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 $C^{\infty }$ 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-$\ell$ 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.

2 figures, 21 pages