A numerical stability analysis for the Einstein–Vlasov system

Abstract

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.

On the Existence of Linearly Oscillating Galaxies

“…In these oscillations the spatial support of the solutions expands and contracts in a time-periodic way, i.e., after perturbation the state starts to pulse. The same behavior was observed numerically for the Vlasov-Poisson system in [56], and again for the Einstein-Vlasov system in [21].…”

“…We recall that the numerical investigations [1,21,56] show that oscillations are possible and consist of periodically repeated expansions and contractions of the configuration in (phase) space. It is important to realize that there exists a one parameter-family of explicit solutions to the non-linear Vlasov-Poisson system, which exhibit exactly this pulsating behavior, the so-called Kurth solutions [39]; for a particular parameter the solution becomes stationary.…”

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

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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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“…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…”

“…The stability question has also been analyzed numerically, cf. [8,18,19] and the references there, where in some cases maximal areal coordinates were used. It turns out that if an unstable steady state is perturbed "towards collapse", then a black hole forms.…”