).

There are so-called wave solutions for (1‘), but they are actually invalid in physics (see §§ 3 & 5) since physical requirements (such as the principle of causality2), the equivalence principle, and so on) are not satisfied. In fact, some of them have been proven to be in disagreement with experiments [9,10]. Their invalid acceptance is due to the incorrect belief3) that the equivalence principle were satisfied by any Lorentz manifold [11].

Moreover, Einstein‘s notion cannot be exact, since it is not localizable [12]. In a field theory, a central problem is the exchange of energy between a particle and the field where the particle is located [13]. Therefore, the gravitational energy-stress must be a tensor (see also Section 4).

2. The Gravitational Wave and Nonexistence of Dynamic Solutions for Einstein‘s Equation

First, a major problem is a mathematical error on the relationship between (1) and its linearization. It was incorrectly believed that the linear Maxwell-Newton Approximation [13]

( c(c(( = - K T(m) (( , where (( = ((( - (((((cd(cd) (3a)

and

(((xi, t) = - (T(((yi, (t - R)]d3y, where R2 =(xi - yi)2 . (3b)

always provides the first-order approximation for equation (1). This belief was verified for the static case only.

For a dynamic4) case, however, this is no longer valid. While the Cauchy data can be arbitrary for (3a), but not for (1). The Cauchy data of (1) must satisfy four constraint equations, G(t = -KT(m)(t (( = x, y, z, t) since G(t contains only first-order time derivatives [8]. This shows that (3a) would be dynamically incompatible5) with equation (1) [10]. Further analysis shows that, in terms of both theory [11] and experiments [13], this mathematical incompatibility is in favor of (3), instead of (1).

In 1957, Fock [14] pointed out that, in harmonic coordinates, there are divergent logarithmic deviations from expected linearized behavior of the radiation. This was interpreted to mean merely that the contribution of the complicated nonlinear terms in the Einstein equation cannot be dealt with satisfactorily following this method and that other approach is needed. Subsequently, vacuum solutions that do not involve logarithmic deviation, were founded by Bondi, Pirani & Robinson [15] in 1959. Thus, the incorrect interpretation appears to be justified and the faith on the dynamic solutions maintained. It was not recognized until 1995 [13] that such a symptom of divergence actually shows the absence of bounded physical dynamic solutions.

In physics, the amplitude of a wave is generally related to its energy density and its source. Equation (3) shows that a gravitational wave is bounded and is related to the dynamic of the source. These are useful to prove that (3), as the first-order approximation for a dynamic problem, is incompatible with equation (1). Its existing wave solutions are unbounded and therefore cannot be associated with a dynamic source [11]. In other words, there is no evidence for the existence of a physical dynamic solution.

With the Hulse-Taylor binary pulsar experiment [16], it became easier to identify that the problem is in (1). Subsequently, it has been shown that (3), as a first-order approximation, can be derived from physical requirements which lead to general relativity [11]. Thus, (3) is on solid theoretical ground and general relativity remains a viable theory. Note, however, that the proof of the nonexistence of bounded dynamic solutions for (1) is essentially independent of the experimental supports for (3).

To prove this, it is sufficient to consider weak gravity since a physical solution must be compatible with Einstein‘s [2] notion of weak gravity (i.e., if there were a dynamic solution for a field equation, it should have a dynamic solution for a related weak gravity [11]). To calculate the radiation, consider further,

G(( ( G(1)(( + G(2)(( , where G(1)(( = (c(c(( + H(1)((, (2b)

H(1)(( ( -(c((((c + (((c( + ((((c(dcd , and ?(((? < 1. (2c)

G(2)(( is at least of second order in terms of the metric elements. For an isolated system located near the origin of the space coordinate system, G(2)(t at large r (= (x2 + y2 + z2 (1/2) is of O(K2/r2) (5,8,17(.

One may obtain some general characteristics of a dynamic solution for an isolated system as follows:

1) The characteristics of some physical quantities of an isolated system:

For an isolated system consisting of particles with typical mass, typical separation , and typical velocities , Weinberg (8( estimated, the power radiated at a frequency ( of order /will be of order

P ((/)624 or P 8/,

since (/is of order 2. The typical deceleration rad of particles in the system owing this energy loss is given by the power P divided by the momentum, or rad 7/. This may be compared with the accelerations computed in Newtonian mechanics, which are of order 2/, and with the post-Newtonian correction of 4/. Since radiation reaction is smaller than the post-Newtonian effects by a factor 3, if (( c, the velocity of light, the neglect of radiation reaction is perfectly justified. This allows us to consider the motion of a particle in an isolated system as almost periodic.

Consider, for instance, two particles of equal mass m with an almost circular orbit in the x-y plane whose origin is the center of the circle (i.e., the orbit of a particle is a circle if radiation are neglected). Thus, the principle of causality [9,10] implies that the metric g(( is weak and very close to the flat metric at distance far from the source and that g(((x, y, z, t‘) is an almost periodic function of t‘ (= t - r/c).

2) The expansion of a bounded dynamic solution g(( for an isolated weak gravitational source:

According (3), a first-order approximation of metric g(((x, y, z, t‘) is bounded and almost periodic since T(( is. Physically, the equivalence principle requires g(( to be bounded [11], and the principle of causality requires g(( to be almost periodic in time since the motion of a source particle is. Such a metric g(( is asymptotically flat for a large distance r, and the expansion of a bounded dynamic solution is:

g(((nx, ny, nz, r, t‘) = ((( +(((k)(nx, ny, nz, t‘)/rk, where n( = x(/r. (4)

3) The non-existence of dynamic solutions:

It follows expansion (4) that the non-zero time average of G(1)(t would be of O(1/r3) due to

((n( = (((( + n( n()/r, (5)

since the term of O(1/r2), being a sum of derivatives with respect to t‘, can have a zero time-average. If G(2)(t is of O(K2/r2) and has a nonzero time-average, consistency can be achieved only if another term of time-average O(K2/r2) at vacuum be added to the source of (1). Note that there is no plane-wave solution for (1‘) [9,18].

