https://doi.org/10.1140/epjc/s10052-017-4712-1
Regular Article - Theoretical Physics
On relativistic generalization of Perelman’s W-entropy and thermodynamic description of gravitational fields and cosmology
1
Heinrich-Wieland-Str. 182, 81735, Munich, Germany
2
National College of Iaşi, 4 Arcu street, 700115, Iasi, Romania
3
Quantum Gravity Research, 101 S. Topanga Canyon Blvd # 1159, Topanga, CA, 90290, USA
4
University “Al. I. Cuza” Iasi, Project IDEI, 700115, Iasi, Romania
5
Flat 4, Brefney house, Fleet street, Ashton-under-Lyne, Lancashire, OL6 7PG, UK
6
Max-Planck-Institute for Physics, Werner-Heisenberg-Institute, Foehringer Ringer 6, 80805, Munich, Germany
7
Institute for Theoretical Physics, Leibniz University of Hannover, Applestrasse 2, 30167, Hannover, Germany
* e-mail: sergiu.vacaru@gmail.com
Received:
27
December
2015
Accepted:
27
February
2017
Published online:
23
March
2017
Using double 
and 
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 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 field theories (relativistic continuum and hydrodynamic models) without an underlying kinetic description, which will be elaborated in other work. The 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 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 work) in Polyakov’s approach.
© The Author(s), 2017
