Multipole moments of bumpy black holes

TL;DR

This paper develops a method to compute and relate the multipole moments of bumpy black holes to their deviations from the Kerr solution, enabling tests of the no-hair theorem through gravitational measurements.

Contribution

It provides a clear mapping between bumpy black hole deviations and Geroch-Hansen moments, including both mass and current moments, extending previous work.

Findings

Derived a method to compute Geroch-Hansen moments for bumpy black holes.

Established a direct link between bumpy black hole parameters and multipole moments.

Extended the formalism to include deviations in current moments.

Abstract

General relativity predicts the existence of black holes, compact objects whose spacetimes depend on only their mass, spin, and charge in vacuum (the "no hair" theorem). As various observations probe deeper into the strong fields of black hole candidates, it is becoming possible to test this prediction. Previous work suggested that such tests can be performed by measuring whether the multipolar structure of black hole candidates has the form that general relativity demands, and introduced a family of "bumpy black hole" spacetimes to be used for making these measurements. These spacetimes have generalized multipoles, where the deviation from the Kerr metric depends on the spacetime's "bumpiness." In this paper, we show how to compute the Geroch-Hansen moments of a bumpy black hole, demonstrating that there is a clean mapping between the deviations used in the bumpy black hole formalism and the Geroch-Hansen moments. We also extend our previous results to define bumpy black holes whose {\it current} moments, analogous to magnetic moments of electrodynamics, deviate from the canonical Kerr value.