Ray-tracing in four and higher dimensional black holes: An analytical approximation

TL;DR

This paper derives accurate analytical approximations for null ray bending angles and time delays in higher-dimensional Schwarzschild--Tangherlini black hole spacetimes, facilitating studies of gravitational lensing in multiple dimensions.

Contribution

It provides the first explicit elementary-function approximations for null ray bending and delay in higher-dimensional black holes, with verified high accuracy across parameter space.

Findings

Analytic expressions closely match numerical results.

Approximation errors are minimal across all tested parameters.

Results applicable to gravitational lensing in higher dimensions.

Abstract

We study null rays propagation in a spacetime of static Schwarzschild--Tangherlini black holes in arbitrary number of dimensions. We focus on the bending angle and the retarded time delay for rays emitted in the vicinity of a black hole and propagating to the infinity. We obtain an analytic expression in terms of elementary functions which approximate the bending angle and time delay in these spacetimes with high accuracy. We analyze the relative error of the developed analytic approximations and show that it is quite small in the complete domain of the parameter space for the rays reaching the infinity and for different number of the spacetime dimensions. Possible applications of the obtained results are briefly discussed.