Global One--Dimensional Quantum General Relativity is the toy model with nontrivial field theoretical content, describing classical one-dimensional massive bosonic fields related to any 3+1 metric, where the dimension is a volume of three-dimensional embedding. In fact it constitutes the midisuperspatial Quantum Gravity model. We use one-particle density operator method in order to building macrostates thermodynamics related with any 3+1 metric. Taking the Boltzmann gas limit, which is given by the energy equipartition law for the Bose-Einstein gas of space quantum states generated from the Bogoliubov vacuum, we receive consistent with General Relativity thermodynamical degrees of freedom number. It confirms that the proposed Quantum Gravity toy model has well-defined classical limit in accordance with classical gravity theory.
Statistical mechanics explains thermodynamics in terms of (quantum) mechanics by equating the entropy of a microstate of a closed system with the logarithm of the number of microstates in the macrostate to which it belongs, but the question `what is a macrostate?' has never been answered except in a vague, subjective, way. However Hawking's discovery of black hole evaporation led to a formula for black hole entropy with no subjective element. In this letter, we argue from this result, together with the assumption that `black hole thermodynamics is just ordinary thermodynamics applied to black …
We study thermodynamics of an ideal gas in Doubly Special Relativity. New type of special functions (which we call Incomplete Modified Bessel functions) emerge. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that DSR thermodynamics is non-perturbative in the SR and massless limits. A stiffer equation of state is found.
The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed. The main building block of the proper relativistic extension of the classical thermodynamics laws is the four-temperature vector β, which plays a major role in the quantum framework and defines a very convenient hydrodynamic frame. The general relativistic thermodynamic equilibrium condition demands βto be a Killing vector field. We show that a remarkable consequence is that all Lie derivatives of all physical observables along the four-temperature flow must then vanish.
Black holes have often provided profound insights into the nature of gravity and the structure of space-time. The study of the mathematical properties of black objects is a major research theme of contemporary theoretical physics. This review presents a comprehensive survey of the various versions of the first law and second law of black hole mechanics in general relativity and beyond. The emphasis is to understand how these laws can constrain the physics beyond general relativity.
We study the black hole thermodynamics and obtain the correction terms for temperature, entropy, and heat capacity of the Schwarzschild black hole, resulting from the commutation relations in the framework of {\it Modified Generalized Uncertainty Principle} suggested by {\it Doubly Special Relativity}.
This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the 'infinite' speed of propagation of thermal disturbances in a dissipative fluid. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a rel…
In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. In the case of scalar fields, a relationship between mass and temperature is also worked out.
The black hole entropy formula applied to local Rindler horizon at each spacetime point has been used in the literature to derive the Einstein field equation as an equation of state of a thermodynamical system of spacetime. In the present paper we argue that due to the key role of causal structure and discrete spacetime in this approach the natural framework is unimodular relativity rather than general relativity. It is shown that the equation of state is trace free unimodular relativity field equation that uniquely determines only the traceless stress tensor. Recent generalization to nonequil…
A general Hamiltonian approach to black hole thermodynamics is used to study entropy and conserved charges for Kerr-AdS solutions in general relativity. These thermodynamic variables are first consistently defined by choosing suitable boundary conditions, and then, they are shown to satisfy the first law of black hole dynamics
There ought to exist a description of quantum field theory which does not depend on an external classical time. To achieve this goal, in a recent paper we have proposed a non-commutative special relativity in which space-time and matter degrees of freedom are treated as classical matrices with arbitrary commutation relations, and a space-time line element is defined using a trace. In the present paper, following the theory of Trace Dynamics, we construct a statistical thermodynamics for the non-commutative special relativity, and show that one arrives at a generalized quantum dynamics in which…
