Photon sphere jet equivalence and the non-uniqueness of black-hole thermodynamics
Abstract
We examine whether scalar eikonal quasinormal-mode data determine black-hole thermodynamics. For static, spherically symmetric, asymptotically flat, nonextremal geometries, we prove that the formal fixed-overtone expansion of a neutral, minimally coupled, massless scalar through order L−N depends only on the (2N+2) jet of the lapse at the unstable photon sphere. For every finite N, real-analytic deformations can preserve this jet and the asymptotic Reissner--Nordström coefficients while changing the surface gravity and Hawking temperature. We also construct a nonanalytic C∞ deformation that is flat at the photon sphere. This deformation preserves the complete formal eikonal series to every algebraic order but changes the horizon position, temperature, and Einstein area entropy. A nonempty charged subfamily satisfies the weak energy condition as an effective Einstein source. Together, these four results establish the new finite-jet and all-orders thermodynamic obstruction. To separate geometric information from action-dependent information, we incorporate a previously established Reissner--Nordström--MOG comparison. The mapped backgrounds have identical normalized metrics, neutral-scalar boundary-value problems, exact scalar spectra, and Hawking temperatures, yet their Wald entropies differ because their curvature couplings differ. The resulting hierarchy shows that local photon-sphere data do not determine horizon thermodynamics and that even exact action-blind scalar data do not determine gravitational entropy. Agreement of the formal eikonal expansions for the deformed metrics does not imply exact finite-ℓ isospectrality. Global differences may appear beyond every algebraic order. Thermodynamic reconstruction from eikonal ringdown is therefore conditional on assumptions about the global metric and the gravitational action.
Related
- Photon sphere for a dyonic black hole corresponding to A2 Toda chain
- Black hole thermodynamics and KK photon quantum corrections in 2D effective dilaton gravity
- Naked Singularities and Black Hole Thermodynamics from Shadows and Photon Rings in the Quantum Oppenheimer-Snyder Model
- Black-Hole Ringdown from Photon-Sphere Modes in an Elastic Analogue
- Conformal boundaries and the photon sphere of AdS black holes
BibTeX
@misc{lobos2026photon,
title = {{Photon sphere jet equivalence and the non-uniqueness of black-hole thermodynamics}},
author = {Nikko John Leo S. Lobos and Emmanuel T. Rodulfo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.20573},
eprint = {2607.20573},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20573},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Photon sphere jet equivalence and the non-uniqueness of black-hole thermodynamics
AU - Nikko John Leo S. Lobos
AU - Emmanuel T. Rodulfo
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.20573
UR - https://arxiv.org/abs/2607.20573
AB - We examine whether scalar eikonal quasinormal-mode data determine black-hole thermodynamics. For static, spherically symmetric, asymptotically flat, nonextremal geometries, we prove that the formal fixed-overtone expansion of a neutral, minimally coupled, massless scalar through order L^−N depends only on the (2N+2) jet of the lapse at the unstable photon sphere. For every finite N, real-analytic deformations can preserve this jet and the asymptotic Reissner--Nordström coefficients while changing the surface gravity and Hawking temperature. We also construct a nonanalytic C^∞ deformation that is flat at the photon sphere. This deformation preserves the complete formal eikonal series to every algebraic order but changes the horizon position, temperature, and Einstein area entropy. A nonempty charged subfamily satisfies the weak energy condition as an effective Einstein source. Together, these four results establish the new finite-jet and all-orders thermodynamic obstruction. To separate geometric information from action-dependent information, we incorporate a previously established Reissner--Nordström--MOG comparison. The mapped backgrounds have identical normalized metrics, neutral-scalar boundary-value problems, exact scalar spectra, and Hawking temperatures, yet their Wald entropies differ because their curvature couplings differ. The resulting hierarchy shows that local photon-sphere data do not determine horizon thermodynamics and that even exact action-blind scalar data do not determine gravitational entropy. Agreement of the formal eikonal expansions for the deformed metrics does not imply exact finite-ℓ isospectrality. Global differences may appear beyond every algebraic order. Thermodynamic reconstruction from eikonal ringdown is therefore conditional on assumptions about the global metric and the gravitational action.
ER -