Perturbation of a Schwarzschild black hole due to a rotating thin disc

TL;DR

This paper develops a closed-form method to analyze how a rotating thin disc perturbs a Schwarzschild black hole, improving convergence over previous multipole expansion techniques and enabling detailed physical interpretation.

Contribution

It generalizes Will's perturbation approach to thin discs and introduces a closed-form Green's function method for better numerical convergence.

Findings

Derived well-converging series for gravitational potential and frame dragging.

Applied method to a constant-density disc with zero black hole angular momentum.

Provided physical interpretation of the dragging effect on the black hole.

Abstract

Will (1974) treated the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to the perturbation by a thin disc (which is more relevant astrophysically), but, due to a rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green's functions represented by the Will's result can be expressed in a closed form (without multipole expansion) which is more useful. In particular, they can be integrated out over the source (thin disc in our case), to yield well converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in the first perturbation order, on the simplest case of a constant-density disc, including physical interpretation of the results in terms of a one-component perfect fluid or a two-component dust on circular orbits about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, being just carried along by the dragging effect of the disc.