Phase velocity and light bending in a gravitational potential

TL;DR

This paper reviews the historical derivation of light bending in gravitational fields, highlighting the roles of the equivalence principle and spacetime curvature, and offers new insights into Einstein's original reasoning.

Contribution

It provides a simplified wave-based explanation of light bending, connecting historical derivations with modern understanding and exploring Einstein's potential insights in 1911.

Findings

The equivalence principle accounts for the Newtonian part of light bending.

Spacetime curvature contributes equally to the total light deflection in GR.

Phase velocity of light varies with gravitational potential, affecting light bending calculations.

Abstract

In this paper we review the derivation of light bending obtained before the discovery of General Relativity (GR). It is intended for students learning GR or specialist that will find new lights and connexions on these historic derivations. Since 1915, it is well known that the observed light bending stems from two contributions : the first one is directly deduced from the equivalence principle alone and was obtained by Einstein in 1911; the second one comes from the spatial curvature of spacetime. In GR, those 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 with 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.