, 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 r
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