;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
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