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Small Black Holes on Cylinders

Troels Harmark

TL;DR

This work provides the complete metric for small neutral, static black holes on a cylinder $\mathbb{R}^{d-1}\times S^1$ by employing the Harmark–Huhans ansatz in the small-mass limit, and uses it to derive first-order corrections to thermodynamics, revealing deviations from $(d+1)$-dimensional Schwarzschild behavior. A Newtonian, near-field analysis yields leading corrections to the metric away from the hole, while a change of coordinates clarifies the near-horizon structure and yields a closed-form description of the full small-hole solution in terms of a function $G(\tilde{\rho})$ with parameter $w$. The results show a nonzero leading relative binding energy $n$ and provide a concrete $(M,n)$ phase-diagram piece for $d=5$, highlighting how black holes on cylinders differ from their flat-space counterparts and informing the black-hole/black-string transition picture. The analysis supports locality of small black holes in locally flat regions and demonstrates the continued relevance of the proposed ansatz for non-maximally symmetric backgrounds, while opening avenues for higher-order corrections and broader phase-structure investigations.

Abstract

We find the metric of small black holes on cylinders, i.e. neutral and static black holes with a small mass in d-dimensional Minkowski-space times a circle. The metric is found using an ansatz for black holes on cylinders proposed in hep-th/0204047. We use the new metric to compute corrections to the thermodynamics which is seen to deviate from that of the (d+1)-dimensional Schwarzschild black hole. Moreover, we compute the leading correction to the relative binding energy which is found to be non-zero. We discuss the consequences of these results for the general understanding of black holes and we connect the results to the phase structure of black holes and strings on cylinders.

Small Black Holes on Cylinders

TL;DR

This work provides the complete metric for small neutral, static black holes on a cylinder by employing the Harmark–Huhans ansatz in the small-mass limit, and uses it to derive first-order corrections to thermodynamics, revealing deviations from -dimensional Schwarzschild behavior. A Newtonian, near-field analysis yields leading corrections to the metric away from the hole, while a change of coordinates clarifies the near-horizon structure and yields a closed-form description of the full small-hole solution in terms of a function with parameter . The results show a nonzero leading relative binding energy and provide a concrete phase-diagram piece for , highlighting how black holes on cylinders differ from their flat-space counterparts and informing the black-hole/black-string transition picture. The analysis supports locality of small black holes in locally flat regions and demonstrates the continued relevance of the proposed ansatz for non-maximally symmetric backgrounds, while opening avenues for higher-order corrections and broader phase-structure investigations.

Abstract

We find the metric of small black holes on cylinders, i.e. neutral and static black holes with a small mass in d-dimensional Minkowski-space times a circle. The metric is found using an ansatz for black holes on cylinders proposed in hep-th/0204047. We use the new metric to compute corrections to the thermodynamics which is seen to deviate from that of the (d+1)-dimensional Schwarzschild black hole. Moreover, we compute the leading correction to the relative binding energy which is found to be non-zero. We discuss the consequences of these results for the general understanding of black holes and we connect the results to the phase structure of black holes and strings on cylinders.

Paper Structure

This paper contains 15 sections, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotically measurable quantities
  4. Ansatz for black hole solution
  5. Finding the Fourier modes and the flat-space limit
  6. Ansatz in new coordinate system
  7. Corrected metric away from black hole
  8. Solving for general Newtonian gravity potential
  9. Corrected metric in $(\tilde{\rho},\tilde{\theta})$ coordinates
  10. Metric for small black holes on cylinders
  11. Corrected thermodynamics
  12. Phase diagram for black holes and black strings
  13. Discussion and conclusions
  14. Thermodynamics in the $(R,v)$ and $(\tilde{\rho},\tilde{\theta})$ coordinates
  15. Derivation of general spherical metric

Figures (1)

  • Figure 1: $(M,n)$ phase diagram for $d=5$ containing the black hole branch, the uniform string branch and the non-uniform string branch of Wiseman.