Phase velocity and light bending in a gravitational potential

Abstract

In this paper we review the derivation of the light bending equation, obtained before the discovery of general relativity (GR). This is intended for students learning about GR, and for specialists who will shed new light on, and make new connections between, these historic derivations. Since 1915 it has been well known that the observed bending of light has two contributory factors: the first is directly deduced from the equivalence principle alone and was obtained by Einstein in 1911; the second comes from the spatial curvature of spacetime. In GR, these two components are equal, but other relativistic theories of gravitation can give different values to those contributions. In this paper we give a simple explanation, based on the wave-particle picture, of why the first term, which relies on the equivalence principle, is identical to the one obtained by a purely Newtonian analysis. In this context of wave analysis, we emphasize that the dependency of the velocity of light on the gravitational potential, as deduced by Einstein, concerns the phase velocity. Then we wonder whether Einstein could have envisaged already, in 1911, the second contribution, and therefore the correct result. We argue that considering a length contraction in the radial direction, along with the time dilation implied by the equivalence principle, could have led Einstein to the correct result.