Geometric Symmetries for the Vanishing of the Black Hole Tidal Love Numbers

TL;DR

This paper uncovers exact geometric symmetries in Schwarzschild black hole perturbations, explaining why tidal Love numbers vanish by revealing underlying SO(3,1) and SO(2) symmetries that organize perturbation behavior.

Contribution

It introduces a unified geometric framework for understanding symmetries in black hole perturbations, clarifying the vanishing of tidal Love numbers across spins 0, 1, and 2.

Findings

Identifies SO(3,1) symmetries in perturbations

Explains vanishing tidal Love numbers via ladder symmetries

Provides a symmetry-based explanation for tidal response coefficients

Abstract

We present a unified geometric perspective on the symmetries underlying the spin 0, 1 and 2 static perturbations around a Schwarzschild black hole. In all cases, the symmetries are exact, each forming an SO(3,1) group. They can be formulated at the level of the action, provided the appropriate field variables are chosen. For spin 1 and 2, the convenient variables are certain combinations of the gauge fields for even perturbations, and dual scalars for odd perturbations. The even and odd sector each has its own SO(3,1) symmetry. In addition, there is an SO(2) symmetry connecting them, furnishing an economical description of Chandrasekhar's duality. When decomposed in spherical harmonics, the perturbations form a non-trivial representation of SO(3,1), giving rise to ladder symmetries which explain the vanishing of the tidal Love numbers. Our work builds on earlier discussions of ladder symmetries, which were formulated in terms of the Newman-Penrose scalar at the level of the Teukolsky equation. Our formulation makes it possible to state the symmetries responsible for the vanishing of the Wilson coefficients characterizing the spin 0, 1 and 2 static tidal response, in the effective point particle description of a black hole.