Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2010-06-08
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
74 pages, 22 figures added, submitted version. The paper has been vastly expanded with a new detailed introduction and with ad
Scientific paper
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.
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