On the basis of the balance equations for energy-momentum, spin, particle and entropy density, an approach is considered which represents a comparatively general framework for special- and general-relativistic continuum thermodynamics. In the first part of the paper, a general entropy density 4-vector, containing particle, energy-momentum, and spin density contributions, is introduced which makes it possible, firstly, to judge special assumptions for the entropy density 4-vector made by other authors with respect to their generality and validity and, secondly, to determine entropy supply and entropy production. Using this entropy density 4-vector, in the second part, material-independent equilibrium conditions are discussed. While in literature, at least if one works in the theory of irreversible thermodynamics assuming a Riemann space-time structure, generally thermodynamic equilibrium is determined by introducing a variety of conditions by hand, the present approach proceeds as follows: For a comparatively wide class of space-time geometries the necessary equilibrium conditions of vanishing entropy supply and entropy production are exploited and, afterwards, supplementary conditions are assumed which are motivated by the requirement that thermodynamic equilibrium quantities have to be determined uniquely.
Non-equilibrium and equilibrium thermodynamics of an interacting component of a special-relativistic multi-component system is discussed by using an entropy identity. The special case of the corresponding free component is considered.
Non-equilibrium and equilibrium thermodynamics of an interacting component in a special-relativistic multi-component system is discussed by use of an entropy identity. The special case of the corresponding free component is considered. Equilibrium conditions and especially the multi-component Killing relation of the 4-temperature are discussed. Two axioms characterize the mixture: additivity of the energy momentum tensors and of the 4-entropies of the components generating those of the mixture. The resulting quantities of a component and of the mixture, energy, energy flux, momentum flux, stre…
We study the thermodynamic properties of the RN-AdS black hole in full phase space and propose a generalized thermodynamic identity. As an example, we use it to find relations of thermodynamical coefficients between the grand canonical and canonical ensembles. We also show, for the first order phase transition, that the usual Maxwell's equal area law should be extended to a new form for the RN-AdS black hole.
Relativistic irreversible thermodynamics is reformulated following the conventional approach proposed by Meixner in the non-relativistic case. Clear separation between mechanical and non-mechanical energy fluxes is made. The resulting equations for the entropy production and the local internal energy have the same structure as the non-relativistic ones. Assuming linear constitutive laws, it is shown that consistency is obtained both with the laws of thermodynamics and causality.
The search for thermodynamic admissibility moreover reveals a fundamental difference between liquids and gases in relativistic fluid dynamics, as the reversible convection mechanism is much simpler for liquids than for gases. In relativistic fluid mechanics, positive entropy production is known to be insufficient for guaranteeing stability. Much more restrictive criteria for thermodynamic admissibility have become available in nonequilibrium thermodynamics. We here perform a linear stability analysis for a model of relativistic hydrodynamics that is based on the GENERIC (general equation for t…
We introduce a class of relativistic fluid states satisfying the relativistic local thermodynamical equilibrium postulate (abbreviated as relativistic (LTE) postulate). States satisfying this postulate, are states "near equilibrium" (a term defined precisely in the course of the paper) and permit us to attach a fictitious "local thermodynamical equilibrium" state that fits event by event the actual fluid state. They single out an admissible class of rest frames relative to which thermodynamical variables like the energy density, thermodynamical pressure, stresses, particle number density (or d…
A general thermodynamic treatment of dissipative relativistic fluids is introduced, where the temperature four vector is not parallel to the velocity field of the fluid. Generic stability and kinetic equilibrium points out a particular thermodynamics, where the temperature vector is parallel to the enthalpy flow vector and the choice of the flow fixes the constitutive functions for viscous stress and heat. The linear stability of the homogeneous equilibrium is proved in a mixed particle-energy flow-frame.
The balance equations of energy-momentum and spin together with Einstein's field equations are investigated by the Liu procedure to find constraints for the constitutive equations in such a way that the Second Law is satisfied. Special cases such as spinless systems and curvature insensitive materials are shortly discussed.
We present a relativistic model describing a thin disk system composed of two fluids. The system is surrounded by a halo in the presence of a non-trivial electromagnetic field. We show that the model is compatible with the variational multi-fluid thermodynamics formalism, allowing us to determine all the thermodynamic variables associated with the matter content of the disk. The asymptotic behaviour of these quantities indicates that the single fluid interpretation should be abandoned in favour of a two-fluid model.
