Einstein-Vlasov System Research Articles

Comments on the paper ‘Static solutions of the Vlasov–Einstein system’ by G. Wolansky

In this note we address the attempted proof of the existence of static solutions to the Einstein-Vlasov system as given in \cite{Wol}. We focus on a specific and central part of the proof which concerns a variational problem with an obstacle. We show that two important claims in \cite{Wol} are incorrect and we question the validity of a third claim. We also discuss the variational problem and its difficulties with the aim to stimulate further investigations of this intriguing problem: to answer the question whether or not static solutions of the Einstein-Vlasov system can be found as local minimizers of an energy-Casimir functional.

Cosmic string and black hole limits of toroidal Vlasov bodies in general relativity

We numerically investigate limits of a two-parameter family of stationary solutions to the Einstein-Vlasov system. The solutions are toroidal and have non-vanishing angular momentum. As one tunes to more relativistic solutions (measured for example by an increasing redshift) there exists a sequence of solutions which approaches the extreme Kerr black hole family. Solutions with angular momentum larger than the square of the mass are also investigated, and in the relativistic limit the near-field geometry of such solutions is observed to become conical in the sense that there is a deficit angle. Such solutions may provide self-consistent models for rotating circular cosmic strings.

The stability of the Minkowski space for the Einstein–Vlasov system

This text serves as an introduction to the article [23] written in collaboration with David Fajman and Jérémie Joudioux, and presented at the Laurent Schwarz Seminar in March 2018. In [23], we establish the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [21, 22]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields.

The massive and massless Einstein-Vlasov system with spherical symmetry

We analyze the asymptotic behavior of small perturbations of Minkowski space caused by the presence of ensembles of massive and massless particles in spherical symmetries. The perturbations we consider are compactly supported in space and momenta.

Self-Similarity Breaking of Cosmological Solutions with Collisionless Matter

In this paper we consider the Einstein-Vlasov system with Bianchi VII$_0$ symmetry. Under the assumption of small data we show that self-similarity breaking occurs for reflection symmetric solutions. This generalizes the previous work concerning the non-tilted fluid case \cite{WHU} to the Vlasov case, and we obtain detailed information about the late-time behaviour of metric and matter terms.

Distribution functions for a family of general-relativistic hypervirial models in the collisionless regime

By considering the Einstein-Vlasov system for static spherically symmetric distributions of matter, we show that configurations with constant anisotropy parameter $\beta$ have, necessarily, a distribution function (DF) of the form $F=l^{-2\beta}\xi(\varepsilon)$, where $\varepsilon=E/m$ and $l=L/m$ are the relativistic energy and angular momentum per unit rest mass, respectively. We exploit this result to obtain DFs for the general relativistic extension of the Hypervirial family introduced by Nguyen and Lingam (2013), which Newtonian potential is given by $\phi(r)=-\phi_o /[1+(r/a)^{n}]^{1/n}$ ($a$ and $\phi_o$ are positive free parameters, $n=1,2,...$). Such DFs can be written in the form $F_{n}=l^{n-2}\xi_{n}(\varepsilon)$. For odd $n$, we find that $\xi_n$ is a polynomial of order $2n+1$ in $\varepsilon$, as in the case of the Hernquist model ($n=1$), for which $F_1\propto l^{-1}\left(2\varepsilon-1\right)\left(\varepsilon-1\right)^2$. For even $n$, we can write $\xi_n$ in terms of incomplete beta functions (Plummer model, $n=2$, is an example). Since we demand that $F\geq 0$ throughout the phase space, the particular form of each $\xi_n$ leads to restrictions for the values of $\phi_o$. For example, for the Hernquist model we find that $0\leq \phi_o \leq2/3$, i.e. an upper bounding value less than the one obtained for Nguyen and Lingam ($0\leq \phi_o \leq1$), based on energy conditions.

The nonvacuum Einstein flow on surfaces of nonnegative curvature

ABSTRACTWe prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ℝ×Σ, where Σ is either 𝕊2 or 𝕋2. For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future geodesically complete. In combination with an earlier result for hyperbolic surfaces we conclude future completeness for the Einstein–Vlasov system in 2+1 dimensions independent of the compact spatial topology for an open set of initial data.

Einstein-Vlasov system in spherical symmetry. II. Spherical perturbations of static solutions

We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation $-\psi_{,tt}=H\psi$, with $H$ containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which $H$ is symmetric, and use the Ritz method to approximate the lowest eigenvalues of $H$ numerically. For two representative background solutions with massless particles we find a single unstable mode with a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinear numerical time evolutions.

Veiled Singularities for the Spherically Symmetric Massless Einstein–Vlasov System

This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L. Vel\'{a}zquez, A class of dust-like self-similar solutions of the massless Einstein-Vlasov system. Annales Henri Poincare 12, 919-964, (2011).

A note on future complete spacetimes with massless outgoing particles

It is shown that asymptotically flat initial data to the spherically symmetric massless Einstein–Vlasov system with outgoing matter has a future complete maximal globally hyperbolic development. This result complements a previous result of Andréasson, Kunze and Rein for the corresponding system with massive particles.

The Nonvacuum Einstein Flow on Surfaces of Negative Curvature and Nonlinear Stability

We prove future nonlinear stability for the Einstein–Vlasov system in 2+1 dimensions, for a manifold of the type \({\Sigma \times \mathbb{R}}\), where \({\Sigma}\) is closed with genus > 1, in the expanding direction. This is the first stability result for the Einstein–Vlasov system for the case of vanishing cosmological constant without symmetry assumptions in the cosmological case. As an essential part of the proof, we introduce a method to obtain decay for the distribution function by the construction of geometric Sobolev-type norms arising from the Sasaki metric on the tangent bundle of spacelike hypersurfaces. Those norms are modified by the implementation of a correction mechanism yielding energy estimates with optimal decay properties of the relevant norms.

