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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., Comment: 74 pages

In 2001 Wolansky \cite{Wol} introduced a particle number-Casimir functional for the Einstein-Vlasov system. Two open questions are associated with this functional. First, a meaningful variational problem should be formulated and the existence of a minimizer to this problem should be established. The second issue is to show that a minimizer, for some choice of the parameters, is a static solution of the Einstein-Vlasov system. In the present work we solve the first problem by proving the existence of a minimizer to the particle number-Casimir functional. On the technical side, it is a main achievement that we are able to bypass the non-compactness of minimizing sequences by new arguments in both $v$-space and $x$-space, which might have several further applications. We note that such compactness results for the Einstein-Vlasov system have been absent in the literature, whereas similar results have been known in the Newtonian case. We also provide arguments which give strong support that minimizers corresponding to small masses are static solutions of the Einstein-Vlasov system. Furthermore, our analysis leads us to propose a new stability criterion for static solutions: We conjecture that a static solution for which the Casimir-binding energy is positive is stable for mass-preserving perturbations., Comment: 51 pages

We rigorously derive the quadrupole formula within the context of the Einstein-Vlasov system. The main contribution of this work is an estimate of the remainder terms, derived from well-defined assumptions, with explicitly stated error terms that depend on the solution's boundedness and decay properties, and the distance to the source. The assumptions are linked to established properties of global solutions of the Einstein-Vlasov system as in \cite{LT}. Prior derivations of the quadrupole formula have relied on post-Newtonian analysis and lacked comparisons with global solution properties. The importance of the no-incoming-radiation condition is emphasized underscoring the need for solutions satisfying this condition. This work thus addresses the limitations of existing results and provides motivation for further research on global solution properties of the Einstein-Vlasov system., Comment: 92 pages. To appear in Dissertationes Mathematicae

Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to non-linear stability that gives this topic its importance. This problem was considered in \cite{Wol}, but as has been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In this work we construct static solutions by solving the Euler-Lagrange equation for the energy density $\rho$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in \cite{Wol}. We then define a density function $f$ on phase space which induces the energy density $\rho$ and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what the author of \cite{Wol} attempted to prove., Comment: 41 pages

We show that there exist steady states of the massless Einstein-Vlasov system which surround a Schwarzschild black hole. The steady states are (thick) shells with finite mass and compact support. Furthermore we prove that an arbitrary number of shells, necessarily well separated, can surround the black hole. To our knowledge this is the first result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numerical investigation about the properties of the shells., Comment: 30 pages, 13 figures

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., Comment: Accepted version. 33 pages; 11 figures

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., Comment: 9 pages

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., Comment: 13 pages

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., Comment: 24 pages

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein-Vlasov system. The solutions are highly relativistic in the sense that the ratio $2m(r)/r$ is close to $8/9$, where $m(r)$ is the Hawking mass and $r$ is the area radius. In 1955 Wheeler constructed, by numerical means, so called idealized spherically symmetric geons, i.e. solutions of the Einstein-Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio $2m/r$ is close to $8/9$, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons.

We construct spherically symmetric, static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant $\Lambda$. The results are divided as follows. For small $\Lambda>0$ we show existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter sources. For $\Lambda<0$ we show via an energy estimate the existence of globally regular solutions which coincide with the Schwarzschild-Anti-deSitter solution in the exterior vacuum region. We also construct solutions with a Schwarzschild singularity at the center regardless of the sign of $\Lambda$. For all solutions considered, the energy density and the pressure components have bounded support. Finally, we point out a straightforward method to obtain a large class of globally non-vacuum spacetimes with topologies $\mathbb R\times S^3$ and $\mathbb R\times S^2\times \mathbb R$ which arise from our solutions using the periodicity of the Schwarzschild-deSitter solution. A subclass of these solutions contains black holes of different masses., Comment: 31 pages, 7 figures

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., Comment: 27 pages, 17 figures. Final version

