Ladder Symmetries of Black Holes and de Sitter Space: Love Numbers and Quasinormal Modes

TL;DR

This paper explores geometric symmetries and ladder structures in black hole and de Sitter perturbations, explaining phenomena like Love numbers and quasinormal modes through group representations and hypergeometric solutions.

Contribution

It uncovers the ladder symmetries in perturbations around black holes and de Sitter space, linking boundary conditions with group representation theory.

Findings

Ladder symmetries explain vanishing black hole Love numbers.

Explicit ladder operators for de Sitter quasinormal modes are provided.

Symmetry structures connect horizon and boundary conditions via hypergeometric relations.

Abstract

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is well known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.