Vanishing Love of Black Holes in General Relativity: From Spacetime Conformal Symmetry of a Two-dimensional Reduced Geometry

TL;DR

This paper explains why black holes have zero Love numbers by revealing a conformal symmetry in a reduced two-dimensional spacetime, connecting supersymmetry and hidden symmetries to the vanishing response of black holes to external perturbations.

Contribution

It uncovers a conformal symmetry structure in the reduced geometry that accounts for the vanishing Love numbers of Schwarzschild and Kerr black holes, linking supersymmetry and hidden symmetries.

Findings

Black holes exhibit vanishing Love numbers due to underlying conformal symmetry.

Supersymmetric algebra explains the pairing of perturbation modes.

The symmetry structure relates to previously known ladder symmetries.

Abstract

We study the underlying structure of the vanishing of the Love numbers of both Schwarzschild and Kerr black holes in terms of spacetime conformal symmetry in a unified manner for the static spin-$s$ fields. The perturbations can be reduced with the harmonic decomposition to a set of infinite static scalar fields in a two-dimensional anti-de Sitter spacetime~$({\rm AdS}_2)$. In the reduced system, each scalar field is paired with another, implying that all multipole modes of the perturbation can be regarded as symmetric partners, which can be understood from the property of the supersymmetry algebra. The generator of the supersymmetric structure is constructed from a closed conformal Killing vector field of ${\rm AdS}_2$. The associated conserved quantity allows one to show no static response, i.e., vanishing of the Love and dissipation numbers. We also discuss the vanishing Love numbers of the Kerr black hole with the nonzero dissipation numbers for the non-axisymmetric perturbations in terms of a radial constant found in a parallel manner as the axisymmetric field case even though the interpretation for the structure is controversial. The symmetric structure corresponds to the ``ladder'' symmetry in Hui et al. [JCAP 01, no.01, 032 (2022)] although the geometrical origin is different. Our ladder operator includes the generators of hidden symmetries in previous works.