Horizon entropy does not by itself determine a force between black holes
Abstract
The Bekenstein-Hawking law fixes the total entropy of a horizon. Membrane, holographic, and entropic-force pictures go further and follow the entropy of individual horizon elements as a configuration changes. We show that this step needs structure that the horizon geometry does not supply. Along a family of horizons labeled by λ, a relabeling of horizon points by a tangential field vA shifts the temperature-weighted entropy variation ∫ T∂λ(dS) exactly by −∫ vADAT dS, so the horizon data determine it only when the temperature is uniform. We test this on two black holes in exact Brill-Lindquist data. At fixed physical masses each horizon area is independent of the separation through five relative orders; the low-order equilibrium force reported earlier comes from holding the bare parameters fixed instead. Weighting the redistribution of area by the nonuniform Noether-charge temperature gives a candidate repulsion equal to (ν/8)(GM/r12)3 times the Newtonian attraction, for total mass M, symmetric mass ratio ν, and coordinate separation r12. As a static interaction it would shift the third post-Newtonian coefficient of the circular-orbit binding energy by 5ν/48, in conflict with general relativity. The identity resolves the conflict. Through fourth post-Newtonian order the candidate lies entirely in the dipole of the area distribution, which the uniform part of the companion's field generates as a Möbius map of the labels, and which vanishes in the area-balanced conformal frame, where the functional begins at fifth order with a coefficient that a further relabeling can still change. The marginally trapped tube of the data and the tidally deformed horizon of a post-Newtonian binary, both of which do supply an identification, transport no dipole at the order the candidate requires.
Related
- How Much Geometry Does Horizon Entropy Determine? A Discourse on Entropy, Gravity, and Observable Quantities
- Area-product universality in multi-horizon black hole entropy
- Bound States Inside the Horizon and Black Hole Entropy
- One Gravity, Many G's: Generalized Horizon Entropy and Gravitational Susceptibility
- Universal framework for horizon-scale tests of gravity with black hole shadows
BibTeX
@misc{wang2026horizon,
title = {{Horizon entropy does not by itself determine a force between black holes}},
author = {Zhi-Wei Wang and Samuel L. Braunstein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31690},
eprint = {2610.31690},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Horizon entropy does not by itself determine a force between black holes
AU - Zhi-Wei Wang
AU - Samuel L. Braunstein
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.31690
UR - https://arxiv.org/abs/2610.31690
AB - The Bekenstein-Hawking law fixes the total entropy of a horizon. Membrane, holographic, and entropic-force pictures go further and follow the entropy of individual horizon elements as a configuration changes. We show that this step needs structure that the horizon geometry does not supply. Along a family of horizons labeled by λ, a relabeling of horizon points by a tangential field v^A shifts the temperature-weighted entropy variation ∫ T∂_λ(dS) exactly by −∫ v^AD_AT dS, so the horizon data determine it only when the temperature is uniform. We test this on two black holes in exact Brill-Lindquist data. At fixed physical masses each horizon area is independent of the separation through five relative orders; the low-order equilibrium force reported earlier comes from holding the bare parameters fixed instead. Weighting the redistribution of area by the nonuniform Noether-charge temperature gives a candidate repulsion equal to (ν/8)(GM/r_12)^3 times the Newtonian attraction, for total mass M, symmetric mass ratio ν, and coordinate separation r_12. As a static interaction it would shift the third post-Newtonian coefficient of the circular-orbit binding energy by 5ν/48, in conflict with general relativity. The identity resolves the conflict. Through fourth post-Newtonian order the candidate lies entirely in the dipole of the area distribution, which the uniform part of the companion's field generates as a Möbius map of the labels, and which vanishes in the area-balanced conformal frame, where the functional begins at fifth order with a coefficient that a further relabeling can still change. The marginally trapped tube of the data and the tidally deformed horizon of a post-Newtonian binary, both of which do supply an identification, transport no dipole at the order the candidate requires.
ER -