It will be shown by contradiction that there is no dynamic solution for (1) with a massive source. Let us define

((( = ((1)(( + ((2)(( ; (i)(( = ((i)(( - ((( (((i)cd (cd), where i = 1, 2 ;

and

(((((1)(( = - K T(m)(( . (6)

Then (1)(( is of a first-order; and ((2)(( is finite. On the other hand, from (1), one has

(((((2)(( + H(1)(( + G(2)(( = 0 . (7)

Note that, for a dynamic case, equation (7) may not be satisfied. If (6) is a first-order approximation, G(2)(( has a nonzero time-average of O(K2/r2) (8(; and thus (2)(( cannot have a solution.

However, if (2)(( is also of the first-order of K, one cannot estimate G(2)(( by assuming that (1)(( provides a first-order approximation. For example, (6) does not provide the first approximation for the static Schwarzschild solution, although it can be transformed to a form such that (6) provides a first-order approximation [11(. According to (7), (2)(( will be a second order term if the sum H(1)(( is of second order. From (2c), this would require (((( being of second order. For weak gravity, it is known that a coordinate transformation would turn (((( to a second order term (can be zero) (8,14,17(. (Eq. [7] implies that (c(c(2)(( - (c((((c + (((c( would be of second order) Thus, it is always possible to turn (6) to become an equation for a first-order approximation for weak gravity.

From the viewpoint of physics, since it has been proven that (3) necessarily gives a first-order approximation [11], a failure of such a coordinate transformation means only that such a solution is not valid in physics. Moreover, for the dynamic of massive matter, experiment [16] supports the fact that Maxwell-Newton Approximation (3) is related to a dynamic solution of weak gravity [13]. Otherwise, not only is Einstein‘s radiation formula not valid, but the theoretical framework of general relativity, including the notion of the plane-wave as an idealization, should be re-examined (see Section 3). In other words, theoretical considerations in physics as well as experiments eliminate other unverified speculations thought to be possible since 1957.

As shown, the difficulty comes from the assumption of boundedness (Section 3), which allows the existence of a bounded first-order approximation, which in turn implies that a time-average of the radiative part of G(2)(( is non-zero (7(. The present method has an advantage over Fock‘s approach to obtaining logarithmic divergence [13,14( for being simple and clear.

In short, according to Einstein‘s radiation formula, a time average of G(2)(t is non-zero and of O(K2/r2) [13(. Although (3) implies G(1)(t is of order K2, its terms of O(1/r2) can have a zero time average because G(1)(t is linear on the metric elements. Thus, (1‘) cannot be satisfied. Nevertheless, a static metric can satisfy (1), since both G(1)(( and G(2)(( are of O(K2/r4) in vacuum. Thus, that a gravitational wave carries energy-momentum does not follow from the fact that G(2)(( can be identified with a gravitational energy-stress (8,17(. Just as G(( , G(2)(( should be considered only as a geometric part. Note that G(t = -KT(m)(t are constraints on the initial data.

In conclusion, in disagreement with the physical requirement, assuming the existence of dynamic solutions of weak gravity for (1) [14,15,19-24( is invalid. This means that the calculations [25,26( on the binary pulsar experiments should, in principle, be re-addressed [12(. This explains also that an attempt by Christodoulou and Klainerman [26( to construct bounded dynamic solutions for G(( = 0 fails to relate to a dynamic source and to be compatible with (3) [28] although their solutions do not imply that a gravitational wave carries energy-momentum.

For a problem such as scattering, although the motion of the particles is not periodic, the problem remains. This will be explained (see Section 4) in terms of the 1995 update of the Einstein equation, due to the necessary existence of gravitational energy-momentum tensor term with an antigravity coupling in the source. To establish the 1995 update equation, the supports of binary pulsar experiments for (3) are needed [13].

3. Gravitational Radiations, Boundedness of Plane-Waves, and the Maxwell-Newton Approximation

An additional piece of evidence is that there is no plane-wave solution for (1). A plane-wave is a spatial-local idealization of a weak wave from a distant source. The plane-wave propagating in the z-direction is a physical model although its total energy is infinite [8,10]. According to (3), one can substitute (t - R) with (t - z) and the other dependence on r can be neglected because r is very large. This results in(((xi, t) becoming a bounded periodic function of (t - z). Since the Maxwell-Newton Approximation provides the first-order, the exact plane-wave as an idealization is a bounded periodic function. Since the dependence of 1/r is neglected, one considers essentially terms of O(1/r2) in G(2)((. In fact, the non-existence of bounded plane-wave for G(( = 0, was proven directly in 1991 [9,18].

In short, Einstein & Rosen [4,29] is essentially right, i.e., there are no wave solutions for R(( = 0. The fact that the existing wave solutions are unbounded also confirms the nonexistence of dynamic solutions. The failure to extend from the linearized behavior of the radiation is due to the fact that there is no bounded physical wave solution for (1) and thus this failure is independent of the method used.

Note that the Einstein radiation formula depends on (3) as a first-order approximation. Thus, metric g(( must be bounded. Otherwise G(( = 0 can be satisfied. For example, the metric of Bondi et al. [15] is

ds2 = exp(2()(d( 2 - d(2) - u2(ch2( (d(2 + d(2) + sh2( cos2( (d(2 - d(2) - 2sh2( sin2( d(d((, (8)

where (, (, ( are functions of u (= ( - ( ). It satisfies the differential equation (i.e., their eq. (2.8(),

2(‘ = u((‘2 + (‘2 sh2(2). (9)

However, metric (8) is not bounded, because this would require the impossibility of u2 The failure of recognizing G(( = 0 as invalid for gravitational waves is due to mistaking (3) as a first-order approximation of (1). Thus, in spite of Einstein‘s discovery [3] and Hogarth‘s conjecture6) [31] on the need of modification, the incompatibility between (1) and (3) was not proven until 1993 [13] after the non-existence of the plane-waves for G(( = 0, has been proven [9,18].

