Franciszek Cudek

We prove that every open connected region of relativistic spacetime (M,g) that encloses a b-incomplete half-curve has an open connected subregion that encloses a b-incomplete half-curve and is also "small" in the following sense: it is the image, under the bundle projection map, of some open region in the (connected) orthonormal frame bundle O+M over that spacetime which is bounded, and whose closure is Cauchy incomplete, with respect to any "natural" distance function on O+M. As a corollary, it follows that every b-incomplete half-curve can be covered by a sequence of singular regions which are images of a sequence of bounded subsets of O+M whose diameter, with respect to any "natural" distance function on O+M, tends to zero. We discuss to what extent these results can be interpreted in favour of the claim that singular structure in classical general relativity is "localizable".

I argue that the singular structure of spacetime is best conceptualized as an extrinsic, non-local, but localizable, property of spacetime regions (of any size), and that there is no unique, best definition of a singular region of spacetime. There is, however, a schema that any adequate definition instantiates. Moreover, the question of what kind of incomplete curves witness singular structure is intimately connected to the question of what kind of curves represent possible trajectories of observers in "physically reasonable" motion.

The hole argument concludes that substantivalism about spacetime entails the radical indeterminism of the general theory of relativity (GR). In this paper, I amend and defend a response to the hole argument first proposed by Butterfield (1989) that relies on the idea of counterpart substantivalism. My amendment clarifies and develops the metaphysical presuppositions of counterpart substantivalism and its relation to various definitions of determinism. My defence consists of two claims. First, contra Weatherall (2018) and others: the hole argument is not a blunder resulting from a mistaken view on how mathematical physics works, and requires a developed (meta)metaphysical response. Second, contra Melia (1999) and others: one can be content with a notion of determinism for GR which is not sensitive to merely haecceitistic differences.