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On Black-Brane Instability In an Arbitrary Dimension

Barak Kol, Evgeny Sorkin

Abstract

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 ${\bf T}^n$, 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 $D\leq 11$.

On Black-Brane Instability In an Arbitrary Dimension

Abstract

The black-hole black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension , treating it as a parameter of the system. We derive the large 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 , the 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 .

Paper Structure

This paper contains 8 sections, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Large dimension asymptotics for $\mu_{GL}$
  3. The negative mode of the $d$-dimensional Schwarzschild
  4. Large $d$ limit
  5. The critical mass
  6. Torus compactification
  7. Indication for critical dimensions with ${\bf T}^n$ compactification
  8. Determination of the critical mass $\mu_{GL}(D)$

Figures (2)

  • Figure 1: The shaded area designate the modular domain of ${\bf T}^2$ (the sides $I$ and $I'$ are identified). The two tachyon instability develops for torii inhabiting the arc $\tau=\exp(i\, \theta)$ with $\pi/3 \le \theta \le 2\pi/3$.
  • Figure 2: The dimension $\tilde{D}^*$ for which $\mu_{GL}(\tilde{D}^*,n) = \mu_{\rm S}(\tilde{D}^*,n)$ as a function of the dimension $n$ of the square torus ${\bf T}^n$. The closest integer dimension above $\tilde{D}^*$ estimates a critical dimension for a change of order in the black brane phase transition. For $n=1$ this estimate is known to be short by about 1 from the actual $D^*$: $D^*_{(n=1)} \simeq 13.5 \simeq \tilde{D}^*_{(n=1)}+1$. For $3 \le n \le 6$ the estimate is around $10D$, making it plausible that the actual critical dimension may be as low as $D^* \leq 11$, where a consistent theory of quantum gravity is believed to exist.