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

Günther, Sebastian; Rein, Gerhard

doi:10.3934/krm.2021040

Cited by 6 publications

(6 citation statements)

References 31 publications

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“…In the singularity-free setting there exist unique, local in-time solutions for smooth, compactly supported initial data together with a continuation criterion [45,46]. Similar results are known in other coordinates, e.g., in maximal-areal coordinates [19] or maximal isotropic coordinates [50]. In the case of a Schwarzschild singularity at the center, it can be shown that the methods from [48] yield a global existence result in Schwarzschild time.…”

Section: The Einstein-vlasov System In Schwarzschild Coordinatesmentioning

confidence: 76%

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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Section: The Einstein-vlasov System In Schwarzschild Coordinatesmentioning

confidence: 76%

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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“…An analogous result holds in maximal areal coordinates, cf. [8], but probably not in Eddington-Finkelstein coordinates, at least not for general smooth data whose support contains the origin. In general, Schwarzschild or maximal areal coordinates are more useful when proving that certain data launch global, geodesically complete solutions, cf.…”

Section: Properties and Comparison Of The Coordinate Systemsmentioning

confidence: 94%

A numerical stability analysis for the Einstein-Vlasov system

Günther,

Körner,

Lebeda

et al. 2020

Preprint

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We investigate stability issues for steady states of the spherically symmetric Einstein-Vlasov system numerically in Schwarzschild, maximal areal, and Eddington-Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle from the viewpoint of our numerical results. The physical reliability of the latter is strengthened by obtaining consistent results in the three different coordinate systems and by the systematic use of dynamically accessible perturbations.

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“…It is also possible that other coordinates adapted to spherical symmetry are more suitable for the stability analysis. We will not pursue this issue but mention maximal areal coordinates as one alternative [35].…”

Section: The Ev Systemmentioning

confidence: 99%

Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas—a review

Rein

2023

Class. Quantum Grav.

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We review stability and instability results for self-gravitating matter distributions, where the matter model is a collisionless gas as described by the Vlasov equation. The focus is on the general relativistic situation, i.e., on steady states of the Einstein-Vlasov system and their stability properties. In order to put things into perspective we include the Vlasov-Poisson system and the relativistic Vlasov-Poisson system into the discussion.

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

Cited by 6 publications

References 31 publications

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

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

A numerical stability analysis for the Einstein-Vlasov system

Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas—a review

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