The entropy of black hole under second-order deviation from equilibrium
Abstract
We investigate the entropy of a dynamical black hole arising from second-order perturbations of a general stationary background with a bifurcate Killing horizon. Using Gaussian null coordinates, we study the geometry of the apparent horizon perturbatively up to second order. Within the covariant phase space formalism, to explore the contribution of matter fields, we introduce a new modified canonical energy, and establish a balance law relating the second-order variation of the entropy to the energy flux entering the black hole. We show that the entropy is given precisely by the area of the apparent horizon at second order when the null energy condition holds for the infalling matter, and that the variation of the entropy also obeys the second law. We also discuss the possibility that the area law continues to hold when the null energy condition is violated.
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BibTeX
@misc{fu2026entropy,
title = {{The entropy of black hole under second-order deviation from equilibrium}},
author = {Wen-Tao Fu and Ming-Fei Ji and Yu-Sen Zhou and Li-Ming Cao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.16757},
eprint = {2606.16757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The entropy of black hole under second-order deviation from equilibrium
AU - Wen-Tao Fu
AU - Ming-Fei Ji
AU - Yu-Sen Zhou
AU - Li-Ming Cao
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2606.16757
UR - https://arxiv.org/abs/2606.16757
AB - We investigate the entropy of a dynamical black hole arising from second-order perturbations of a general stationary background with a bifurcate Killing horizon. Using Gaussian null coordinates, we study the geometry of the apparent horizon perturbatively up to second order. Within the covariant phase space formalism, to explore the contribution of matter fields, we introduce a new modified canonical energy, and establish a balance law relating the second-order variation of the entropy to the energy flux entering the black hole. We show that the entropy is given precisely by the area of the apparent horizon at second order when the null energy condition holds for the infalling matter, and that the variation of the entropy also obeys the second law. We also discuss the possibility that the area law continues to hold when the null energy condition is violated.
ER -