Kaluza–Klein gravity and cosmology emerging from G. Perelman’s entropy functionals and quantum geometric information flows (original) (raw)
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2021
We elaborate on quantum geometric information flows, QGIFs, and emergent (modified) Einstein-Maxwell and Kaluza-Klein, KK, theories formulated in Lagrange-Hamilton and general covariant variables. There are considered nonholonomic deformations of Grigory Perelman's F- and W-functionals (originally postulated for Riemannian metrics) for describing relativistic geometric flows, gravity and matter field interactions, and associated statistical thermodynamic systems. We argue that the concept of Perelman W-entropy presents more general and alternative possibilities to characterize geometric flow evolution, GIF, and gravity models than the Bekenstein-Hawking and another area-holographic type entropies. Formulating the theory of QGIFs, a set of fundamental geometric, probability, and quantum concepts, and methods of computation, are reconsidered for curved spacetime and (relativistic) phase spaces. Such generalized metric-affine spaces are modeled as nonholonomic Lorentz manifolds, (c...
eb 2 00 1 Gravitational Entropy and Quantum Cosmology
2001
We investigate the evolution of different measures of “Gravitational Entropy” in Bianchi type I and Lemâitre-Tolman universe models. A new quantity behaving in accordance with the second law of thermodynamics is introduced. We then go on and investigate whether a quantum calculation of initial conditions for the universe based upon the Wheeler-DeWitt equation supports Penrose’s Weyl Curvature Conjecture, according to which the Ricci part of the curvature dominates over the Weyl part at the initial singularity of the universe. The theory is applied to the Bianchi type I universe models with dust and a cosmological constant and to the Lemâitre-Tolman universe models. We investigate two different versions of the conjecture. First we investigate a local version which fails to support the conjecture. Thereafter we construct a non-local entity which shows more promising behaviour concerning the conjecture.
arXiv (Cornell University), 2019
We elaborate on quantum geometric information flows, QGIFs, and emergent (modified) Einstein-Maxwell and Kaluza--Klein, KK, theories formulated in Lagrange-Hamilton and general covariant variables. There are considered nonholonomic deformations of Grigory Perelman's F- and W-functionals (originally postulated for Riemannian metrics) for describing relativistic geometric flows, gravity and matter field interactions, and associated statistical thermodynamic systems. We argue that the concept of Perelman W-entropy presents more general and alternative possibilities to characterize geometric flow evolution, GIF, and gravity models than the Bekenstein--Hawking and another area--holographic type entropies. The geometric and classical and quantum thermodynamics methods allow us to understand and describe important classical and quantum physical properties of more general classes of exact solutions in modified gravity and geometric flow theories. Formulating the theory of QGIFs, a set of fundamental geometric, probability and quantum concepts, and methods of computation, are reconsidered for curved spacetime and (relativistic) phase spaces. Such generalized metric-affine spaces are modelled as nonholonomic Lorentz manifolds, (co) tangent Lorentz bundles and associated vector bundles. Using geometric and entropic and thermodynamic values, we define QGIF versions of the von Neumann entropy, relative and conditional entropy, mutual information etc. There are analyzed certain important inequalities and possible applications of G. Perelman and related entanglement and Rényi entropies to theories of KK QGIFs and emergent gravitational and electromagnetic interactions.
The European Physical Journal C
Using double 2 + 2 and 3 + 1 nonholonomic fibrations on Lorentz manifolds, we extend the concept of W-entropy for gravitational fields in general relativity (GR). Such F-and W-functionals were introduced in the Ricci flow theory of three dimensional (3-d) Riemannian metrics by Perelman (the entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159). Non-relativistic 3-d Ricci flows are characterized by associated statistical thermodynamical values determined by Wentropy. 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 field theories (relativistic continuum and hydrodynamic models) without an underlying kinetic description, which will be elaborated in other work. 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 spec-Sergiu I. Vacaru: two DAAD fellowship affiliations.
2013
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 * Address for contact: Flat 4 Brefney house, Fleet street, Ashton-under-Lyne, OL6 7PG, the UK † two DAAD fellowship visiting affiliations in Germany, where the paper was performed 1 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.