We study some properties of static spherically symmetric elastic bodies in general relativity using both analytical and numerical tools. The materials considered belong to the class of John elastic materials and reduce to perfect fluids when the rigidity parameter is set to zero. We find numerical support that such elastic bodies exist with different possible shapes (balls, single shells and multiple shells) and that their gravitational redshift can be very large ($z\approx 2.8$) without violating the dominant energy condition. Moreover we show that the elastic body has finite radius even in the case when the constitutive equation of the elastic material is a perturbation of a polytropic fluid without finite radius, thereby concluding that such fluids are structurally unstable within the larger class of elastic matter models under study., Comment: 25 pages, 11 figures. The title has been changed to match the published version. References added

The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions; the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T^3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T^2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T^3-Gowdy symmetry to this list of requirements, we obtain C^0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter., Comment: 72 pages. Revised version to appear in JEMS

The existence of stationary solutions to the Einstein-Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat non-vacuum spacetimes. If angular momentum is allowed to be non-zero, the system of equations to solve contains one semilinear elliptic equation which is singular on the axis of rotation. This can be handled very efficiently by recasting the equation as one for an axisymmetric unknown on $\R^5$., Comment: 29 pages. arXiv admin note: substantial text overlap with arXiv:1006.1225

We consider charged spherically symmetric static solutions of the Einstein-Maxwell equations with a positive cosmological constant $\Lambda$. If $r$ denotes the area radius, $m_g$ and $q$ the gravitational mass and charge of a sphere with area radius $r$ respectively, we find that for any solution which satisfies the condition $p+2p_{\perp}\leq \rho,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, $\rho\geq 0$ is the energy density, and for which $0\leq \frac{q^2}{r^2}+\Lambda r^2\leq 1,$ the inequality $\frac{m_g}{r} \leq 2/9+\frac{q^2}{3r^2}-\frac{\Lambda r^2}{3}+2/9\sqrt{1+\frac{3q^2}{r^2}+3\Lambda r^2}$ holds. We also investigate the issue of sharpness, and we show that the inequality is sharp in a few cases but generally this question is open., Comment: 12 pages. Revised version to appear in Class. Quant. Grav

Initial data for the spherically symmetric Einstein-Vlasov system is constructed whose past evolution is regular and whose future evolution contains a black hole. This is the first example of initial data with these properties for the Einstein-matter system with a "realistic" matter model. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass $\open{m}=\open{m}(r)$ and the area radius $r$ is arbitrarily small everywhere, such that a black hole forms in the evolution. This result is in a sense analogous to the result for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time $t\in (-\infty,\infty)$., Comment: 30 pages. Revised version to appear in Annales Henri Poincar\'e

The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein--Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on non-relativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein--Vlasov system. Since then many theorems on global properties of solutions to this system have been established., Comment: Published version http://www.livingreviews.org/lrr-2011-4

The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{RRS} for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in \cite{AR1}. In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as $3m\leq r.$ This removes an additional assumption made in \cite{A1}. Our result in maximal-isotropic coordinates is analogous to the result in \cite{R1}, but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general., Comment: 25 pages. To appear in Ann. Henri Poincar\'e

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system., Comment: 32 pages

We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is analogous to what is found in certain Vaidya spacetimes. The analysis is carried out in Eddington-Finkelstein coordinates., Comment: 27

We consider spherically symmetric static solutions of the Einstein equations with a positive cosmological constant $\Lambda,$ which are regular at the centre, and we investigate the influence of $\Lambda$ on the bound of M/R, where M is the ADM mass and R is the area radius of the boundary of the static object. We find that for any solution which satisfies the energy condition $p+2p_{\perp}\leq\rho,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, and $\rho\geq 0$ is the energy density, and for which $0\leq \Lambda R^2\leq 1,$ the inequality \frac{M}{R}\leq\frac29-\frac{\Lambda R^2}{3}+\frac29 \sqrt{1+3\Lambda R^2}, holds. If $\Lambda=0$ it is known that infinitely thin shell solutions uniquely saturate the inequality, i.e. the inequality is sharp in that case. The situation is quite different if $\Lambda>0.$ Indeed, we show that infinitely thin shell solutions do not generally saturate the inequality except in the two degenerate situations $\Lambda R^2=0$ and $\Lambda R^2=1$. In the latter situation there is also a constant density solution, where the exterior spacetime is the Nariai solution, which saturates the inequality, hence, the saturating solution is non-unique. In this case the cosmological horizon and the black hole horizon coincide. This is analogous to the charged situation where there is numerical evidence that uniqueness of the saturating solution is lost when the inner and outer horizons of the Reissner-Nordstr\"{o}m solution coincide., Comment: 14 pages; Improvements and corrections, published version

