|
|
In physics, the amplitude of a wave is generally related to its energy density and its source. Equation (3) shows that a gravitational wave is bounded and is related to the dynamic of the source. These are useful to prove that (3), as the first-order approximation for a dynamic problem, is incompatible with equation (1). Its existing "wave" solutions are unbounded and therefore cannot be associated with a dynamic source [11]. In other words, there is no evidence for the existence of a physical dynamic solution.
With the Hulse-Taylor binary pulsar experiment [16], it became easier to identify that the problem is in (1). Subsequently, it has been shown that (3), as a first-order approximation, can be derived from physical requirements which lead to general relativity [11]. Thus, (3) is on solid theoretical ground and general relativity remains a viable theory. Note, however, that the proof of the nonexistence of bounded dynamic solutions for (1) is essentially independent of the experimental supports for (3).
To prove this, it is sufficient to consider weak gravity since a physical solution must be compatible with Einstein's [2] notion of weak gravity (i.e., if there were a dynamic solution for a field equation, it should have a dynamic solution for a related weak gravity [11]). To calculate the radiation, consider further,
G(( ( G(1)(( + G(2)(( , where G(1)(( = (c(c(( + H(1)((, (2b)
H(1)(( ( -(c((((c + (((c( + ((((c(dcd , and ?(((? << 1. (2c)
G(2)(( is at least of second order in terms of the metric elements. For an isolated system located near the origin of the space coordinate system, G(2)(t at large r (= (x2 + y2 + z2 (1/2) is of O(K2/r2) (5,8,17(.
One may obtain some general characteristics of a dynamic solution for an isolated system as follows:
1) The characteristics 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). |
|
