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Simultaneity in Minkowski Spacetime: From Uniqueness to Arbitrariness

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Abstract

Malament (Noûs 11:293–300, 1977) proved a certain uniqueness theorem about standard synchrony, also known as Poincaré-Einstein simultaneity, which has generated many commentaries over the years, some of them contradictory. We think that the situation called for some clarification. After reviewing and discussing some of the literature involved, we prove two results which, hopefully, will help clarifying this debate by filling the gap between the uniquess of Malament’s theorem, which allows the observer to use very few tools, and the complete arbitrariness of a time coordinate in full-fledged Relativity theory. In the spirit of Malament’s theorem, and in opposition to most of its commentators, we emphasize explicit definability of simultaneity relations, and give only constructive proofs. We also explore what happens when we reduce to “purely local” data with respect to an observer.

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Notes

  1. To avoid any problem with gender, we have decided that the observer will be an asexual robot. Such an observer has the additional advantage of being possibly eternal, which is helpful in mathematical arguments.

  2. Since the preprint [7] appeared during the revision process of this paper, we could not include it in the discussion.

  3. Malament calls them ‘causal automorphisms’, but it needs to be emphasized that they do not preserve the causal order relation, but only the symmetric causal connectibility relation.

  4. He did not give a self-contained proof though, giving only an example and referring to Robb [9] for the general case. For the ease of the reader, we will give a complete proof in Sect. 3 of this paper.

  5. Here non-trivial means different from the identity and the universal relation.

  6. Ben-Yami uses the notation σ instead of ⪯.

  7. In fact, he uses both Malament’s and Giulini’s criteria, but Malement’s one suffices.

  8. It is maybe ironical that this proof is enlightening precisely because it relies heavily on good old euclidean geometry and Poincaré-Einstein simultaneity…

  9. The converse is true but is not needed in the sequel.

  10. It is an interesting exercise to prove that \(\psi\in\operatorname{Aut}_{X,L_{X}}\) if and only if ψ=ϕ f ∘T∘τ, where ϕ f is as above, T is a “twist”, that is a bijection of \(\mathcal{M}\) to itself which globally preserves all the 3-spheres centered on X, and τ is a translation in the direction of X.

References

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  1. EPF, 3 bis, rue Lakanal, 92330, Sceaux, France

    Fabien Besnard

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  1. Fabien Besnard
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Besnard, F. Simultaneity in Minkowski Spacetime: From Uniqueness to Arbitrariness. Found Phys 42, 1115–1134 (2012). https://doi.org/10.1007/s10701-012-9649-5

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