4. Gravitational Radiation and the 1995 update of the Einstein Equation

In general, (3) is actually an approximation of the 1995 update of the Einstein equation [13],

G(( ( R(( - g((R = - K (T(m)(( - t(g)(((, (10)

where t(g)(( is the energy-stress tensors for gravity. Then,

((T(m)(( = 0, and ((t(g)(( = 0. (11)

Equation (11) implies that the equivalence principle would be satisfied. From (10), the equation in vacuum is

G(( ( R(( - g((R = K t(g)(( . (10‘)

Note that t(g)(( is equivalent to G(2)(( (and Einstein‘s gravitational pseudotensor) in terms of his radiation formula. The fact that t(g)(( and G(2)(( are related under some circumstances does not cause G(2)(( to be an energy-stress nor t(g)(( a geometric part, just as G(( and T(( must be considered as distinct in (1).

When gravitational wave is present, the gravitational energy-stress tensor t(g)(( is non-zero. Thus, a gravitational radiation does carry energy-momentum as physics requires. This explains also that the absence of an anti-gravity coupling which is determined by Einstein‘s radiation formula, is the physical reason that the 1915 Einstein equation (1) is incompatible with radiation.

Note that the radiation of the binary pulsar can be calculated without detailed knowledge of t(g)((. From (10‘), the approximate value of t(g)(( at vacuum can be calculated through G((/K as before since the first-order approximation of g(( can be calculated through (3). In view of the facts that Kt(g) (( is of the fifth order in a post-Newtonian approximation, that the deceleration due to radiation is of the three and a half order in a post-Newtonian approximation [8] and that the perihelion of Mercury was successfully calculated with the second-order approximation from (1), the orbits of the binary pulsar can be calculated with the second-order post-Newtonian approximation of (10) by using (1) (see also Section 6). Thus, the calculation approaches of Damour and Taylor [25,26] would be essentially valid except that they did not realize the crucial fact that (3) is actually an approximation of the update equation (10) [13].

In light of the above, the Hulse-Taylor experiments support the anti-gravity coupling being crucial to the existence of the gravitational wave [10,13], and (3) being an approximation of weak waves generated by massive matter. Thus, it has been experimentally verified that (1) is incompatible with radiation.

It should be noted also that the existence of an anti-gravity coupling7) means the energy conditions in the singularity theorems [6,17] are not valid at least for a dynamic situation. Thus, the existence of singularity is not certain, and the claim of inevitably breaking of general relativity is actually baseless since these singularity theorems have been proven to be unrealistic in physics. As pointed out by Einstein [2], his equation may not be valid for very high density of field and matter. In short, the singularity theorems show only the breaking down of theories of the Wheeler-Hawking school, which are actually different3) from general relativity.

The theories of this school, in addition to making crucial mistakes in mathematics as shown in this paper (see also [11,28]), differ from general relativity in at least the following important aspects:

1) They reject an anti-gravity coupling7), which is considered as highly probable by Einstein himself.

2) They implicitly replaced Einstein‘s equivalence principle in physics3) with merely the mathematical requirement of the existence of local Minkowski spaces [5,6].

3) They, do not consider physical principles [9-11,28] (see also Section 5), such as the principle of causality, the coordinate relativistic causality, the correspondence principle and etc. of which the satisfaction is vital for a physical space, which models reality, such that Einstein‘s equivalence principle can be applicable.

Thus, in spite of currently declaring their theories as the development of general relativity, these theories actually contradict crucial features that are indispensable in Einstein‘s theory of general relativity. More importantly, in the development of their so-called orthodox theory, they implicitly violate physical principles that took generations to establish. As a result, Einstein‘s theory has been unfairly considered as irrelevant in the eyes of many physicists.

Of course, the exact form of t(g)(( is important for the investigation of high density of field. However, it seems, the physics of very high density of field and matter is not yet mature enough at present to allow a definitive conclusion. For instance, it is unclear what influence the discovery of quarks and gluons in particle physics would have on the evolution of stars. It is known that atomic physics supports the notion of white-dwarf stars, and that nuclear physics leads to the notion of neutron stars.

5. Physically Invalid Unbounded Gravitational Waves and the Principle of Causality

To my mind there must be at the bottom of it all, not an equation, but an utterly simple idea. And to me that idea, when we finally discover it, will be so compelling, so inevitable, that we will say to one another, ‘Oh, how beautiful. How could it have been otherwise?‘ -- J. A. Wheeler [32].

It seems, the principle of causality2) (i.e., phenomena can be explained in terms of identifiable causes) [9,10] would be qualified as Wheeler‘s utterly simple idea. Being a physicist, his notion of beauty should be based on compelling and inevitability, but would not be based on some perceived mathematical ideas. It will be shown that the principle of causality is useful in examining validity of accepted wave solutions.

According to the principle of causality, a wave solution must be related to a dynamic source, and therefore is not just a time-dependent metric. A time-dependent solution, which can be obtained simply by a coordinate transformation, may not be related to a dynamic source8) [33]. Even in electrodynamics, satisfying the vacuum equation can be insufficient. For instance, the electromagnetic potential solution A0[exp(t - z)2] (A0 is a constant), is not valid in physics because one cannot relate such a solution to a dynamic source. Thus, as shown in Section 4, a solution free of singularities may not be physically valid.

A major problem in general relativity is that the equivalence principle has not been understood adequately [11,34]. Since a Lorentz manifold was mistaken as always valid, physical principles were often not considered. For instance, the principle of causality was neglected such that a gravitational wave was not considered as related to a dynamic source, which may not be just the source term in the field equation [8,35].

