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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), 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). |
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