Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we
extend the concept of W-entropy for gravitational fields in the general
relativity, GR, theory. Such F- and W-functionals were introduced in the Ricci
flow theory of three dimensional, 3-d, Riemannian metrics by G. Perelman,
arXiv: math.DG/0211159. Nonrelativistic 3-d Ricci flows are characterized by
associated statistical thermodynamical values determined by W--entropy.
Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are
considered for models with local thermodynamical equilibrium and separation of
dissipative and non-dissipative processes in relativistic hydrodynamics. The
approach is elaborated in the framework of classical filed theories
(relativistic continuum and hydrodynamic models) without an underlying kinetic
description which will be elaborated in other works. The 3+1 splitting allows
us to provide a general relativistic definition of gravitational entropy in the
Lyapunov-Perelman sense. It increases monotonically as structure forms in the
Universe. We can formulate a thermodynamic description of exact solutions in GR
depending, in general, on all spacetime coordinates. A corresponding 2+2
splitting with nonholonomic deformation of linear connection and frame
structures is necessary for generating in very general form various classes of
exact solutions of the Einstein and general relativistic geometric flow
equations. Finally, we speculate on physical macrostates and microstate
interpretations of the W-entropy in GR, geometric flow theories and possible
connections to string theory (a second unsolved problem also contained in
Perelman's works) in the Polyakov's approach.Comment: latex2e, v4 is an accepted to EPJC substantial extension of a former
letter type paper on 10 pages to a research article on 41 pages; a new author
added, the paper's title and permanent and visiting affiliations were
correspondingly modified; and new results, conclusions and references are
provide