Effective refractive index tensor for weak field gravity

TL;DR

This paper introduces a tensorial description of gravitational lensing in weak fields, highlighting conditions under which the effective refractive index is isotropic or anisotropic, with formulas for practical computation.

Contribution

It derives simple formulas for the effective refractive index tensor in weak gravitational fields, extending the understanding from isotropic to anisotropic cases.

Findings

Refractive index tensor simplifies to a scalar in isotropic stress conditions.

Anisotropic stresses lead to tensorial, anisotropic refractive indices.

Formulas provided for calculating the tensor in various gravitational scenarios.

Abstract

Gravitational lensing in a weak but otherwise arbitrary gravitational field can be described in terms of a 3 x 3 tensor, the "effective refractive index". If the sources generating the gravitational field all have small internal fluxes, stresses, and pressures, then this tensor is automatically isotropic and the "effective refractive index" is simply a scalar that can be determined in terms of a classic result involving the Newtonian gravitational potential. In contrast if anisotropic stresses are ever important then the gravitational field acts similarly to an anisotropic crystal. We derive simple formulae for the refractive index tensor, and indicate some situations in which this will be important.