Probing the anisotropic velocity of light in a gravitational field: another test of general relativity

A corollary of general relativity that the velocity of light is anisotropic in a gravitational field has received little attention so far. It is shown that this anisotropy can be directly probed by an experiment initially proposed for testing the one-way velocity of light. The proposed experiment constitutes another test of general relativity since the anisotropy in the propagation of light in a gravitational field is not a new effect but follows from general relativity. This experiment is also an indirect test of the local constancy of the velocity of light which is a key concept of the standard interpretation of general relativity. PACS numbers: 04.80.Cc, 04.20.Cv Although there exist two clear indications that the propagation of light in the vicinity of massive bodies is anisotropic the bending of light and the zero velocity of light which tries to leave a gravitationally collapsing body (a black hole) the anisotropy in the velocity of light in a gravitational field has been only barely mentioned [1], [2]. If, however, the average velocity of light toward and away from a gravitational center between two points of different gravitational potential is calculated, it becomes clear that the two velocities are not the same. In general, the anisotropy in the propagation of light in a non-inertial frame of reference (accelerating or supported in a gravitational field) can be demonstrated by using a version of the classical thought experiment [3] involving two Einstein elevators one accelerating with an uniform acceleration a and one at rest in a gravitational field of strength g = −a. Consider an elevator accelerating with an acceleration a = |a| which represents a non-inertial (accelerating) reference frame N. A light ray is emitted from a point D on the elevator’s wall (at equal distances h from the floor and the ceiling) and propagates in a direction perpendicular to the elevator’s acceleration toward a point B on the opposite wall (also situated at equal distances h from the floor and the ceiling; the distance between D and B is h too). Due to the accelerated motion of the elevator the ray bends (for an observer in the elevator) and arrives not at the middle point B but at a point B displaced at a distance δ from B toward the floor. This is the original thought experiment considered by Einstein. This is also an experiment that allows an observer in N (in the elevator) to determine from within N that it is not an inertial frame. In order to calculate the anisotropic velocity of light in N, in addition to the horizontal light ray propagating from point D toward point B, let us consider two other light rays propagating vertically along the line of acceleration which are emitted from two points on the wall where point B is situated. The two light rays are emitted simultaneously with the horizontal ray in N from two points A and C separated by a distance 2h, one from point C (at the elevators floor) toward B (in the direction of the acceleration), and the other from point A (at the ceiling) in the opposite direction (also toward B). They will not meet at the middle point B because during the time t = h/c, the light rays travel toward B, N will move at a distance δ = at/2 as measured in the co-moving (instantaneous) inertial reference frame. In N the three rays will meet at a point B which is displaced from the middle point B (in the direction opposite to the acceleration) by the same distance determined in the instantaneous inertial frame: