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

In backgrounds with compact dimensions there may exist several phases for black objects including the black-hole and the black-string, and 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, and the a scalar charge is defined and used as an improved order parameter which puts both the black hole and the black string at finite values. Predictions for small black holes are presented. The integrated first law (Smarr’s formula) is derived and will be used to estimate the “overall numerical error”. Expressions for physical quantities in terms of the numerical ones are supplied. Techniques include “method of equivalent charges”, dimensional reduction, analytic perturbation for small black holes, and free energy.