Evgeny Sorkin

Critical dimension in the black-string phase transition

In spacetimes with compact dimensions, there exist several black object solutions including the black hole and the black string. They may become unstable depending on their relative size and the length scales in 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 nonuniform strings around the marginally stable uniform string. First, we compute the instability mass for a large range of dimensions and find that it follows an exponential law gamma(d), where gamma<1 is a constant. Then we determine that there is a critical dimension, d(*)=13, such that for d</=d(*) the phase transition is of first order, while for d>d(*) it is, surprisingly, of higher order.

Black-brane instability in an arbitrary dimension

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.

Caged black holes: Black holes in compactified space-times. 2. 5-d numerical implementation

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.

Nonuniform black strings in various dimensions

The nonuniform black-strings branch, which emerges from the critical Gregory-Laflamme string, is numerically constructed in dimensions 6{<=}D{<=}11 and extended into the strongly nonlinear 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 conelike 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.

Formation and evaporation of charged black holes

We investigate the dynamical formation and evaporation of a spherically symmetric charged black hole. We study the self-consistent one loop order semiclassical back reaction problem. To this end the mass evaporation is modeled by an expectation value of the stress-energy tensor of a neutral massless scalar field, while the charge is not radiated away. We observe the formation of an initially nonextremal black hole which tends toward the extremal black hole M=Q, emitting a Hawking radiation. If the discharge due to the instability of a vacuum-to-pair creation in strong electric fields also occurs, then the black hole discharges and evaporates simultaneously and decays regularly until the scale where the semiclassical approximation breaks down. We calculate the rates of the mass and the charge loss and estimate the lifetime of the decaying black holes.

Choptuik's scaling in higher dimensions

We extend Choptuiks scaling phenomenon found in general relativistic critical gravitational collapse of a massless scalar field to higher dimensions. We find that in the range 4{<=}D{<=}11 the behavior is qualitatively similar to that discovered by Choptuik. In each dimension we obtain numerically the universal numbers associated with the critical collapse: the scaling exponent {gamma} and the echoing period {delta}. The behavior of these numbers with increasing dimension seems to indicate that {gamma} reaches a maximum and {delta} a minimum value around 11{<=}D{<=}13. These results and their relation to the black hole-black string system are discussed.

Striped order in AdS/CFT correspondence

We study the formation of inhomogeneous order in the Einstein-Maxwell-axion system, dual to a 2+1 dimensional field theory that exhibits a spontaneously generated current density, momentum density and modulated scalar operator. Below the critical temperature, the Reissner-Nordstrom-AdS black hole becomes unstable and stripes form in the bulk and on the boundary. The bulk geometry possesses striking geometrical features, including a modulated horizon that tends to pinch off as T -> 0. On a domain of fixed length, we find a second order phase transition to the striped solution in each of the grand canonical, canonical and microcanonical ensembles, with modulated charges that grow and saturate as we lower the temperature and descend into the inhomogeneous phase. For the black hole on an infinite domain, a similar second order transition occurs, and the width of the dominant stripe increases in the zero temperature limit.

LG (Landau–Ginzburg) in GL (Gregory–Laflamme)

This paper continues the study of the Gregory–Laflamme instability of black strings, or more precisely of the order of the transition, being either first or second order, and the critical dimension which separates the two cases. First, we describe a novel method based on the Landau–Ginzburg perspective for the thermodynamics that somewhat improves the existing techniques. Second, we generalize the computation from a circle compactification to an arbitrary torus compactification. We explain that the critical dimension cannot be lowered in this way, and moreover, in all cases studied the transition order depends only on the number of extended dimensions. We discuss the richer phase structure that appears in the torus case.

An axisymmetric generalized harmonic evolution code

We describe the first axisymmetric numerical code based on the generalized harmonic formulation of the Einstein equations, which is regular at the axis. We test the code by investigating gravitational collapse of distributions of complex scalar field in a Kaluza-Klein spacetime. One of the key issues of the harmonic formulation is the choice of the gauge source functions, and we conclude that a damped-wave gauge is remarkably robust in this case. Our preliminary study indicates that evolution of regular initial data leads to formation both of black holes with spherical and cylindrical horizon topologies. Intriguingly, we find evidence that near threshold for black hole formation the number of outcomes proliferates. Specifically, the collapsing matter splits into individual pulses, two of which travel in the opposite directions along the compact dimension and one which is ejected radially from the axis. Depending on the initial conditions, a curvature singularity develops inside the pulses.

Initial data for black holes and black strings in 5D

We explore time-symmetric hypersurfaces containing apparent horizons of black objects in a 5D spacetime with one coordinate compactified on a circle. We find a phase transition within the family of such hypersurfaces: the horizon has different topology for different parameters. The topology varies from S3 to S2 x S1. This phase transition is discontinuous--the topology of the horizon changes abruptly. We explore the behavior around the critical point and present a possible phase diagram.