Ricci Flow Conjugated Initial Data Sets for Einstein Equations

TL;DR

This paper introduces a Ricci flow-based conjugation method for initial data sets in Einstein's equations, establishing entropy functionals, mode expansion, and stability properties, enabling comparison and averaging of data sets in general relativity.

Contribution

It develops a novel Ricci flow conjugation framework for Einstein data sets, including entropy analysis, spectral mode expansion, and stability insights, advancing geometric analysis in relativity.

Findings

Existence of entropy functionals with monotonicity properties

Development of a spectral mode expansion for data comparison

Ricci flow conjugation preserves energy conditions and enables localized averaging

Abstract

We discuss a natural form of Ricci--flow conjugation between two distinct general relativistic data sets given on a compact $n\geq 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense. We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge--DeRham operator. This mode expansion allows to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data. In particular this shows that Ricci flow conjugation entails a form of $L^2$ averaging of one data set with respect to the other with a number of desiderable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals which allow the analysis of its entropic stability.