Hidden symmetries for tidal Love numbers: generalities and applications to analogue black holes

TL;DR

The paper investigates why tidal Love numbers vanish for four-dimensional Schwarzschild black holes and shows that analogue acoustic black holes (ABHs) exhibit a parallel ladder structure in their perturbation equations. It develops a unified symmetry-based framework, tying ladder operators to closed conformal Killing vectors in an effective $AdS_2$-like geometry, Möbius transformations of a trivializing coordinate, and Darboux/intertwining constructions that connect perturbations across multipole levels. In ABHs, TLNs vanish for multipoles $\ell=4n$ and $\ell=3+4n$ due to two interconnected ladders, with a small ladder linking $\ell=0$ and $\ell=3$; near-horizon and asymptotic analyses reveal an $SL(2,\mathbb{R})$ structure that underpins these results. The work also shows that adding a mass term (e.g., from vorticity) destroys the ladder mechanism, limiting the universality of TLN vanishing, and outlines future extensions to spinning ABHs and broader integrability questions.

Abstract

Tidal Love numbers characterize the conservative, static response of compact objects to external tidal fields. Remarkably, these quantities vanish identically for asymptotically flat black holes in four-dimensional General Relativity. This behavior has been attributed to hidden symmetries -- both geometric and algebraic -- governing perturbations in these space-times. Interestingly, a similar vanishing of selected multipolar Love numbers arises in the context of supersonic acoustic flows. These systems share several key features with black holes in General Relativity, such as the presence of an effective acoustic horizon and a wave equation describing linear excitations. In this work, we explore a symmetry-based connection between the two frameworks and demonstrate that the ladder symmetries observed in acoustic black holes can be traced to structural properties of the underlying wave equation, mirroring those found in general relativistic black hole space-times.

Figures (1)

• Figure 1: Schematic map of the relations among the hidden symmetries behind the vanishing of the TLNs for Schwarzschild black holes in $D = 4$ dimensions. In this paper, we show that a similar diagram also exists for ABHs in the static limit (except that closed CKVs are associated to a 1+1 effective geometry rather than ${\rm AdS}_2$).