(4a) should be valid for dynamic problems. Also, an implicit assumption of Einstein calculation is that the gravitational effects due to the light itself, is negligible. To address this issue theoretically, would be complicated and is beyond the scope of this paper [9]. Here, this negligibility is justified from the viewpoint of practical observations only.
Now, according to the geodesic eq. (2), one has d2x /ds2 = 0 for x( (= x, y, z) since (gtt/(x( ( 0. Thus, the gravitational force is non-zero, and the equivalence principle would be applicable. (For the non-applicable cases, please see 壯5-7.) Consider an observer P at (x0, y0, z0, t0) in a ree falling" state,
dx/ds = dy/ds = dz/ds = 0. (5)
According to the equivalence principle and eq. (1), state (5) implies the time dt and dT are related by
c2(1 - )dt2 = ds2 = c2dT2 (6)
since the local coordinate system is attached to the observer P (i.e., dX = dY = dZ = 0 in eq. [1]). This is the time dilation of metric (4b). Eq. (6) shows that the gravitational red shifts are related to gtt, and is compatible with his 1911 derivation [2]. Moreover, since the space coordinates are orthogonal to dt, at (x0, y0, z0, t0), for the same ds2, eq. (6) implies [3]
(1 + )(dx2 + dy2 + dz2) = dX2 + dY2 + dZ2 . (7)
On the other hand, the law of the propagation of light is characterized by the light-cone condition,
ds2 = 0. (8)
Then, to the first order approximation, the velocity of light is expressed in our selected coordinates S by
= c(1 - ). (9)
It is crucial to note that the light speed (9), for an observer P1 attached to the system S at (x0, y0, z0), is smaller than c; and this condition is required by the coordinate relativistic causality for a physically 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)
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