As a point of departure it is suggested that Quantum Cosmology is a topological concept independent from metrical constraints. Methods of continuous topological evolution and topological thermodynamics are used to construct a cosmological model of the present universe, using the techniques based upon Cartan's theory of exterior differential systems. Thermodynamic domains, which are either Open, Closed, Isolated, or in Equilibrium, can be put into correspondence with topological systems of Pfaff topological dimension 4, 3, 2 and 1. If the environment of the universe is assumed to be a physical …
In [1], the thermal equilibrium of static, spherically symmetric perfect fluids in General Relativity was studied. I would like to elaborate three points relevant to the results of [1]. The first point is only a clarification, summarized in theorem 1 below, of results that appear in [1]. The following two points correct the error in [1], stating that the condition for thermodynamic stability, found in [1], is referring to the microcanonical ensemble, while it was referring to the canonical one. In theorems 2 and 3, specific cases for which equivalence of dynamical and thermodynamic stability h…
Doubly special relativity (DSR), with both an invariant velocity and an invariant length scale, elegantly preserves the principle of relativity between moving observers, and appears as a promising candidate of the quantum theory of gravity. We study the modifications of photon gas thermodynamics in the framework of DSR with an invariant length $|λ|$, after properly taking into account the effects of modified dispersion relation, upper bounded energy-momentum space, and deformed integration measure. We show that with a positive $λ$, the grand partition function, the energy density, the specific…
We critically revisit the definition of thermal equilibrium, in its operational formulation, provided by standard thermodynamics. We show that it refers to experimental conditions which break the covariance of the theory at a fundamental level and that, therefore, it cannot be applied to the case of moving bodies. We propose an extension of this definition which is manifestly covariant and can be applied to the study of isolated systems in special relativity. The zeroth law of thermodynamics is, then, proven to establish an equivalence relation among bodies which have not only the same tempera…
The deep connection between gravitational dynamics and horizon thermodynamics leads to several intriguing features both in general relativity and in Lanczos-Lovelock theories of gravity. Recently in arXiv:1312.3253 several additional results strengthening the above connection have been established within the framework of general relativity. In this work we provide a generalization of the above setup to Lanczos-Lovelock gravity as well. To our expectation it turns out that most of the results obtained in the context of general relativity generalize to Lanczos-Lovelock gravity in a straightforwa…
An effective action is obtained for the area and mass aspect of a thin shell of radiating self-gravitating matter. On following a mini-superspace approach, the geometry of the embedding space-time is not dynamical but fixed to be either Minkowski or Schwarzschild inside the shell and Vaidya in the external space filled with radiation. The Euler-Lagrange equations of motion are discussed and shown to entail the expected invariance of the effective Lagrangian under time-reparametrization. They are equivalent to the usual junction equations and suggest a macroscopic quasi-static thermodynamic des…
We construct a generally-covariant formulation of non-equilibrium thermodynamics in General Relativity. We find covariant entropic forces arising from gradients of the entropy density, and a corresponding non-conservation of the energy momentum tensor in terms of these forces. We also provide a Hamiltonian formulation of General Relativity in the context of non-equilibrium phenomena and write the Raychaudhuri equations for a congruence of geodesics. We find that a fluid satisfying the strong energy condition could avoid collapse for a positive and sufficiently large entropic-force contribution…
In \cite{Bahamonde:2019zea}, a spherically symmetric black hole (BH) was derived from the quadratic form of $f(T)$. Here we derive the associated energy, invariants of curvature, and torsion of this BH and demonstrate that the higher-order contribution of torsion renders the singularity weaker compared with the Schwarzschild BH of general relativity (GR). Moreover, we calculate the thermodynamic quantities and reveal the effect of the higher--order contribution on these quantities. Therefore, we derive a new spherically symmetric BH from the cubic form of $f(T)=T+ε\Big[\frac{1}{2}αT^2+\frac{1}…
