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

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 $ω$ 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 $ω\to 0$ 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 $ω$ in large $D$ - which we announce in this paper.