  In a recent work [arXiv:2307.13489 [gr-qc]], 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 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 to express one EMD in terms of the others, thus demonstrating their
equivalence.