In this paper we carefully reexamine the various framworks existing in the field of relativistic thermodynamics. We scrutinize in particular the different conceptual foundations of notions like the relativistic work, heat force, moving heat and relativistic temperature. As to the latter notion we argue that, as in ordinary thermodynamics, relativistic absolute temperature should be introduced operationally via relativistic Carnot processes. We exhibit the more implicit or even hidden tacit preassumptions being made and point to a couple of gaps, errors and inconclusive statements in some of th…
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 …
In four space-time dimensions General Relativity can be non-trivially deformed. Deformed theories continue to describe two propagating degrees of freedom, as GR. We study Euclidean black hole thermodynamics in these deformations. We use the recently developed formulation that works with SO(3) connections as well as certain matrices M of auxiliary fields. We show that the black hole entropy is given by one quarter of the horizon area as measured by the Lie algebra valued two-form MF, where F is the connection curvature. This coincides with the horizon area as measured by the metric only for the…
On the basis of horizon thermodynamics we study the thermodynamic stability of black holes constructed in general relativity and Gauss-Bonnet gravity. In the framework of horizon thermodynamics there are only five thermodynamic variables $E,P,V,T,S$. It is not necessary to consider concrete matter fields, which may contribute to the pressure of black hole thermodynamic system. In non-vacuum cases, we can derive the equation of state, $P=P(V,T)$. According to the requirements of stable equilibrium in conventional thermodynamics, we start from these thermodynamic variables to calculate the heat …
I show that the gravitational dynamics in a bulk region of space can be connected to a thermodynamic description in the boundary of that region, thereby providing clear physical interpretations of several mathematical features of classical general relativity: (1) The Noether charge contained in a bulk region, associated with a specific time evolution vector field, has a direct thermodynamic interpretation as the gravitational heat content of the boundary surface. (2) This result, in turn, shows that all static spacetimes maintain holographic equipartition; in these spacetimes, the number of de…
Thermodynamic acceptability of spherically symmetric perfect-fluid solutions in general relativity
Al-Farabi Kazakh National University（阿拉伯·法拉比哈萨克国立大学）；Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Dipartimento di Fisica and ICRA, Università di Roma “La Sapienza”（罗马第一大学物理系及ICRA）
We investigate the stability of motion and the thermodynamics in the case of spherically symmetric solutions in $f(T)$ gravity using the perturbative approach. We consider small deviations from general relativity and we extract charged black hole solutions for two charge profiles, namely with or without a perturbative correction in the charge distribution. We examine their asymptotic behavior, we extract various torsional and curvature invariants, and we calculate the energy and the mass of the solutions. Furthermore, we study the stability of motion around the obtained solutions, by analyzing…
A particular framework for Quantum Gravity is the Doubly Special Relativity (DSR) formalism that introduces a new observer independent scale, the Planck energy. Our aim in this paper is to study the effects of this energy upper bound in relativistic thermodynamics. We have explicitly computed the modified equation of state for an ideal fluid in the DSR framework. In deriving our result we exploited the scheme of treating DSR as a non-linear representation of the Lorentz group in Special Relativity.
We discuss head-on collisions of neutron stars and disks of dust ("galaxies") following the ideas of equilibrium thermodynamics, which compares equilibrium states and avoids the description of the dynamical transition processes between them. As an always present damping mechanism, gravitational emission results in final equilibrium states after the collision. In this paper we calculate selected final configurations from initial data of colliding stars and disks by making use of conservation laws and solving the Einstein equations. Comparing initial and final states, we can decide for which ini…
In this thesis, the connections between thermodynamics and general relativity are explored. We review the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. We then move to a more fundamental question: can we still defin…
Black hole thermodynamics of the Kerr-AdS spacetime in three-dimensional general relativity is analyzed using a Hamiltonian approach. The values of the conserved charges and entropy, obtained by a proper treatment of the AdS asymptotic conditions, are shown to satisfy the first law of black hole dynamics.
