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General Relativity and the Ricci Flow
by Mohammed Alzain
Submission summary
Authors (as registered SciPost users):  Mohammed Alzain 
Submission information  

Preprint Link:  scipost_202108_00019v1 (pdf) 
Date submitted:  20210810 19:44 
Submitted by:  Alzain, Mohammed 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci flow equation with the HamiltonJacobi equation for general relativity.
Current status:
Editorial decision:
For Journal SciPost Physics Core: Reject
(status: Editorial decision fixed and (if required) accepted by authors)
Submission & Refereeing History
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Reports on this Submission
Report
In this manuscript, the author first discusses the Ricci flow which is an evolution equation for the metric of a smooth manifold driven by its Ricci curvature. This flow has been used successfully to prove the Poincare conjecture in mathematics.
The author then describes several attempts to incorporate the Ricci flow in physics by modifying the Einstein equation. The modifications are totally ad hoc and the author then outlines several problems with these modifications. The author then discusses the HamitonJacobi functional for Einstein gravity, and introduces in a completely ad hoc manner, eqn (9) in the manuscript which is the evolution of this functional under a new flow parameter \lambda, the flow being driven by the scalar curvature. This is then put back in the HamiltonJacobi equation. This whole procedure is poorly motivated, and this ad hoc modification of the HamiltonJacobi equation is not even analyzed to work out how the resulting theory differs from Einstein gravity and what are its consequences. This is unsuitable for publication in SciPost Physics.
Weaknesses
The paper requires more careful work given the aim of discovering new physics.
Report
I cannot recommend the paper for publication given its ambitious claim. If one takes Equations (9) and (14) and (11) once cannot reconcile the results. R is a function of $g_{\mu \nu}$ and from the Ansatz (11) I can't see why (9) would be true, unless R is a constant, and that is a very restrictive set of geometries.
Requested changes
The author reformulate the action using a more careful analysis.
Author: Mohammed Alzain on 20210911 [id 1748]
(in reply to Report 1 on 20210909)The observation made by the referee that R must be a constant for the ansatz (11) to be true is accurate, this is indeed a very restrictive set of geometries but it must be noted that it has been proposed for the sole purpose of finding a solution. The constancy of R is a necessary requirement for the validity of the ansatz (11) but not a necessary requirement for the validity of the equation (9). Equation (9) remains valid without the requirement that R must be a constant unless the ansatz (11) is imposed.
Author: Mohammed Alzain on 20210913 [id 1749]
(in reply to Report 2 on 20210912)I would just like to point out that the old quantum theory, although empirically well motivated, was a set of ad hoc modifications. My goal was to introduce the concept of time evolution in canonical general relativity in a way that is mathematically consistent and does not violate the principles of general relativity, and the report did not indicate that there is a mathematical inconsistency or a violation of the principles of relativity.