Black Holes Lessons from Multipole Ratios

TL;DR

This paper develops methods to calculate multipole moments of various black holes, revealing invariant ratios that constrain deviations from Kerr geometry, with implications for gravitational wave observations.

Contribution

It introduces a detailed calculation framework for gravitational multipoles of general black holes and microstate geometries, identifying finite ratios that remain invariant under deformations.

Findings

Many multipoles vanish for Kerr and BPS black holes.

Deformation of black holes reveals finite ratios of multipoles.

Ratios computed via different methods agree for certain black holes.

Abstract

We explain in detail how to calculate the gravitational mass and angular momentum multipoles of the most general non-extremal four-dimensional black hole with four magnetic and four electric charges. We also calculate these multipoles for generic supersymmetric four-dimensional microstate geometries and multi-center solutions. Both for Kerr black holes and BPS black holes many of these multipoles vanish. However, if one embeds these black holes in String Theory and slightly deforms them, one can calculate an infinite set of ratios of vanishing multipoles which remain finite as the deformation is taken away, and whose values are independent of the direction of deformation. For supersymmetric black holes, we can also compute these ratios by taking the scaling limit of multi-center solutions, and for certain black holes the ratios computed using the two methods agree spectacularly. For the Kerr black hole, these ratios pose strong constraints on the parameterization of possible deviations away from the Kerr geometry that should be tested by future gravitational wave interferometers.