We consider the thermodynamic properties of the constant curvature black hole solution recently found by Banados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are no…
We analyze the compatibility between the geometrodynamics and thermodynamics of a binary mixture of perfect fluids which describe inhomogeneous cosmological models. We generalize the thermodynamic scheme of general relativity to include the chemical potential of the fluid mixture with non-vanishing entropy production. This formalism is then applied to the case of Szekeres and Stephani families of cosmological models. The compatibility conditions turn out to impose symmetry conditions on the cosmological models in such a way that only the limiting case of the Friedmann-Robertson-Walker model re…
Physical spacetime geometry follows from some effective thermodynamics of quantum states of all fields and particles described in frames of General Relativity. In the sense of pure field theoretical Einstein's point of view on gravitation the thermodynamic information is actually quantum gravity. We propose new realization of this old idea by studying the canonical 3+1 Dirac-ADM approach to pseudo-Riemannian (Lorentzian) manifold of General Relativity. We derive the Wheeler-DeWitt theory as the Global One-Dimensional classical field theory of the Bose field associated with embedded 3-space, wh…
The formulation of a dynamical theory of General Relativity, including matter, is viewed as a problem of coupling Einstein's theory of pure gravity, formulated as an action principle, to an independently chosen and well defined field theory of matter. It is well known that this is accomplished in a most natural way when the matter theory is formulated as a relativistic, Lagrangian field theory. Special matter models of this type have been available; here a thermodynamical model that allows for general flows is used. A problem that is of even older date, one that was pursued vigorously by leadi…
There is an intriguing analogy between the gravitational dynamics of the horizons and thermodynamics. In case of general relativity, as well as for a wider class of Lanczos-Lovelock theories of gravity, it is possible to interpret the field equations near any spherically symmetric horizon as a thermodynamic identity TdS = dE + PdV. We study this approach further and generalize the results to two more generic cases within the context of general relativity: (i) stationary axis-symmetric horizons and (ii) time dependent evolving horizons. In both the cases, the near horizon structure of Einstein …
It has been shown in an earlier work [arXiv:1303.1535] that there exists a pair of canonically conjugate variables $(f^{ab},N^a_{bc})$ in general relativity which also act as thermodynamically conjugate variables on any horizon. In particular their variations $(f^{bc}δN^a_{bc}, N^a_{bc}δf^{bc})$, which occur in the surface term of the Einstein-Hilbert action, when integrated over a null surface, have direct correspondence with $(SδT,TδS)$ where $(T,S)$ are the temperature and entropy. We generalize these results to Lanczos-Lovelock models in this paper. We identify two such variables in Lanczo…
Analytical solutions of nonminimally coupled scalar field cosmology corresponding to critical scalar field values constitute a potential challenge to the recent first-order thermodynamics of scalar-tensor gravity (a formalism picturing general relativity as the zero-temperature equilibrium state for modified gravity). The critical solutions are unstable with respect to homogeneous perturbations, hence unphysical.
Black holes in general relativity coupled with NEDs surrounded by PFDM: thermodynamics, epicyclic oscillations, QPOs, and shadow
The Assam Royal Global University（阿萨姆皇家全球大学）；New Uzbekistan University（新乌兹别克斯坦大学）；Institute of Fundamental and Applied Research, National Research University TIIAME（基础与应用研究所，国立研究大学TIIAME）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Tashkent State Technical University（塔什干国立技术大学）
We consider the stability of black holes within both classical general relativity and the semiclassical thermodynamic description. In particular, we study linearised perturbations and their contribution to the gravitational partition function, addressing technical issues for charged (Reissner-Nordstrom) and rotating (Kerr-AdS) black holes. Exploring the connection between classical and thermodynamic stability, we find classical instabilities of Myers-Perry black holes and bifurcations to new black hole families.
The thermodynamic equilibrium condition for a static self-gravitating fluid in the Einstein theory is defined by the Tolman-Ehrenfest temperature law, $T{\sqrt {g_{00}(x^{i})}} = constant$, according to which the proper temperature depends explicitly on the position within the medium through the metric coefficient $g_{00}(x^{i})$. By assuming the validity of Tolman-Ehrenfest "pocket temperature", Klein also proved a similar relation for the chemical potential, namely, $μ{\sqrt {g_{00}(x^{i})}} = constant$. In this letter we prove that a more general relation uniting both quantities holds regar…
In this paper, we study thermostatistical properties of a photon gas in the framework of two deformed special relativity models defined by the cosmological coordinatizations of the de Sitter (dS) and anti-de Sitter (AdS) momentum spaces. The dS model is a doubly special relativity theory in which an ultraviolet length scale is invariant under the deformed Lorentz transformations. For the case of AdS model, however, the Lorentz symmetry breaks at the high energy regime. We show that the existence of a maximal momentum in dS momentum space leads to maximal pressure and temperature at the thermod…
We revisit the analogy between a minimally coupled scalar field in general relativity and a perfect fluid, correcting previous identifications of effective temperature and chemical potential. This provides a useful complementary picture for the first-order thermodynamics of scalar-tensor gravity, paving the way for the Einstein frame formulation (which eluded previous attempts) and raises interesting questions to further develop the analogy.
