Black Hole Scattering and Integrability: A Hyperboloidal Approach

TL;DR

The paper addresses integrable structures in black hole scattering within Schwarzschild perturbations by employing a hyperboloidal foliation that connects the horizon to null infinity. It introduces a weak adjoint Lax pair, yielding an energy-dependent Schrödinger operator and an infinite hierarchy of isospectral flows whose bulk dynamics are governed by hyperboloidal KdV structures; the flows are tied to conserved quantities via variational derivatives, ensuring isospectrality. A Hamiltonian framework is developed for the first few flows, with explicit forms for the first two Hamiltonians and open questions about extending to all orders. Overall, the work provides a dispersive-integrable description of BH dynamics that separates bulk and boundary contributions and supports a wave-mean flow interpretation of strong gravity phenomena near black holes.

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.