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

TL;DR

This work identifies an exact geometric origin for the vanishing static tidal Love numbers of Schwarzschild black holes across spin $s=0,1,2$ by unveiling melodic conformal Killing vector symmetries that generate an $SO(3,1)$ ladder structure. Each spin sector (even/odd) admits its own $SO(3,1)$ symmetry, with an $SO(2)$ Chandrasekhar duality linking sectors; these symmetries act at the level of gauge fields or dual scalars in appropriately chosen 3D effective metrics. The ladder structure connects perturbations at different angular momentum $\ell$, with a ground-state solution that is horizon-regular and grows at infinity, ensuring the absence of the tidal response term $r^{-(\ell+1)}$ and thus $\lambda_\ell=0$ for all $\ell$. The framework extends to a worldline EFT formulation, where the large-$r$ bulk symmetries forbid Love-number operators, and provides a path to generalizations to Kerr and higher dimensions, offering a geometric mechanism for BH tidal responses and a practical tool for EFT analyses.

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.