Effective metric descriptions of quantum black holes - Manuel Del Piano; Stefan Hohenegger; Francesco Sannino (European Physical Journal C: Particles and Fields)
Abstract In a recent work (Del Piano et al. in Phys Rev D 109(2):024045, 2024), we have described spherically symmetric and static quantum black holes as deformations of the classical Schwarzschild metric that depend on the physical distance to the horizon. We have developed a framework that allows us to compute the latter in a self-consistent fashion from the deformed geometry, in the vicinity of the horizon. However, in this formalism, the distance can be replaced by other physical quantities, e.g. curvature invariants such as the Ricci- or Kretschmann scalar. Here, we, therefore, define a more general framework, which we call an effective metric description (EMD), that captures the deformed geometry based on a generic physical quantity. We develop in detail the Ricci- and Kretschmann scalar EMD, in particular demonstrating how to compute the geometry in a self-consistent manner. Moreover, we provide explicit relations that allow us to express one EMD  in terms of the others, thus demonstrating their equivalence.