We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that 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 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 that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. Introduction 1.1 History and preliminaries Hyperarithmetic Computations in MH Spacetimes 2.1 Generalising SADn regions 2.2 The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime CiteULike Connotea Del.icio.us What's this? (shrink)

In this note, we cast doubt on the requirement of spacetime inextendibility; it is not at all clear that our universe is “as large as it can be.”.

A number of models of general relativity seem to contain “holes” that are thought to be “physically unreasonable.” One seeks a condition to rule out these models. We examine a number of possibilities already in use. We then introduce a new condition: epistemic hole-freeness. Epistemic hole-freeness is not just a new condition—it is new in kind. In particular, it does not presuppose a distinction between space-times that are “physically reasonable” and those that are not.

Stephen Hawking, among others, has proposed that the topological stability of a property of space-time is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single ‘right’ topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Robert Geroch and some propositions of my own show that the main candidates—and (...) each possible choice, to some extent—faces the horns of a no-go result. I suggest that instead of trying to decide what the ‘right’ topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology. (shrink)

Spacetime substantivalism leads to a radical form of indeterminism within a very broad class of spacetime theories which include our best spacetime theory, general relativity. Extending an argument from Einstein, we show that spacetime substantivalists are committed to very many more distinct physical states than these theories' equations can determine, even with the most extensive boundary conditions.

We address the question of whether it is possible to operate a time machine by manipulating matter and energy so as to manufacture closed timelike curves. This question has received a great deal of attention in the physics literature, with attempts to prove no-go theorems based on classical general relativity and various hybrid theories serving as steps along the way towards quantum gravity. Despite the effort put into these no-go theorems, there is no widely accepted definition of a time machine. (...) We explain the conundrum that must be faced in providing a satisfactory definition and propose a resolution. Roughly, we require that all extensions of the time machine region contain closed timelike curves; the actions of the time machine operator are then sufficiently “potent” to guarantee that closed timelike curves appear. We then review no-go theorems based on classical general relativity, semi-classical quantum gravity, quantum field theory on curved spacetime, and Euclidean quantum gravity. Our verdict on the question of our title is that no result of sufficient generality to underwrite a confident “yes” has been proven. Our review of the no-go results does, however, highlight several foundational problems at the intersection of general relativity and quantum physics that lend substance to the search for an answer. (shrink)

The standard theory of computation excludes computations whose completion requires an infinite number of steps. Malament-Hogarth spacetimes admit observers whose pasts contain entire future-directed, timelike half-curves of infinite proper length. We investigate the physical properties of these spacetimes and ask whether they and other spacetimes allow the observer to know the outcome of a computation with infinitely many steps.

I discuss candidates for definitions of determinism in the context of general relativistic spacetimes, and argue that a definition which does not make recourse to any particular region of spacetime should be preferred over alternatives; one such notion is discussed in detail in the light of various physical examples. The emerging picture of determinism is a pluralist one: sometimes there is no unique way of making our intuitive concept of determinism precise. Instead, what is crucial for assessment of determinism of (...) the theory are auxiliary conditions under which it counts as deterministic or indeterministic, and justification for using them in a given situation. (shrink)

I discuss candidates for definitions of determinism in the context of general relativistic spacetimes, and argue that a definition which does not make recourse to any particular region of spacetime should be preferred over alternatives; one such notion is discussed in detail in the light of various physical examples. The emerging picture of determinism is a pluralist one: sometimes there is no unique way of making our intuitive concept of determinism precise. Instead, what is crucial for assessment of determinism of (...) the theory are auxiliary conditions under which it counts as deterministic or indeterministic, and justification for using them in a given situation. (shrink)

I discuss candidates for definitions of determinism in the context of general relativistic spacetimes, and argue that a definition which does not make recourse to any particular region of spacetime should be preferred over alternatives; one such notion is discussed in detail in the light of various physical examples. The emerging picture of determinism is a pluralist one: sometimes there is no unique way of making our intuitive concept of determinism precise. Instead, what is crucial for assessment of determinism of (...) the theory are auxiliary conditions under which it counts as deterministic or indeterministic, and justification for using them in a given situation. (shrink)

Much controversy surrounds the question of what ought to be the proper definition of 'singularity' in general relativity, and the question of whether the prediction of such entities leads to a crisis for the theory. I argue that a definition in terms of curve incompleteness is adequate, and in particular that the idea that singularities correspond to 'missing points' has insurmountable problems. I conclude that singularities per se pose no serious problem for the theory, but their analysis does bring into (...) focus several problems of interpretation at the foundation of the theory often ignored in the philosophical literature. (shrink)

In this paper we aim to disentangle the thesis that the future is open from theses that often get associated or even conflated with it. In particular, we argue that the open future thesis is compatible with both the unrestricted principle of bivalence and determinism with respect to the laws of nature. We also argue that whether or not the future (and indeed the past) is open has no consequences as to the existence of (past and) future ontology.

Indeed, this is the first serious book-length study of the subject by a philosopher of science.

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

This exploration of the global structure of spacetime within the context of general relativity examines the causal and singular structures of spacetime, revealing some of the curious possibilities that are compatible with the theory, such as `time travel' and `holes' of various types. Investigations into the epistemic and modal structures of spacetime highlight the difficulties in ruling out such possibilities, unlikely as they may seem at first. The upshot seems to be that what counts as a `physically reasonable' spacetime structure (...) in modern physics is far from clear. (shrink)