Early universe thermostatistics in curved momentum spaces
Physical review, 2016
The theories known as doubly special relativity are introduced in order to take into account an observer-independent length scale and the speed of light in the framework of special relativity. These theories can be generally formulated on the de Sitter and also recently proposed anti-de Sitter momentum spaces. In the context of these theories, we study the statistical mechanics and to do this, we consider the natural measure on the corresponding extended phase space. The invariant measure on the space of distinct microstates is obtained by restriction of the natural measure of the extended phase space to the physical phase space through the disintegration theorem. Having the invariant measure, one can study the statistical mechanics in an arbitrary ensemble for any doubly special relativity theory. We use the constructed setup to study the statistical properties of four doubly special relativity models. Applying the results to the case of early universe thermodynamics, we show that one of these models that is defined by the cosmological coordinatization of anti-de Sitter momentum space, implies a finite total number of microstates. Therefore, without attribution to any ensemble density and quite generally, we obtain entropy and internal energy bounds for the early radiation dominated universe. We find that while these results cannot be supported by the standard Friedmann equations, they indeed are in complete agreement with the nonsingular effective Friedmann equations that arise in the context of loop quantum cosmology.
Annals of Physics, 2020
We investigate gravity models emerging from nonholonomic (subjected to non-integrable constraints) Ricci flows. Considering generalizations of G. Perelman's entropy functionals, relativistic geometric flow equations, nonholonomic Ricci soliton and equivalent (modified) Einstein equations are derived. There are studied nonholonomic configurations which allow explicit modeling of entropic scenarios for gravity and dark matter (in the E. Verlinde approach and/or other variants). It is shown that using the anholonomic frame deformation method, the systems of nonlinear partial differential equations for geometric flow evolution of nonlinear stationary gravitations systems can be decoupled and integrated in general forms. In this and a series of partner works, we elaborate on stationary models of emergent gravity with quasi-periodic gravitational, matter fields and dark energy/matter structure. Such configurations cannot be described thermodynamically using the concept of Bekenstein-Hawking entropy if area-entropy, holographic or duality relations are not involved. Nevertheless, generalizing G. Perelman statistic thermodynamic approach to models of relativistic Ricci flows and nonholonomic solitons, we can compute respective thermodynamic variables for all types of gravitational and matter field configurations and their geometric evolution. Nonholonomic deformations of the F-and W-entropy considered and relativistic thermodynamic models are studied in more general cases when physically important solutions with quasi-periodic and pattern forming structure are found in modified gravity theories (MGT) and general relativity (GR).
2021
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 nonl...
Quantum cosmic models and thermodynamics
Classical and Quantum Gravity, 2008
The current accelerating phase of the evolution of the universe is considered by constructing most economical cosmic models that use just general relativity and some dominating quantum effects associated with the probabilistic description of quantum physics. Two of such models are explicitly analyzed. They are based on the existence of a sub-quantum potential and correspond to a generalization of the spatially flat exponential model of de Sitter space. The thermodynamics of these two cosmic solutions is discussed, using the second principle as a guide to choose which among the two is more feasible. The paper also discusses the relativistic physics on which the models are based, their holographic description, some implications from the classical energy conditions, and an interpretation of dark energy in terms of the entangled energy of the universe. PACS numbers: 95.36.+x, 98.80.-k
The European Physical Journal Plus
We elaborate on statistical thermodynamics models of relativistic geometric flows as generalizations of G. Perelman and R. Hamilton theory centred around C. Carathéodory axiomatic approach to thermodynamics with Pfaffian differential equations. The anholonomic frame deformation method, AFDM, for constructing generic off-diagonal and locally anisotropic cosmological solitonic solutions in the theory of relativistic geometric flows and general relativity is developed. We conclude that such solutions can not be described in terms of the Hawking-Bekenstein thermodynamics for hypersurface, holographic, (anti) de Sitter and similar configurations. The geometric thermodynamic values are defined and computed for nonholonomic Ricci flows, (modified) Einstein equations, and new classes of locally anisotropic cosmological solutions encoding solitonic hierarchies.