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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 < constant. Note that an unbounded function of u, f(u) grows anomaly large as time ( goes by.
It should be noted also that metric (8) is only a plane, but not a periodic function because a smooth periodic function must be bounded. This unboundedness is a symptom of unphysical solutions because they cannot be related to a dynamic source (see also [9,11]). Note that solution (8) can be used to construct a smooth one-parameter family of solutions [11] although solution (8) is incompatible with Einstein's notion of weak gravity [2].
In 1953, questions were raised by Schiedigger [30] as to whether gravitational radiation has any well-defined existence. 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]. |
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