Quantifying closeness between black hole spacetimes: a superspace approach

TL;DR

This paper develops a geometric framework using superspace to quantify how similar or different black hole spacetimes are, enabling systematic comparison of various black hole solutions and their deviations from the Kerr metric.

Contribution

It introduces a superspace construction for stationary black holes with multiple hairs, providing a new method to measure geometric differences between black hole metrics.

Findings

Strengthens the relevance of certain deviation parameters for astrophysical observations.

Provides a systematic way to quantify non-Kerr black hole geometries.

Demonstrates the application of superspace distances to black hole metrics.

Abstract

The set of all metrics that can be placed on a given manifold defines an infinite-dimensional `superspace' that can itself be imbued with the structure of a Riemannian manifold. Geodesic distances between points on Met$(M)$ measure how close two different metrics over $M$ are to one another. Restricting our attention to only those metrics that describe physical black holes, these distances may therefore be thought of as measuring the level of geometric similarity between different black hole structures. This allows for a systematic quantification of the extent to which a black hole, possibly arising as an exact solution to a theory of gravity extending general relativity in some way, might be `non-Kerr'. In this paper, a detailed construction of a superspace for stationary black holes with an arbitrary number of hairs is carried out. As an example application, we are able to strengthen a recent claim made by Konoplya and Zhidenko about which deviation parameters describing a hypothetical, non-Schwarzschild black hole are likely to be most relevant for astrophysical observables.