Black-Hole First Laws and Horizon Constraints: A Differential Rank Criterion
Abstract
Thermodynamic rewritings of horizon equations depend on which relations are varied and which have already been imposed. We formulate a differential rank criterion that makes these choices explicit. The construction uses a stationary metric family together with an independently marked radial surface, physical charges determined by the metric parameters, and specified extensions of the horizon thermodynamic quantities. A first-law covector is pointwise representable by selected constraints precisely when it lies in the span of their differentials. We distinguish this algebraic condition from a smooth identity in a neighborhood. If regular constraints already define the complete physical family, pointwise representability follows from the first law itself; a neighborhood identity additionally depends on the chosen extensions. With explicit prescriptions, Kerr--Newman, rotating BTZ black holes in new massive gravity, and the new-type NMG black hole all exhibit exact alignment with their horizon constraints. By contrast, substituting an exact metric family into a field-equation component can annihilate that residual before any horizon condition is tested. We also give an explicit entropy--volume comparison and specify the role of fixed ensemble data in free-energy variations. The framework separates consequences of the established first law from additional claims about selected constraints and their extensions.
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BibTeX
@misc{kim2026blackhole,
title = {{Black-Hole First Laws and Horizon Constraints: A Differential Rank Criterion}},
author = {Gunn Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16571},
eprint = {2609.16571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16571},
}
RIS
TY - UNPB
TI - Black-Hole First Laws and Horizon Constraints: A Differential Rank Criterion
AU - Gunn Kim
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16571
UR - https://arxiv.org/abs/2609.16571
AB - Thermodynamic rewritings of horizon equations depend on which relations are varied and which have already been imposed. We formulate a differential rank criterion that makes these choices explicit. The construction uses a stationary metric family together with an independently marked radial surface, physical charges determined by the metric parameters, and specified extensions of the horizon thermodynamic quantities. A first-law covector is pointwise representable by selected constraints precisely when it lies in the span of their differentials. We distinguish this algebraic condition from a smooth identity in a neighborhood. If regular constraints already define the complete physical family, pointwise representability follows from the first law itself; a neighborhood identity additionally depends on the chosen extensions. With explicit prescriptions, Kerr--Newman, rotating BTZ black holes in new massive gravity, and the new-type NMG black hole all exhibit exact alignment with their horizon constraints. By contrast, substituting an exact metric family into a field-equation component can annihilate that residual before any horizon condition is tested. We also give an explicit entropy--volume comparison and specify the role of fixed ensemble data in free-energy variations. The framework separates consequences of the established first law from additional claims about selected constraints and their extensions.
ER -