Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential φ would be c0
(1 + φ/c0
²), where c0
is the nominal speed of light in the absence of gravity. In geometrical units we define c0
= 1, so Einstein's 1911 formula can be written simply as c = 1 + φ. However, this formula for the speed of light (not to mention this whole approach to gravity) turned out to be incorrect, as Einstein realized during the years leading up to 1915 and the completion of the general theory. In the general theory of relativity the speed of light in a gravitational field cannot be represented by a simple scalar field of c values. Instead, the "speed of light" at a each point depends on the direction of the light ray through that point
– and also on the choice of coordinate systems – so we can't generally talk about the value of c at a given point in a non-vanishing gravitational field
. However, if we consider just radial light rays near a spherically symmetrical (and non- rotating) mass, and if we agree to use a specific set of coordinates, namely those in which the metric coefficients are independent of t, then we can read a formula analogous to Einstein's 1911 formula directly from the Schwarzschild metric. The result differs from the 1911 formula by a factor of 2...
Now that we have derived the Schwarzschild metric, we can easily correct the "speed of light" formula that Einstein gave in 1911. A ray of light always travels along a null trajectory, i.e., with dt = 0, and for a radial ray we have dθ and dπ both equal to zero, so the equation for the light ray trajectory through spacetime, in Schwarzschild coordinates (which are the only spherically symmetrical ones in which the metric is independent of t) is simply:
from which we get:
where the ± sign just indicates that the light can be going radially inward or outward. (Note that we're using geometric units, so c = 1.) In the Newtonian limit the classical gravitational potential at a distance r from mass m is φ = -m/r, so if we let cr
= dr/dt denote the radial speed of light in Schwarzschild coordinates, we have:
which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term. Thus, as φ becomes increasingly negative (i.e., as the magnitude of the potential increases), the radial "speed of light" cr
defined in terms of the Schwarzschild parameters t and r is reduced to less than the nominal value of c. The factor of 2 relative to the equation of 1911 arises because in the full theory there is gravitational length contraction as well as time dilation. Of course, the length contraction doesn’t affect the gravitational redshift, which is purely a function of the time dilation, so the redshift prediction of 1911 remains valid. Only the radial speed of light (in terms of Schwarzschild coordinates) is changed.
On the other hand, if we define the tangential speed of light at a distance r from a gravitating mass center in the equatorial plane (θ = π/2) in terms of the Schwarzschild coordinates as ct
= r(dθ/dt), then the metric divided by (dt)² immediately gives:Thus, we again find that the "velocity of light" is reduced in a region with a strong gravitational field, but this speed is the square root of the radial speed at the same point,
and to the first order in m/r this is the same as Einstein's 1911 formula, although it is understood now to signify just the tangential speed. This illustrates the fact that the general theory doesn't lead to a simple scalar field of c values. The effects of gravitation can only be accurately represented by a tensor field.