We consider the grand canonical thermodynamics of a noninteracting scalar field in a static spacetime. We take the nonrelativistic limit of thermodynamic quantities in a way that leaves the curved structure of the background geometry intact. Using Mellin transform and heat kernel techniques we obtain asymptotic expansions of thermodynamic quantities appropriate for the analysis of Bose-Einstein condensation. We apply our results to investigate gravitational effects on the Bose-Einstein condensation for a scalar field in a finite volume. We also analyze the boundary effects on the depletion coe…
We show that the covariant Raychaudhuri identity describing kinematic characteristics of space-time admits a representation involving a geometrical scalar $ξ$ which, depending on circumstances, might be related to, e.g., relativistic temperature or cosmological scalar field. With an appropriately chosen spacetime deformation tensor (fixing the symmetry of a problem under consideration), such scalarization opens a wide scope for physical applications. We consider few such applications including dynamics of cosmological (anti)scalar background, non-variational deduction of the field equations, s…
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 …
Non-equilibrium and equilibrium thermodynamics of an interacting component in a relativistic multi-component system is discussed covariantly by exploiting an entropy identity. The special case of the corresponding free component is considered. Equilibrium conditions and especially the multi-component Killing relation of the 4-temperature are discussed. Two axioms characterize the mixture: additivity of the energy momentum tensors and additivity of the 4-entropies of the components generating those of the mixture. The resulting quantities of a single component and of the mixture as a whole, ene…
A generally relativistic theory of thermodynamics is developed, based on four main physical principles: heat is a local form of energy, therefore described by a thermal energy tensor; conservation of mass, equivalent to conservation of heat, or the local first law; entropy is a local current; and non-destruction of entropy, or the local second law. A fluid is defined by the thermostatic energy tensor being isotropic. The entropy current is related to the other fields by certain equations, including a generalised Gibbs equation for the thermostatic entropy, followed by linear and quadratic term…
Previous work has demonstrated that the gravitational field equations in all Lanczos-Lovelock models imply a thermodynamic identity TdS=dE+PdV (where the variations are interpreted as changes due to virtual displacement along the affine parameter) in the near-horizon limit in static spacetimes. Here we generalize this result to any arbitrary null surface in an arbitrary spacetime and show that certain components of the Einstein's equations can be expressed in the form of the above thermodynamic identity. We also obtain an explicit expression for the thermodynamic energy associated with the nul…
This article extends the single-fluid relativistic irreversible thermodynamics theory of {\it M{ü}ller}, {\it Israel} and {\it Stewart} (hereafter the {MIS} theory) to a multi-fluid system with inherent species interactions. This is illustrated in a two-fluid toy-model where an effective complex 4-velocity plays the role of a primary dynamical parameter. We find that an observer who resides in the {\it real}-part of this universe will notice that their knowledge of the universe parametrized using {\it real}, rather than {\it imaginary}, quantities are insufficient to fully determine properties…
The recent monumental detection of gravitational waves by LIGO, the subsequent detection by the LIGO/VIRGO observatories of a binary neutron star merger seen in the gravitational wave signal $GW170817$,the first photo of the event horizon of the supermassive black hole at the center of the $M87$ galaxy released by the EHT telescope and the ongoing experiments on Relativistic Heavy Ion Collisions at the BNL and at the CERN, demonstrate that we are witnessing the second golden era of observational relativistic gravity. These new observational breakthroughs, although in the long run would influen…
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…
A unified first law of black-hole dynamics and relativistic thermodynamics is derived in spherically symmetric general relativity. This equation expresses the gradient of the active gravitational energy E according to the Einstein equation, divided into energy-supply and work terms. Projecting the equation along the flow of thermodynamic matter and along the trapping horizon of a blackhole yield, respectively, first laws of relativistic thermodynamics and black-hole dynamics. In the black-hole case, this first law has the same form as the first law of black-hole statics, with static perturbati…
It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near…
In this paper we simulate a two dimensional relativistic ideal gas by implementing a relativistic elastic binary collision algorithm. We show that the relativistic gas faithfully obeys Jüttner's speed distribution function. Furthermore, using this numeric simulation in conjunction with the energy equipartition theorem for a relativistic gas, we conclude that a moving system appears cooler.
It has been shown three different views in relativistic thermodynamics can be derived from the basic formulation proposed by van Kampen and Israel. The way to decompose energy-momentum into the reversible and irreversible parts is not uniquely determined, and different choices result in different views. The effect of difference in the definition of a finite volume is also considered.
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）
Extended theories are widely used in the literature to describe relativistic fluids. The motivation for this is mostly due to the causality issues allegedly present in the first order in the gradients theories. However, the decay of fluctuations in the system is also at stake when first order theories that couple heat with acceleration are used. This paper shows that although the introduction of the Maxwell-Cattaneo equation in the description of a simple relativistic fluid formally eliminates the generic instabilities identified by Hiscock and Lindblom in 1985, the hypothesis on the order of …
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.
