Ricci-flow-conjugated initial data sets for Einstein equations.(English)Zbl 1258.83008

Summary: 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 one 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 natural form of $$L^2$$ parabolic averaging of one data set with respect to the other with a number of desirable 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, that allow the analysis of its entropic stability.

MSC:

 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 53Z05 Applications of differential geometry to physics 35L15 Initial value problems for second-order hyperbolic equations 94A17 Measures of information, entropy
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