The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

The gravitational field equations in general relativity (GR) consist of a sophisticated system of nonlinear partial differential equations. Solving such equations in some generic off-diagonal forms is usually a hard analytic or numeric task. Physically important solutions in GR were constructed using diagonal ansatz for metrics with maximum 4 independent coefficients. The Einstein equations can be solved in exact or parametric forms determined by some integration constants for corresponding assumptions on spherical or cylindric spacetime symmetries. The anholonomic frame and connection deformation method allows us to construct generic off-diagonal solutions described by 6 independent coefficients of metrics depending, in general, on all spacetime coordinates. New types of exact and parametric solutions are determined by generating and integration functions and (effective) generating sources. They may describe vacuum gravitational and matter fields solitonic hierarchies; locally anisotropic polarizations of physical constants for black holes, wormholes, black toruses, or cosmological solutions; various types of off-diagonal deformations of horizons etc. The additional degrees of freedom (related to off-diagonal coefficients) can be used to describe dark energy and dark matter configurations and elaborate locally anisotropic cosmological scenarios. In general, the generic off-diagonal solutions do not involve certain hypersurface or holographic configurations and can’t be described in the framework of the Bekenstein-Hawking thermodynamic paradigm. We argue that generalizing the concept of G. Perelman’s entropy for relativistic Ricci flows allows us to define and compute geometric thermodynamic variables for all possible classes of solutions in GR.