Metamorphoses of a photon sphere

TL;DR

This paper investigates how the photon sphere around a Schwarzschild black hole becomes a fractal basin boundary under external quadrupolar distortion, revealing chaotic null geodesic behavior.

Contribution

It introduces the study of fractal basin boundaries and chaos in null geodesics around distorted black holes, extending understanding beyond the standard Schwarzschild case.

Findings

Photon sphere becomes a fractal basin boundary under quadrupolar distortion

Fractal dimension and uncertainty exponent depend on quadrupole moment

Chaotic behavior of null geodesics is demonstrated

Abstract

There are circular planar null geodesics at $r=3M$ around a Schwarzschild black hole of mass $M$. These geodesics form a photon sphere. Null geodesics of the Schwarzschild space-time which do not form the photon sphere are either escape to null infinity or get captured by the black hole. Thus, from the dynamical point of view, the photon sphere represents a smooth basin boundary that separates the basins of escape and capture of the dynamical system governing the null geodesics. Here we consider a Schwarzschild black hole distorted by an external, static, and axisymmetric quadrupolar gravitational field. We study null geodesics around such a black hole and show that the photon sphere transforms into a fractal basin boundary that indicates chaotic behavior of the null geodesics. We calculate the box-counting fractal dimension of the basin boundary and the related uncertainty exponent, which depend on the value of the quadrupole moment.