Eikonal Quality-Factor Parameterization of Model-Conditional Static Black-Hole Thermodynamics

Abstract

We study how one complex quasinormal frequency can parameterize the geometry and model-conditional thermodynamics of a prescribed family of static, spherically symmetric black holes. For a minimally coupled massless test scalar in the eikonal limit, the ratio $χ=ω_Rτ=ω_R/ω_I=2Q$ removes the overall mass scale. If $χ$ is injective in one dimensionless parameter, that parameter and the geometric scale can be reconstructed within the chosen family. The normalized metric then fixes the horizon and Hawking temperature, whereas Wald entropy requires the gravitational action. For the RN-form family, $χ$ determines $u=q^2$ but not the sign of charge. The constant-coupling static MOG metric is exactly RN under $\mathcal M=(1+α)m_{\rm MOG}$ and $u=α/(1+α)$; consequently, the complete minimally coupled scalar spectrum is identical on mapped backgrounds for every multipole and overtone. The two assignments have equal temperature but different action-dependent entropy. Chebyshev and WKB--Padé calculations quantify the finite-multipole error, with a maximum eikonal ratio error of $2.1\times10^{-3}$ on the tested $l=2$ RN grid. The construction is therefore a model-dependent inverse, not a theory selector. Applying it to gravitational-wave observations requires the appropriate tensor, vector, and scalar perturbation sectors beyond the test-field eikonal approximation.