Publication Date

2019-07-30

Availability

Open access

Embargo Period

2019-07-30

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2019-03-20

First Committee Member

Gregory Galloway

Second Committee Member

Pengzi Miao

Third Committee Member

Ming-Liang Cai

Fourth Committee Member

Orlando Alvarez

Abstract

Ever since the realization, from the Hawking-Penrose singularity theorems, that singularities in spacetime can develop under generic circumstances, the question has been considered as to what extent general relativity is a classically deterministic theory. The essence of Penrose's strong cosmic censorship conjecture [23] is that, indeed, general relativity is deterministic. Put in rough physical terms, under reasonable physical conditions, spacetime should not develop naked singularities, that is to say, no singularity (due e.g. to curvature blow-up) should ever be visible to any observer. Such singularities would undermine the predictive ability of general relativity. More modern statements of the strong cosmic censorship conjecture focus on the Cauchy problem for the Einstein equations. Strong cosmic censorship conjecture: The maximal globally hyperbolic development of 'generic initial data' for the Einstein equations is inextendible as a 'suitably regular' Lorentzian manifold. Formulating a precise statement of the strong cosmic censorship conjecture is itself a challenge because one needs to make precise the phrases 'generic initial data' and 'suitably regular Lorentzian manifold'. Understanding the latter is where general relativity in low regularity and in particular (in-)extendibility results become significant. In [9], Dafermos and Luk show that the conjecture is false when 'suitably regular' is taken to mean a Lorentzian manifold with a C0 metric. Prior to recent work of Sbierski [25], very little had been done to address the issue of the extendibility (or not) of Lorentzian manifolds with metrics at lower regularity. In [25] Sbierski develops methods for establishing the C0-inextendibility of Lorentzian manifolds, which he uses to prove the C0-inextendibility of Minkowski space and the maximally analytic Schwarzschild spacetime. In chapter one of this thesis we review the properties of C0 spacetimes. In chapter two we establish C0-inextendibility results applicable to the asymptotic regions of black hole spacetimes where future timelike completeness is assumed to hold. We then show how these techniques can be applied to establish the C0-inextendibility of Minkowski, de Sitter, and anti-de Sitter spaces. In chapter three we show that a class of k = -1 inflationnary FLRW spacetimes dubbed 'Milne-like' are in fact C0-extendible (i.e. they extend through the big bang). We prove that a certain subclass of these spacetimes also do not admit curvature singularities. We also show that the cosmological constant appears as an initial condition for Milne-like spacetimes and that these spacetimes have a notion of Lorentz invariance. In chapter four we give brief overviews of other results obtained in the smooth spacetime category. We establish a theorem linking cosmological singularities to 3-manifold topology. We prove the invisibility of (weakly) trapped surfaces in asymptotically de Sitter spacetimes. Lastly, we establish an existence result for Cauchy surfaces with constant mean curvature from a spacetime curvature condition.

Keywords

spacetime; big bang; inextendibility

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