Since the principle of causality was not understood adequately, solutions with arbitrary nonphysical parameters were accepted as valid [34]. Similarly, Misner, Thorne & Wheeler [5], assumed that the metric due to an electromagnetic plane-wave is invariant with respect to a rotation whose axis is in the direction of propagation. Consequently, in addition to the fact that the polarization is incorrect, Misner et al. were not aware of that, in disagreement with what they stated, such a metric cannot be bounded. Such unbounded solutions disagree with experiments [10,11].

Among the existing so-called wave solutions, not only Einstein‘s equivalence principle but the principle of causality is not satisfied because they cannot be related to a dynamic source. (However, a source term in an equation, though related to, may not necessarily represent the physical cause [9,34].) Here, examples of accepted gravitational waves are shown as actually invalid in physics.

1. Let us examine the cylindrical waves of Einstein & Rosen [29]. In cylindrical coordinates, (, (, and z,

ds2 = exp(2( - 2()(dT2 - d(2) - (2exp(-2()d(2 - exp(2()dz2 (12)

where T is the time coordinate, and ( and ( are functions of ( and T. They satisfy the equations

((( + (1/()(( - (TT = 0, (( = ([((2 + (T2], and (T = 2((((T. (13)

Rosen [36] consider the energy-stress tensor T(( that has cylindrical symmetry. He found that

T44 + t44 = 0, and T4l + t4l = 0 (14)

where t(( is Einstein‘s gravitational pseudotensor, t4l is momentum in the radial direction.

However, Weber & Wheeler [37] argued that these results are meaningless since t(( is not a tensor. They further pointed out that the wave is unbounded and therefore the energy is undefined. Moreover, they claimed metric (12) satisfying the equivalence principle and speculated that the energy flux is non-zero.

Their claim shows an inadequate understanding of the equivalence principle. To satisfy this principle requires that a time-like geodesic must represent a physical free fall. This means that all (not just some) physical requirements are necessarily satisfied. Thus, the equivalence principle may not be satisfied in a Lorentz Manifold [11,35], which implies only the necessary condition of the mathematical existence of a co-moving local Minkowski space along a time-like geodesic. It will be shown that manifold (12) cannot satisfy coordinate relativistic causality. Moreover, as pointed out earlier, an unbounded wave is unphysical.

Weber and Wheeler‘s arguments for unboundedness are complicated, and they agreed with Fierz‘s analysis, based on (13), that ( is a strictly positive where ( = 0 [37]. However, it is possible to see that there is no physical wave solution in a simpler way. Gravitational red shifts imply that gtt ( 1 [2]; and

-g(( ( gtt , -g((/(2 ( gtt , and -gzz ( gtt , (15a)

are implies by coordinate relativistic causality. Thus, according to these constraints, from metric (12) one has

exp(2() ( 1 and exp (2() ( exp(4(). (15b)

Equation (15) implies that gtt ( 1 and that ( ( 0. However, this also means that the condition ( > 0 cannot be met. Thus, this shows again that there is no physical wave solution for G(( = 0.

Weber and Wheeler are probably the earliest to show the unboundedness of a wave solution for G(( = 0. Nevertheless, due to their inadequate understanding of the equivalence principle, they did not reach a valid conclusion. It is ironic that they therefore criticized Rosen who come to a valid conclusion, though with dubious reasoning.

2. Robinson and Trautman [38] dealt with a metric of spherical gravitational waves for G(( = 0. However, their metric has the same problem of unboundedness and having no dynamic source connection. This confirms further that the cause of this problem is intrinsically physical in nature. Their metric has the following form:

ds2 = 2d(d( + (K - 2H( - 2m/()d(2 - (2p-2{[d( + ((q/(()d(]2 + [d( +((q/(()d(]2}, (16a)

where m is a function of ( only, p and q are functions of (, (, and (,

H = p-1(p/(( + p(2p-1q/(((( - pq (2p-1/(((( , (16b)

and K is the Gaussian curvature of the surface ( = 1, ( = constant,

K = p2((2/((2 + (2/((2)ln p. (16c)

For this metric, the empty-space condition G(( = 0 reduces to

(2q/((2 + (2q/((2 = 0, and (2K/((2 + (2K/((2 = 4p-2((/(( - 3H)m. (17)

To see this metric has no dynamic connection, let us examine their special case as follows:

ds2 = 2d(d( - 2Hd(2 - d(2 - d(2, and (H/(( = (2H/((2 + (2H/((2 = 0. (18)

This is a plane-fronted wave [39] derived from metric (16) by specializing

p = 1 + ((2 + (2)K(()/4. (19a)

substituting

( = (-2 + (-1, ( = (, ( = (2, ( = (2, q = (4, (19b)

where ( is constant, and taking the limit as ( tends to zero [38]. Although (18) is a Lorentz metric, there is a singularity on every wave front where the homogeneity conditions

(3H/((3 = (3H/((3 = 0. (20)

are violated [38]. Obviously, this is also incompatible with Einstein‘s notion of weak gravity [2]. A problem in current theory is its rather insensitivity toward theoretical self-consistency [9,13,35,40-42].

3. To illustrate the non-existence of a bounded radiating physical solution further, let us examine a recent solution of R(( = 0, the cylindrical symmetry solution of Au, Fang & To [43]. Their metric is

ds2 = N2(c2dt2 - dz2) - L2d(2 - M2(2d(2 (21)

where

N2 = (-4exp(-4((d() exp(2n1), L2 = (-8(1 + (()2exp(-6((d(),

and

M2 = exp(2((d() where n1= n1(ct - z), and ( = ((()

are respectively arbitrary functions of (ct - z) and of (. The function n1(ct - z) makes N2 a propagating wave. If solution (21) were a physical solution, M should be a bounded function of (, i.e.,

exp(2((d() However, this also means that N is not bounded for small (. Moreover, if light velocity is not larger than its vacuum velocity c, one should have N2/L2 and N2/M2 ( 1. It thus follows that

(1 + (()2 ( (4 exp(2((d()exp(2n1), and exp(6((d() ( exp(2n1) (-4. (23)

Hence,

(1/( + ()2 ( (2/3 exp (8n1/3) and (2 > ( O((2/3). (24)

Thus, condition (24) is also inconsistent with condition (22). In summary, solution (21) is also not a physical solution and is unbounded in contrast to as required by the principle of causality.

