Black Hole Scattering and Integrability: A Hyperboloidal Approach

Abstract

Integrability structures are known to play a key role in one-dimensional scattering. In the Schwarzschild gravitational context, the analysis emphasizing the role of the so-called Darboux covariance and its intimate connection with KdV conserved quantities was recently introduced by Lenzi & Sopuerta. In a second stage, together with Jaramillo, this led in particular to the identification of the structural role of the "KdV-Virasoro-Schwarzian derivative" triangle in this problem. Such a gravitational scattering description dwells naturally on a Cauchy foliation of the spacetime. In the following, we first review--for the Schwarzschild background--this problem in a hyperboloidal foliation scheme, where the infinitesimal time generator of the dynamics is a non-selfadjoint operator. Then, we explore the underlying integrability features through a Lax-pair formulation. Specifically, the main results presented here are i) the explicit proposal of a weak Lax-pair, valid under suitable conditions involving fields at null infinity, with ii) the construction of the associated infinite sequence of isospectral flows. From a broader perspective, the very form of the non-selfadjoint infinitesimal time operator, which neatly separates into two components corresponding to bulk and boundary structures, paves the way for the description of the gravitational dynamics in terms of a "semi-direct action" of bulk degrees of freedom onto boundary degrees of freedom. This is akin to the "wave-mean flow" approach for black hole strong-gravity dynamics recently proposed in this line of research.

arXiv comment4 pages, no figure. Prepared for the proceedings of the 24th International Conference on General Relativity and Gravitation (GR24), Glasgow, 14-18 July 2025