On the logarithmic Love number of black holes beyond general relativity

TL;DR

This paper derives a formula for calculating the logarithmic Love number of black holes, revealing that modifications to classical solutions generally induce non-zero logarithmic responses, with implications for black hole metrics beyond general relativity.

Contribution

The paper provides a direct formula for the logarithmic Love number based on the perturbation equation, clarifying conditions for its non-zero value in modified black hole spacetimes.

Findings

Any perturbative modification to Schwarzschild or Reissner-Nordström black holes results in non-zero logarithmic Love numbers.

Explicit solutions beyond general relativity can have zero logarithmic Love numbers if perturbativity is not assumed.

The method extends known results for the Hayward metric, demonstrating its broad applicability.

Abstract

Tidal Love numbers and other response coefficients of black holes sometimes exhibit a logarithmic dependence on scale, or 'running'. We clarify that this coefficient is directly calculable from the structure of the equation obeyed by the field perturbation, and requires no knowledge of the full solution. The derived formula allows us to establish some general results on the existence of logarithmic running. In particular, we show that any static and spherically symmetric spacetime that modifies the Schwarzschild or Reissner-Nordstr\"om solutions in a perturbative way must have non-zero logarithmic Love numbers. This applies for instance to all regular black hole metrics. On the other hand, our analysis highlights the importance of the perturbativity assumption: without it, we find explicit black hole solutions beyond general relativity with exactly zero running. We also illustrate the advantage of our method by recovering and extending the known results for the Hayward metric.