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Naked singularities in self-similar gravitational collapse: Stability properties of the Cauchy horizon

Duffy, Emily (2011) Naked singularities in self-similar gravitational collapse: Stability properties of the Cauchy horizon. PhD thesis, Dublin City University.

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Abstract

The background of this thesis is the cosmic censorship hypothesis, which states that the gravitational collapse of physically reasonable matter should not result in the formation of naked singularities. In the absence of a proof of this hypothesis, much effort has been directed towards examining spacetimes which contain naked singularities, in an attempt to determine the nature of the cosmic censor. One area of particular interest is the study of perturbations in naked singularity spacetimes. Should perturbations of a spacetime diverge on the Cauchy horizon associated with the naked singularity, then this spacetime can be ruled out as a serious counter-example to cosmic censorship. In this thesis we examine the behaviour of general linear perturbations of the class of self-similar Lemaître-Tolman-Bondi spacetimes which contain a naked singularity. The perturbations naturally split into two classes, odd and even parity, which we consider in turn. For the odd parity perturbation, we first identify a single gauge invariant scalar which describes the perturbation and obeys an inhomogeneous wave equation. We then show that a perturbation which evolves from initially regular data remains finite on the Cauchy horizon. Finiteness is demonstrated by considering the behaviour of suitable energy norms of the perturbation (and pointwise values of these quantities) on natural spacelike hypersurfaces. For the even parity perturbations, we first show that a particular average of the state variable describing the perturbations generically diverges at the Cauchy horizon. Using this, we show that the L^p-norm of the perturbations also diverges, for 1 ≤ p ≤ ∞. This divergence has a characteristic form that depends only on the background spacetime. By combining these results with an extension of odd parity methods, as well as some theorems from real analysis, we can demonstrate that the perturbations generically diverge pointwise on the Cauchy horizon. A general perturbation is a sum of odd and even perturbations; our results therefore indicate that a general perturbation diverges on the Cauchy horizon. This result supports the cosmic censorship hypothesis.

Item Type:Thesis (PhD)
Date of Award:November 2011
Refereed:No
Additional Information:The accompanying Mathematica notebook includes calculations required for Section 5.2 of the thesis.
Supervisor(s):Nolan, Brien
Uncontrolled Keywords:spacetime; perturbations; cosmic censorship hypothesis
Subjects:Mathematics > Relativity
Mathematics > Differential equations
DCU Faculties and Centres:DCU Faculties and Schools > Faculty of Science and Health > School of Mathematical Sciences
Use License:This item is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 3.0 License. View License
Funders:IRCSET
ID Code:16591
Deposited On:01 Dec 2011 11:28 by Brien Nolan. Last Modified 01 Dec 2011 11:28

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