Black hole perturbation theory and multiple polylogarithms

TL;DR

This paper develops a new analytical approach to black hole perturbation theory in Schwarzschild (anti) de Sitter backgrounds, revealing a recursive structure involving multiple polylogarithms and analyzing various perturbations and boundary conditions.

Contribution

It introduces an alternative method for perturbation analysis in negative cosmological constant cases, uncovering a recursive structure with multiple polylogarithms.

Findings

New recursive structure involving multiple polylogarithms.

Detailed analysis of scalar gravitational perturbations and hydrodynamic limit.

Method applicable to small and large black holes with various boundary conditions.

Abstract

We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail.