C\u3csup\u3e0\u3c/sup\u3e-(in)extendibility of Spacetimes

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\u27s 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 \u27generic initial data\u27 for the Einstein equations is inextendible as a \u27suitably regular\u27 Lorentzian manifold. Formulating a precise statement of the strong cosmic censorship conjecture is itself a challenge because one needs to make precise the phrases \u27generic initial data\u27 and \u27suitably regular Lorentzian manifold\u27. 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 \u27suitably regular\u27 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 \u27Milne-like\u27 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