Collisionless Equilibria in General Relativity: Stable Configurations beyond the First Binding Energy Maximum

Günther, Sebastian; Straub, Christopher; Rein, Gerhard

doi:10.3847/1538-4357/ac0eef

Cited by 12 publications

(27 citation statements)

References 28 publications

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“…The dynamical instability of a self-gravitating sphere in the context of GR has been explored long ago by [23,24] and Zel'dovich & Podurets (1966) [25], via the study of pulsation equation and binding energy, respectively. It was found that the turning point of fractional binding energy [26,27] is very close to the result using the pulsation equation [28]. Chandrasekhar's criteron [23,24] provides a sufficient condition triggering the black hole formation.…”

Section: Introductionmentioning

confidence: 55%

On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-dimensional Spacetime

Feng¹

2021

Preprint

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In this note, I derive the Chandrasekhar instability of a fluid sphere in (N +1)-dimensional Schwarzschild-Tangherlini spacetime and take the homogeneous (uniform energy density) solution for illustration. Qualitatively, the effect of positive (negative) cosmological constant tends to destabilize (stabilize) the sphere. In the absence of cosmological constant, the privileged position of (3+1)-dimensional spacetime is manifest in its own right. As it is the marginal dimensionality in which a monatomic ideal fluid sphere is stable but not too stable to trigger the onset of gravitational collapse. Furthermore, it is the unique dimensionality that can accommodate stable hydrostatic equilibrium with positive cosmological constant. However, given the current cosmological constant observed no stable configuration can be larger than 10 20 M ⊙ . On the other hand, in (2+1) dimensions it is too stable to collapse either in the context of Newtonian Gravity (NG) or Einstein's General Relativity (GR). In GR, the role of negative cosmological constant is crucial not only to guarantee fluid equilibrium (decreasing monotonicity of pressure) but also to have the Bañados-Teitelboim-Zanelli (BTZ) solution. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk with mass 0 < M ≤ 0.5 to collapse into a naked singularity, which supports the Cosmic Censorship Conjecture. However, the relativistic instability can be triggered for a homogeneous disk with mass 0.5 < M ≲ 0.518 under causal limit, which implies that BTZ holes of mass M BTZ > 0 could emerge from collapsing fluid disks under proper conditions. The implicit assumptions and implications are also discussed.

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Section: Introductionmentioning

confidence: 55%

On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-dimensional Spacetime

Feng¹

2021

Preprint

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“…However, recent numerical evidence [20] strongly contradicts this hypothesis and shows that stability behaviors can be much more diverse than previously thought. As already stated by Ipser and Thorne [27], new versatile criteria are needed in order to gain more understanding of stability issues in general relativity.…”

Section: Steady States and Previous Stability Resultsmentioning

confidence: 89%

A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole

Günther¹,

Rein²,

Straub³

2022

Preprint

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We develop a Birman-Schwinger principle for the spherically symmetric, asymptotically flat Einstein-Vlasov system. It characterizes stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert-Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman-Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

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“…

We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [3,8,18,19], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap.

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mentioning

confidence: 99%

“…The system as stated above has been used both in analytical and numerical investigations, cf. [3,8,18,19], even though the switch to Cartesian coordinates is not done in all of these papers, and the momentum variable v is sometimes replaced by…”

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The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation

Günther¹,

Rein²

2022

KRM

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<p style='text-indent:20px;'>We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.</p>

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Collisionless Equilibria in General Relativity: Stable Configurations beyond the First Binding Energy Maximum

Cited by 12 publications

References 28 publications

On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-dimensional Spacetime

On the Dynamical Instability of Monatomic Fluid Spheres in (N+1)-dimensional Spacetime

A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole

The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation

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