ON INCOMPATIBILITY OF GRAVITATIONAL RADIATION
WITH THE 1915 EINSTEIN EQUATION
理学论文 >> 物理学 >> 论文正文?ON INCOMPATIBILITY
OF GRAVITATIONAL RADIATION WITH THE 1915 EINSTEIN
EQUATION
ON INCOMPATIBILITY OF GRAVITATIONAL RADIATION
WITH THE 1915 EINSTEIN EQUATION
作者；潇霖 发布时间；2003-3-18
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ON INCOMPATIBILITY OF GRAVITATIONAL RADIATION
WITH THE 1915 EINSTEIN EQUATION
Applied and Pure Research Institute
17 Newcastle Drive, Nashua, NH 03060
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 ?(((? 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() ( 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 g