Conventionalism in general relativity?: formal existence proofs and Reichenbach's theorem {\theta} in context

TL;DR

This paper clarifies the logical structure of existence proofs and Reichenbach's theorem in general relativity, showing how assumptions influence the possibility of alternative geometries and proposing a systematic approach to explore spacetime theories.

Contribution

It disambiguates the targets of existence claims and Reichenbach's theorem, demonstrating the absence of a strong no-go theorem and extending the analysis to torsionful spacetimes.

Findings

Reichenbach's theorem theta is not supported by a strong no-go theorem.

Explicitly breaking assumptions allows for alternative geometries, including torsionful spacetimes.

The proof can be used to systematically explore the space of spacetime theories.

Abstract

Weatherall and Manchak (2014) show that, under reasonable assumptions, Reichenbachean universal effects, constrained to a rank-2 tensor field representation in the geodesic equation, always exist in non-relativistic gravity but not so for relativistic spacetimes. Thus general relativity is less susceptible to underdetermination than its Newtonian predecessor. D\"urr and Ben-Menahem (2022) argue these assumptions are exploitable as loopholes, effectively establishing a (rich) no-go theorem. I disambiguate between two targets of the proof, which have previously been conflated: the existence claim of at least one alternative geometry to a given one and Reichenbach's (in)famous ``theorem theta", which amounts to a universality claim that any geometry can function as an alternative to any other. I show there is no (rich) no-go theorem to save theorem theta. I illustrate this by explicitly breaking one of the assumptions and generalising the proof to torsionful spacetimes. Finally, I suggest a programmatic attitude: rather than undermining the proof one can use it to systematically and rigorously articulate stronger propositions to be proved, thereby systematically exploring the space of alternative spacetime theories.