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Exact black hole solutions in gravity with a background Kalb-Ramond field

Jia-Zhou Liu, Shan-Ping Wu, Shao-Wen Wei, Yu-Xiao Liu

TL;DR

The paper analyzes gravity nonminimally coupled to a Kalb-Ramond field that spontaneously violates Lorentz symmetry, deriving exact static four-dimensional black-hole solutions for three cases and a rotating three-dimensional BTZ-like black hole. Using the Iyer-Wald formalism, it computes Noether charges and Wald entropy, obtaining extended first laws and Smarr relations that remain valid despite Lorentz-violating corrections, with the Kalb-Ramond background non-dispersing at the horizon. The authors emphasize the correct treatment of the curvature–Kalb-Ramond coupling term, showing it cannot be absorbed into a redefinition of the gravitational constant and leads to dimensionless Lorentz-violating parameters $ ext{l}_{1}$ and $ ext{l}_{2}$ that modulate the solutions. The results reveal small but potentially observable deviations from GR in strong gravity regimes, with implications for black-hole shadows, quasinormal modes, and extreme mass-ratio inspirals.

Abstract

In this work, we derive exact solutions for four-dimensional static spherically symmetric black holes and three-dimensional rotating black holes within a Lorentz-violating gravity theory. In this framework, Lorentz symmetry is spontaneously broken when a nonminimally coupled Kalb-Ramond tensor field acquires a nonzero vacuum expectation value. Building upon these solutions, we investigate the thermodynamic properties of the black holes using the Iyer-Wald formalism. Our findings reveal that the standard first law of thermodynamics and the Smarr relation remain valid for black holes in the presence of the Kalb-Ramond field.

Exact black hole solutions in gravity with a background Kalb-Ramond field

TL;DR

The paper analyzes gravity nonminimally coupled to a Kalb-Ramond field that spontaneously violates Lorentz symmetry, deriving exact static four-dimensional black-hole solutions for three cases and a rotating three-dimensional BTZ-like black hole. Using the Iyer-Wald formalism, it computes Noether charges and Wald entropy, obtaining extended first laws and Smarr relations that remain valid despite Lorentz-violating corrections, with the Kalb-Ramond background non-dispersing at the horizon. The authors emphasize the correct treatment of the curvature–Kalb-Ramond coupling term, showing it cannot be absorbed into a redefinition of the gravitational constant and leads to dimensionless Lorentz-violating parameters and that modulate the solutions. The results reveal small but potentially observable deviations from GR in strong gravity regimes, with implications for black-hole shadows, quasinormal modes, and extreme mass-ratio inspirals.

Abstract

In this work, we derive exact solutions for four-dimensional static spherically symmetric black holes and three-dimensional rotating black holes within a Lorentz-violating gravity theory. In this framework, Lorentz symmetry is spontaneously broken when a nonminimally coupled Kalb-Ramond tensor field acquires a nonzero vacuum expectation value. Building upon these solutions, we investigate the thermodynamic properties of the black holes using the Iyer-Wald formalism. Our findings reveal that the standard first law of thermodynamics and the Smarr relation remain valid for black holes in the presence of the Kalb-Ramond field.
Paper Structure (11 sections, 72 equations)

This paper contains 11 sections, 72 equations.

Table of Contents

  1. INTRODUCTION
  2. THE KALB-RAMOND MODEL FOR SPONTANEOUS LORENTZ SYMMETRY BREAKING
  3. STATIC SPHERICALLY SYMMETRIC BLACK HOLE SOLUTIONS
  4. Case A: $V(X)=\frac{\lambda}{2}X^2$ and $\Lambda=0$
  5. Case B: $V(X)=\frac{\lambda}{2}X$ and $\Lambda\neq0$
  6. Case C: $V(X)=\frac{\lambda}{2}X^2$ and $\Lambda\neq0$
  7. ROTATING BTZ-LIKE BLACK HOLE SOLUTION
  8. THERMODYNAMICS
  9. Three-Dimensional Case ($D=3$)
  10. Four-Dimensional Case ($D=4$)
  11. CONCLUSIONS