We study the problem of heat conduction in general relativity by using Carter's variational formulation. We write the creation rates of the entropy and the particle as combinations of the vorticities of temperature and chemical potential. We pay attention to the fact that there are two additional degrees of freedom in choosing the relativistic analog of Cattaneo equation for the parts binormal to the caloric and the number flows. Including the contributions from the binormal parts, we find a $\textit{new}$ heat-flow equations and discover their dynamical role in thermodynamic systems. The bene…
We investigate whether inertial thermometers moving in a thermal bath behave as being hotter or colder. This question is directly related to the classical controversy concerning how temperature transforms under Lorentz transformations. Rather than basing our arguments on thermodynamical hypotheses, we perform straightforward calculations in the context of relativistic quantum field theory. For this purpose we use Unruh-DeWitt detectors, since they have been shown to be reliable thermometers in semi-classical gravity. We believe that our discussion helps in definitely clarifying this issue.
It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as a universal form $dE = TdS + WdV$ at apparent horizon, where $E$ and $V$ are the matter energy and volume inside the apparent horizon (the energy $E$ is the same as the Misner-Sharp energy in the case of Einstein general relativity), $W=(ρ-P)/2$ is the work density and $ρ$ and $P$ are energy density and pressure of the matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon has associated entropy $S= A/4G$ and temperature $T = κ/ 2π$ in Einst…
We examine the consistency of the thermodynamics of irrotational and non-isentropic perfect fluids complying with matter conservation by looking at the integrability conditions of the Gibbs-Duhem relation. We show that the latter is always integrable for fluids of the following types: (a) static, (b) isentropic (admits a barotropic equation of state), (c) the source of a spacetime for which $r\ge 2$, where $r$ is the dimension of the orbit of the isometry group. This consistency scheme is tested also in two large classes of known exact solutions for which $r< 2$, in general: perfect fluid Szek…
A promising theory in modifying general relativity by violating the ordinary energy-momentum conservation law in curved spacetime is the Rastall theory of gravity. In this theory, geometry and matter fields are coupled to each other in a non-minimal way. Here, we study thermodynamic properties of some black hole solutions in this framework, and compare our results with those of general relativity. We demonstrate how the presence of these matter sources may amplify effects caused by the Rastall parameter in thermodynamic quantities. Our investigation also shows that black holes with radius smal…
This SpringerBrief is based on a masters course on black hole thermodynamics and the black hole information problem taught by Dieter Lüst during the summer term 2017 at the Ludwig-Maximilians-Universität in Munich; it was written by Ward Vleeshouwers. It provides a short introduction to general relativity, which describes gravity in terms of the curvature of space-time, and examines the properties of black holes. These are central objects in general relativity which arise when sufficient energy is compressed into a finite volume, so that even light cannot escape its gravitational pull. We will…
Some very interesting solutions of the field equations of Einstein's general theory of relativity have been constructed in the framework of nonlinear electrodynamics. In particular, magnetically charged black hole solutions in the framework of exponential nonlinear electrodynamics have been obtained in general relativity in Kruglov (2017). Using this approach a magnetically charged non-singular black hole spacetime in the framework of exponential electrodynamics in some modified theory of gravity has been constructed in this letter. The metric describing asymptotic non-singular magnetized blac…
During the past 30 years, research in general relativity has brought to light strong hints of a very deep and fundamental relationship between gravitation, thermodynamics, and quantum theory. The most striking indication of such a relationship comes from black hole thermodynamics, where it appears that certain laws of black hole mechanics are, in fact, simply the ordinary laws of thermodynamics applied to a system containing a black hole. This article will review the present status of black hole thermodynamics and will discuss some of the related unresolved issues concerning gravitation, therm…
