haracteristics of some physical quantities of an isolated system:
For an isolated system consisting of particles with typical mass, typical separation , and typical velocities , Weinberg (8( estimated, the power radiated at a frequency ( of order /will be of order

P " ((/)624 or P "8/,

since (/is of order 2. The typical deceleration rad of particles in the system owing this energy loss is given by the power P divided by the momentum, or rad "7/. This may be compared with the accelerations computed in Newtonian mechanics, which are of order 2/, and with the post-Newtonian correction of 4/. Since radiation reaction is smaller than the post-Newtonian effects by a factor 3, if (( c, the velocity of light, the neglect of radiation reaction is perfectly justified. This allows us to consider the motion of a particle in an isolated system as almost periodic.
Consider, for instance, two particles of equal mass m with an almost circular orbit in the x-y plane whose origin is the center of the circle (i.e., the orbit of a particle is a circle if radiation are neglected). Thus, the principle of causality [9,10] implies that the metric g(( is weak and very close to the flat metric at distance far from the source and that g(((x, y, z, t') is an almost periodic function of t' (= t - r/c).
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] sup

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