Abstract
We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman’s functionals and related entropic values used for deriving relativistic geometric evolution flow equations. For self-similar configurations, such equations describe generalized Ricci solitons defining modified Einstein equations. We analyse possible connections between relativistic models of nonholonomic Ricci flows and emergent modified gravity theories. We prove that corresponding systems of nonlinear partial differential equations, PDEs, for entropic flows and modified gravity posses certain general decoupling and integration properties. There are constructed new classes of exact and parametric solutions for nonstationary configurations and locally anisotropic cosmological metrics in modified gravity theories and general relativity. Such solutions describe scenarios of nonlinear geometric evolution and gravitational and matter field dynamics with pattern-forming and quasiperiodic structure and various space quasicrystal and deformed spacetime crystal models. We analyse new classes of generic off-diagonal solutions for entropic gravity theories and show how such solutions can be used for explaining structure formation in modern cosmology. Finally, we speculate why the approaches with Perelman–Lyapunov type functionals are more general or complementary to the constructions elaborated using the concept of Bekenstein–Hawking entropy.
Highlights
The most inspiring ideas in recent development of the gravity theory and cosmology are on the emergent thermodynamic nature of the spacetime geometry, when the Einstein equations can be derived using area-entropy formulas for horizons of black holes, BH, [1,2,3] and from supposed elastic properties of gravity [5,6,7]
2.3.1 1-d relativistic time QC structures defined by a generating function q = q(xi, y3, τ ) subjected to the condition that it is a solution of an evolution equation with conserved dynamics
We study self-similar configurations of nonholonomic geometric flows resulting in entropic Ricci solitons
Summary
The most inspiring ideas in recent development of the gravity theory and cosmology are on the emergent thermodynamic nature of the spacetime geometry, when the Einstein equations can be derived using area-entropy formulas for horizons of black holes, BH, [1,2,3] and from supposed elastic properties of gravity [5,6,7]. 4, we develop and apply the anholonomic frame deformation method, AFDM, [32,38,40,58,63,64] in order to prove general decoupling properties and integrability of nonholonomic geometric flow and Ricci soliton equations encoding elastic and quasiperiodic spacetime and (effective) matter fields properties. Such solutions are described by generic off-diagonal metrics, and generalized connections, depending on all spacetime coordinates and temperature like parameters via general classes of generating functions and (effective) sources of entropic gravity and matter fields.
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