Vlasov Matter Research Articles

The conformal Einstein field equations with massless Vlasov matter

We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.

Future attractors of Bianchi types II and V cosmologies with massless Vlasov matter

It is shown that the generalized Collins–Stewart radiation and Milne solutions are attractors of the massless Einstein–Vlasov system for Bianchi types II and V spacetimes, respectively. The proof is based on an energy method and bootstrap argument which are used to determine the decay rates of the perturbations away from the attractors.

Uniqueness of static, isotropic low-pressure solutions of the Einstein\u2013Vlasov system

In Beig and Simon (Commun Math Phys 144:373–390, 1992) the authors prove a uniqueness theorem for static solutions of the Einstein–Euler system which applies to fluid models whose equation of state fulfills certain conditions. In this article it is shown that the result of Beig and Simon (1992) can be applied to isotropic Vlasov matter if the gravitational potential well is shallow. To this end we first show how isotropic Vlasov matter can be described as a perfect fluid giving rise to a barotropic equation of state. This Vlasov equation of state is investigated, and it is shown analytically that the requirements of the uniqueness theorem are met for shallow potential wells. Finally the regime of shallow gravitational potential is investigated by numerical means. An example for a unique static solution is constructed, and it is compared to astrophysical objects like globular clusters. Finally we find numerical indications that solutions with deep potential wells are not unique.

Rotating clouds of charged Vlasov matter in general relativity

The existence of stationary solutions of the Einstein–Vlasov–Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof relies on the methods of Andréasson et al (2014 Commun. Math. Phys. 329 787–808) where a similar result is obtained for uncharged particles. Among the solutions constructed in this article there are rotating and non-rotating ones. Static solutions exhibit an electric but no magnetic field. In the case of rotating solutions, in addition to the electric field, a purely poloidal magnetic field is induced by the particle current. The existence of toroidal components of the magnetic field turns out to be not possible in this setting.

Kantowski–Sachs cosmology with Vlasov matter

We analyse the Kantowski–Sachs cosmologies with Vlasov matter of massive and massless particles using dynamical systems analysis. We show that generic solutions are past and future asymptotic to the non-flat locally rotationally symmetric Kasner vacuum solution. Furthermore, we establish that solutions with massive Vlasov matter behave like solutions with massless Vlasov matter towards the singularities.

Existence of Static Solutions of the Einstein--Vlasov--Maxwell System and the Thin Shell Limit

In this article the static Einstein--Vlasov--Maxwell system is considered in spherical symmetry. This system describes an ensemble of charged particles interacting by general relativistic gravity and Coulomb forces. First, a proof for local existence of solutions around the center of symmetry is given. Then, by virtue of a perturbation argument, global existence is established for small particle charges. The method of proof yields solutions with matter quantities of bounded support---among other classes, shells of charged Vlasov matter. As a further result, the limit of infinitesimally thin shells as solution of the Einstein--Vlasov--Maxwell system is proven to exist for arbitrary values of the particle charge parameter. In this limit the inequality which has been obtained by Andréasson in [Comm. Math. Phys., 288 (2009), pp. 715--730], and which bounds the mass-to-radius ratio by a constant and the charge-to-radius ratio, becomes sharp. However, in this limit the charge terms in the inequality are shown to tend to zero.

Future asymptotic behavior of three-dimensional spacetimes with massive particles

We consider non-vacuum initial data for the three-dimensional Einstein equations coupled to Vlasov matter composed of massive particles, on an arbitrary compact Cauchy hypersurface without boundary. We show that conservation of the total mass implies future completeness of the corresponding maximal development in the isotropic case, independent of the topology. This behavior is fundamentally different from the vacuum case and also from the same model in higher dimensions. In particular, we find that a positive mass of particles in three dimensions avoids recollapse of the spatial geometry. Finally, we construct similar solutions for the Einstein-dust system and describe to what extent the construction fails for massless matter models.

