The extent of computation in Malament-Hogarth spacetimes

Abstract

We analyse the extent of possible computations following Hogarth in Malament-Hogarth (MH) spacetimes, and Etesi and N\'emeti in the special subclass containing rotating Kerr black holes. Hogarth had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Nemeti had shown that some \forall \exists relations on natural numbers which are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n \in H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime which is thus a universal constant of the space-time M. Theorem C. Assuming the (modest and standard) requirement that space-time manifolds be paracompact and Hausdorff, for any MH spacetime M there will be a countable ordinal upper bound, w(M), on the complexity of questions in the Borel hierarchy resolvable in it.