Einstein-Vlasov System Research Articles

Future of Bianchi I magnetic cosmologies with kinetic matter

We show under the assumption of small data that solutions to the Einstein-Vlasov system with a pure magnetic field and Bianchi I symmetry isotropise and tend to dust solutions. We also obtain the decay rates for the main variables. This generalises part of the work (LeBlanc 1997 Class. Quantum Grav. 14 2281–301) concerning the future behaviour of orthogonal perfect fluids with a linear equation of state in the presence of a magnetic field to the Vlasov case.

EVStabilityNet: predicting the stability of star clusters in general relativity

We present a deep neural network which predicts the stability of isotropic steady states of the asymptotically flat, spherically symmetric Einstein–Vlasov system in Schwarzschild coordinates. The network takes as input the energy profile and the redshift of the steady state. Its architecture consists of a U-Net with a dense bridge. The network was trained on more than ten thousand steady states using an active learning scheme and has high accuracy on test data. As first applications, we analyze the validity of physical hypotheses regarding the stability of the steady states.

Hoop and weak cosmic censorship conjectures for the axisymmetric Einstein-Vlasov system

Hoop and weak cosmic censorship conjectures for the axisymmetric Einstein-Vlasov system

Static Solutions to the Spherically Symmetric Einstein–Vlasov System: A Particle Number–Casimir Approach

Static Solutions to the Spherically Symmetric Einstein–Vlasov System: A Particle Number–Casimir Approach

Einstein-Vlasov system with equal-angular momenta in AdS5

We investigate solutions of the five-dimensional rotating Einstein-Vlasov system with an $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry group. In a five-dimensional spacetime, there are two independent planes of rotation; thus, considering $U(1)$ symmetry on each rotation plane, we may impose an $R\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry to a stationary spacetime. Furthermore, when the values of the two angular momenta are equal to each other, the spatial symmetry gets enhanced to $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry, and the spacetime has a cohomogeneity-1 structure. Imposing the same symmetry to the distribution function of the particles of which the Vlasov system consists, the distribution function can be dependent on three mutually independent and commutative conserved charges for particle motion [energy, total angular momentum on $SU(2)$ and $U(1)$ angular momentum]. We consider the distribution function which exponentially depends on the $U(1)$ angular momentum and reduces to the thermal equilibrium state in spherical symmetry. Then, in this paper, we numerically construct solutions of the asymptotically anti--de Sitter Einstein-Vlasov system.

On almost Ehlers–Geren–Sachs theorems

We show assuming small data that massless solutions to the reflection symmetric Einstein–Vlasov system with Bianchi VII0 symmetry which are not locally rotational symmetric, can be arbitrarily close to and will remain close to isotropy as regards to the shear. However in general the shear will not tend to zero and the Hubble normalised Weyl curvature will blow up. This generalises the work (Nilsson et al 2000 Class. Quantum Grav. 17 3119–34; Wainwright et al 1999 Class. Quantum Grav. 16 2577–98), which considered a non-tilted radiation fluid to the massless Vlasov case. This represents another example of the fact that almost Ehlers–Geren–Sachs theorems do not hold in general and that collisionless matter behaves differently than a perfect fluid.

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

<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>], 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. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.</p>

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.

Collisionless Equilibria in General Relativity: Stable Configurations beyond the First Binding Energy Maximum

Abstract 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 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.

Characteristic Cauchy problem on the light cone for the Einstein–Vlasov system in temporal gauge

This article is concerned with the Cauchy problem on the initial light cone for geometric-transport equations in general relativity when temporal gauge is considered. A novel hierarchy of characteristic initial data constraints is highlighted which includes the standard Gauss Codazzi’s constraints equations and temporal-gauge-dependent constraints, and gauge-preservation is established. The global resolution of the constraints is studied, a large class of initial data sets is deduced from suitable free data and their behavior at the vertex of the cone is examined. For initial data induced by smooth functions on the light cone, a short time solution is established for the Einstein–Vlasov system in a neighborhood of the tip of the cone.

Asymptotic Stability of Minkowski Space-Time with Non-compactly Supported Massless Vlasov Matter

We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein–Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.

Static Spherically Symmetric Einstein-Vlasov Bifurcations of the Schwarzschild Spacetime

We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and of the effective potential energy for static spacetimes close to Schwarzschild. This provides an alternative approach to the construction of static solutions to the Einstein-Vlasov system in a neighbourhood of black hole vacuum spacetimes made in Rein (in Mathematical proceedings of the cambridge philosophical society. Cambridge University Press, Cambridge, vol 115, pp 559–570, 1994).

Dynamics of gravitational collapse in the axisymmetric Einstein–Vlasov system

We numerically investigate the dynamics near black hole formation of solutions to the Einstein–Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2 + 1) + 1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from non-stationary initial data and exhibit type I critical behaviour. In particular, we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.

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

We prove 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 [FJS15; FJS17]. 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.

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.

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.

Sharp Asymptotic Behavior of Solutions of the 3d Vlasov\u2013Maxwell System with Small Data

We study the asymptotic properties of the small data solutions of the Vlasov–Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain, as in Fajman et al. (The Stability of the Minkowski space for the Einstein-Vlasov system, 2017. arXiv:1707.06141), optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations.

On the future of solutions to the massless Einstein–Vlasov system in a Bianchi I cosmology

We show that massless solutions to the Einstein-Vlasov system in a Bianchi I space-time with small anisotropy, i.e. small shear and small trace-free part of the spatial energy momentum tensor, tend to a radiation fluid in an Einstein-de Sitter space-time with the anisotropy $\Sigma^a_b\Sigma^b_a$ and $\tilde{w}^i_j \tilde{w}^j_i$ decaying as $O(t^{-\frac12})$.

Isotropization of slowly expanding spacetimes

We show that the homogeneous, massless Einstein-Vlasov system with toroidal spatial topology and diagonal Bianchi type I symmetry for initial data close to isotropic data isotropizes towards the future and in particular asymptotes to a radiative Einstein-de Sitter model. We use an energy method to obtain quantitative estimates on the rate of isotropization in this class of models.

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.