Conventionalism in General Relativity?: Formal Existence Proofs and Reichenbach’s Theorem θ in Context

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. Dürr and Ben-Menahem (2022) argue that 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 generalizing the proof to torsionful spacetimes. Finally, I suggest adopting 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.