Schwarzschild black hole encircled by a rotating thin disc: Properties of perturbative solution

TL;DR

This paper develops a practical, closed-form perturbative solution for a Schwarzschild black hole encircled by a rotating thin disc, analyzing how the disc's gravity affects the black hole's horizon and orbital properties.

Contribution

It provides a closed-form Green function approach for perturbations caused by extended sources, enabling detailed analysis of black hole-disc systems beyond previous methods.

Findings

Disc gravity influences black hole horizon geometry

No ergosphere appears at linear order

Orbital radii are affected by the disc's gravitational field

Abstract

In ApJ (1974), Will solved the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring, using Green's functions expressed as infinite-sum expansions in multipoles and in the small mass and rotational parameters. In a previous paper (ApJS 2017, Paper I), we expressed the Green functions in closed form containing elliptic integrals, leaving just summation over the mass expansion. Such a form is more practical for numerical evaluation, but mainly for generalizing the problem to extended sources where the Green functions have to be integrated over the source. We exemplified the method by computing explicitly the first-order perturbation due to a slowly rotating thin disc lying between two finite radii. After finding basic parameters of the system -- mass and angular momentum of the black hole and of the disc -- we now add further properties, namely those which reveal how the disc gravity influences geometry of the black-hole horizon and those of circular equatorial geodesics (specifically, radii of the photon, marginally bound and marginally stable orbits). We also realize that, in the linear order, no ergosphere occurs and the central singularity remains point-like, and check the implications of natural physical requirements (energy conditions and subluminal restriction on orbital speed) for the single-stream as well as counter-rotating double-stream interpretations of the disc.