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

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.Comment: 12 pages, 11 figures, text added to the figures along the abscissa and ordinate axes, typing errors corrected, some new text added in the main text also. Accepted for publication in Physical Review