Asymptotic safety in quantum gravity and diffeomorphic non-isometric metric solutions to the Schwarzschild metric

Abstract

We revisit the construction of diffeomorphic but not isometric metric solutions to the Schwarzschild metric. These solutions require the introduction of non-trivial areal–radial functions and are characterized by the key property that the radial horizon’s location is displaced continuously towards the singularity (r = 0). In the limiting case scenario the location of the singularity and horizon merges and any in-falling observer hits a null singularity at the very moment they cross the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. This construction allows us to borrow the results over the past two decades pertaining to the study of the renormalization group improvement of Einstein’s equations, which was based on the possibility that quantum Einstein gravity may be non-perturbatively renormalizable and asymptotically safe because of the presence of interacting (non-Gaussian) ultraviolet fixed points. The particular areal–radial function that eliminates the interior of a black hole, and furnishes a truly static metric solution everywhere, is used to establish the desired energy-scale relation k = k(r), which is obtained from the k (energy) dependent modifications to the running Newtonian coupling G(k), cosmological constant Λ(k), and space–time metric gij,(k)(x). (Anti) de Sitter – Schwarzschild metrics are also explored as examples. We conclude with a discussion of the role that asymptotic safety may have in the geometry of phase spaces (cotangent bundles of space–time) (i.e., in establishing a quantum space–time geometry or classical phase geometry correspondence gij,(k)(x) ↔ gij(x, E)).