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Conventionalism in general relativity?: formal existence proofs and Reichenbach's theorem θ in context

Ruward Mulder

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. Dürr 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.

Conventionalism in general relativity?: formal existence proofs and Reichenbach's theorem θ in context

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. Dürr 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.

Paper Structure

This paper contains 13 sections, 2 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weatherall and Manchak's relativistic proof as a no-go theorem
  3. Universality, not existence: most assumptions are not loopholes to theorem $\theta$
  4. Generalising to torsionful connections
  5. A programme: systematically exploring the space of spacetime theories
  6. Logical structure of the "go theorem" and its targets
  7. $\neg$(TOPO)
  8. $\neg$(RIEM)
  9. $\neg$(FORCE)
  10. $\neg$(CONF)
  11. $\neg$(NORM)
  12. Discussion: Conjecture $\theta$ and conceiving of alternatives rigorously
  13. Appendix: proof of Proposition 3

Key Result

Proposition 2

[The relativistic case]—Let ($M$, $g_{ab}$) be a relativistic spacetime, let $\tilde{g}_{ab}= \Omega^2 g_{ab}$ be a metric conformally equivalent to $g_{ab}$, and let $\nabla$ and $\widetilde{\nabla}$ be the Levi-Civita derivative operators compatible with $g_{ab}$ and $\tilde{g}_{ab}$, respectively

Figures (1)

  • Figure 1: The space of relativistic spacetimes and some inhabitants.

Theorems & Definitions (2)

  • Proposition 2
  • Proposition 3