Geodesic motion around a distorted static black hole

TL;DR

This paper investigates how external quadrupole distortions affect geodesic motion around a Schwarzschild black hole, revealing conditions for stable orbits and how distortions can stabilize particle and light trajectories.

Contribution

It provides a detailed analysis of equatorial geodesics in a distorted Schwarzschild black hole, focusing on the effects of quadrupole moments on orbit stability and ISCO locations.

Findings

Finite stable orbits exist only within specific quadrupole moment ranges.

Distortions with negative small quadrupole moments stabilize orbits closer to the horizon.

A null ISCO exists at a specific negative quadrupole moment.

Abstract

In this paper we study geodesic motion around a distorted Schwarzschild black hole. We consider both timelike and null geodesics which are confined to the black hole's equatorial plane. Such geodesics generically exist if the distortion field has only even interior multipole moments and so the field is symmetric with respect to the equatorial plane. We specialize to the case of distortions defined by a quadrupole Weyl moment. An analysis of the effective potential for equatorial timelike geodesics shows that finite stable orbits outside the black hole are possible only for $q\in(q_{\text{min}}, q_{\text{max}}]$, where $q_{\text{min}}\approx-0.0210$ and $q_{\text{max}}\approx2.7086\times10^{-4}$, while for null equatorial geodesics a finite stable orbit outside the black hole is possible only for $q\in[q_{\text{min}},0)$. Moreover, the innermost stable circular orbits (ISCOs) are closer to the distorted black hole horizon than those of an undistorted Schwarzschild black hole for $q\in(q_{\text{min}},0)$ and a null ISCO exists for $q=q_{\text{min}}$. These results shows that an external distortion of a negative and sufficiently small quadrupole moment tends to stabilize motion of massive particles and light.