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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() < C12 (22)
for some constant C1. 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. |
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