We construct, by numerical means, static solutions of the spherically symmetric Einstein-Vlasov-Maxwell system and investigate various features of the solutions. This extends a previous investigation \cite{AR1} of the chargeless case. We study the possible shapes of the energy density profile as a function of the area radius when the electric charge of an individual particle is varied as a parameter. We find profiles which are multi-peaked, where the peaks are separated either by vacuum or a thin atmosphere, and we find that for a sufficiently large charge parameter there are no physically meaningful solutions. Furthermore, we investigate if the inequality \sqrt{M}\leq \frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^2}{3R}}, derived in \cite{An2}, is sharp within the class of solutions to the Einstein-Vlasov-Maxwell system. Here M is the ADM mass, Q the charge, and R the area radius of the boundary of the static object. We find two classes of solutions with this property, while there is only one in the chargeless case. In particular we find numerical evidence for the existence of arbitrarily thin shell solutions to the Einstein-Vlasov-Maxwell system. Finally, we consider one parameter families of steady states, and we find spirals in the mass-radius diagram for all examples of the microscopic equation of state which we consider., Comment: 25 pages, 16 figures

Given 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. In the present paper 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 we 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., Comment: 22 pages

We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general context, and we include arguments which show that the spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain a black hole in the sense of suitable mathematical definitions of these concepts which are available in the literature., Comment: 35 pages

In a recent paper by Giuliani and Rothman \cite{GR}, the problem of finding a lower bound on the radius $R$ of a charged sphere with mass M and charge Q

This paper has been withdrawn since its content is included in the more general version arXiv:0804.1882, Comment: This paper has been withdrawn since its content is included in the more general version arXiv:0804.1882

We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system. We find explicit conditions on the initial data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2M, the timelike lines $r=c\in [0,2M]$ are incomplete, and for r>2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We also give examples of such initial data with the additional property that the solutions exist for all $r\geq 0$ and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities., Comment: 36 pages. A corollary on global existence in Schwarzschild coordinates for data which are not small is added together with minor modifications

In 1959 Buchdahl \cite{Bu} obtained the inequality $2M/R\leq 8/9$ under the assumptions that the energy density is non-increasing outwards and that the pressure is isotropic. Here $M$ is the ADM mass and $R$ the area radius of the boundary of the static body. The assumptions used to derive the Buchdahl inequality are very restrictive and e.g. neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider \textit{any} static solution of the spherically symmetric Einstein equations for which the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $p_T,$ satisfy $p+2p_T\leq\Omega\rho, \Omega>0,$ and we show that $$\sup_{r>0}\frac{2m(r)}{r}\leq \frac{(1+2\Omega)^2-1}{(1+2\Omega)^2},$$ where $m$ is the quasi-local mass, so that in particular $M=m(R).$ We also show that the inequality is sharp. Note that when $\Omega=1$ the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with $p\geq 0$ which satisfies the dominant energy condition satisfies the hypotheses with $\Omega=3.$, Comment: 28 pages, typos corrected and the analogy in section 4 is presented slightly differently

We prove a new global existence result for the asymptotically flat, spherically symmetric Einstein-Vlasov system which describes in the framework of general relativity an ensemble of particles which interact by gravity. The data are such that initially all the particles are moving radially outward and that this property can be bootstrapped. The resulting non-vacuum spacetime is future geodesically complete., Comment: 16 pages

