The formation of black holes in spherically symmetric gravitational collapse

Andréasson, Håkan; Kunze, Markus; Rein, Gerhard

doi:10.1007/s00208-010-0578-3

Cited by 20 publications

(58 citation statements)

References 38 publications

(55 reference statements)

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“…Park [13] has given a first existence proof for static and spherical elastic bodies in general relativity. More recently, these works on Vlasov-Einstein systems and elastic bodies have been extended to non-spherical and rotating systems [14,15]. However, since all this work is based on the implicit function theorem, the deviations from sphericity and staticity are of unknown smallness, and therefore it is unclear how far the results apply to realistic astrophysical objects.…”

Section: Introductionmentioning

confidence: 99%

A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

Pfister¹

2011

Class. Quantum Grav.

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To cite this version:Herbert Pfister. A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity. Classical and Quantum Gravity, IOP Publishing, 2011, 28 (7) AbstractIn comparison to previous existence proofs for static and spherically symmetric perfect fluid stars in general relativity the new proof applies to a more general class of equations of state: In the star's interior we allow for piecewise Lipschitz continuous functions, in this way including the physically important case of phase transitions. Near the star's surface we allow for even more general functions, thereby including a large class of polytropic equations of state. Furthermore, the proof technique proceeds along standard techniques of functional analysis (Banach's fixed point theorem), and therefore applies in a similar manner to static stars in Newtonian gravity, and perhaps to rotating Newtonian and Einsteinian stars. In detail, the Einstein field equations for static perfect fluid stars are transformed to a system of coupled nonlinear integral equations being valid equally in the matter region and in the vacuum exterior. These integral equations are interpreted as a mapping in a Banach space. With the standard iteration technique, beginning with appropriate start functions, it is proven that the mapping has a unique fixed point, and that the solutions have appropriate regularity properties determined by the properties of the equation of state. The Introduction gives an overview of earlier work on such systems, on the question of sphericity of static fluid stars, and on possible extensions of the above methods to rotating Newtonian

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Section: Introductionmentioning

confidence: 99%

A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

Pfister¹

2011

Class. Quantum Grav.

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“…[1,3,4,6,12,[18][19][20]. Global existence for small initial data has been proved in [18] and [12] for massive and massless particles respectively.…”

Section: Introductionmentioning

confidence: 94%

“…Moreover, a proof of global existence would be of great importance due to its relation to the weak cosmic censorship conjecture, cf. [1,4,5,9,11,13]. A third motivation for our interest in this problem is the fact that global existence for general data have been obtained for the Vlasov-Poisson system, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Regularity Results for the Spherically Symmetric Einstein–Vlasov System

Andréasson

2010

Ann. Henri Poincaré

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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 3m ≤ 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.

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“…Among these were abstracts by Andréasson on the formation of black holes in spherically symmetric gravitational collapse [4], Isenberg on the AVTD behavior of T 2 symmetric solutions of the Einstein vacuum equations, Narita on cylindrically symmetric expanding spacetimes, Reula, [34], showing that the Robinson-Trautman-Maxwell equations do not constitute a well-posed initial value problem, and Valiente Kroon on a characterization of Schwarzschild initial data sets [26].…”

Section: Postersmentioning

confidence: 99%

Summary of session A3: mathematical studies of the field equations

Andersson¹

2008

Class. Quantum Grav.

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In this report I will give a summary of some of the main topics covered in session A3, mathematical studies of the field equations, at GRG18, Sydney. Unfortunately, due to length constraints, some of the topics covered at the session will be very briefly mentioned or left out altogether. The summary is mainly based on extended abstracts submitted by the speakers and some of those who presented posters at the session. I would like to thank all participants for their contributions and help with this summary.PACS numbers: 04.20.−q, 04.25.−g, 04.40.−b The Buchdahl inequalityThe Buchdahl inequality, which is included in most textbooks on general relativity, states that for a static, self-gravitating body, 2M/R 8/9,where M is the ADM mass and R the area radius of the boundary of the static body. The proof by Buchdahl, cf [16] of (1) assumed that the energy density is non-increasing outwards and that the pressure is isotropic. A bound on 2M/R has an immediate observational consequence: if 2M/R < 8/9 then the gravitational red shift is less than 2 but if 2M/R approaches 1 the red shift is unbounded. The assumptions used to derive the inequality are very restrictive, and as, e.g. pointed out by Guven andÓ Murchhada [27] neither of them hold in a simple soap bubble and they do not approximate any known topologically stable field configuration. In addition to the restrictions implied by the hypotheses made by Buchdahl there are two other disadvantages associated with this inequality: it refers to the boundary of the body (i.e. the interior is excluded) and the solution which gives equality in the inequality is the Schwarzschild interior solution which has constant energy density and for which the pressure blows up at the center. In particular it violates the dominant energy condition. Andréasson explained recent work [2], where all four restrictions described above are eliminated. He considered matter models for which the energy density ρ and the radial pressure p is non-negative and which satisfies p + 2p T ρ, where is a non-negative constant and p T is the tangential pressure, and showed thatHere m is the quasi-local mass so that M = m(R). The case = 1 gives the bound 2m/r 8/9. Among the matter models which satisfy the conditions stated are Vlasov matter and matter which satisfies the dominant energy condition and has non-negative pressure. Inequality (2) is sharp in the sense of measures and there are examples which come arbitrarily close to saturating the inequality. Andréasson has recently generalized the inequality to the case of charged spheres [3]. Initial-boundary value problemsA long standing problem in the analytical and numerical study of the Einstein equations is the choice of boundary conditions for boundaries near 'infinity'. As is well known, the Einstein equations in harmonic coordinates reduces to a quasilinear system of wave equations. It was proved in previous work by Kreiss and Winicour cf [36] that the initial-boundary value problem is well posed for such systems with the Sommerfeld outgoin...

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The formation of black holes in spherically symmetric gravitational collapse

Cited by 20 publications

References 38 publications

A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

Regularity Results for the Spherically Symmetric Einstein–Vlasov System

Summary of session A3: mathematical studies of the field equations

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