Black Hole Scattering and Partition Functions

TL;DR

This paper explores how scattering phases encode the density of states around black holes, linking them to quasinormal modes and greybody factors, and proposes a renormalization approach to handle divergences in the partition function.

Contribution

It introduces a method to extract finite density of states from scattering phases, connecting Euclidean path integrals, quasinormal modes, and greybody factors in black hole physics.

Findings

Scattering phases encode the density of states for black holes.

Renormalized DOS relates to quasinormal modes and greybody factors.

The approach applies to various black hole spacetimes, including BTZ and Nariai.

Abstract

When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by "renormalizing" the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler space, and the renormalized DOS captures the quasinormal modes for the scalar. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We comment on possible connections with recent works from an algebraic point of view.