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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 interpretati