Einstein's Hole Argument and Weyl's Field-body Relationalism

Abstract

Einstein’s hole problem concerns the nature of causality in the General Theory of Relativity (GTR). This paper introduces a number of formal concepts and distinc- tions which are necessary for a clear understanding of the nature of causality in GTR. In particular, the distinctions between formal, theoretic and physical coordinates are introduced as well as the distinction between model and symmetry transform- ations. Utilizing the notion of local diffeomorphisms which are globally defined locally invertible maps, it is made explicit that model diffeomorphisms and passive coordinate transformations are mathematically equivalent. This, it is argued, decis- ively undercuts the claims by Earman and Norton that a spacetime substantivalist view is faced with ‘radical local indeterminism’ for a range of modern spacetime theor- ies, including GTR. Additional epistemic and ontological difficulties in Earman’s and Norton’s accounts of Einstein’s hole argument are discussed, and it is argued that these difficulties underscore the need to adopt the ontological position called ‘field-body relationalism’, a position forcefully advanced byHermannWeyl. Thepaper concludes with a discussion ofWeyl’s critique of body-relationalism, Weyl’s argument for the necessity of the inertial field (guiding field), and a modern re-formulation of New- ton’s laws of motion that explicitly takes account of Weyl’s field-body-relationalist ontology.