Physics of Relativistic Perfect Fluids (original) (raw)
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Relativistic perfect fluids in local thermal equilibrium
General Relativity and Gravitation, 2017
Every evolution of a fluid is uniquely described by an energy tensor. But the converse is not true: an energy tensor may describe the evolution of different fluids. The problem of determining them is called here the inverse problem. This problem may admit unphysical or non-deterministic solutions. This paper is devoted to solve the inverse problem for perfect energy tensors in the class of perfect fluids evolving in local thermal equilibrium (l.t.e.). The starting point is a previous result (Coll and Ferrando in J Math Phys 30: 2918-2922, 1989) showing that thermodynamic fluids evolving in l.t.e. admit a purely hydrodynamic characterization. This characterization allows solving this inverse problem in a very compact form. The paradigmatic case of perfect energy tensors representing the evolution of ideal gases is studied in detail and some applications and examples are outlined.
Relativistic thermodynamics of perfect fluids
Cornell University - arXiv, 2022
The relativistic continuity equations for the extensive thermodynamic quantities are derived based on the divergence theorem in Minkowski space outlined by Stückelberg. This covariant approach leads to a relativistic formulation of the first and second laws of thermodynamics. The internal energy density and the pressure of a relativistic perfect fluid carry inertia, which leads to a relativistic coupling between heat and work. The relativistic continuity equation for the relativistic inertia is derived. The relativistic corrections in the Euler equation and in the continuity equations for the energy and momentum are identified. This relativistic theoretical framework allows a rigorous derivation of the relativistic transformation laws for the temperature, the pressure and the chemical potential based on the relativistic transformation laws for the energy density, the entropy density, the mass density and the number density.
On the thermodynamics of simple non-isentropic perfect fluids in general relativity
Classical and Quantum Gravity, 1995
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 ≥ 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 Szekeres solutions (classes I and II). In none of these cases, the Gibbs-Duhem relation is integrable, in general, though specific particular cases of Szekeres class II (all complying with r < 2) are identified for which the integrability of this relation can be achieved. We show that Szekeres class I solutions satisfy the integrability conditions only in two trivial cases, namely the spherically symmetric limiting case and the Friedman-Roberson-Walker (FRW) cosmology. Explicit forms of the state variables and equations of state linking them are given explicitly and discussed in relation to the FRW limits of the solutions. We show that fixing free parameters in these solutions by a formal identification with FRW parameters leads, in all cases examined, to unphysical temperature evolution laws, quite unrelated to those of their FRW limiting cosmologies.
Temperature Evolution Law of Imperfect Relativistic Fluids
General Relativity and Gravitation, 2002
The first-order general relativistic theory of a generic dissipative (heatconducting, viscous, particle-creating) fluid is rediscussed from a unified covariant frame-independent point of view. By generalizing some previous works in the literature, we derive a formula for the temperature variation rate, which is valid both in Eckart's (particle) and in the Landau-Lifshitz (energy) frames. Particular attention is paid to the case of gravitational particle creation and its possible cross-effect with the bulk viscosity mechanism. * Electronic address: raimundo@dfte.ufrn.br † Electronic address:
Variational aspects of relativistic field theories, with application to perfect fluids
Annals of Physics, 1977
By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully cova.riant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor Py. By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for FPV and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) + (c)l. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einstein's equations having Iw4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Euler's equations. We discuss their meaning and give what we consider to be the "minimally constrained" principle. A discussion of one constraint, "preservation of particle identity," from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.
The Thermodynamics for Relativistic Multi-Fluid Systems. (arXiv:1906.01331v3 [gr-qc] UPDATED)
arXiv General Relativity and Quantum Cosmology, 2020
This article extends the single-fluid relativistic irreversible thermodynamics theory of Müller, Israel and 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 real-part of this universe will notice that their knowledge of the universe parametrized using real, rather than imaginary, quantities are insufficient to fully determine properties such as the total energy density, pressure or entropy, In fact, such an observer will deduce the existence of some negative energy that affects the expansion of their perceived real universe.
The Thermodynamics of Relativistic Multi-Fluid Systems
Letters in High Energy Physics
This article extends the single-fluid relativistic irreversible thermodynamics theory of Müller, Israel and 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 real-part of this universe will notice that their knowledge of the universe parametrized using real, rather than imaginary, quantities are insufficient to fully determine properties such as the total energy density, pressure or entropy, In fact, such an observer will deduce the existence of some negative energy that affects the expansion of their perceived real universe.
The Thermodynamics for Relativistic Multi-Fluid Systems
arXiv (Cornell University), 2019
This article extends the single-fluid relativistic irreversible thermodynamics theory of Müller, Israel and 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 real-part of this universe will notice that their knowledge of the universe parametrized using real, rather than imaginary, quantities are insufficient to fully determine properties such as the total energy density, pressure or entropy, In fact, such an observer will deduce the existence of some negative energy that affects the expansion of their perceived real universe.
Dynamical systems approach to relativistic spherically symmetric static perfect fluid models
Classical and Quantum Gravity, 2003
We investigate relativistic spherically symmetric static perfect fluid models in the framework of the theory of dynamical systems. The field equations are recast into a regular dynamical system on a 3dimensional compact state space, thereby avoiding the non-regularity problems associated with the Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space thus obtained is used to derive qualitative features and to prove theorems about mass-radius properties. The perfect fluids we discuss are described by barotropic equations of state that are asymptotically polytropic at low pressures and, for certain applications, asymptotically linear at high pressures. We employ dimensionless variables that are asymptotically homology invariant in the low pressure regime, and thus we generalize standard work on Newtonian polytropes to a relativistic setting and to a much larger class of equations of state. Our dynamical systems framework is particularly suited for numerical computations, as illustrated by several numerical examples, e.g., the ideal neutron gas and examples that involve phase transitions. We will see in Sec. 3 that, e.g., C 2 is sufficient, although this restriction can be weakened. Below we show how to handle even less restrictive situations like phase transitions. 7 However, note that ρ stands for the rest-mass density in the Newtonian case. 8 The region where m(r) < 0 can be analyzed with the same dynamical systems methods that are going to be used in the following. The treatment turns even out to be considerably simpler, cf. the case of Newtonian perfect fluids . If a solution to (3) satisfies 1 − 2Gm/(rc 2 ) > 0 initially at r = r 0 for initial data (m 0 , p 0 ), then this condition holds everywhere. This has been proved (for regular solutions), e.g., in [1]. Within the dynamical systems formulation this result can be established quite easily as we will see in Sec. 4. Solutions violating the condition 1 − 2Gm/(rc 2 ) > 0 could be treated with the dynamical systems methods presented in this paper as well. However, we refrain from a discussion of such solutions here.
Journal of Mathematical Physics, 1997
The Gibbs-Duhem equation dU + pdV = TdS imposes restrictions on the perfect fluid solutions of Einstein equations that have a one-dimensional symmetry group or no symmetry at all. In this paper, we investigate the restrictions imposed on the Stephani Universe and on the two classes of models found by Szafron. Upon the Stephani Universe and the β = / 0 class of Szafron symmetries are forced. We find the most general subcases of the β = 0 model of Szafron that are consistent with the Gibbs-Duhem equation and have no symmetry.