Regarding the ‘Hole Argument’ and the ‘Problem of Time’

Abstract

The canonical formalism of general relativity affords a particularly interesting characterisation of the infamous hole argument. It also provides a natural formalism in which to relate the hole argument to the problem of time in classical and quantum gravity. In this paper we examine the connection between these two much discussed problems in the foundations of spacetime theory along two interrelated lines. First, from a formal perspective, we consider the extent to which the two problems can and cannot be precisely and distinctly characterised. Second, from a philosophical perspective, we consider the implications of various responses to the problems, with a particular focus upon the viability of a 'deflationary' attitude to the relationalist/substantivalist debate regarding the ontology of spacetime. Conceptual and formal inadequacies within the representative language of canonical gravity will be shown to be at the heart of both the canonical hole argument and the problem of time. Interesting and fruitful work at the interface of physics and philosophy relates to the challenge of resolving such inadequacies. *

Reopening the Hole Argument

“…In any case, the apparently controversial Hole Argument is clarified by placing it in the context of an uncontroversial theorem due to Choquet-Bruhat-Geroch (1969) on the existence and uniqueness of maximal globally hyperbolic solutions to the Einstein field 12 This suggestion was subsequently somewhat weakened in Weatherall (2021) and is also challenged by, e.g., Roberts (2020), Gryb and Thébault (2022), and Pooley and Read (2022).…”

Regarding ‘Leibniz Equivalence’

“…The general principle that Weatherall proposes is thus: we should be sure to adopt an adequate equivalence relation "given by the mathematics used in formulating those models" [27, p. 331] 10 . Earman and Norton's manifold substantivalist is making an illegitimate step in using the manifold identity 1 M to judge the equivalence of (M, g ab ) and (M, * g ab ) , according to this view.…”

“…This ambiguity is briefly discussed by Gryb and Thébault[10] in reference toWeatherall [27] and in an earlier version of this paper[22] 3. The concept of a 'representational capacity' was applied to human cognition at least as early as Kant's Critique of Cure Reason (B72), and has been recently adopted by philosophers of scientific representation [c.f 28,30]…”