General symmetries: From homogeneous thermodynamics to black holes (original) (raw)
Related papers
Quasi-homogeneous thermodynamics and black holes
Journal of Mathematical Physics, 2003
We propose a generalized thermodynamics in which quasi-homogeneity of the thermodynamic potentials plays a fundamental role. This thermodynamic formalism arises from a generalization of the approach presented in paper [1], and it is based on the requirement that quasi-homogeneity is a non-trivial symmetry for the Pfaffian form δQrev. It is shown that quasi-homogeneous thermodynamics fits the thermodynamic features of at least some self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is suggested by black hole thermodynamics. Then, some existing results involving self-gravitating systems are also shortly discussed in the light of this thermodynamic framework. The consequences of the lack of extensivity are also recalled. We show that generalized Gibbs-Duhem equations arise as a consequence of quasi-homogeneity of the thermodynamic potentials. An heuristic link between this generalized thermodynamic formalism and the thermodynamic limit is also discussed.
The geometry of thermodynamics
2008
We present a review of the main aspects of geometrothermodynamics, an approach which allows us to associate a specific Riemannian structure to any classical thermodynamic system. In the space of equilibrium states, we consider a Legendre invariant metric, which is given in terms of the fundamental equation of the corresponding thermodynamic system, and analyze its geometric properties in the case of the van der Waals gas, and black holes. We conclude that the geometry of this particular metric reproduces the thermodynamic behavior of the van der Waals gas, and the Reissner-Nordstr\"om black hole, but it is not adequate for the thermodynamic description of Kerr black holes.
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).
Convexity and symmetrization in relativistic theories
Continuum Mechanics and Thermodynamics, 1990
There is a strong motivation for the desire to have symmetric hyperbolic field equations in thermodynamics, because they guarantee well-posedness of Cauchy problems. A generic quasi-linear first order system of balance laws --in the non-relativistic case --can be shown to be symmetric hyperbolic, if the entropy density is concave with respect to the variables. In relativistic thermodynamics this is not so. This paper shows that there exists a scalar quantity in relativistic thermodynamics whose concavity guarantees a symmetric hyperbolic system. But that quantity --we call it --h --is not the entropy, although it is closely related to it. It is formed by contracting the entropy flux vector --h a with a privileged time-like congruence ~'~.
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.
Thermodynamic geometry demystified
2019
Following our earlier works [1,2], we introduce a new simple metric form for thermodynamic geometry. The new thermodynamic geometry (NTG) indicates correctly a one-to-one correspondence between curvature singularities and phase transitions. For a non-homogeneous thermodynamic potential, by considering a phantom RN-AdS black hole, the NTG formalism represents a one-to-one correspondence between curvature singularities and phase transitions. Working with the NTG metric neatly excludes all unphysical points that were generated in other geometric formulations of thermodynamics such as the geometrothermodynamics (GTD). We show that the NTG is conformally related to GTD. However, the conformal transformation is singular at unphysical points that were generated by the GTD.
On geometry of phenomenological thermodynamics
2018
We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symplectic geometry, and argue that it catches its geometric essence in a more profound and clearer way than the popular odd-dimensional contact structure description. Among the advantages are a number of conceptual clarifications: the geometric role of internal energy (not made as an independent variable), the lattice of potentials, and the gauge interpretation of the theory.
Cornell University - arXiv, 2022
This paper systematically explores new classes of black hole (BH) solutions in nonassociative and noncommutative gravity by focusing on features that generalize to higher dimensions. The theories we study are modelled on (co) tangent Lorentz bundles (i.e. eight dimensional phase spaces) with a star product structure determined by R-flux deformations in string theory. The nonassociative vacuum Einstein equations involve real and complex effective sources with coefficients which are proportional to the Planck and string constants or their products. We develop the anholonomic frame and connection deformation method, and prove that such systems of nonlinear partial differential equations can be decoupled and integrated in general form for a generic off-diagonal ansatz for symmetric and non-symmetric metrics as well as for generalized (non) linear connection structures. The coefficients of such metrics may depend on all phase space coordinates (space-time coordinates plus energy-momentum). Conditions are given when the generating and integration functions and effective sources define two classes of physically important exact and parametric solutions with R-flux sources related via nonlinear symmetries to effective cosmological constants: (1) 6D Tangherlini BHs, which are star product and R-flux distorted to quasi-stationary configurations and 8D black ellipsoids (Bes) and BHs; (2) nonassocitative space-time and co-fiber space double BH and/or BE configurations generalizing Schwarzschild-de Sitter metrics. We argue that the concept of Bekenstein-Hawking entropy is applicable only for very special classes of nonassociative BHs encoding conventional horizons and (anti) de Sitter configurations. Finally, we show how analogs of the relativistic G. Perelman W-entropy and related geometric thermodynamic variables can be defined and computed for general classes of off-diagonal solutions encoding nonassociative R-flux deformations.