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2) The expansion of a bounded dynamic solution g(( for an isolated weak gravitational source:
According (3), a first-order approximation of metric g(((x, y, z, t') is bounded and almost periodic since T(( is. Physically, the equivalence principle requires g(( to be bounded [11], and the principle of causality requires g(( to be almost periodic in time since the motion of a source particle is. Such a metric g(( is asymptotically flat for a large distance r, and the expansion of a bounded dynamic solution is:
g(((nx, ny, nz, r, t') = ((( +(((k)(nx, ny, nz, t')/rk, where n( = x(/r. (4)
3) The non-existence of dynamic solutions:
It follows expansion (4) that the non-zero time average of G(1)(t would be of O(1/r3) due to
((n( = (((( + n( n()/r, (5)
since the term of O(1/r2), being a sum of derivatives with respect to t', can have a zero time-average. If G(2)(t is of O(K2/r2) and has a nonzero time-average, consistency can be achieved only if another term of time-average O(K2/r2) at vacuum be added to the source of (1). Note that there is no plane-wave solution for (1') [9,18].
It will be shown by contradiction that there is no dynamic solution for (1) with a massive source. Let us define
((( = ((1)(( + ((2)(( ; (i)(( = ((i)(( - ((( (((i)cd (cd), where i = 1, 2 ;
and
(((((1)(( = - K T(m)(( . (6)
Then (1)(( is of a first-order; and ((2)(( is finite. On the other hand, from (1), one has
(((((2)(( + H(1)(( + G(2)(( = 0 . (7)
Note that, for a dynamic case, equation (7) may not be satisfied. If (6) is a first-order approximation, G(2)(( has a nonzero time-average of O(K2/r2) (8(; and thus (2)(( cannot have a solution.
However, if (2)(( is also of the first-order of K, one cannot estimate G(2)(( by assuming that (1)(( provides a first-order approximation. For example, (6) does not provide the first approximation for the static Schwarzschild solution, although it can be transformed to a form such that (6) provides a first-order approximation [11(. According to (7), (2)(( will be a second order term if the sum H(1)(( is of second order. From (2c), this would require (((( being of second order. For weak gravity, it is known that a coordinate transformation would turn (((( to a second order term (can be zero) (8,14,17(. (Eq. [7] implies that (c(c(2)(( - (c((((c + (((c( would be of second order) Thus, it is always possible to turn (6) to become an equation for a first-order approximation for weak gravity.
From the viewpoint of physics, since it has been proven that (3) necessarily gives a first-order approximation [11], a failure of such a coordinate transformation means only that such a solution is not valid in physics. Moreover, for the dynamic of massive matter, experiment [16] supports the fact that Maxwell-Newton Approximation (3) is related to a dynamic solution of weak gravity [13]. Otherwise, not only is Einstein's radiation formula not valid, but the theoretical framework of general relativity, including the notion of the plane-wave as an idealization, should be re-examined (see Section 3). In other words, theoretical considerations in physics as well as experiments eliminate other unverified speculations thought to be possible since 1957. |
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