Optical Approach to Gravitational Redshift

Abstract

An optical approach begins by interpreting the gravitational redshift resulting to a change in the relative velocity of light due to the medium of propagation in the gravitational field. The discussion continues by pointing out an agreement in structure between the equation for rays in geometrical optics and the geodesic equation of general relativity. From their comparison we learn that the path of rays should be given by the relation $ds^2=n^2(r)dr^2+r^2d\theta^2$, not by $ds^2=dr^2+r^2d\theta^2$, in a medium with spherical symmetry of refractive index $n(r)$. The development of an optical analogy suggests introducing $n^2(r)$ in place of $g_{rr}$ as an optical version of the Schwarzschild metric. In form and content, $n^2(r)$ is different from $g_{rr}$. The optical point of view replaces the general-relativity explanations in terms of time and gravitation.