Title:

Predictability, computability, and spacetime

The thesis falls in three chapters: (1) Spacetime; (2) Predictability; (3) Computability. The investigations are based on classical general relativity. So Chapter 1 provides a sketch of the background theory required for the following two chapters. Topics covered include: the Einstein field equations, causal structure, and handy hints on how to make new spacetimes from old. I also outline a simple singularity theorem and provide some statements of the cosmic censorship hypothesis. Chapter 2 begins by drawing a distinction between the notions of determinism and predictability. There follows a critique of Geroch's paper on relativistic prediction. A new definition of a predictable event is advanced, and some results regarding the global structure of spacetimes possessing such an event are given. Various apparently distinct generalisations of predictable spacetimes are considered and shown to be equivalent. Predictable spacetimes are sought among solutions to the Einstein field equations but not found. I create a worry that predictable spacetimes yield paradox; the worry is soothed. The connections with singularities are then investigated. Naked singularities are found to upset predictability, as expected. But the existence of predictable events is found to necessitate singularities, which is unexpected. Chapter 3 begins with Pitowsky's idea about how to perform a computation supertask in a relativistic spacetime. A fundamental flaw is corrected, and this leads to the definition of what is known as a MalamentHogarth (MH) spacetime. MH spacetimes are shown to be nonglobally hyperbolic, possess a noncompact slice, and to be prone to infinite blueshifts. The rest of chapter is concerned with the bearing of MH spacetimes on the concept of Computability. Membership of a recursively enumerable set is shown to be decidable by a single Turing machine (TM) operating in any MH spacetime. A strict power hierarchy of other TMbased computers is shown to map precisely into the Kleene arithmetic hierarchy. Arithmetic is decidable by an even more powerful TMbased computer. There are radical implications for the concept of Computability; an analogy is found in the modern concept of Geometry.
