Gravitostatic black hole Love numbers from Membrane Paradigm at large $D$

TL;DR

This paper calculates the static gravitational response (Love numbers) of black holes in large dimensions using the Membrane Paradigm, providing a foundation for future frequency-dependent analyses.

Contribution

It introduces a method to compute black hole Love numbers at large D using the Membrane Paradigm, extending previous static results to dynamic cases.

Findings

Computed gravitostatic Love numbers at large D

Established a framework for frequency-dependent Love number calculations

Opened pathways for extending Love number computations to all frequencies

Abstract

Black holes, like any other object - respond to external fields like background electric and gravitational fields. At lowest order, the response of a black hole to, (e.g.) a gravitational wave in the $l^{th}$ spherical harmonic mode is to develop a generalized `quadrupole moment' in the same symmetry channel. The induced moment of the black hole is proportional to the external forcing field. The constant of proportionality between the field and the response is a function of the frequency $\omega$ of the incident perturbation. These constants of proportionality are sometimes called `Love' numbers in the study of astrophysical black holes. Using the `Large $D$ Membrane Paradigm' formalism, we compute gravitostatic black hole `Love' numbers (at $\omega \to 0$ limit), in large $D$. This work opens the road of using Large $D$ Membrane Paradigm formalism to extend the computation of these black hole `Love' numbers for all values of $\omega$ in large $D$ - which we announce in this paper.