arXiv subjects

Gerhard Rein

Publications and source records attributed to Gerhard Rein.

At least 19 recordsLinked to original sources

arXiv2024

Steady states of the spherically symmetric Vlasov-Poisson system as fixed points of a mass-preserving algorithm

We give a new proof for the existence of spherically symmetric steady states to the Vlasov-Poisson system, following a strategy that has been used successfully to approximate axially symmetric solutions numerically, both to the Vlasov-Poisson system and to the Einstein-Vlasov system. There are several reasons why a mathematical analysis of this numerical scheme is important. A generalization of the present result to the case of flat axially symmetric solutions would prove that the steady states obtained numerically in \cite{AR3} do exist. Moreover, in the relativistic case the question whether a steady state can be obtained by this scheme seems to be related to its dynamical stability. This motivates the desire for a deeper understanding of this strategy.

math.AP↗

arXiv2024

Oppenheimer-Snyder type collapse for a collisionless gas

In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf.~ \cite{OS}. In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein-Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlev\'{e}-Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework.

gr-qc↗

arXiv2024

Quantitative phase mixing for Hamiltonians with trapping

We prove quantitative decay estimates of macroscopic quantities generated by the solutions to linear transport equations driven by a general family of Hamiltonians. The associated particle trajectories are all trapped in a compact region of phase-space and feature a non-degenerate elliptic stagnation point. The analysis covers a large class of Hamiltonians generated by the radially symmetric compactly supported equilibria of the gravitational Vlasov-Poisson system. Working in radial symmetry, our analysis features both the 1+2-dimensional case and the harder 1+1-dimensional case, where all the particles have the same value of the modulus of angular momentum. The latter case is also of importance in both the plasma physics case and two dimensional incompressible fluid flows.

math.AP↗

arXiv2023

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

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.

gr-qc↗

arXiv2023

Damping versus oscillations for a gravitational Vlasov-Poisson system

We consider a family of isolated inhomogeneous steady states to the gravitational Vlasov-Poisson system with a point mass at the centre. They are parametrised by the polytropic index $k>1/2$, so that the phase space density of the steady state is $C^1$ at the vacuum boundary if and only if $k>1$. We prove the following sharp dichotomy result: if $k>1$ the linear perturbations Landau damp and if $1/2< k\le1$ they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with $k>1$ is the first such result for the gravitational Vlasov-Poisson system. The key step in the proof is to show that no embedded eigenvalues exist in the essential spectrum of the linearised system.

math.AP↗

arXiv2022

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

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.

math.AP↗

arXiv2021

Strong Lagrangian solutions of the (relativistic) Vlasov-Poisson system for non-smooth, spherically symmetric data

We prove a local existence and uniqueness result for the non-relativistic and relativistic Vlasov-Poisson system for data which need not even be continuous. The corresponding solutions preserve all the standard conserved quantities and are constant along their pointwise defined characteristic flow so that these solutions are suitable for the stability analysis of not necessarily smooth steady states. They satisfy the well-known continuation criterion and are global in the non-relativistic case. The only unwanted requirement on the data is that they be spherically symmetric.

math.AP↗

arXiv2021

Collisionless equilibria in general relativity: stable configurations beyond the first binding energy maximum

We numerically study the stability of collisionless equilibria in the context of general relativity. More precisely, we consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in Schwarzschild and in maximal areal coordinates. Our results provide strong evidence against the well-known binding energy hypothesis which states that the first local maximum of the binding energy along a sequence of isotropic steady states signals the onset of instability. We do however confirm the conjecture that steady states are stable at least up to the first local maximum of the binding energy. For the first time, we observe multiple stability changes for certain models. The equations of state used are piecewise linear functions of the particle energy and provide a rich variety of different equilibria.

gr-qc↗

arXiv2021

On the existence of linearly oscillating galaxies

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

math.AP↗

arXiv2020

A numerical stability analysis for the Einstein-Vlasov system

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.

gr-qc↗

arXiv2020

On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states

If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator $\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In the Vlasov-Poisson case we also determine the kernel of this operator.

math-ph↗

arXiv2018

Stability and instability of self-gravitating relativistic matter distributions

We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift $\kappa>0$. We prove their linear instability when $\kappa$ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich \& Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, \& Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

gr-qc↗

arXiv2017

The asymptotic behavior of solutions to the repulsive $n$-body problem

The $n$-body problem with a purely repulsive Coulomb interaction is considered. It is shown that for large times $t$ the distance between any two particles grows linearly in $t$. The trajectory of each particle is asymptotically a straight line with a fixed velocity which is different for different particles.

math.CA↗

arXiv2016

Oscillating solutions of the Vlasov-Poisson system -- A numerical investigation

Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov-Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along one-parameter families of polytropic steady states we establish an Eddington-Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.

math-ph↗

arXiv2016

Mass-radius spirals for steady state families of the Vlasov-Poisson system

We consider spherically symmetric steady states of the Vlasov-Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley-Dickens, and related models this mass-radius relation takes the form of a spiral.

math-ph↗

arXiv2015

Gravitational collapse and the Vlasov-Poisson system

A self-gravitating homogeneous ball of a fluid with pressure zero where the fluid particles are initially at rest collapses to a point in finite time. We prove that this gravitational collapse can be approximated arbitrarily closely by suitable solutions of the Vlasov-Poisson system which are known to exist globally in time.

math-ph↗

arXiv2015

On the rotation curves for axially symmetric disk solutions of the Vlasov-Poisson system

A large class of flat axially symmetric solutions to the Vlasov-Poisson system is constructed with the property that the corresponding rotation curves are approximately flat, slightly decreasing or slightly increasing. The rotation curves are compared with measurements from real galaxies and satisfactory agreement is obtained. These facts raise the question whether the observed rotation curves for disk galaxies may be explained without introducing dark matter. Furthermore, it is shown that for the ansatz we consider stars on circular orbits do not exist in the neighborhood of the boundary of the steady state.

astro-ph.GA↗

arXiv2014

On the small redshift limit of steady states of the spherically symmetric Einstein-Vlasov system and their stability

Families of steady states of the spherically symmetric Einstein-Vlasov system are constructed, which are parametrized by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov-Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analyzed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

gr-qc↗