4. To illustrate an invalid source and an intrinsic non-physical space, consider the following metric ,

ds2 = du dv + Hdu2 - dxi dxi, where H = hij(u)xi xj (25)

where u = ct - z, v = ct + z, x = x1 and y = x2, hii(u) ( 0, and hij = hji [44]. This metric satisfies the harmonic gauge. The cause of metric (25) can be an electromagnetic plane wave. Metric (25) satisfies

((( (((( (tt = -2{hxx(u) + hyy(u)} where ((( = g(( - (((. (26)

However, this does not mean that causality is satisfied although metric (25) is related to a dynamic source. It will be shown that (25) is not a physical solution because physical principles are violated.

A light trajectory satisfies ds2 = 0 [2]. For a light in the z-direction (i.e. dx = dy = 0), one obtains

dz/dt = c or -c (1 + H)/(1 - H); but H ( 0 (27)

would fail since hii(u) ( 0 ; and so coordinate relativistic causality would also fail. Thus, a formal satisfaction of the conservation law due to ((G(( ( 0, is inadequate to ensure the validity of (1).

Moreover, the gravitational force is related to (ztt = (1/2)(H/(t. There are arbitrary non-physical parameters (the choice of origin) that are unrelated to the cause (a plane wave). Apparently, believing that any Lorentz manifold is valid in physics, Penrose [44] over-looked the physical requirements, in particular the principle of causality. Experimentally, being unbounded, metric (25) is also incompatible with the calculation of light bending and classical electrodynamics.

These examples confirm that there is no bounded wave solution for (1) although a time-dependent solution need not be logarithmic divergent [14]. A fundamental reason for the boundedness of a dynamic solution for gravity, is the equivalence principle [11]. This would mean that the hyperboloid solution in Friedmann‘s theory might not be valid in general relativity (see Appendix).

6. Conclusions and Discussions

In general relativity, the existence of gravitational wave is a crucial test of the field equation. Thus, an important question is: what does the gravitational field of a radiating asymptotically Minkowskian system look like? Without experimental inputs, to answer this question would be very difficult.

Einstein [2] proposed the linearized theory for a weak radiating gravitational field. But, Bondi [24] commented, it is never entirely clear whether solutions derived by the usual method of linear approximation necessarily correspond in every case to exact solutions, or whether there might be spurious linear solutions which are not in any sense approximations to exact ones. Thus, in calculating gravitational waves from the Einstein equation, problems are considered as due to the method rather than inherent in the equations.

Physically, it is natural to continue assuming Einstein‘s notion of weak gravity is valid. (Boundedness, though a physical requirement, may not be mathematically compatible to a nonlinear field equation. But, no one except perhaps Gullstrand [40,41], expected the nonexistence of dynamic solutions.) The complexity of the Einstein equation makes it very difficult to have a close form. Thus, it is necessary that a method of expansion should be used to examine the problem of weak gravity, if one expects such an expansion to be valid.

A factor which contributes to this faith is that ((G(( ( 0 implies ((T(m)(( = 0, the energy-momentum conservation law. However, this is only necessary but not sufficient for a dynamic solution. Although the 1915 equation gives an excellent description of planetary motion, including the advance of the perihelion of Mercury, this is essentially a test-particle theory, in which the reaction of the test particle is neglected. Thus, the so obtained solutions are not dynamic solutions. As pointed out by Gullstrand [41,45] such a solution may not be obtainable as a limit of a dynamic solution. Nevertheless, Einstein, Infeld, and Hoffmann [22] incorrectly assumed the existence of bounded dynamic solution and deduced the geodesic equation from the 1915 equation. Recently, Feymann [23] made the same incorrect assumption that a physical requirement would be unconditionally applicable to a mathematical equation.

The nonlinear nature of Einstein equation certainly gives surprises. In 1959, Fock [14] pointed out that, in harmonic coordinates, there are divergent logarithmic deviations from expected linearized behavior of the radiation. After the discovery that some vacuum solutions are not logarithmic divergent [15], the inadequacy of Einstein‘s equation was not recognized. Instead, the method of calculation was mistaken as the problem.

To avoid the appearance of logarithms, Bondi et al. [24] and Sachs [46] introduced a new approach to gravitational radiation theory. They used a special type of coordinate system, and instead of assuming an asymptotic expansion in the gravitational coupling constant (, they assume the existence of an asymptotic expansion in inverse power of the distance r (from the origin where the isolated source is located in r ( a, which is a positive constant). The approach of Bondi-Sachs was clarified by the geometrical ‘conformal‘ reformulation of Penrose [47].

However, this approach is unsatisfactory, because it rests on a set of assumptions that have not been shown to be satisfied by a sufficiently general solution of the inhomogeneous Einstein field equation [48]. In other words, this approach provides only a definition of a class of space-times that one would like to associate to radiative isolated systems, neither the global consistency nor the physical appropriateness of this definition has been proven. Moreover, perturbation calculations have given some hints of inconsistency between the Bondi-Sachs-Penrose definition and some approximate solution of the field equation. Not less important, it seems a priori difficult to relate to the source located within r ( a [48].

