Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thermodynamics of Black Holes in Two (and Higher) Dimensions

Daniel Grumiller, Robert McNees

TL;DR

This work develops a robust, semi-classical framework for black hole thermodynamics in two-dimensional dilaton gravity by constructing an improved Euclidean action Gamma that is finite on-shell and variationally well-posed. By embedding BHs in a finite cavity and coupling to a thermal reservoir, the authors formulate a canonical ensemble, derive universal relations like S = 2π X_h and a quasilocal first law, and analyze stability and phase structure in broad model classes. They apply the formalism to notable 2D string-theory backgrounds (Witten BH and the Exact String BH) and demonstrate how α' corrections and boundary counterterms modify thermodynamics, including a vanishing Gibbs-Duhem relation for certain cases and a finite Hagedorn-like limit for ESBH. The framework also extends to higher dimensions via spherical reductions, reproducing Schwarzschild, Schwarzschild–AdS, and BTZ thermodynamics within the same quasi-local approach, and clarifies the role of the dilaton charge as the natural horizon/area proxy. Overall, the paper provides a universal, geometrically grounded route to black hole thermodynamics across 2D models, string-theory backgrounds, and certain higher-dimensional reductions, with clear guidelines for stability and phase transitions.

Abstract

A comprehensive treatment of black hole thermodynamics in two-dimensional dilaton gravity is presented. We derive an improved action for these theories and construct the Euclidean path integral. An essentially unique boundary counterterm renders the improved action finite on-shell, and its variational properties guarantee that the path integral has a well-defined semi-classical limit. We give a detailed discussion of the canonical ensemble described by the Euclidean partition function, and examine various issues related to stability. Numerous examples are provided, including black hole backgrounds that appear in two dimensional solutions of string theory. We show that the Exact String Black Hole is one of the rare cases that admits a consistent thermodynamics without the need for an external thermal reservoir. Our approach can also be applied to certain higher-dimensional black holes, such as Schwarzschild-AdS, Reissner-Nordstrom, and BTZ.

Thermodynamics of Black Holes in Two (and Higher) Dimensions

TL;DR

This work develops a robust, semi-classical framework for black hole thermodynamics in two-dimensional dilaton gravity by constructing an improved Euclidean action Gamma that is finite on-shell and variationally well-posed. By embedding BHs in a finite cavity and coupling to a thermal reservoir, the authors formulate a canonical ensemble, derive universal relations like S = 2π X_h and a quasilocal first law, and analyze stability and phase structure in broad model classes. They apply the formalism to notable 2D string-theory backgrounds (Witten BH and the Exact String BH) and demonstrate how α' corrections and boundary counterterms modify thermodynamics, including a vanishing Gibbs-Duhem relation for certain cases and a finite Hagedorn-like limit for ESBH. The framework also extends to higher dimensions via spherical reductions, reproducing Schwarzschild, Schwarzschild–AdS, and BTZ thermodynamics within the same quasi-local approach, and clarifies the role of the dilaton charge as the natural horizon/area proxy. Overall, the paper provides a universal, geometrically grounded route to black hole thermodynamics across 2D models, string-theory backgrounds, and certain higher-dimensional reductions, with clear guidelines for stability and phase transitions.

Abstract

A comprehensive treatment of black hole thermodynamics in two-dimensional dilaton gravity is presented. We derive an improved action for these theories and construct the Euclidean path integral. An essentially unique boundary counterterm renders the improved action finite on-shell, and its variational properties guarantee that the path integral has a well-defined semi-classical limit. We give a detailed discussion of the canonical ensemble described by the Euclidean partition function, and examine various issues related to stability. Numerous examples are provided, including black hole backgrounds that appear in two dimensional solutions of string theory. We show that the Exact String Black Hole is one of the rare cases that admits a consistent thermodynamics without the need for an external thermal reservoir. Our approach can also be applied to certain higher-dimensional black holes, such as Schwarzschild-AdS, Reissner-Nordstrom, and BTZ.

Paper Structure

This paper contains 38 sections, 255 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Semi-Classical Approximation of the Dilaton Gravity Path Integral
  3. Equations of Motion
  4. Black Holes
  5. Variational Properties of the Action
  6. The Hamilton-Jacobi Boundary Term
  7. Path Integral Based on the Improved Action
  8. Thermodynamics in the Canonical Ensemble
  9. Dilaton Charge and Free Energy
  10. Entropy and Area
  11. Dilaton Chemical Potential and Surface Pressure
  12. Internal Energy and First Law
  13. Specific Heat at Constant Dilaton Charge and Fluctuations
  14. Free Enthalpy and Gibbs-Duhem Violation
  15. Miscellaneous Thermodynamical Quantities
  16. ...and 23 more sections

Figures (4)

  • Figure 1: Heat capacity for the Schwarzschild BH in dimension $d+1 \geq 4$.
  • Figure 2: Heat capacity for the Schwarzschild-AdS BH in $d+1$ dimensions, with the cavity wall sent to infinity. The horizontal axis is the horizon radius $r_h$, and the vertical axis is the $X_c \to \infty$ limit of $C_{D}$.
  • Figure 3: Summary of Dilaton Gravity models, adapted from Grumiller:2006rc.
  • Figure 4: Dilaton Gravity models with Maxwell fields.