Approximate analytical calculations of photon geodesics in the Schwarzschild metric

TL;DR

This paper introduces approximate analytical formulas for photon paths in Schwarzschild spacetime, including light bending, delay, and solid angle, with applications to astrophysics.

Contribution

It presents the first analytical approximation for the solid angle in Schwarzschild spacetime, enhancing previous empirical models.

Findings

The new equations are accurate within specified ranges.

They simplify calculations of photon trajectories near compact objects.

Applications demonstrate practical usefulness in astrophysics.

Abstract

We develop a method for deriving approximate analytical formulae to integrate photon geodesics in a Schwarzschild spacetime. Based on this, we derive the approximate equations for light bending and propagation delay that have been introduced empirically. We then derive for the first time an approximate analytical equation for the solid angle. We discuss the accuracy and range of applicability of the new equations and present a few simple applications of them to known astrophysical problems.