A classical (non-quantum-mechanical) relativistic ideal gas in thermodynamic equilibrium in a uniformly accelerated frame of reference is studied using Gibbs's microcanonical and grand canonical formulations of statistical mechanics. Using these methods explicit expressions for the particle, energy and entropy density distributions are obtained, which are found to be in agreement with the well known results of the relativistic formulation of Boltzmann's kinetic theory. Explicit expressions for the total entropy, total energy and rest mass of the gas are obtained. The position of the center of …
A kinetic theory of relativistic gases in a two-dimensional space is developed in order to obtain the equilibrium distribution function and the expressions for the fields of energy per particle, pressure, entropy per particle and heat capacities in equilibrium. Furthermore, by using the method of Chapman and Enskog for a kinetic model of the Boltzmann equation the non-equilibrium energy-momentum tensor and the entropy production rate are determined for a universe described by a two-dimensional Robertson-Walker metric. The solutions of the gravitational field equations that consider the non-equ…
We discuss the concept of local thermodynamical equilibrium in relativistic hydrodynamics in flat spacetime in a quantum statistical framework without an underlying kinetic description, suitable for strongly interacting fluids. We show that the appropriate definition of local equilibrium naturally leads to the introduction of a relativistic hydrodynamical frame in which the four-velocity vector is the one of a relativistic thermometer at equilibrium with the fluid, parallel to the inverse temperature four-vector β, which then becomes a primary quantity. We show that this frame is the most appr…
We investigate cosmological density perturbations in a covariant and gauge-invariant formalism, incorporating relativistic causal thermodynamics to give a self-consistent description. The gradient of density inhomogeneities splits covariantly into a scalar part, equivalent to the usual density perturbations, a rotational vector part that is determined by the vorticity, and a tensor part that describes the shape. We give the evolution equations for these parts in the general dissipative case. Causal thermodynamics gives evolution equations for viscous stress and heat flux, which are coupled to …
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…
We present a new approach to the problem of the thermodynamical equilibrium of a quantum relativistic fluid in a curved spacetime in the limit of small curvature. We calculate the mean value of local operators by expanding the four-temperature Killing vector field in Riemann normal coordinates about the same spacetime point and we derive corrections with respect to the flat spacetime expressions. Thereby, we clarify the origin of the terms proportional to Riemann and Ricci tensors introduced in general hydrodynamic expansion of the stress-energy tensor.
In a relativistic context, the main purpose of Extended Irreversible Thermodynamics (EIT) is to generalize the principles of non-equilibrium thermodynamics to the domain of fluid dynamics. In particular, the theory aims at modelling any diffusion-type process (like heat as diffusion of energy, viscosity as diffusion of momentum, charge-conductivity as diffusion of particles) directly from thermodynamic laws. Although in Newtonian physics this task can be achieved with a first-order approach to dissipation (i.e. Navier-Stokes-Fourier like equations), in a relativistic framework the relativity o…
We present an extension of relativistic single-particle distribution function for weakly interacting particles at local thermodynamical equilibrium including spin degrees of freedom, for massive spin 1/2 particles. We infer, on the basis of the global equilibrium case, that at local thermodynamical equilibrium particles acquire a net polarization proportional to the vorticity of the inverse temperature four-vector field. The obtained formula for polarization also implies that a steady gradient of temperature entails a polarization orthogonal to particle momentum. The single-particle distributi…
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.
The approach of extended irreversible thermodynamics consists of promoting the dissipative fluxes to non-equilibrium thermodynamic variables. In a relativistic context, this naturally leads to the formulation of the theory of Israel and Stewart (1979), which is, to date, one of the most successful theories for relativistic dissipation. Although the generality of the principle makes it applicable to any dissipative fluid, a connection of the Israel-Stewart theory with microphysics has been established, through kinetic theory, only for the case of ideal quantum gases. By performing a convenient …
In discussing fundamentals of general-relativistic irreversible continuum thermodynamics, this theory is shown to be characterized by the feature that no thermodynamical degrees of freedom are ascribed to gravitation. However, accepting that black hole thermodynamics seems to oppose this harmlessness of gravitation one is called on consider other approaches. Therefore, in brief some gravitational and thermodynamical alternatives are reviewed.
It is shown that the description of a relativistic fluid at local thermodynamic equilibrium depends on the particular quantum stress-energy tensor operator chosen, e.g., the canonical or symmetrized Belinfante stress-energy tensor. We argue that the Belinfante tensor is not appropriate to describe a relativistic fluid whose macroscopic polarization relaxes slowly to thermodynamic equilibrium and that a spin tensor, like the canonical spin tensor, is required. As a consequence, the description of a polarized relativistic fluid involves an extension of relativistic hydrodynamics including a new …
Equations of gravity when projected on spacetime horizons resemble Navier-Stokes equation of a fluid with a specific equation of state [1-3]. We show that this equation of state describes massless ideal relativistic gas. We use these results, and build an explicit and simple molecular model of the fluid living on the Schwarzschild and Reissner-Nordström black hole horizons. For the spin zero Bose gas, our model makes two predictions: (i) The horizon area/entropy is quantized as given by Bekenstein's quantization rule, (ii) The model explains the correct type of proportionality between horizon …
The linear stability of the homogeneous equilibrium of non-relativistic fluids with mass flux and special relativistic fluids with the absolute value of the energy vector as internal energy is investigated. It is proved that the equilibrium is asymptotically stable in both cases due to purely thermodynamic restrictions; the only requirements are the thermodynamic stability and the nonnegativity of the transport coefficients.
