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Speed of Light in a Uniform Field
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The metric in a uniform gravitational field is, by the equivalence principle, identical to the metric in a accelerated frame
[Eq. 1] ds2 = -c2(1 + gz/c2)2dt2 + dx2 + dy2 + dz2
where c is the speed of like in a vacuum in a Minkowski frame of reference.
Since light moves on null geodesice, i.e. we set ds2 = 0 in [Eq. 1]
[Eq. 2] -c2(1 + gz/c2)2dt2 + dx2 + dy2 + dz2 = 0
Divide through by dt2, and substitute vx = dx/dt, vy = dy/dt, vz = dz/dt, v2 = vx2 + vy2 + vz2
-c2(1 + gz/c2)2 + dx2/dt2 + dy2/dt2+ dz2/dt2 = -c2(1 + gz/c2)2dt2 + (dx/dt)2 + (dy/dt)2+ (dz/dt)2 = 0
-c2(1 + gz/c2)2 + vx2 + vy2 + vz2 = -c2(1 + gz/c2)2 + v2 = 0
v2 = c2(1 + gz/c2)2
[Eq. 3] v = (1 + gz/c2)c
Let F = gz. Note that we choose the arbitrary constant, that is usually associated with a Newtonian potential, to be zero.
[Eq. 4] v = (1 + F/c2)c
This is exactly the result obtained by Einstein in 1907 [1]
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References
[1] On the Relativity Principle and the Conclusions Drawn from It, Albert Einstein, Jahrbuch der Radioaktivitat und Electronik 4 (1907)