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We show that there exist steady states of the spherically symmetric 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. [ABSTRACT FROM AUTHOR]

In this note we address the attempted proof of the existence of static solutions to the Einstein–Vlasov system as given in Wolansky (Arch Ration Mech Anal 156:205–230, 2001). 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 Wolansky (2001) are incorrect and we question the validity of a third claim. We also discuss the variational problem and its difficulties with the aim of stimulating further investigation of this intriguing problem in particular answering the question of whether or not static solutions of the Einstein–Vlasov system can be found as local minimizers of an energy-Casimir functional. [ABSTRACT FROM AUTHOR]

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 2 m( 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 2 m / 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. [ABSTRACT FROM AUTHOR]

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 T³-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 T²-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T³-Gowdy symmetry to this list of requirements, we obtain C0-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. [ABSTRACT FROM AUTHOR]

We construct spherically symmetric static solutions to the Einstein-Vlasov system with nonvanishing cosmological constant Λ. The results are divided as follows. For small Λ > 0 we show the existence of globally regular solutions which coincide with the Schwarzschild-deSitter solution in the exterior of the matter regions. For Λ < 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 Λ. For all solutions considered, the energy density and the pressure components have bounded support. Finally, we point out a straightforward method for obtaining a large class of global, nonvacuum spacetimes with topologies ℝ x S² and ℝ x S² x ℝ which arise from our solutions as a result of using the periodicity of the Schwarzschild-deSitter solution. A subclass of these solutions contains black holes of different masses. [ABSTRACT FROM AUTHOR]

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 disc 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 neighbourhood of the boundary of the steady state. [ABSTRACT FROM AUTHOR]

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 $${\mathbb{R}^5}$$ . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 m̊= m̊ (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 (Christodoulou Commun Pure Appl Math 44:339-373, ) for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time $${t \in (-\infty,\infty)}$$ . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 center in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in Rein et al. (Commun Math Phys 168:467-478, 1995) 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 Andréasson and Rein (Math Proc Camb Phil Soc 149:173-188, 2010). 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 3 m ≤ r. This removes an additional assumption made in Andréasson (Indiana Univ Math J 56:523-552, 2007). Our result in maximal-isotropic coordinates is analogous to the result in Rendall (Banach Center Publ 41:35-68, 1997), but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general. [ABSTRACT FROM AUTHOR]

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. 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. [ABSTRACT FROM AUTHOR]

In a recent paper by Giuliani and Rothman [17], the problem of finding a lower bound on the radius R of a charged sphere with mass M and charge Q < M is addressed. Such a bound is referred to as the critical stability radius. Equivalently, it can be formulated as the problem of finding an upper bound on M for given radius and charge. This problem has resulted in a number of papers in recent years but neither a transparent nor a general inequality similar to the case without charge, i.e., M ≤ 4 R/9, has been found. In this paper we derive the surprisingly transparent inequality The inequality is shown to hold for any solution which satisfies p + 2 pT ≤ ρ, where p ≥ 0 and pT are the radial- and tangential pressures respectively and ρ ≥ 0 is the energy density. In addition we show that the inequality is sharp, in particular we show that sharpness is attained by infinitely thin shell solutions. [ABSTRACT FROM AUTHOR]

Abstract: In 1959 Buchdahl [H.A. Buchdahl, General relativistic fluid spheres, Phys. Rev. 116 (1959) 1027–1034] obtained the inequality 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 for instance neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider any static solution of the spherically symmetric Einstein equations for which the energy density , and the radial and tangential pressures and satisfy , , and we show that where m is the quasi-local mass, so that in particular . We also show that the inequality is sharp under these assumptions. Note that when the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with which satisfies the dominant energy condition satisfies the hypotheses with . [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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 2 M/ 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 ≤ Ω ρ, 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 2 M/ 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 2 M/ 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 2 M/ 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 2 M/ 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. [ABSTRACT FROM AUTHOR]

In a previous work [1] matter models such that the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ω ρ, Ω ≥ 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 → 1, then the limit supremum of 2 M/ R 1 was shown to be bounded by ((2Ω + 1)2 − 1)/(2Ω + 1)2. In this paper we show that the hypothesis that R 1/ R 0 → 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 2 M/ R 1 is bounded, but that the limit is ((2Ω + 1)2 − 1)/(2Ω + 1)2 = 8/9, since Ω = 1 for Vlasov matter. Thus, static shells of Vlasov matter can have 2 M/ R 1 arbitrary close to 8/9, which is interesting in view of [3], where numerical evidence is presented that 8/9 is an upper bound of 2 M/ R 1 of any static solution of the spherically symmetric Einstein-Vlasov system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 () 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. [ABSTRACT FROM AUTHOR]

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. 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 a good comprehension of kinetic theory in general relativity. [ABSTRACT FROM AUTHOR]

Background: Adequate performance of trauma team activation (TTA) criteria is important in order to accurately triage trauma patients. The Swedish National Trauma Triage Criteria (SNTTC) consists of 29 criteria that trigger either a Trauma Alert, the highest level of TTA, or a Trauma Response. This study aimed to evaluate the SNTTC and its accuracy in predicting a severely injured patient in a multicenter setting.Methods: A cohort study in Sweden involving six trauma receiving hospitals. Data was collected from the Swedish Trauma Registry. Some 626 patients were analyzed with regard to the specific criteria used to initiate the TTA, injury severity with New Injury Severity Score (NISS) and emergency interventions. Sensitivity, specificity, positive predictive value (PPV) and positive likelihood ratio (LR+) of the criteria were calculated, as well as undertriage and overtriage.Results: All 29 criteria of SNTTC had a sensitivity > 80% for identifying a severely injured patient. The 16 Trauma Alert Criteria had a lower sensitivity of 62.6% but higher LR+ (3.5 vs all criteria 1.4), specificity (82.3 vs 39.1%) and PPV (55.4 vs 37.6%) and the highest accuracy (AUC 0.724). When using only the six physiological criteria, sensitivity (44.8%) and accuracy (AUC 0.690) decreased while LR+ (6.7), specificity (93.3%) and PPV (70.2%) improved.Conclusion: SNTTC is efficient in identifying severely injured patients. The current set of criteria exhibits the best sensitivity compared to other examined combinations and no additional criterion was found to improve the protocol enough to promote a change. [ABSTRACT FROM AUTHOR]