Self-regularized entropy: What does black hole entropy predict for tests of Kerr no-hair theorem?

Abstract

We compute the canonical, or brick-wall, entropy of a massless scalar field in a quantum black hole model whose strong field exterior is described phenomenologically by the static $q$-metric, also known as the Zipoy-Voorhees metric. This geometry is an exact vacuum deformation of Schwarzschild with a small quadrupolar distortion parameter, $q$. Using WKB counting of trapped near horizon cavity modes, we show that this deformation changes the near horizon density of states so that the usual Schwarzschild brick-wall ultraviolet divergence is self-regularized, eliminating the need for an ad hoc proper distance cutoff within the perturbative regime studied here. Treating the Hawking temperature and Bekenstein-Hawking entropy of a Schwarzschild black hole of the same mass as external thermodynamic benchmarks, we obtain an analytic entropy-motivated deformation scale, $|q|\sim 0.2$, across the stellar-to-supermassive black hole mass range. Through a stationary extension, this scale maps phenomenologically onto percent-to-tens-of-percent violations of the Kerr multipole relations, providing observational targets for ngEHT imaging, LISA extreme mass ratio inspirals, and third generation ground based gravitational wave tests.