In spacetimes with compact dimensions there exist several black object solutions including the black-hole and the black-string. These solutions may become unstable depending on their relative size and the relevant length scale set by the compact dimensions. The transition between these solutions raises puzzles and addresses fundamental questions such as topology change, uniquenesses and cosmic censorship. Here, we consider black strings wrapped over the compact circle of a d-dimensional cylindrical spacetime. We construct static perturbative non-uniform string solutions around the instability point of a uniform string. First we compute the instability mass for a large range of dimensions, d, and find that it follows essentially an exponential law γ d , where γ is a constant. Then we determine that there is a critical dimension, d * = 13, such that for d ≤ d * the phase transition between the uniform and the non-uniform strings is of first order, while for d > d * , it is, surprisingly, of higher order.

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.

The black-hole black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension D, treating it as a parameter of the system. We derive the large D asymptotics of the critical, i.e. marginally stable, string following an earlier numerical analysis. For a background with an arbitrary compactification manifold we give an expression for the critical mass of a corresponding black brane. This expression is completely explicit for T n , the n dimensional torus of an arbitrary shape. An indication is given that by employing a higher dimensional torus, rather than a single compact dimension, the total critical dimension above which the nature of the black-brane black-hole phase transition changes from sudden to smooth could be as low as D ≤ 11.

The nonuniform black strings branch, which emerges from the critical GregoryLaflamme string, is numerically constructed in dimensions 6 ≤ D ≤ 11 and extended into the strongly non-linear regime. All the solutions are more massive and less entropic than the marginal string. We find the asymptotic values of the mass, the entropy and other physical variables in the limit of large horizon deformations. By explicit metric comparison we verify that the local geometry around the "waist" of our most nonuniform solutions is cone-like with less than 10% deviation. We find evidence that in this regime the characteristic length scale has a power-law dependence on a parameter along the branch of the solutions, and estimate the critical exponent.

We describe the first convergent numerical method to determine static black hole solutions (with S 3 horizon) in 5d compactified spacetime. We obtain a family of solutions parametrized by the ratio of the black hole size and the size of the compact extra dimension. The solutions satisfy the demanding integrated first law. For small black holes our solutions approach the 5d Schwarzschild solution and agree very well with new theoretical predictions for the small corrections to thermodynamics and geometry. The existence of such black holes is thus established. We report on thermodynamical (temperature, entropy, mass and tension along the compact dimension) and geometrical measurements. Most interestingly, for large masses (close to the Gregory-Laflamme critical mass) the scheme destabilizes. We interpret this as evidence for an approach to a physical tachyonic instability. Using extrapolation we speculate that the system undergoes a first order phase transition.