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For a dynamic4) case, however, this is no longer valid.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.