A Dialogue of Multipoles: Matched Asymptotic Expansion for Caged Black Holes

TL;DR

This paper develops an analytic perturbation theory for black holes in compact dimensions using matched asymptotic expansions, providing leading corrections to the metric and physical properties, and introduces a novel 'dialogue of multipoles' approach.

Contribution

It introduces a new perturbation method based on matched asymptotic expansions for caged black holes, enabling analytic calculations of corrections to their metrics and properties.

Findings

Derived the leading correction to the black hole metric.

Calculated the first correction to the area-temperature relation.

Determined static perturbations of Schwarzschild black holes in higher dimensions.

Abstract

No analytic solution is known to date for a black hole in a compact dimension. We develop an analytic perturbation theory where the small parameter is the size of the black hole relative to the size of the compact dimension. We set up a general procedure for an arbitrary order in the perturbation series based on an asymptotic matched expansion between two coordinate patches: the near horizon zone and the asymptotic zone. The procedure is ordinary perturbation expansion in each zone, where additionally some boundary data comes from the other zone, and so the procedure alternates between the zones. It can be viewed as a dialogue of multipoles where the black hole changes its shape (mass multipoles) in response to the field (multipoles) created by its periodic "mirrors", and that in turn changes its field and so on. We present the leading correction to the full metric including the first correction to the area-temperature relation, the leading term for black hole eccentricity and the "Archimedes effect". The next order corrections will appear in a sequel. On the way we determine independently the static perturbations of the Schwarzschild black hole in dimension d>=5, where the system of equations can be reduced to "a master equation" - a single ordinary differential equation. The solutions are hypergeometric functions which in some cases reduce to polynomials.