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The Entropy of Dynamical Black Holes

Stefan Hollands, Robert M. Wald, Victor G. Zhang

TL;DR

This work defines a locally computable dynamical entropy S[ℂ] for cross-sections of black-hole horizons in arbitrary diffeomorphism-covariant gravity, built from a horizon Noether charge Q[ξ] and a horizon boundary term B_H. In stationary backgrounds, S reduces to the Iyer–Wald entropy, while in dynamical settings GR yields a leading GR correction S[ℂ] = A[ℂ]/4 − (1/4) ∫_{ℂ} V ϑ ε^{(n−2)}, tying entropy to the apparent-horizon area locally. The paper proves a physical-process first law (κ/2π) ΔS = ΔE for perturbations with external matter and analyzes second-law behavior via the modified canonical energy, showing GR positivity and potential theory-dependence for vacuum perturbations. It also relates the new entropy to the Dong–Wall proposal and links the generalized second law to QNEC in semiclassical gravity, introducing a modified matter entropy to recover a local GSL aligned with quantum energy conditions.

Abstract

We propose a new formula for the entropy of a dynamical black hole$-$valid to leading order for perturbations off of a stationary black hole background$-$in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in $n$ dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In general relativity, our formula for the entropy of a dynamical black hole is $1/4$ of the horizon area plus a term involving the integral of the expansion of the null generators of the horizon, which we show is $1/4$ of the area of the apparent horizon to leading order. Our formula for entropy in a general theory of gravity obeys a "local physical process version" of the first law of black hole thermodynamics. For first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the second law is obeyed at leading order if and only if the "modified canonical energy flux" is positive (as is the case in general relativity but presumably would not hold in general theories). We obtain a general relationship between our formula for the entropy of a dynamical black hole and a formula proposed independently by Dong and by Wall. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter.

The Entropy of Dynamical Black Holes

TL;DR

This work defines a locally computable dynamical entropy S[ℂ] for cross-sections of black-hole horizons in arbitrary diffeomorphism-covariant gravity, built from a horizon Noether charge Q[ξ] and a horizon boundary term B_H. In stationary backgrounds, S reduces to the Iyer–Wald entropy, while in dynamical settings GR yields a leading GR correction S[ℂ] = A[ℂ]/4 − (1/4) ∫_{ℂ} V ϑ ε^{(n−2)}, tying entropy to the apparent-horizon area locally. The paper proves a physical-process first law (κ/2π) ΔS = ΔE for perturbations with external matter and analyzes second-law behavior via the modified canonical energy, showing GR positivity and potential theory-dependence for vacuum perturbations. It also relates the new entropy to the Dong–Wall proposal and links the generalized second law to QNEC in semiclassical gravity, introducing a modified matter entropy to recover a local GSL aligned with quantum energy conditions.

Abstract

We propose a new formula for the entropy of a dynamical black holevalid to leading order for perturbations off of a stationary black hole backgroundin an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In general relativity, our formula for the entropy of a dynamical black hole is of the horizon area plus a term involving the integral of the expansion of the null generators of the horizon, which we show is of the area of the apparent horizon to leading order. Our formula for entropy in a general theory of gravity obeys a "local physical process version" of the first law of black hole thermodynamics. For first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the second law is obeyed at leading order if and only if the "modified canonical energy flux" is positive (as is the case in general relativity but presumably would not hold in general theories). We obtain a general relationship between our formula for the entropy of a dynamical black hole and a formula proposed independently by Dong and by Wall. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter.
Paper Structure (22 sections, 3 theorems, 203 equations, 1 figure)

This paper contains 22 sections, 3 theorems, 203 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lagrangian Formalism, Killing Horizons, Tetrad Choice, and Gauge Conditions
  3. Lagrangians, Noether Charge, and the Fundamental Identity
  4. Tetrad Choice on a Bifurcate Killing Horizon
  5. Gauge Conditions on Perturbations
  6. The Pullback of the Symplectic Potential is a Total Variation for Perturbations of a Stationary Black Hole
  7. Dynamical Black Hole Entropy
  8. General Definition of Dynamical Black Hole Entropy
  9. Dynamical Black Hole Entropy in General Relativity
  10. Entropy Form for $L(R_{abcd})$ Theories
  11. The Physical Process First Law and the Second Law
  12. Perturbations with External Matter Sources
  13. Vacuum Perturbations
  14. Relationship with the Dong--Wall Entropy
  15. The Generalized second law and QNEC
  16. ...and 7 more sections

Key Result

Theorem 1

Let $(\mathcal{M}, g_{ab})$ be an n-dimensional spacetime with a bifurcate Killing horizon $\mathcal{H}$ with horizon Killing field $\xi^a$. Let $\delta g_{ab}$ be an arbitrary smooth perturbation of $g_{ab}$ satisfying the gauge conditions gaugecon1 and gaugecon2. Let $\alpha^a$ be a vector field o where the the tensors $T_{(i)}^{a b_1 \cdots b_i cd} = T_{(i)}^{a (b_1 \cdots b_i) (cd)}$ are smoot

Figures (1)

  • Figure 1: Cross section continuity Chen2022 of $S[\mathcal{C}]$ means that $S[{\mathcal{C}}']$ approaches $S[{\mathcal{C}}]$ for a cut ${\mathcal{C}}'$ tending to a cut $\mathcal{C}$, even if ${\mathcal{C}}'$ is very wiggly as illustrated.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2: Physical process version of the first law Gao_Wald_2001
  • proof
  • Remark
  • Definition 1
  • Lemma 1
  • proof