Stationarity of extremum entropy fluid bodies in general relativity

Schiffrin, Joshua

doi:10.1088/0264-9381/32/18/185011

Cited by 8 publications

(12 citation statements)

References 26 publications

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“…This is confirmed by the fact that for k → 0 the Wilson model is equivalent to a polytrope of index n = 7/2 > 3 that is canonically unstable (the corresponding gaseous spheres are dynamically unstable with respect to the Euler-Poisson equation) while being microcanonically stable (see Ref [131]. for more details) 23. Actually, this remains a conjecture because it has not been proven mathematically that isothermal spheres become unstable after the turning point of fractional binding energy.…”

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confidence: 84%

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Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Chavanis

2020

Eur. Phys. J. Plus

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We study the statistical mechanics of classical self-gravitating systems confined within a box of radius R in general relativity. It has been found that the caloric curve T∞(E) has the form of a double spiral whose shape depends on the compactness parameter ν = GN m/Rc 2 . The double spiral shrinks as ν increases and finally disappears when νmax = 0.1764. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as ν decreases. Using a normalization Λ = −ER/GN 2 m 2 and η = GN m 2 /RkBT∞ appropriate to the nonrelativistic limit, and considering ν → 0, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization M = GM/Rc 2 and B = Rc 4 /GN kBT∞ appropriate to the ultrarelativistic limit, and considering ν → 0, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the selfgravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONIn a preceding paper [1] (Paper I), relying on previous works on the subject [2-25], we have developed a general formalism to study the statistical mechanics of self-gravitating systems in general relativity. This formalism is valid for an arbitrary form of entropy. The statistical equilibrium state is obtained by maximizing the entropy S at fixed mass-energy M c 2 and particle number N . The extremization problem yields the Tolman-Oppenheimer-Volkoff (TOV) equations [2,26] expressing the condition of hydrostatic equilibrium together with the Tolman-Klein relations [2,27]. The precise form of entropy determines the distribution function and the equation of state of the system. For illustration, in Paper I, we considered a system of self-gravitating fermions. In that case, the statistical equilibrium state is obtained by maximizing the Fermi-Dirac entropy at fixed mass-energy and particle number. This yields the relativistic Fermi-Dirac distribution function. In the present paper, we consider the statistical mechanics of classical self-gravitating particles. 1 In that case, the statistical equilibrium state is obtained by maximizing the Boltzmann entropy at fixed mass-energy and particle number. This yields the relativistic Maxwell-Boltzmann (or Maxwell-Juttner) distribution function. We start by a brief history of the subject giving an exhaustive list of references. 2 The statistical mechanics of nonrelativistic classical self-gravitating systems was developed in connection with the dynamical evolution of globular clusters (see...

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confidence: 84%

“…( 24) and (25) determine the Lagrange multipliers β(r) and α(r), or the thermodynamical variables T (r) and µ(r), in terms of (r) and n(r). Substituting the Maxwell-Juttner distribution function (23) into Eq. ( 1), and using Eq.…”

Section: Statistical Mechanics Of General Relativistic Classical Part...mentioning

confidence: 99%

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Chavanis

2020

Eur. Phys. J. Plus

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“…The equation that describes hydrostatic equilibrium in General Relativity is called the Tolman-Oppenheimer-Volkoff equation (TOV equation). It may be derived from Einstein's equations for a perfect gas [9,10] and expresses a maximum entropy state [11][12][13][14][15][16].…”

Section: Tolman-oppenheimer-volkoff Equationmentioning

confidence: 99%

Gravitational Instability Caused by the Weight of Heat

Roupas

2019

Symmetry

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Thermal energy points towards a disordered, completely uniform state act to counter gravity's tendency to generate order and structure through gravitational collapse. It is, therefore, expected to contribute to the stabilization of a self-gravitating, classical ideal gas over collapse. However, I identified an instability that always occurs at sufficiently high energies: the high-energy or relativistic gravothermal instability. I argue here that this instability presents an analogous core-halo structure as its Newtonian counterpart, the Antonov instability. The main difference is that in the former case the core is dominated by the gravitation of thermal energy and not rest mass energy. A relativistic generalization of Antonov's instability-the low-energy gravothermal instability-also occurs. The two turning points, which make themselves evident as a double spiral of the caloric curve, approach each other as relativistic effects become more intense and eventually merge in a single point. Thus, the high and low-energy cases may be realized as two aspects of a single phenomenon-the gravothermal instability-which involves a core-halo separation and an intrinsic heat flow. Finally, I argue that the core formed during a core-collapse supernova is subject to the relativistic gravothermal instability if it becomes sufficiently hot and compactified at the time of the bounce. In this case, it will continue to collapse towards the formation of a black hole.

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“…Gao extended their work to an arbitrary perfect fluid in a static spherical geometry [16], in which they only used some thermodynamical relations. After that, this principle has been widely studied by the researchers [17][18][19][20][21][22]. Their results showed the consistency between the ordinary thermodynamics of fluid and gravitational dynamics.…”

Section: Introductionmentioning

confidence: 92%

Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory

Jiang,

Fang,

Gao

2021

Preprint

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Thermodynamics plays an important role in gravitational theories. It is a principle independent of the gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system in a general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of perfect fluid inside a compact region for fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. This means that the entropy principle of the fluid stores the same information as the gravitational equation in a static configuration. Our proof is universal and holds for any diffeomorphism-covariant purely gravitational theories, such as Einstein gravity, f (R) gravity, Lovelock gravity, f (Gauss-Bonnet) gravity and Einstein-Weyl gravity. Our result shows the consistency between the ordinary thermodynamics and the gravitational dynamics.

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Stationarity of extremum entropy fluid bodies in general relativity

Cited by 8 publications

References 26 publications

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Gravitational Instability Caused by the Weight of Heat

Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory

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