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First Law for Nonsingular Black Holes in 2D Dilaton Gravity

Peng Yu, Yuan Zhong

Abstract

A central issue in the thermodynamics of nonsingular black holes is the apparent violation of the first law. In this work, we use 2D dilaton gravity as a simple theoretical setting to study this issue. We systematically construct a broad class of nonsingular black hole solutions with metric function $A(x)=f(x)+c$, through a procedure that is considerably simpler than in higher-dimensional theories. Using the Iyer-Wald covariant phase space formalism, we derive the correct energy formula and establish a consistent first law for this entire class of solutions. The apparent first-law violation in a previous work is caused by an incorrect choice of energy. Our energy formula agrees, up to a normalization factor for the asymptotic Killing vector, with the Casimir function in 2D dilaton gravity, confirming its interpretation as the physical black hole energy. Our results clarify the correct first law for 2D nonsingular black holes and may provide insights into the first law of nonsingular black holes in higher dimensions.

First Law for Nonsingular Black Holes in 2D Dilaton Gravity

Abstract

A central issue in the thermodynamics of nonsingular black holes is the apparent violation of the first law. In this work, we use 2D dilaton gravity as a simple theoretical setting to study this issue. We systematically construct a broad class of nonsingular black hole solutions with metric function , through a procedure that is considerably simpler than in higher-dimensional theories. Using the Iyer-Wald covariant phase space formalism, we derive the correct energy formula and establish a consistent first law for this entire class of solutions. The apparent first-law violation in a previous work is caused by an incorrect choice of energy. Our energy formula agrees, up to a normalization factor for the asymptotic Killing vector, with the Casimir function in 2D dilaton gravity, confirming its interpretation as the physical black hole energy. Our results clarify the correct first law for 2D nonsingular black holes and may provide insights into the first law of nonsingular black holes in higher dimensions.
Paper Structure (15 sections, 96 equations, 2 figures)

This paper contains 15 sections, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Nonsingular black hole solutions
  3. A general discussion
  4. Analytical solutions
  5. Solution with sine-Gordon kink-like metric function
  6. Solution with arctan kink-like metric function
  7. Causal structure analysis
  8. The first law
  9. General formalism
  10. Entropy formula
  11. Energy formula
  12. First law
  13. Casimir function in generic 2D dilaton gravity
  14. Conclusions and outlook
  15. Symplectic potential and Noether charge for our 2D dilaton gravity model

Figures (2)

  • Figure 1: Metric functions $A(x)$ (left panel) and Ricci scalars $R(x)$ (right panel) for the nonsingular black hole solutions with $\varphi^4$ kink (dotted), sine-Gordon kink (solid), and arctan kink (dashed) metric profiles. The parameters are chosen as $c=-\pi/4$ for the sine-Gordon solution and $c=0$ for the $\varphi^4$ and arctan solutions.
  • Figure 2: Penrose diagram of the maximally extended spacetime in Eq. \ref{['eq:metric_UV_appendix']}. The null lines $U=0$ and $V=0$ form a bifurcate Killing horizon, separating the spacetime into two exterior regions $I$ and $III$, the black-hole interior $II$, and the white-hole interior $IV$.