The Einstein–Vlasov scalar field system with Gowdy symmetry in the expanding direction

We prove the global existence theorem for spacetimes with Gowdy symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of Gowdy symmetric arbitrary (in size) initial data with Vlasov and scalar field matter (see Choquet-Bruhat and Geroch 1969 Commun. Math. Phys. 44 344–57) exists globally in the future (expanding) time direction.

On gravitational collapse and cosmic censorship for collisionless matter

The weak cosmic censorship conjecture is a central open problem in classical general relativity. Under the assumption of spherical symmetry, Christodoulou has investigated the conjecture for two different matter models; a scalar field and dust. He has shown that the conjecture holds true for a scalar field but that it is violated in the case of dust. The outcome of the conjecture is thus sensitive to which model is chosen to describe matter. Neither a scalar field nor dust are realistic matter models. Collisionless matter, or Vlasov matter, is a simple matter model but can be considered to be realistic in the sense that it is used by astrophysicists. The present status on the weak cosmic censorship conjecture for the Einstein–Vlasov system is reviewed here.

On the area of the symmetry orbits of cosmological spacetimes with toroidal or hyperbolic symmetry

We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of $T^2$-symmetric initial data with positive cosmological constant $\Lambda >0$, in the vacuum or with Vlasov matter, may be covered by a global areal foliation with the area of the symmetry orbits tending to zero in the contracting direction. We then prove the same result for surface symmetric spacetimes in the hyperbolic case with Vlasov matter and $\Lambda \ge 0$. In all cases, there is no restriction on the size of initial data.

Hamiltonian dynamics of spatially-homogeneous Vlasov-Einstein systems

We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry. For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter. The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi Type I models with cold, counter-streaming Vlasov matter potentials analogous to the curvature potentials of corresponding vacuum models.

Bianchi cosmologies with anisotropic matter: Locally rotationally symmetric models

The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following: (i) There exist matter models, compatible with the standard energy conditions, such that solutions of Bianchi type IX (closed cosmologies) need not necessarily recollapse; there is an open set of forever expanding solutions. (ii) Generic type IX solutions associated with a matter model like Vlasov matter exhibit oscillatory behavior toward the initial singularity. This behavior differs significantly from that of vacuum/perfect fluid cosmologies; hence "matter matters". Finally, we indicate that our methods can probably be extended to treat a number of open problems, in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.

The asymptotic behaviour in Schwarzschild time of Vlasov matter in spherically symmetric gravitational collapse

AbstractGiven a static Schwarzschild spacetime of ADM mass M, it is well known that no ingoing causal geodesic starting in the outer domain r > 2M will cross the event horizon r = 2M in finite Schwarzschild time. We show that in gravitational collapse of Vlasov matter this behaviour can be very different. We construct initial data for which a black hole forms and all matter crosses the event horizon as Schwarzschild time goes to infinity, and show that this is a necessary condition for geodesic completeness of the event horizon. In addition to a careful analysis of the asymptotic behaviour of the matter characteristics our proof requires a new argument for global existence of solutions to the spherically symmetric Einstein–Vlasov system in an outer domain, since our initial data have non-compact support in the radial momentum variable and previous methods break down.

On the Buchdahl Inequality for Spherically Symmetric Static Shells

A classical result by Buchdahl [6] shows that for static solutions of the spherically symmetric Einstein equations, the ADM mass M and the area radius R of the boundary of the body, obey the inequality 2M/R ≤ 8/9. The proof of this inequality rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl’s hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R 0, R 1], R 0 > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1. We show a Buchdahl type inequality for shells which are thin; given an \(\epsilon < 1/4\) there is a κ > 0 such that 2M/R 1 ≤ 1 − κ when \(R_1/R_0 \leq 1 + \epsilon\). It is also shown that for a sequence of solutions such that R 1/R 0 → 1, the limit supremum of 2M/R 1 of the sequence is bounded by ((2Ω + 1)2 − 1)/(2Ω + 1)2. In particular if Ω = 1, which is the case for Vlasov matter, the bound is 8/9. The latter result is motivated by numerical simulations [3] which indicate that for non-isotropic shells of Vlasov matter 2M/R 1 ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R 1/R 0 → 1. In [1] a sequence of shells of Vlasov matter is constructed with the properties that R 1/R 0 → 1, and that 2M/R 1 equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in [1] the Vlasov equation is important.