There are two other main classes of approach: 1) the post-Newtonian approaches (1/c expansions) and the post-Minkowskian approaches (K expansions). The post-Newtonian approaches are fraught with serious internal consistency problems [48] because they often lead, in higher approximations, to divergent integrals. The post-Minkowskian approach is an extension of the linearization, one may expect that there are some problems related to divergent logarithmic deviations [14]. Moreover, it has unexpectedly been found that perturbative calculations on radiation actually depend on the approach chosen [49]. Mathematically, this non-uniqueness shows, in disagreement with (3), that a dynamic solution of (1) is unbounded.

Based on the binary pulsar experiments, it is proven that the Einstein equation does not have any dynamic solution even for weak gravity [13]. Mathematically, however, the proof that is aimed directly to the nonexistence of a dynamic solution is independent of the experimental supports for (3). This long process is, in part, due to theoretical consistency were inadequately considered [9,10,13,35]. Moreover, it was not recognized that boundedness of a wave is crucial for its association with a dynamic source. These inadequacies allowed acceptance of unphysical time-dependent solutions as physical waves (Sections 3 & 5).

Although non-linearity of the 1915 Einstein equation was new, in view of impressive observational confirmations, it seemed natural to assume that gravitational waves would be produced. Moreover, gravitational radiation is often considered as due to the acceleration in a geodesic alone [50-52]. It is remarkable that in 1936 Einstein and Rosen [4] are the first to discover this problem of excluding the gravitational wave. However, without clear experimental evidence, it was difficult to make an appropriate modification.

From studying the gravity of electromagnetic waves, it was also clear that Einstein equation must be modified [11,18]. However, the Hulse and Taylor binary pulsar experiments, which confirm Hogarth‘s 1953 conjecture6) [31,35], are indispensable for verifying the necessity of the anti-gravity coupling in general relativity [10,13]. In addition to experimental supports, the Maxwell-Newton Approximation can be derived from physical principles, and the equivalence principle also implies boundedness of a normalized metric in general relativity [11]. A perturbative approach cannot be fully established for (1) simply because there are no bounded dynamic solutions10), which must, owing to radiation, be associated with an anti-gravity coupling.

Nevertheless, Christodoulou and Klainerman [27] claimed to have constructed bounded gravitational (unverified) waves. Obviously, their claim is incompatible with the findings of others. Furthermore, their presumed solutions are incompatible with Einstein‘s radiation formula and are unrelated to dynamic sources [10,11]. Thus, they simply have mistaken5) an unphysical assumption (which does not satisfy physical requirements) as a wave [28].

Within the theoretical framework of general relativity, however, the gravitational field of a radiating asymptotically Minkowskian system is given by the Maxwell-Newton Approximation [13]. With the need of rectifying the 1915 Einstein equation established, the exact form of t(g)(( in the equation of 1995 update [13] is an important problem since a dynamic solution that gives an approximation for the perihelion of Mercury remains unsolved [41]. Moreover, the update equation shows that the singularity theorems prove only the breaking down of theories of the Wheeler-Hawking school3), but not general relativity (see Section 4). Experimentally, the Maxwell-Newton Approximation would be further tested by the Gravity Probe-B gyroscopes [53] on the precessions. This analysis suggests that further confirmation of this Approximation and thus the equivalence principle is expected.

Appendix: Dynamic Space-Time, Space-Time Coordinate System, and the Big Bang Theory

The equivalence principle, in a certain sense, is a non-local property, since its physics is whether the geodesic represents a physical free fall [11]. Thus, one must consider beyond the mathematical tangent space, that is, mathematical local Minkowski spaces. To determine whether a manifold solution can be diffeomorphic to a physical space is a difficult problem and physical requirements are needed [10].

In physics, the frame of reference is often chosen to be best for the problem. If a valid physical solution cannot be found, the difficult is usually not due to the coordinates. In addition, as a practical approximate means, a Galilean transformation can be used in some class of problems. Thus, that a certain coordinate system is useful for some calculations does not mean that the coordinate system is, in principle, realizable.

For a practical problem, in spite of talks about coordinates cannot be chosen a priori, general relativity is actually not an exception11). For instance, in the Schwarzschild static solution, the frame of reference is chosen a priori and the radial r is (x2 + y2 + z2)1/2. This frame of reference is used to access the amount of light bending. In the problem of light bending, the total field (space-time metric) should be time-dependent, but r as a variable would be the same if the frame of reference does not change.

Nevertheless, in cosmology, there are time-dependent solutions that do not involve a coordinate system chosen a priori, nor gravitational radiation. However, one should note also that all the cosmological models are based on idealizations that have not been established beyond reasonable doubt [32,54]. For this reason alone, such examples are unsuitable for our discussion on a fundamental problem of realistic situations. However, some discussions on this subject are needed, since it is claimed that the big bang theory is based on general relativity [32,55].

It is generally assumed [55] that the energy-momentum tensor in the universe today is that of a uniform gas with zero pressure. The galaxies may be regarded as the ‘particles‘ out of which this gas is made, and since the velocities of the galaxies do not deviate much from uniform expansion, we can neglect the ‘pressure‘ of the gas of galaxies. ... The Friedmann models assumed homogeneous, isotropic models of the universe with mass density but with zero pressure. A difficult in cosmology is that many usual physical requirements, on which a judgment of physical validity depends, are probably not applicable.

Nevertheless, some discussions may be helpful in clarifying coordinate relativistic causality. To discuss the Friedmann model, one must first accept essentially by faith that the mass distribution of the whole universe is homogeneous and isotropic. One must decide also modeling a galaxy as a particle is consistent with the normal understanding of Einstein‘s equivalence principle. Then, in Cartesian coordinates,

ds2 = d(2 - 2((){dx2 + dy2 + dz2}, (A1)

the Robertson-Walker geometry, is believed to be appropriate. Then, the Einstein equation (1) with source energy tensor T(( = u(u(+ P(u(u( - g(() leads to the following general evolution equations [17]:

3 = -4((( + 3P) (A2)

and

32/2 = 8(( - 3k/2, (A3)

where ( is the mass density, and P is the pressure. For different values of k, there are different types of solutions: k = +1 for the 3-sphere, k = 0 for the flat space, and k = -1 for the hyperboloid. For k = -1, 2(() is unbounded [17] and is therefore incompatible with the equivalence principle [11].

