A Microscopic Model for the Black hole - Black string Phase Transition

TL;DR

We address the black hole–black string phase transition in spacetimes with a compact transverse circle and develop a microscopic model by mapping neutral gravity solutions to near-extremal D1–D5 configurations via boosts and dualities. The core idea is a fractionation mechanism in which the nonextremal energy splits into BH-like and BS-like excitations, controlled by the D1–D5 product and a partition parameter, reproducing the three gravity phases (BH, uniform BS, non-uniform BS) and capturing the Gregory-Laflamme transition qualitatively. The study relates gravity to a microscopic CFT picture, deriving entropy and tension relations, and identifies tensions missing in the BH phase due to neglected interactions, which are discussed through a mixed gravity–CFT approach. The results illuminate brane fractionation in horizon-changing transitions and provide a framework for understanding how microstates reorganize during such phase changes, while highlighting areas—such as precise GL energies and BH tension—where higher-order interactions must be incorporated for full quantitative agreement.

Abstract

Computations in general relativity have revealed an interesting phase diagram for the black hole - black string phase transition, with three different black objects present for a range of mass values. We can add charges to this system by `boosting' plus dualities; this makes only kinematic changes in the gravity computation but has the virtue of bringing the system into the near-extremal domain where a microscopic model can be conjectured. When the compactification radius is very large or very small then we get the microscopic models of 4+1 dimensional near-extremal holes and 3+1 dimensional near-extremal holes respectively (the latter is a uniform black string in 4+1 dimensions). We propose a simple model that interpolates between these limits and reproduces most of the features of the phase diagram. These results should help us understand how `fractionation' of branes works in general situations.

Figures (10)

• Figure 1: (a) A small black hole in a space with a compact circle of length $L$ (b) The horizon distorts when the mass is increased so that its radius becomes comparable to $L$ (c) At still larger masses we get a black string which wraps uniformly around the compact circle.
• Figure 2: The phase diagram for (a) 4 noncompact space directions and (b) 3 noncompact space directions. The vertical axis measures the dimensionless relative tension and and the horizontal axis gives a dimensionless mass parameter. The solid line denotes the uniform black string, the dashed line denotes the black hole, and the dot-dashed line denotes the non-uniform black string.
• Figure 3: The entropies for (a) 4 noncompact space directions and (b) 3 noncompact space directions. The vertical axis gives the ratio of the entropy to the entropy of the uniform black string, and the horizontal axis gives a dimensional mass. The three kinds of lines label the three phases in the same way as in Fig.(\ref{['fig:nVSmu_gravity']})
• Figure 4: (a) The entropy is plotted on the vertical axis, as a function of $x, \epsilon_1$, for $\epsilon=0.5$, a value below the point where the three phase structure appears; there is only one maximum at the boundary of the parameter space (b) The entropy graph for $\epsilon=2.4$, a value at which there are three phases; there are two maxima at the boundary of the parameter space and a saddle point in the middle.
• Figure 5: The phase diagram predicted by the leading order microscopic model. The solid line is the uniform black string. The black hole branch is a horizonal (dashed) line that overlaps with the x-axis -- the tension is zero. The dot-dashed line is the non-uniform string branch.