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4. Derivation of the Maxwell-Newton Approximation for Massive Matter
For massive matter, it has been proven [7] that eq. (4a) is dynamically incompatible with eq. (3). The binary pulsar experiments [31] make it necessary to modify eq. (1) to a 1995 update version,
Gab ( Rab -R gab = -K[T(m)ab - t(g)ab]. (10a)
and
(cT(m)cb = (ct(g)cb = 0, (10b)
where t(g)ab is the gravitational energy-stress tensor. The first order approximation of eq. (10a) is
(c(cab = -KT(m)ab (10c)
Eq. (10c) is called the Maxwell-Newton Approximation [7] and is equivalent to eq. (4a).
The above modification is based on the facts that, as a first order approximation, eq. (10c) is supported by experiments [7,19] and that it is the natural extension from Newtonian theory. However, one may argue that this is not yet entirely satisfactory since it has not been shown rigorously that eq. (10c) is compatible with general relativity. In particular, one might still argue [32] that the wave component in gat (for a = x, y, z, t) as artificially induced by the harmonic gauge.
It will be shown that the Maxwell-Newton Approximation (10c) can be rigorously derived from the equivalence principle and related physical principles that lead to general relativity. Since linear eq. (10c) is supported by experiments, to reaffirm the validity of general relativity, one must show clearly that eq. (10c) is compatible with the theoretical framework of relativity. Thus, such a proof of eq. (10c) not only provides a theoretical foundation for eq. (10) but also reaffirms general relativity.
In general relativity [2] there are three basic assumptions namely: 1) the principle of equivalence; 2) the principle of covariance (as will be shown necessarily be restricted to space-time coordinate systems which are compatible with the equivalence principle.) and 3) the field equation whose source can be modified. Note that eq. (10c) is invariant with respect to the Lorentz transformations. Moreover, eq. (10c) is compatible with the notion of weak gravity. Thus, eq. (10c) as an approximation for a specified coordinate system, is compatible with the requirement of covariance and compatibility with weak gravity. It remains to show that eq. (10c) is derivable from the equivalence principle.
The equivalence principle and the principle of general relativity imply that the geodesic equation (2) is the equation of motion for a neutral particle [2,3]. In comparison with Newton theory, Einstein [2] obtains the gravitational potential,
( " c2g00/2. (11)
Since ( satisfies the Poisson equation (( = 4(((, according to the correspondence principle, one has the field equation, (g00/2 = 4((c-2T00, where T00 "(, the mass density and ( is the coupling constant.
Then, according to special relativity and the Lorentz invariance, one has
(c( cgab = (c( c (ab = -4((c-2((T(m)ab + ((m)(ab(, (12a)
where
( + ( = 1, (m) = (cd T(m)cd , (12b)
T(m)ab is the tensor for massive matter, (ab is the Minkowski metric, and ( and ( are constants. Eq. (12) is a field equation for the first order approximation (as assumed) for weak gravity of moving particles. An implicit gauge condition is that the flat metric (ab is the asymptotic limit at infinity. To have the exact equation, since the left hand side of eq. (12a) does not satisfy the covariance principle, one must search for a tensor whose difference from (c( c (ab/2 is of second order in (c-2. |
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