The rate of change of R (the distance between two isotropic observers at time () is

v= HR , (A4)

where H(() =/is identified with Hubble‘s constant. This means, however, the constant is time-dependent. Note, however, the observed red shifts may not be due to the Doppler effect alone [11,54,56].

However, within the above constraint, a model-independent feature of (() is

(()(( ( ( = 0; (A5a)

and

((()n(() = constant, where n ( 3 (A5b)

On the other hand, ds2 = 0 could imply that the light speed in the x-direction would be

(A6)

Thus, (A5a) and (A6) lead to a result that the light speed could be larger than c. Thus, it seems, either that coordinate relativistic causality could be violated or metric (A1) would be invalid.

Nevertheless, one must be careful because things are not that simple. For ds2 = 0 leads to a light speed in vacuum. However, in the Friedmann model, when a(() is very small, according to (A5b), not only there is no vacuum but the mass density ((() would be too large for the light to go through. Thus, the argument that leads to (A6) breaks down. Moreover, to justify the Robertson-Walker geometry, the effects of gravitational radiation should have been shown to be negligible at least for the assumed early universe. The existence of gravitational radiation, as pointed out by Lorentz and Wheeler [1], is due to the theory of relativity. Thus, it is also not clear that Friedmann‘s solution must be deduced from general relativity.

In reality, a galaxy is not a particle, the mass distribution is not homogeneous, and a light speed has nothing to do with Friedmann‘s modeling. Thus, it is clearly unsuitable for a discussion on fundamental questions. Now, it should be clear also that the Big Bang theory, though can be related to (1), depends on too many dubious assumptions (see also [32,54]) for the claim of being a consequence of general relativity. (Also, in view of the idealizations, the possibility of deriving eqs. (A2) and (A3) from another equation cannot be rule out.) Nevertheless, this discussion illustrates also the importance of the equivalence principle.

Acknowledgments

This paper is dedicated to Professor J. E. Hogarth of Queen‘s University, Kingston, Ontario, Canada, who conjectured in 1953 the nonexistence of dynamic solutions for the 1915 Einstein equation. The author wishes to express his appreciation to Professor Xin Yu for the hospitality of the Hong Kong Polytechnic University where substantial of this work was done in 1995. The author gratefully acknowledges stimulating discussions with Dr. H. C. Chan, Professor C. Au, Professor J. E. Hogarth, Professor S. A. Lamb, Professor P. Morrison, and Professor H. Nicolai. The author wishes to thank the referees for valuable comments and pointing out useful literature; and Ms. P. Ma for the French abstract. The author is indebted to Mr. David P. Chan and Mr. Richard C. Y. Hui for their supports and hospitality while in Hong Kong. This publication is supported by Innotec Design, Inc., U.S.A.

ENDNOTES

1) Some authors prefer, different from Einstein, to define K = 8( (c-4 [55]. Then, the four velocity u( would be defined as cdx(/ds, where ds2 = g(( dx(dx( such that equation (1) remains the same.

2) The time-tested assumption that phenomena can be explained in terms of identifiable causes is called the principle of causality. This is the basis of relevance for all scientific investigations. The principle of causality implies that any parameter in a physical solution must be related to some physical causes.

3) This explicit reinterpretation of Einstein‘s equivalence principle (based on Pauli‘s misinterpretation that Einstein objected [57]) as just the signature of Lorentz metric was advocated by Synge [58] earlier and Friedman9) currently. Recently, it has been proven that such a reduction is inconsistent with Einstein‘s own interpretation and physical principles [11,35,57] as well as in disagreement with experiments including the Michelson-Morley experiment [59]. However, the advocates disregard all these inconsistencies because, owing to their inadequate understanding of physics at the fundamental level, they believe that a coordinate system (including its metric) has no physical meaning [60]. (Moreover, following the step of Fock [61], Ohanian, and Ruffini openly declared in their book [55], which is endorsed by Wheeler, that both of Einstein‘s equivalence principle and the principle of relativity are invalid.) Nevertheless, this seemingly exceedingly ingenious defense collapses because the observed gravitational red shifts unequivocally imply that their interpretation is invalid in physics.

4) A dynamic metric solution in gravity is related to the dynamics of its source matter. A dynamic source, according to relativity, would generate gravitational radiation [1]. For the perihelion of Mercury and the deflection of light, the metric is a static solution although solutions of the test particles are calculated. It was believed that the influence of a test particle to the metric could also be calculated with (1). However, as suspected by Gullstrand [40,41] and conjectured by Hogarth6) [31], the truth is the opposite.

5) K. Kuchar [62] claimed to have proved that the initial condition of Einstein‘s equation (1) can be approximated by the initial condition of the linear equation (3) by using a power series expansion. Note, however, that the only valid case of such a power series expansion is a non-dynamic solution (see Sections 2-4). Thus, he has proven only that the properties are true in an unintended void set. Such a basic mistake is essentially repeated 20 years later by Christodoulou and Klainerman [27] for claiming the existence of bounded radiative solutions (see Section 6). Nevertheless, the Editorial Board of Quantum and Classical Gravity [63], unlike the book review [64] and the Editor of GRG [65], considered these invalid claims as proofs. Moreover, a solution relating to a dynamic source by an equation alone, as suggested by Klainerman and Nicolò [66], is insufficient because such a solution may still violate other physical requirements (see Section 5).

6) Hogarth conjectured that, for an exact solution of the two-particle problem, the energy-momentum tensor did not vanish in the surrounding space and would represent the energy of gravitational radiation.

