Addressing issues in defining the Love numbers for black holes

TL;DR

This paper develops an analytic method to compute the tidal response of black holes, clarifies the definitions of Love numbers in relativistic contexts, and explores their values for nonrotating and rotating black holes.

Contribution

It introduces a new analytic approach to calculate black hole tidal responses and compares different definitions of Love numbers in relativistic regimes.

Findings

Nonrotating black holes have zero tidal Love numbers under both definitions.

For rotating black holes, different definitions yield zero or purely imaginary Love numbers.

The method accounts for all linear frequency terms, improving upon previous approaches.

Abstract

We present an analytic method for calculating the tidal response function of a nonrotating and a slowly rotating black hole from the Teukolsky equation in the small-frequency and near horizon limit. We point out that in the relativistic context, there can be two possible definitions of the tidal Love numbers and the dissipative part that arises from the tidal response function. Our results suggest that both of these definitions predict zero tidal Love numbers for a nonrotating black hole. On the other hand, for a slowly rotating black hole in a generic tidal environment, these two definitions of the tidal Love numbers do not coincide. While one procedure suggests zero tidal Love numbers, the other procedure gives purely imaginary tidal Love numbers. As expected, the dissipative terms differ as well. We emphasize that in our analysis, we keep all the terms linear in the frequency, unlike previous works in the literature. Following this, we propose a procedure to calculate the tidal response function -- and hence the Love numbers -- for an arbitrarily rotating black hole.