Caged Black Holes: Black Holes in Compactified Spacetimes I -- Theory

TL;DR

This work addresses black objects in spacetimes with a single compact dimension, where phases include uniform and non-uniform black strings and black holes, and where phase transitions probe topology change and Cosmic Censorship. It develops a numerical framework based on a conformal gauge ansatz, the method of equivalent sources for asymptotics, and thermodynamic consistency checks via the first law $\,dm = T\, dS + \tau\, d\hat{L}$ and Smarr's formula, augmented by dimensional reduction that yields a scalar charge $b$ as an additional order parameter. The paper provides explicit mappings between asymptotic data and physical charges through a $2\times2$ linear relation between $(a,b)$ and $(m,\tau)$, analyzes small black holes (where $\tau\to0$) and derives precise horizon observables such as $A$, $\kappa$, and horizon eccentricity, including dimension-specific predictions. Together, these results establish a robust, gauge-consistent toolkit to explore the phase structure and topology-change scenarios of black holes in compactified spacetimes, setting the stage for forthcoming numerical 5d solutions and comparisons with GL-transition predictions.

Abstract

In backgrounds with compact dimensions there may exist several phases of black objects including the black-hole and the black-string. The phase transition between them raises puzzles and touches fundamental issues such as topology change, uniqueness and Cosmic Censorship. No analytic solution is known for the black hole, and moreover, one can expect approximate solutions only for very small black holes, while the phase transition physics happens when the black hole is large. Hence we turn to numerical solutions. Here some theoretical background to the numerical analysis is given, while the results will appear in a forthcoming paper. Goals for a numerical analysis are set. The scalar charge and tension along the compact dimension are defined and used as improved order parameters which put both the black hole and the black string at finite values on the phase diagram. Predictions for small black holes are presented. The differential and the integrated forms of the first law are derived, and the latter (Smarr's formula) can be used to estimate the ``overall numerical error''. Field asymptotics and expressions for physical quantities in terms of the numerical ones are supplied. Techniques include ``method of equivalent charges'', free energy, dimensional reduction, and analytic perturbation for small black holes.