7) The possibility of having an anti-gravity coupling was formally mentioned by Pauli [12]. In a different way, such a possibility was actually first mentioned by Einstein [67] in 1921. He wrote in Geometry and Experience, But, if the universe is finite, there is a second deviation from Newtonian theory, which, in the language of Newtonian theory, may be expressed thus: the gravitational field is such as if it were produced, not only by the ponderable masses, but in addition by a mass-density of negative sign, distributed uniformly through out space. He also firmly believed in such a possibility. However, it was not recognized that an anti-gravity coupling is crucial for a dynamic solution [9,13]. On the other hand, Hawking and Penrose [6,17] had implicitly assumed, in their singularity theorems, the impossibility of an anti-gravity coupling. A rather common erroneous ground to reject the existence an antigravity coupling is due to a misinterpretation of the equivalence of mass and energy in the energy-mass conservation law E = mc2 [68]. For instance, Fock [61] claimed, We saw that to any energy E one should ascribe a mass m = E/c2 and to every mass one should ascribe an energy E = mc2. However, this is inconsistent with general relativity with a tensor field. According to Einstein [69], only the latter is valid. Einstein stated, Now we can reverse the relation and say that an increase of E in the amount of energy must be accompanied by an increase of E/c2 in the mass. I can easily supply energy to the mass - for instance, if I heat it by ten degrees. He also wrote For a mass increase to be measurable, the change of energy per mass unit must be enormously large. The key word in Einstein‘s statements is increase. Thus, E/c2 is related to an increment of mass to massive matter. However, this does not mean that in general any kind of energy E has a related mass E/c2, as Fock claimed. He also remarked, Also, the law permits us to calculate in advance, from precisely determined atomic weights, just how much energy will be released with any atomic disintegration we have in mind. The law says nothing, of course, as to whether-or how - the disintegration reaction can be brought about.

8) A traditional viewpoint of the Physics Department of MIT is that general relativity must be understood in terms of physics [8].

9) John L. Friedman, Divisional Associate Editor of Phys. Rev. Letts., officially claims The existence of local Minkowski space has replaced the equivalence principle that initially motivated it. Note that he also clarifies that the theory of the Wheeler-Hawking [5,6] school is not really general relativity by using the word replaced (Feb. 17, 2000).

10) Max Planck once remarked, A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. Fortunately, it seems, mathematics is an exception to his rule.

11) Einstein [70] once remarked, If you want to find out anything from the theoretical physicists about the methods they use, I advise you stick to one principle, don‘t listen to their words, fix your attention on their deeds.

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Résumé

Il est démontré que l‘équation 1915 d‘Einstein est incompatible avec la notion physique où une onde emporte la vitesse à l‘énergie. Cette preuve est compatible avec l‘approximation Maxwell-Newton, qui formule que l‘équation du champ linéaire pour la gravité faible, est soutenue par les expériences binaires pulsars. Pour les problèmes dynamiques, l‘équation du champ linéaire est indépendante on plus d‘être incompatible avec l‘équation d‘Einstein. L‘équation linéaire, comme première approximation, nécessite l‘existence de l‘onde gravitationnelle faible. Il faut qu‘elle soit liée dans l‘amplitude et soit en rapport avec les dynamiques de la source de la radiation. En raison de la négligence aux associations physiques importantes, et une compréhension insuffisante du principe d‘équivalence, les solutions non physiques auraient causées des interprétations erronées des ondes gravitationnelles. Théoriquement, et tel que suggéré par Einstein et Rosen, il est conclu que la solution concernant une onde physique gravitationnelle pour l‘équation 1915 n‘existe pas. Cette conclusion est soutenue par les analyses des ondes plates contre les solutions ondes exactes. De plus, les conceptions de la radiation des pulsars binaires de Damour et de Taylor seraient valides seulement si elles sont une approximation de l‘équation mise à jour de 1995. L‘équation mise à jour montre que les théorèmes singuliers peuvent seulement l‘analyse des théories de Wheeler-Hawking, mais pas la relativité générale. De plus, il est à remarquer que quelques attestations de Lorentz sont parmi celles qui ne sont pas d‘accord avec des faits expérimentaux bien connus.

Subj: Re: CQG/104867/PAP and Comments on Christodoulou & Kla

Date: 10/6/01 2:13:33 PM Eastern Daylight Time

From: Chungylo

To: cqg@ioppublishing.co.uk, jehogarth@home.com, DZG@vega.bac.pku.edu.cn, philmorr@MIT.EDU, qmy@wuhan.cngb.com, seri@math.princeton.edu, demetri@math.princeton.edu, jawheeler@pupgg.princeton.edu, joe_wisnovsky@pupress.princeton.edu, beiglboeck@springer.de

File: C&K.ZIP (96253 bytes)

Debra Wills

Publishing Administrator

Classical and Quantum Gravity

E-mail: cqg@ioppublishing.co.uk

Dear Ms. Wills:

I have informed you two years ago that your board report will be responded in my published papers. Now, the two papers concerned your board report have been published. They are : C. Y. Lo, Physics Essays, 13 (1), 109-120 (March 2000); and C. Y. Lo, Physics Essays, 13 (4), 527-539 (Dec. 2000).

In the first paper, I show that the solutions constructed by Christodoulou and Klainerman are proven to be physically incorrect. Mathematically, their claims are invalid simply because the proof is incomplete. Your Journal is referred to in reference [44]. The second paper pointed out that the claims of Kuchar are also incorrect, and the proof is invalid because it is not applicable to a dynamical case. Also, the more recent paper of Klainerman and Nicolo published in your journal is also commented. My over all criticism to your journal on these issues are (in endnote [5]) that your journal considered these invalid claims as proofs.

I assume that your board would be interested in reading my papers. For your convenience, the electronic files of these two papers are attached. Any comments the board of your journal may have will be greatly appreciated. Thank you.

Sincerely yours,

C. Y. Lo