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Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three articles investigating the asymptotic behaviour of cosmological spacetimes with symmetries arising in Mathematical General Relativity.

In Paper A and B, we consider spacetimes with Bianchi symmetry and where the matter model is that of a perfect fluid. We investigate the behaviour of such spacetimes close to the initial singularity ('Big Bang'). In Paper A, we prove that the Strong Cosmic Censorship conjecture holds in non-exceptional Bianchi class B spacetimes. Using expansion-normalised variables, we further show detailed asymptotic estimates. In Paper B, we prove similar estimates in the case of stiff fluids.

In Paper C, we consider T2-symmetric spacetimes satisfying the Einstein equations for a non-linear scalar field. To given initial data, we show global existence and uniqueness of solutions to the corresponding differential equations for all future times. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we investigate in detail the asymptotic behaviour towards the future. We prove that the Cosmic No-Hair conjecture holds for solutions satisfying an additional a priori estimate, an estimate which we show to hold in T3-Gowdy symmetry.

Abstract [sv]

Denna avhandling består av tre artiklar som undersöker det asymptotiska beteendet hos kosmologiska rumstider med symmetrier som uppstår i Matematisk Allmän Relativitetsteori.

I Artikel A och B studerar vi rumstider med Bianchi symmetri och där materiemodellen är en ideal fluid. Vi undersöker beteendet av sådana rumstider nära ursprungssingulariteten ('Big Bang'). I Artikel A bevisar vi att den Starka Kosmiska Censur-förmodan håller för icke-exceptionella Bianchi klass B-rumstider. Med hjälp av expansions-normaliserade variabler visar vi detaljerade asymptotiska uppskattningar. I Artikel B visar vi liknande uppskattningar för stela fluider.

I Artikel C betraktar vi T2-symmetriska rumstider som uppfyller Einsteins ekvationer för ett icke-linjärt skalärfält. För givna begynnelsedata visar vi global existens och entydighet av lösningar till motsvarande differentialekvationer för all framtid. I det speciella fallet med en konstant potential, en situation som motsvarar ett linjärt skalärfält på en bakgrund med en positiv kosmologisk konstant, undersöker vi i detalj det asymptotiska beteendet mot framtiden. Vi visar att den Kosmiska Inget-Hår-förmodan håller för lösningar som uppfyller en ytterligare a priori uppskattning, en uppskattning som vi visar gäller i T3-Gowdy-symmetri.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2017. , p. 39
Series
TRITA-MAT-A ; 2017:06
Keywords [en]
Cosmic Censorship, Cosmic No-Hair, Big Bang, symmetry, Bianchi, T2-symmetry, Gowdy, general relativity, cosmology, late time, initial time, asymptotic behaviour, perfect fluid, scalar field, Einstein equations
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-220400ISBN: 978-91-7729-632-4 (print)OAI: oai:DiVA.org:kth-220400DiVA, id: diva2:1167807
Public defence
2018-02-19, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20171220

Available from: 2017-12-20 Created: 2017-12-19 Last updated: 2017-12-20Bibliographically approved
List of papers
1. Strong Cosmic Censorship in orthogonal Bianchi class B perfect fluid and vacuum models
Open this publication in new window or tab >>Strong Cosmic Censorship in orthogonal Bianchi class B perfect fluid and vacuum models
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curvature invariants such as the Kretschmann scalar is a generic property. The only spacetimes where this scalar remains bounded exhibit local rotational symmetry or are of plane wave equilibrium type.

We further investigate the qualitative behaviour of solutions towards the initial singularity. To this end, we work in the expansion-normalised variables introduced by Hewitt-Wainwright and show that a set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subarc of the Kasner parabola. We further give an explicit construction enabling the translation between these variables and geometric initial data to Einstein's equations.

Keywords
cosmic censorship, Bianchi, class B, spatially homogeneous, prefect fluid, vacuum, curvature, blow-up
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-220393 (URN)
Note

QC 20171219

Available from: 2017-12-19 Created: 2017-12-19 Last updated: 2017-12-19Bibliographically approved
2. Orthogonal Bianchi B stiff fluids close to the initial singularity
Open this publication in new window or tab >>Orthogonal Bianchi B stiff fluids close to the initial singularity
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In our previous article [Rad16], we investigated the asymptotic behaviour of orthogonal Bianchi class B perfect fluids close to the initial singularity and proved the Strong Cosmic Censorship conjecture in this setting. In several of the statements, the case of a stiff fluid had to be excluded. The present paper fills this gap.

We work in expansion-normalised variables introduced by Hewitt-Wainwright and find that solutions converge, but show a convergence behaviour very different from the non-stiff case: All solutions tend to limit points in the two-dimensional Jacobs set. A set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subset of the Jacobs set.

Keywords
perfect fluid, stiff fluid, expansion-normalised variables, initial singularity, Bianchi B, homogeneous spacetimes
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-220397 (URN)
Note

QC 20171219

Available from: 2017-12-19 Created: 2017-12-19 Last updated: 2017-12-19Bibliographically approved
3. On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimes
Open this publication in new window or tab >>On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider spacetimes solving the Einstein non-linear scalar field equations with T2-symmetry and show that they admit an areal time foliation in the expanding direction. In particular, we prove global existence and uniqueness of solutions to the corresponding system of evolution equations for all future times. The only assumption we have to make is that the potential is a non-negative smooth function.

In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we achieve detailed asymptotic estimates for the different components of the spacetime metric. This result holds for all T3-Gowdy symmetric metrics and extends to certain T2-symmetric ones satisfying an a priori decay property. Building upon these asymptotic estimates, we show future causal geodesic completeness and prove the Cosmic No-Hair conjecture.

Keywords
No-Hair, no hair, scalar field, T2, T3, Gowdy, symmetry, geodesic, complete, areal foliation
National Category
Geometry Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-218325 (URN)
Note

QC 20171218

Available from: 2017-12-11 Created: 2017-12-11 Last updated: 2017-12-19Bibliographically approved

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