Einstein-Vlasov System Research Articles

Static solutions of the spherically symmetric Vlasov–Einstein system

AbstractWe consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.

Hamiltonian Structure of the Vlasov-Einstein System and the Problem of Stability for Spherical Relativistic Star Clusters

The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics is generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function ƒ (a noncanonical variable) being the conserved ADM mass-energy H ADM. This facilitates a geometric understanding of the evolution of ƒ in an infinite-dimensional phase space, providing thereby a natural interpretation of the constraints associated with conservation of phase space. This geometric interpretation also facilitates the derivation of improved criteria for linear stability by focusing on dynamically accessible perturbations δƒ which satisfy all the constraints of phase space conservation. An explicit expression is derived for the energy δ (2) H ADM associated with an arbitrary spherical phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if δ (2) H ADM is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems. Intuition derived from simple finite models clarifies several features of the Vlasov-Einstein system; for example, how, negative energy modes preclude necessary and sufficient conditions for stability and why, unlike the Newtonian case, the existence of negative energy perturbations for some static, isotropic equilibrium apparently signals the Onset of a linear instability. An Appendix exhibits the construction of a completely covariant bracket which generates the Vlasov-Einstein system for arbitrary configurations in a form independent of any assumed 3 + 1 splitting.

A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

In a previous paper two of the authors (G. R. and A. D. R.) showed that there exist global, classical solutions of the spherically symmetric Vlasov-Einstein system for small initial data. The present paper continues this investigation and allows also large initial data. It is shown that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The result adds weight to the conjecture that cosmic censorship holds if one replaces dust as matter model for which naked singularities do form by a collisionless gas described by the Vlasov equation. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.

The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system

It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition. A central role in the proof is played by energy estimates in unweighted Sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation.

The Newtonian limit of the spherically symmetric Vlasov-Einstein system

We prove that spherically symmetric solutions of the Vlasov-Einstein system with a fixed initial value converge to the corresponding solution of the Vlasov-Poisson system if the speed of lightc is taken as a parameter and tends to infinity. The convergence is uniform on compact time intervals with convergence rate 1/c 2. Thus the classical Vlasov-Poisson system appears as the Newtonian limit of the general relativistic Vlasov-Einstein system in a spherically symmetric setting.

Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data

We show that global asymptotically flat singularity-free solutions of the spherically symmetric Vlasov-Einstein system exist for all initial data which are sufficiently small in an appropriate sense. At the same time detailed information is obtained concerning the asymptotic behaviour of these solutions. A key element of the proof which is also of intrinsic interest is a local existence theorem with a continuation criterion which says that a solution cannot cease to exist as long as the maximum momentum in the support of the distribution function remains bounded. These results are contrasted with known theorems on spherically symmetric dust solutions.

Hamiltonian formulation of the linearized perturbation equations for the Vlasov-Einstein system of relativistic stellar dynamics

Relativistic star clusters are often modeled as static, spherically symmetic solutions to the Vlasov-Einstein system. It is shown here that, for spherical disturbances, the equations for linearized perturbations about such a static equilibrium can be cast into a Hamiltonian form using relativistic analogues of Morrison's bracket for the Vlasov-Poisson system and the energy functional of Lynden-Bell and Sanitt for a linearized perturbation.

Self-similar analysis of Vlasov-Einstein equations in spherical symmetry

view Abstract Citations (1) References (16) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Self-similar analysis of Vlasov-Einstein equations in spherical symmetry Munier, A. ; Burgan, J. R. ; Feix, M. ; Fijalkow, E. Abstract The Vlasov-Einstein system of equations is studied from the point of view of group transformations. Continuous groups are shown to generalize the usual infinitesimal treatment of the metric tensor to the case of a distribution function. Reduced equations are obtained, leading to a time-dependent analytical solution, which yields as a limiting case the Schwarzschild metric. The problem of a purely radial motion of null particles is discussed and leads to an expression for the redshift in a nonstatic, inhomogeneous spacetime. Publication: The Astrophysical Journal Pub Date: March 1980 DOI: 10.1086/157824 Bibcode: 1980ApJ...236..970M Keywords: Einstein Equations; Relativity; Space-Time Functions; Transformations (Mathematics); Vlasov Equations; Cosmology; Group Theory; Invariance; Red Shift; Schwarzschild Metric; Tensor Analysis; Astrophysics full text sources ADS |

Stationary macroscopic motions of a relativistic gas and its relation to the symmetries of the gravitational field

On the basis of an investigation of the Einstein-Vlasov self-consistent system of equations, a classification is given of the gravitational fields due to a collisionless gravitating gas, using the groups of the motions. It is shown that the time-like Killing vector, directed along the vector of the macroscopic gas velocity, is the center of the group of motions Gr.