Generalised time functions and finiteness of the Lorentzian distance

We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco and Moretti.Comment: 22 pages. Some imprecisions clarified compared to first versio

A Correspondence Between Distances and Embeddings for Manifolds: New Techniques for Applications of the Abstract Boundary

We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use the Abstract Boundary to describe the structure of the `edge' of our manifold without resorting to structures external to the manifold itself. This is particularly important in the study of singularities within General Relativity where singularities lie on this `edge'. The ability to talk about the same objects, e.g., singularities, via different structures provides alternative routes for investigation which can be invaluable in the pursuit of physically motivated problems where certain types of information are unavailable or difficult to use.Comment: 23 page

COFFEE -- An MPI-parallelized Python package for the numerical evolution of differential equations

COFFEE (ConFormal Field Equation Evolver) is a Python package primarily developed to numerically evolve systems of partial differential equations over time using the method of lines. It includes a variety of time integrators and finite differencing stencils with the summation-by-parts property, as well as pseudo-spectral functionality for angular derivatives of spin-weighted functions. Some additional capabilities include being MPI-parallelisable on a variety of different geometries, HDF data output and post processing scripts to visualize data, and an actions class that allows users to create code for analysis after each timestep.Comment: 12 pages, 1 figure, accepted to be published in Software

An Algebraic Proof of Quillen's Resolution Theorem for K_1

In his 1973 paper Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of a paper by Nenashev, we are able to give an algebraic proof of Quillen's Resolution Theorem for K_1 of an exact category.Comment: 30 pages, 0 figures, uses xypic.st

Generalizations of the Abstract Boundary singularity theorem

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from $C^1$ curves to locally Lipschitz curves.Comment: 24 page

Foundations of and Applications for the Abstract Boundary Construction for Space-Time

The original content of this thesis is comprised of three parts. First, we investigate the foundations of the Abstract Boundary. We start by presenting a one-to-one correspondence between the set of envelopments and a subset of the set of distances on our manifold. This correspondence allows us to define the Abstract Boundary in terms of mathematical structures defined on the manifold, rather than having to use structures additional to the manifold. We take the ideas used in the correspondence and generalise the Abstract Boundary to be applicable to any first countable topological space. Then, using the correspondence and the generalisation we give two alternative constructions for the Abstract Boundary. These new methods of construction allow us to bring many new tools to the analysis of the Abstract Boundary and thus enrich the subject and provide new avenues for research. Second, we discuss how the limiting behaviour of curves relates to the Abstract Boundary. We restrict our attention to the manifold itself and give a classification of the behaviour of curves via the number of limit points they possess. As an application of the classification we weaken the causality assumption of the Abstract Boundary singularity theorem. As an illustration of the problems that curves in a certain class of the classification can cause we give a definition of causality for Abstract Boundary points. In the process of doing so we generalise the distinguishing and strong causality conditions for the boundaries of envelopments and the Abstract Boundary itself. Third, we investigate the link between the Penrose-Hawking singularity theorems and the Krolak strong curvature condition. We review the singularity theorems and analyse their proofs to determine what can be said about the predicted incomplete geodesics. We see that the conclusions that can be made and the criteria for the Krolak strong curvature condition do not mesh easily. For this reason we present two necessary and sufficient conditions for a geodesic to satisfy the Krolak strong curvature condition, that provide a link between the conclusions and the Krolak condition. The result is that we need to investigate the limiting behaviour of jacobi fields along conjugate point free geodesics. Hence we provide a preliminary result showing that maximal extension of the metric places real constraints on the behaviour of parallelly propagated frames. This material provides some interesting results, and opens the door to a number of new problems