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Analytic approximation to 5 dimensional Black Holes with one compact dimension

D. Karasik, C. Sahabandu, P. Suranyi, L. C. R. Wijewardhana

TL;DR

This work develops a controlled analytic framework to construct small 5D black holes on a cylinder by expanding in $\epsilon=(\mu/L)^2$ and matching asymptotic and near-horizon solutions. The authors compute the metric up to second order in $\epsilon$ (fourth order in $\mu/L$), obtain corrected thermodynamics (entropy and temperature), and derive the horizon geometry, including a prolate deformation. They analyze the black hole–black string transition via phase-diagram considerations and horizon-topology arguments, identifying a transition region around $\epsilon\sim0.1$ and revealing a close interplay between mass, tension, and horizon filling. The results solidify the picture that small black holes on cylinders are well-described by a two-region matching method and provide quantitative benchmarks for the transition to nonuniform strings.

Abstract

We study black hole solutions in $R^4\times S^1$ space, using an expansion to fourth order in the ratio of the radius of the horizon, $μ$, and the circumference of the compact dimension, $L$. A study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at $ε(μ/L)^2\simeq0.1$. At the same value of $ε$ the entropies of the uniform black string and of the black hole are approximately equal.

Analytic approximation to 5 dimensional Black Holes with one compact dimension

TL;DR

This work develops a controlled analytic framework to construct small 5D black holes on a cylinder by expanding in and matching asymptotic and near-horizon solutions. The authors compute the metric up to second order in (fourth order in ), obtain corrected thermodynamics (entropy and temperature), and derive the horizon geometry, including a prolate deformation. They analyze the black hole–black string transition via phase-diagram considerations and horizon-topology arguments, identifying a transition region around and revealing a close interplay between mass, tension, and horizon filling. The results solidify the picture that small black holes on cylinders are well-described by a two-region matching method and provide quantitative benchmarks for the transition to nonuniform strings.

Abstract

We study black hole solutions in space, using an expansion to fourth order in the ratio of the radius of the horizon, , and the circumference of the compact dimension, . A study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at . At the same value of the entropies of the uniform black string and of the black hole are approximately equal.

Paper Structure

This paper contains 32 sections, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The general method
  3. The equations
  4. The boundary conditions.
  5. Matching the asymptotic and near solutions
  6. Physical parameters
  7. Zeroth order in $\epsilon$
  8. First order in $\epsilon$
  9. The asymptotic metric
  10. The near solution
  11. Conditions at $\psi=\pi/2$
  12. Conditions at $\psi=0$
  13. Evenness in $\mu$
  14. Requirement of a Killing horizon
  15. Matching the near and the asymptotic solutions
  16. ...and 17 more sections

Figures (4)

  • Figure 1: $(\mu/\rho)^{2n}(\rho/L)^{2k}$ terms as points in the $(n,k)$ grid. The horizontal (vertical) lines describe the asymptotic (near) solution in increasing order of $\epsilon$. The matching is done at the intersecting points denoted by circles.
  • Figure 2: In the intermediate region the asymptotic solution is expanded in $\rho/L$, while the near solution is expanded in $\mu/\rho$. The table shows the sequence of calculating terms in the asymptotic and near solutions. Terms connected by an arrow are identified by the double expansion. A similar table has been presented in gorbonos
  • Figure 3: Phase diagram for the uniform black string (dashed green line), non-uniform black string (blue line), and black hole branches (solid red continued in dashed red in the region where the black hole turns into a nonuniform black string).
  • Figure 4: The entropy (in units of $L^{3}/G_{5}$) of the uniform black string (dashed green line), non-uniform black string ( dashed red line), and black hole branches (solid red line).