Speed of Light in a Gravitational Field

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Speed of Light in a Uniform Gravitational field

The metric in a uniform gravitational field is, according to the equivalence principle, identical to the metric in a uniformly accelerating frame of reference [1]

where c is the speed of like in a vacuum in a Minkowski frame of reference. Since light moves on null geodesics we set ds² = 0

Divide through by dt2 to obtain

Now substitute v_x  º dx/dt, v_y º dy/dt, v_z  º dz/dt, v2 º v_x ² + v_y² + v_z ²

Let F = gz. Note that we choose the arbitrary additive constant that is usually associated with a Newtonian potential, to be zero. Our final result is obtained by taking the square root of both sides of this expression

This is exactly the result obtained by Einstein in 1907 [2]. The speed of light in Eq. (5) is known as the coordinate speed of light. Eq. (4) states that as light rises in a uniform gravitational the coordinate speed will increase. If the light travels in the opposite direction the coordinate speed of light will decrease.

Speed of Light in a Schwarzschild Gravitational Field

The gravitational field of a spherical body is described by the Schwarzschild metric

As above we set ds² = 0 in Eq. (1)

Divide through by dt² , and substitute vr  º dr/dt, v_q º dq/dt, v_f º df/dt

If the light is radial, i.e. v_q = v_f = 0, then Eq. (7) simplifies to

If we now take the square root of both sides of Eq. (9) then we have the speed of light that is traveling radially in a Schwarzschild geometry

Therefore as light gets closer to the center of the gravitating object the coordinate speed of light slows down.

References;

[1] The Principle of Equivalence, F. Rohrlich, Annals of Physics, 22, 160-191 (1963).
[2] The Principle of Relativity, Lorentz, Einstein, Minkowski, Weyl, Dover Pub. See article which starts on page 99, i.e. On the Influence of Gravitation on the Propagation of Light.

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