A new approach to static numerical relativity and its application to Kaluza–Klein black holes

Abstract

We propose a framework for solving the Einstein equation for static and Euclidean metrics. First, we address the issue of gauge-fixing by borrowing from the Ricci-flow literature the so-called DeTurck trick, which renders the Einstein equation strictly elliptic and generalizes the usual harmonic-coordinate gauge. We then study two algorithms, Ricci-flow and Newton's method, for solving the resulting Einstein–DeTurck equation. We illustrate the use of these methods by studying localized black holes and non-uniform black strings in five-dimensional Kaluza–Klein theory, improving on previous calculations of their thermodynamic and geometric properties. We study spectra of various operators for these solutions, in particular finding the negative modes of the Lichnerowicz operator. We classify the localized solutions into two branches that meet at a minimum temperature. We find good evidence for a merger between the localized and non-uniform solutions. We also find a narrow window of localized solutions that possess negative modes yet have positive specific heat.