Reopening the Hole Argument

TL;DR

This paper explores the Hole Argument in general relativity, relating it to a key theorem on solution uniqueness, and discusses its implications for determinism, gauge symmetries, and philosophical interpretations of spacetime.

Contribution

It clarifies the relation between the Hole Argument and the Choquet-Bruhat-Geroch theorem, emphasizing the role of isometries and proposing amendments to the definition of determinism in GR.

Findings

The theorem links the Hole Argument to solution uniqueness in GR.

Isometries in GR are gauge symmetries, similar to Poincaré transformations.

The paper discusses philosophical implications for the interpretation of spacetime and determinism.

Abstract

This expository paper relates the Hole Argument in general relativity (GR) to the well-known theorem of Choquet-Bruhat and Geroch (1969) on the existence and uniqueness of globally hyperbolic solutions to the Einstein field equations. Like the Earman-Norton (1987) version of the Hole Argument (which is originally due to Einstein), this theorem exposes the tension beween determinism and some version of spacetime substantivalism. But it seems less vulnerable to the campaign by Weatherall (2018) and followers to close the Hole Argument on the basis of ``mathematical practice'', since the theorem only talks about isometries and hence does not make the pointwise identifications via diffeomorphisms that Weatherall objects to. Among other implications of the theorem for the philosophy of GR, we reconsider Butterfield's (1987) influential definition of determinism. This should be amended if its goal is to express the idea that \GR\ is deterministic in the absence of Cauchy horizons, although its original form does capture the way GR is indeterministic in their presence! Furthermore, in GR isometries come out as gauge symmetries, as do Poincar'e transformations in special relativity. Finally, I discuss some implications of the theorem for the philosophy of science: accepting the determinism horn still requires a choice between Frege-style abstractionism and Hilbert-style structuralism; and, within the latter, between structural realism and empiricist structuralism (which I favour).