Probing effective black hole deformations

Abstract In recent works (M.D. Piano, S. Hohenegger, F. Sannino, Quantum black hole physics from the event horizon. Phys. Rev. D 109(2), 024045 (2024). https://doi.org/10.1103/PhysRevD.109.024045 . arXiv:2307.13489 [gr-qc], M.D. Piano, S. Hohenegger, F. Sannino, Effective metric descriptions of quantum black holes. Eur. Phys. J. C 84(12), 1273 (2024). https://doi.org/10.1140/epjc/s10052-024-13609-5 . arXiv:2403.12679 [gr-qc]), a framework has been developed to describe (quantum) deformed, spherically symmetric and static black holes in four dimensions. The key idea of this so-called Effective Metric Description (EMD) is to parametrise deformations of the classical Schwarzschild geometry by two functions that depend on a physical quantity and which are calculated in a self-consistent way as series expansions in the vicinity of the horizon. In this work we further strengthen this framework by first demonstrating that the corresponding series expansion coefficients can be completely and uniquely determined from measurements that are accessible for observers outside of the event horizon: we propose a Gedankenexperiment, consisting of probes following a free-falling trajectory that send signals to a stationary observer and show how an EMD can be constructed from suitable telemetric data. Furthermore, by linking the expansion coefficients of the EMD to the invariant eigenvalues of the energy momentum tensor, we determine a system of physical fields that provides an effective Einstein equation for the deformed black hole geometry. In the case of a simplified geometry and assuming that the metric deformations are small, we can write the leading order of the physical fields in a closed form in the metric functions. We illustrate our results at the example of the Hayward space-time.