y realizable space-time coordinate system (see §6). Observer P1 shares the same frame of reference with the sun, and the velocity of light is clearly frame-dependent, but restricted.
This difference from c is due to gravity (or the curved space) together with the equivalence principle. The observer P is in a free falling frame of reference and thus would not experience the gravitational force as P1. Note that eq. (9) is consistent with eqs. (6) and (7) which are due to the equivalence principle. A reason for deriving eq. (6) and eq. (7) is that if the metric of a manifold does not satisfy the equivalence principle, ds2 = 0 would lead to an incorrect light velocity (see §5-7). Thus, not only eq. (6), which leads to gravitational red shifts, but also eq. (9) is a test of the equivalence principle.
Einstein [3] wrote, ?e can therefore draw the conclusion from this, that a ray of light passing near a large mass is deflected." Thus, Einstein has demonstrated that the equivalence principle requires that a space-time coordinates system must have a physical meaning; and a space-time coordinate system cannot be just any Gaussian coordinate system. It seems, Einstein [2] chose this calculation method to clarify his statements on the equivalence principle. In many textbooks [12,13,21-23], derivation of the coordinate light speed is circumvented, and the deflection angle is obtained directly. But, such a manipulation has not really achieved a derivation independent of the coordinate system since a particular type is needed to define the angle.
However, although Einstein emphasized the importance of satisfying the equivalence principle, he did not discuss what could go wrong. For instance, if the requirement of asymptotically flat were not used, one could obtain a solution, which does not satisfy the equivalence principle. Another interesting question is whether the equivalence principle is satisfied if ((tt = 0 (( = x, y, z). What has been missing is a discussion on the validity of the geodesic representing a physical free fall. Understandably, such a discussion was not provided since the validity of (4b) can be decided only through observations. This illustrates also that to see whether the equivalence principle is satisfied, one must consider beyond the Einstein equation (see §5).


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.


In Riemannian geometry, it has been proven [12] that the curvature tensor ?(((( is the only tensor that can be constructed from the metric tensor and its first and second derivatives, and is linear in the second derivatives." Einstein identified the Ricci curvature tensor Rab (( R(a(b) as the required