Cauchy problem on a characteristic cone for the Einstein–Vlasov system: (I) The initial data constraints

In this paper, one considers a Cauchy problem with data on a characteristic cone for the Einstein–Vlasov system in temporal gauge. One highlights gauge-dependent constraints that, supplemented by the standard constraints i.e. the Hamiltonian and the momentum constraints, define the full set of constraints for the considered setting. One studies their global resolution from a suitable choice of some free data, the behavior of the deduced initial data at the vertex of the cone, and the preservation of the gauge.

Einstein-Vlasov system in spherical symmetry: Reduction of the equations of motion and classification of single-shell static solutions in the limit of massless particles

We express the Einstein-Vlasov system in spherical symmetry in terms of a dimensionless momentum variable $z$ (radial over angular momentum). This regularises the limit of massless particles, and in that limit allows us to obtain a reduced system in independent variables $(t,r,z)$ only. Similarly, in this limit the Vlasov density function $f$ for static solutions depends on a single variable $Q$ (energy over angular momentum). This reduction allows us to show that any given static metric which has vanishing Ricci scalar, is vacuum at the centre and for $r>3M$ and obeys certain energy conditions uniquely determines a consistent $f=\bar k(Q)$ (in closed form). Vice versa, any $\bar k(Q)$ within a certain class uniquely determines a static metric (as the solution of a system of two first-order quasilinear ODEs). Hence the space of static spherically symmetric solutions of Einstein-Vlasov is locally a space of functions of one variable. For a simple 2-parameter family of functions $\bar k(Q)$, we construct the corresponding static spherically symmetric solutions, finding that their compactness is in the interval $0.7\lesssim \rm max_r(2M/r)\le 8/9$. This class of static solutions includes one that agrees with the approximately universal type-I critical solution recently found by Akbarian and Choptuik (AC) in numerical time evolutions. We speculate on what singles it out as the critical solution found by fine-tuning generic data to the collapse threshold, given that AC also found that {\em all} static solutions are one-parameter unstable and sit on the threshold of collapse.

On the Evolution Problem for the Einstein-Vlasov System

In this work, we solve the evolution problem associated with the Einstein-Vlasov system, with initial data specified on two transversally intersecting null hypersurfaces. The main existence and uniqueness result of the paper is obtained by a contracting mapping principle, combined with Sobolev inequalities and Moser estimates as well as energy inequalities for first order and second order linear hyperbolic systems. The whole investigation is conducted in appropriate weighted Sobolev spaces.

On axisymmetric and stationary solutions of the self-gravitating Vlasov system

Axisymmetric and stationary solutions are constructed to the Einstein–Vlasov and Vlasov–Poisson systems. These solutions are constructed numerically, using finite element methods and a fixed-point iteration in which the total mass is fixed at each step. A variety of axisymmetric stationary solutions are exhibited, including solutions with toroidal, disk-like, spindle-like, and composite spatial density configurations, as are solutions with non-vanishing net angular momentum. In the case of toroidal solutions, we show for the first time, solutions of the Einstein–Vlasov system which contain ergoregions.

The Cylindrical Symmetry Einstein-Vlasov System with Charged Particles in Expansion

We prove the property of being global in time for the solution of the Cauchy problem that corresponds to the cylindrically symmetric Einstein-Vlasov-Maxwell Scalar field (EVMS) system with general (in size) data. It is shown that if there is no singularity at the axis of symmetry, then the local solutions given by Choquet-Bruhat (Ann. Inst. Fourier 21(3):181–201, 2) can be extended to the global one. We notice that these results generalize those already known for the uncharged case.

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

AbstractFamilies of steady states of the spherically symmetric Einstein–Vlasov system are constructed, which are parametrised 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 analysed 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.

Future global nonlinear stability of surface symmetric solutions of the Einstein–Vlasov system with a cosmological constant

We show future global nonlinear stability of surface symmetric solutions of the Einstein–Vlasov system with a positive cosmological constant. Estimates of higher derivatives of the metric and the matter terms are obtained using an inductive argument. In a recent research monograph Ringström shows future nonlinear stability of (not necessarily symmetric) solutions of the Einstein–Vlasov system with a nonlinear scalar field if certain local estimates on the geometry and the matter terms are fulfilled. We show that these assumptions are satisfied at late times for the case under consideration here which together with Cauchy stability leads to our main conclusion.

Critical collapse in the spherically symmetric Einstein-Vlasov model

We solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scaling exponent, which is yet to be understood. Additionally, we clarify the role that angular momentum plays in the critical behaviour in the massless case.

The Fractional Kinetic Einstein-Vlasov System and its Implications in Bianchi Spacetimes Geometry

The main purpose of this work is to introduce the basic concepts and global properties of the fractional Einstein-Vlasov equation based on the fractional calculus of variations, mainly the fractional actionlike variational approach. We believe that kinetic theory in non-curved spacetimes is fundamental to a good understanding of kinetic theory in general relativity. Besides, the fractional calculus of variations has proved recently to be an important mathematical field of research which has been applied successfully to a broad range of physical and mathematical researches. We expect therefore that the merge of both fields will bring some new insights to general relativity and accordingly to its cosmological and astrophysical implications. Based on the new fractional settings, some cosmological applications are discussed in this work mainly within the aspects of Bianchi spacetimes geometry.