Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows

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.

A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime

“…The idea of a geometry having an entropy is counter-intuitive but is a consequence of the systematic consideration of the complementary nature of information and entropy that has become accepted since Shannon's seminal introduction of his "information entropy" [2]. We should note that the concept of a "geometric entropy" is already well accepted in quantum gravity contexts (see, for example, Vacaru et al [3]) but is not usually treated as an intrinsic property of geometrical structures, although perhaps Quevedo's "Geometrothermodynamics" [4] is an exception. We should also note not only that Wang et al [5] have recently specifically investigated the entropy production of certain natural growth processes but also that Pearson et al [6] have recently shown that a chronometer's accuracy is proportional to its entropy production.…”

The Entropy Production of Galaxies

“…The idea of a geometry having an entropy is counter-intuitive, but is a consequence of the systematic consideration of the complementary nature of information and entropy that has become accepted since Shannon's seminal introduction of his "information entropy" [2]. We should note that the concept of a "geometric entropy" is already well accepted in quantum gravity contexts (see for example Vacaru et al [3]), but is not usually treated as an intrinsic property of geometrical structure, although perhaps Quevedo's "Geometrothermodynamics" [4] is an exception. We should also note that Wang et al [5] have recently specifically investigated the entropy production of certain natural growth processes.…”