It is shown that the so-called generic instabilities that appear in the framework of relativistic linear irreversible thermodynamics, describing the fluctuations of a simple fluid close to equilibrium, arise due to the coupling of heat with hydrodynamic acceleration which appears in Eckart's formalism of relativistic irreversible thermodynamics. Further, we emphasize that such behavior should be interpreted as a contradiction to the postulates of linear irreversible thermodynamics (LIT), namely a violation of Onsager's hypothesis on the regression of fluctuations, and not as fluid instabilitie…
We investigate the quantum thermodynamical properties of localised relativistic quantum fields, and how they can be used as quantum thermal machines. We study the efficiency and power of energy transfer between the classical gravitational degrees of freedom, such as the energy input due to the motion of boundaries or an impinging gravitational wave, and the excitations of a confined quantum field. We find that the efficiency of energy transfer depends dramatically on the input initial state of the system. Furthermore, we investigate the ability of the system to extract energy from a gravitatio…
I argue that the field equations of any theory of gravity which is diffeomorphism invariant must be expressible as a thermodynamic identity, TdS=dE around any event in the spacetime. This fact can be demonstrated explicitly (and rather easily) if: (a) one accepts the Noether current of the theory as providing the definition for local entropy density and (b) one is allowed to introduce the local notions of a Rindler frame, acceleration horizon and a Killing vector (related to translation in Rindler time) around any event. It is conceptually incorrect - in general - to invert this argument and o…
In the light of recent results relating dynamic and thermodynamic stability of relativistic stars and black holes, we re-examine the relationship between "turning points"---i.e., extrema of thermodynamic variables along a one-parameter family of solutions---and instabilities. We give a proof of Sorkin's general result---showing the existence of a thermodynamic instability on one side of a turning point---that does not rely on heuristic arguments involving infinite dimensional manifold structure. We use the turning point results to prove the existence of a dynamic instability of black rings in …
We analyse the thermodynamics of a quantum system in a trajectory of constant velocity that interacts with a static thermal bath. The latter is modeled by a massless scalar field in a thermal state. We consider two different couplings of the moving system to the heat bath, a coupling of the Unruh-DeWitt type and a coupling that involves the time derivative of the field. We derive the master equation for the reduced dynamics of the moving quantum system. It has the same form with the quantum optical master equation, but with different coefficients that depend on velocity. This master equation h…
It is shown that different pairs of stress-energy and spin tensors of quantum relativistic fields related by a pseudo-gauge transformation, i.e. differing by a divergence, imply different mean values of physical quantities in thermodynamical nonequilibrium situations. Most notably, transport coefficients and the total entropy production rate are affected by the choice of the spin tensor of the relativistic quantum field theory under consideration. Therefore, at least in principle, it should be possible to disprove a fundamental stress-energy tensor and/or to show that a fundamental spin tensor…
Thermodynamic coefficients in third-order relativistic fluid dynamics
Nelson Mandela University（纳尔逊·曼德拉大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（理论与计算科学国家研究所）；University of Johannesburg（约翰内斯堡大学）
The internal layers of neutron stars are expected to contain several superfluid components that can significantly affect their dynamics. The description of such objects should rely on hydrodynamic models in which it is possible to unambiguously assign the value of the thermodynamic variables from microscopic calculations of the properties of matter. In this work we consider the phenomenological approach to multifluids modelling championed by Carter and, studying the relaxation of the system towards equilibrium, we assign a precise thermodynamic interpretation to its variables. We show that in …
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 …
Motivated by Aoki et al.'s recent research on conserved charges and entropy current, we investigated the conservation of relativistic Ertel's current, which has received little attention outside the field of geophysical fluid dynamics. Ertel's charge is an important indicator of the correlation between vortex vectors and entropy gradient fields in Earth's meridional heat transport. We first show that in the generalized Hamiltonian structure of baroclinic fluids, the duality between the total energy and the Casimir as a function of Ertel's charge plays an important role in the nonrelativistic c…
The relativistic transformation rule for temperature is a subject under debate for more than 110 years. Several incompatible proposals exist in the literature, but a final resolution is still missing. In this work, we reconsider the problem of relativistic transformation rules for a number of thermodynamic parameters, including temperature, chemical potential, pressure, entropy and enthalpy densities for a relativistic perfect fluid using relativistic kinetic theory. The analysis is carried out in a fully relativistic covariant manner, and the explicit transformation rules for the above quanti…
