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

TL;DR

This paper discusses the theoretical background and numerical methods for studying black holes in compactified spacetimes, focusing on phase transitions between black holes and black strings, with results to be detailed in future work.

Contribution

It introduces a framework for numerical analysis of black holes in compactified spaces, defining new order parameters and setting the stage for future detailed results.

Findings

Predictions for small black holes in compactified spacetimes.

Development of numerical techniques including equivalent charges and perturbation methods.

Establishment of phase diagram parameters for black hole and black string phases.

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.