Charging the Love numbers: Charged scalar response coefficients of Kerr-Newman black holes

TL;DR

This paper calculates the exact scalar response coefficients, known as Love numbers, for Kerr-Newman black holes, revealing a discontinuity between neutral and charged scalars and uncovering an $SL(2, ext{R})$ symmetry in the near-zone dynamics.

Contribution

It provides the first exact computation of charged scalar Love numbers for Kerr-Newman black holes, including their dependence on electric charge and symmetry properties.

Findings

Love numbers vanish for neutral scalars.

Charged scalar Love numbers are inversely proportional to charge in the small charge limit.

An $SL(2, ext{R})$ symmetry governs the near-zone response.

Abstract

We study the electrically-charged massless scalar response to the general Kerr-Newman black hole and obtain the exact (complex) response coefficients, whose real parts are the Love numbers. We find there is a curious discontinuity between the neutral and charged scalar. The Love numbers vanish for the neutral scalar, but are inversely proportional to the electric charge of the scalar in the small charge limit. We study the near-zone of the dynamical equation with non-vanishing frequency $\omega$, and show that the system have an $SL(2,\mathbb{R})$ Love-like symmetry. The symmetry selects a special frequency $\omega_{\rm cr}$, which vanishes in the neutral case, so that the real part of the response coefficient vanishes identically. We also generalize the results into higher dimensions to some extend.