Deflection of light due to spheroidal oblate static objects

TL;DR

This paper theoretically analyzes how the ellipticity of massive objects affects light deflection, providing a series expression that extends previous spherical models and highlights the significance of shape in gravitational lensing.

Contribution

It introduces a converging series expression for light deflection by oblate objects, generalizing the Schwarzschild solution to include ellipticity effects.

Findings

Ellipticity causes a small but potentially significant increase in light deflection.

The derived formula reduces to the Schwarzschild case when ellipticity is zero.

Ellipticity effects can surpass those caused by celestial rotation in certain scenarios.

Abstract

Deflection of light due to massive objects was predicted by Einstein in his General Theory of Relativity. This deflection of light has been calculated by many researchers in past, for spherically symmetric objects. But, in reality, most of these gravitating objects are not spherical instead they are ellipsoidal ( oblate) in shape. The objective of the present work is to study theoretically the effect of this ellipticity on the trajectory of a light ray. Here, we obtain a converging series expression for the deflection of a light ray due to an ellipsoidal gravitating object, characterised by an ellipticity parameter. As a boundary condition, by setting the ellipticity parameter to be equal to zero, we get back the same expression for deflection as due to Schwarzschild object. It is also found that the additional contribution in deflection angle due to this ellipticity though small, but could be typically higher than the similar contribution caused by the rotation of a celestial object. Therefore for a precise estimate of the deflection due to a celestial object, the calculations presented here would be useful.