Gravitational perturbation induced by a rotating ring around a Kerr black hole

TL;DR

This paper analyzes the linear gravitational perturbation of a Kerr black hole caused by a rotating ring, extending previous Schwarzschild results using the Teukolsky formalism and Hertz potential methods.

Contribution

It presents a detailed computation of Kerr black hole perturbations induced by a rotating ring, incorporating lower multipole modes and gauge adjustments for a smooth gravitational field.

Findings

Perturbed Weyl scalars $\psi_0$ and $\psi_4$ obtained from Teukolsky equation.

Perturbed metric constructed in a radiation gauge using Hertz potential.

Resulting gravitational field is smooth outside the ring except on the equatorial plane.

Abstract

The linear perturbation of a Kerr black hole induced by a rotating massive circular ring is discussed by using the formalism by Teukolsky, Chrzanowski, Cohen and Kegeles. In these formalism, the perturbed Weyl scalars, $\psi_0$ and $\psi_4$, are first obtained from the Teukolsky equation. The perturbed metric is obtained in a radiation gauge via the Hertz potential. The computation can be done in the same way as in our previous paper, in which we considered the perturbation of a Schwarzschild black hole induced by a rotating ring. By adding lower multipole modes such as mass and angular momentum perturbation which are not computed by the Teukolsky equation, and by appropriately setting the parameters which are related to the gauge freedom, we obtain the perturbed gravitational field which is smooth except on the equatorial plane outside the ring.