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We propose a general framework for solving the Einstein equation for staticand Euclidean metrics. First, we address the issue of gauge-fixing by borrowingfrom the Ricci-flow literature the so-called DeTurck trick, which renders theEinstein equation strictly elliptic and generalizes the usualharmonic-coordinate gauge. We then study two algorithms, Ricci-flow andNewton's method, for solving the resulting Einstein-DeTurck equation. Weillustrate the use of these methods by studying localized black holes andnon-uniform black strings in five-dimensional Kaluza-Klein theory, improving onprevious calculations of their thermodynamic and geometric properties. We studyspectra of various operators for these solutions, in particular findingnegative modes of the Lichnerowicz operator. We classify the localizedsolutions into two branches that meet at a minimum temperature. We find goodevidence for a merger between the localized and non-uniform solutions. We alsofind a narrow window of localized solutions that possess negative modes yetappear to have positive specific heat.

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