On the Asymptotic Behavior of Static Perfect Fluids

Andersson, Lars; Burtscher, Annegret

doi:10.1007/s00023-018-00758-z

Cited by 13 publications

(19 citation statements)

References 80 publications

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“…Moreover, there are cases with zero bare mass m 0 = 0 and conical singularities that do admit a transformation to spatially conformally flat coordinates (2.1) if we allow ρ 0 = 0 and possibly interpret the Bray mass using the more general definition of [9, section 3.2] via approximation of regular ones. Examples are an explicit singular solution of the astrophysically important Tolmann-Oppenheimer-Volkoff equation, studied already by Chandrasekhar [10] and others [1,35] in view of its asymptotics and discussed below, and also the Hoffmann spacetime discussed in [25,43]. Furthermore, it is clear that the main ideas developed in this paper are not restricted to static spherically symmetric spacetimes and are adaptable to more general situations.…”

Section: Discussionmentioning

confidence: 88%

Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

Burtscher

Kiessling

Tahvildar-Zadeh

2021

Class. Quantum Grav.

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The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a 'matter field' as the source of spacetime curvature, this identity implies the physical laws of energy and momentum conservation for the 'matter field'. The present work inquires into whether such a Bianchi identity can still hold in a weak sense for spacetimes with curvature singularities associated with timelike singularities in the 'matter field'. Sufficient conditions that establish a distributional version of the twice-contracted second Bianchi identity are found. In our main theorem, a large class of spherically symmetric static Lorentzian metrics with timelike one-dimensional singularities is identified, for which this identity holds. As an important first application we show that the well-known Reissner-Weyl-Nordström spacetime of a point charge does not belong to this class, but that Hoffmann's spacetime of a point charge with negative bare mass in the Born-Infeld electromagnetic vacuum does.

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Section: Discussionmentioning

confidence: 88%

Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

Burtscher

Kiessling

Tahvildar-Zadeh

2021

Class. Quantum Grav.

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“…Due to the contracted Bianchi identities, ∇ µ G µν = 0, Eq. ( 3) is a consequence of Einstein's field equations (2), so in principle it is sufficient to solve Eq. (2).…”

Section: Field Equations and Static Spherically Symmetric Ansatzmentioning

confidence: 99%

“…where (x µ ) = (t, r, ϑ, ϕ) are spherical coordinates and Φ and Ψ are functions of the radius coordinate r only which will be determined by the field Eqs. (2,3). Note that when Φ = Ψ = 0, the metric (9) reduces to the Minkowski metric in spherical coordinates.…”

Section: Field Equations and Static Spherically Symmetric Ansatzmentioning

confidence: 99%

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Static spherical perfect fluid stars with finite radius in general relativity: a review

Nambo

Sarbach

2021

Rev. Mex. Fis. E

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In this article, we provide a pedagogical review of the Tolman-Oppenheimer-Volkoff (TOV) equation and its solutions which describe static, spherically symmetric gaseous stars in general relativity. Our discussion starts with a systematic derivation of the TOV equation from the Einstein field equations and the relativistic Euler equations. Next, we give a proof for the existence and uniqueness of solutions of the TOV equation describing a star of finite radius, assuming suitable conditions on the equation of state characterizing the gas. We also prove that the compactness of the gas contained inside a sphere centered at the origin satisfies the well-known Buchdahl bound, independent of the radius of the sphere. Further, we derive the equation of state for an ideal, classical monoatomic relativistic gas from statistical mechanics considerations and show that it satisfies our assumptions for the existence of a unique solution describing a finite radius star. Although none of the results discussed in this article are new, they are usually scattered in different articles and books in the literature; hence it is our hope that this article will provide a self-contained and useful introduction to the topic of relativistic stellar models.

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“…A description of the two body problem in general relativity which goes beyond approximations is challenging in part because fairly little is known about the dynamics of extended bodies. 1 A physically relevant example of such a body is a neutron star. 2 While the question of the correct equation of state has been subject to much debate, 3 we focus here on an idealized description in the context of Christodoulou's two-phase model [9], and are interested in the dynamical stability of even one such body in spherical symmetry.…”

Section: Introductionmentioning

confidence: 99%

On “Hard Stars” in General Relativity

Fournodavlos

Schlue

2019

Ann. Henri Poincaré

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We study spherically symmetric solutions to the Einstein-Euler equations which model an idealized relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets the speed of sound equal to the speed of light. We demonstrate the existence of a 1-parameter family of static solutions, or "hard stars," and describe their stability properties: First, we show that small stars are a local minimum of the mass energy functional under variations which preserve the total number of particles. In particular, we prove that the second variation of the mass energy functional controls the "mass aspect function." Second, we derive the linearisation of the Euler-Einstein system around small stars in "comoving coordinates," and prove a uniform boundedness statement for an energy, which is exactly at the level of the variational argument. Finally, we exhibit the existence of time periodic solutions to the linearised system, which shows that energy boundedness is optimal for this problem.

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On the Asymptotic Behavior of Static Perfect Fluids

Cited by 13 publications

References 80 publications

Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

Weak second Bianchi identity for static, spherically symmetric spacetimes with timelike singularities

Static spherical perfect fluid stars with finite radius in general relativity: a review

On “Hard Stars” in General Relativity

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