Axial tidal Love numbers of black holes in matter environments

TL;DR

This paper investigates the axial tidal Love numbers of Schwarzschild black holes surrounded by matter, using analytic and numerical methods, emphasizing the impact of matter distribution on tidal responses.

Contribution

It introduces a general framework for calculating tidal Love numbers in matter environments, including anisotropic fluids, and explores the effects of density profiles lacking compact support.

Findings

Density profiles without compact support produce logarithmic terms in perturbations.

The asymptotic structure of matter affects the definition of tidal observables.

Analytic and numerical methods agree on the influence of matter on Love numbers.

Abstract

We study the axial (magnetic) tidal Love numbers of a Schwarzschild black hole surrounded by a spherically symmetric matter distribution. While the formalism developed here is general, we specialize to the case of anisotropic fluids as a proxy for dark matter distributions, computing the Love numbers for different density profiles of astrophysical interest. We employ two complementary methods: a small-compactness expansion, yielding closed-form analytic expressions, and direct numerical integration of the perturbation equations. We discuss the connection between different formulations of the fluid perturbations and the resulting Love numbers. We further show that density profiles lacking compact support generically produce logarithmic terms in the asymptotic expansion of the perturbation variable, which obstruct the standard tidal matching procedure and whose origin we trace to the absence of a strictly vacuum exterior. Our findings highlight the importance of controlling the asymptotic structure of the matter distribution when defining tidal observables for black holes dressed by matter, and provide a general framework that can be applied to other spherically symmetric environments.