**Authors:** Carlos Castro

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 infalling observer hits a null singularity at the very moment he/she crosses 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 to borrow the results over the past two decades pertaining the study of the Renormalization Group (RG) improvement of Einstein's equations which was based on the possibility that Quantum Einstein Gravity might be non-perturbatively renormalizable and asymptotically safe due to 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 $\Lambda (k) $ and spacetime metric $g_{ij, (k) } (x)$. (Anti) de Sitter-Schwarzschild metrics are also explored as examples. We conclude with a discussion of the role that Asymptotic Safety might have in the geometry of phase spaces (cotangent bundles of spacetime); i.e. namely, in establishing a quantum spacetime geometry/classical phase geometry correspondence $g_{ij, (k) } (x) \leftrightarrow g_{ij} (x, E) $.

**Comments:** 18 Pages.

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[v1] 2017-01-29 04:30:35

[v2] 2017-02-02 04:00:25

[v3] 2017-03-05 01:22:50

[v4] 2017-04-24 01:32:44

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