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Invariant Renormalization-Group improvement
by Aaron Held
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Renormalization-Group (RG) improvement has been frequently applied to capture the effect of quantum corrections on cosmological and black-hole spacetimes. This work utilizes an algebraically complete set of curvature invariants to establish that: On the one hand, RG improvement at the level of the metric is coordinate-dependent. On the other hand, a newly proposed RG improvement at the level of curvature invariants is coordinate-independent. Spherically-symmetric and axially-symmetric black-hole spacetimes serve as physically relevant examples.
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the author describes an interesting generalization to the RG-improvement procedure in gravity. It greatly improves the potential of the method because it address the issue of coordinate dependence in the procedure of RG-improvement. It thus greatly enlarges the potential application of the method (besides simple spacetimes). I think the paper should be published in your journal. I recommend its publication after the following points have been clarified. A) The structure of the induced energy-momentum tensor is not discussed. In particular the violation of the weak-energy condition or dominant energy conditions produced by the metric of the type (33). B) What is the causal structure of the resulting spacetime? does this depend on the choice of the K_i? C) What is the explicit form of the resulting metric near r=0? (in particular it seems that the limiting curvature hypothesis by ref.3 is automatically satisfied in this case).
In fact in ref.16 the RG improvement was obtained by means of the proper-distance of a free-falling observer along the radial geodesic. But the latter behaves as (the inverse) of the Coulomb component of the Weyl tensor. It seems to me that near the singularity all the RG-improvements are equivalent/universal. Can the author comment on this?
A minor note: the first use of eq.(3) in the context of RG-improvement has appeared in ref., see in particular eq.(15), which is precisely eq.(3) of the paper. I would recommend to mention this in the main text. (note that the Petrov classification has been used, instead of the one presented in this work).