Using both numerical and analytical tools we study various features of static, spherically symmetric solutions of the Einstein-Vlasov system. In particular, we investigate the possible shapes of their mass-energy density and find that they can be multi-peaked, we give numerical evidence and a partial proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r < 8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both isotropic {\em and} anisotropic--, and we give numerical evidence and a partial proof for the conjecture that for any given microscopic equation of state--both isotropic {\em and} anisotropic--the resulting one-parameter family of static solutions generates a spiral in the radius-mass diagram., Comment: 34 pages, 18 figures, LaTex

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., Comment: 20 pages, Latex

A classical result by Buchdahl \cite{Bu1} shows that for static solutions of the spherically symmetric Einstein-matter system, the total ADM mass M and the area radius R of the boundary of the body, obey the inequality $2M/R\leq 8/9.$ 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 $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1.$ We show a Buchdahl type inequality for shells which are thin; given an $\epsilon<1/4$ there is a $\kappa>0$ such that $2M/R_1\leq 1-\kappa$ when $R_1/R_0\leq 1+\epsilon.$ It is also shown that for a sequence of solutions such that $R_1/R_0\to 1,$ the limit supremum of $2M/R_1$ of the sequence is bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In particular if $\Omega=1,$ which is the case for Vlasov matter, the boumd is $8/9.$ The latter result is motivated by numerical simulations \cite{AR2} which indicate that for non-isotropic shells of Vlasov matter $2M/R_1\leq 8/9,$ and moreover, that the value 8/9 is approached for shells with $R_1/R_0\to 1$. In \cite{An2} a sequence of shells of Vlasov matter is constructed with the properties that $R_1/R_0\to 1,$ 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 \cite{An2} the Vlasov equation is important., Comment: 13 pages, Latex. Second statement in Theorem 1 corrected

The stability features of steady states of the spherically symmetric Einstein-Vlasov system are investigated numerically. We find support for the conjecture by Zeldovich 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., Comment: 29 pages, 10 figures

The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given., Comment: 32 pages, latex

The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, {\it i.e.} to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models ({\it i.e.} fluid models). This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity., Comment: 40 pages, updated version, to appear in Living Reviews in Relativity

We show that deletion of the loss part of the collision term in all physically relevant versions of the Boltzmann equation, including the relativistic case, will in general lead to blowup in finite time of a solution and hence prevent global existence. Our result corrects an error in the proof given in Ref. [12], where the result was announced for the classical hard sphere case; here we give a simpler proof which applies much more generally., Comment: 10 pages, 2 figures

Classical solutions of the spherically symmetric Nordstr\"{o}m-Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordstr\"{o}m-Vlasov system is its relation to the Einstein-Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordstr\"{o}m-Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis., Comment: 8 pages, LaTeX

The global structure of solutions of the Einstein equations coupled to the Vlasov equation is investigated in the presence of a two-dimensional symmetry group. It is shown that there exist global CMC and areal time foliations. The proof is based on long-time existence theorems for the partial differential equations resulting from the Einstein-Vlasov system when conformal or areal coordinates are introduced., Comment: 31 pages

The main purpose of this article is to guide the reader to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades where the main focus has been on nonrelativistic- and special relativistic physics, e.g. to model the dynamics of neutral gases, plasmas and Newtonian self-gravitating systems. In 1990 Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (e.g. fluid models). The first part of this paper gives an introduction to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental in order to get a good comprehension of kinetic theory in general relativity., Comment: 31 pages. This article has been submitted to Living Rev. Relativity (http://www.livingreviews.org)

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed., Comment: 23 pages LaTeX2e

A global existence theorem, with respect to a geometrically defined time, is shown for Gowdy symmetric globally hyperbolic solutions of the Einstein-Vlasov system for arbitrary (in size) initial data. The spacetimes being studied contain both matter and gravitational waves., Comment: Latex, 40 pages, submitted to CMP