On Static Shells and the Buchdahl Inequality for the Spherically Symmetric Einstein-Vlasov System

In a previous work \cite{An1} matter models such that the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega+1)^2-1)/(2\Omega+1)^2=8/9,$ since $\Omega=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.

An investigation of the Buchdahl inequality for spherically symmetric static shells

A classical result by Buchdahl [9] shows that a class of static spherically symmetric solutions of the Einstein equations obey the inequality 2M/R ≤ 8/9, where M is the total ADM mass and R the area radius of the body. Buchdahl's proof rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl's hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R0, R1], R0 > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1. Note that this inequality holds with Ω = 3 for any matter model which satisfies the dominant energy condition. We show a Buchdahl type inequality for shells which are thin; given an ϵ < 1/4 there is a κ > 0 such that 2M/R1 ≤ 1 − κ when R1/R0 1 + ϵ. It is also shown that for a sequence of solutions such that R1/R0 → 1, the limit supremum of 2M/R1 of the sequence is bounded by ((2Ω + 1)2 − 1)/(2Ω + 1)2 (which equals 8/9 if Ω = 1). We emphasize that no field equations for the matter are used for this result. However, in the second part we consider the Einstein-Vlasov system and use the matter field equation to construct a family of static solutions with the property that R1/R0 → 1. We also show that for this sequence the value 8/9 of 2M/R1 is attained in the limit (note that Ω = 1 for Vlasov matter). The energy density of this sequence get more and more peaked, which should be contrasted to the solution for which 2M/R = 8/9 in Buchdahl's original work where ρ is constant and p blows up at r = 0. Clearly, this solution cannot satisfy the inequality p = q ≤ Ωρ, in particular it violates the dominant energy condition.

On the nature of initial singularities for solutions of the Einstein–Vlasov-scalar field system with surface symmetry

Global existence results in the past time direction of cosmological models with collisionless matter and a massless scalar field are presented. It is shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached. In the case without Vlasov matter, the singularity is velocity dominated and the generalized Kasner exponents converge at each spatial point as the singularity is approached.

A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system

The stability features of steady states of the spherically symmetric Einstein–Vlasov system are investigated numerically. We find support for the conjecture by Zel'dovich and Novikov that the binding energy maximum along a steady state sequence signals the onset of instability, a conjecture which we extend to and confirm for non-isotropic states. The sign of the binding energy of a solution turns out to be relevant for its time evolution in general. We relate the stability properties to the question of universality in critical collapse and find that for Vlasov matter universality does not seem to hold.

Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations

We investigate the dynamics of spatially homogeneous solutions of the Einstein–Vlasov equations with Bianchi type I symmetry by introducing a new formulation that allows an efficient use of dynamical systems methods. We find that all models are forever expanding and that they isotropize towards the future; towards the past there exists a singularity—we identify and describe all possible past asymptotic states. In this context, we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. The past asymptotic structure illustrates that the dynamics of Vlasov matter models differs significantly from that of perfect fluid models.

On the area of the symmetry orbits in T2 symmetric spacetimes with Vlasov matter

This paper treats the global existence question for a collection of general relativistic collisionless particles, all having the same mass. The spacetimes considered are globally hyperbolic, with the Cauchy surface a 3-torus. Furthermore, the spacetimes considered are isometrically invariant under a two-dimensional group action, the orbits of which are spacelike 2-tori. It is known from previous work that the area of the group orbits serves as a global time coordinate. In the present work it is shown that the area takes on all positive values in the maximal Cauchy development.