Time travel and coincidence-free local dynamical theories


I criticize Lockwood’s solution to the “paradoxes” of time travel, thus endorsing Lewis’s more conservative position. Lockwood argues that only in the context of a 5D space-time-actuality manifold is the possibility of time travel compatible with the Autonomy Principle (according to which global constraints cannot override what is physically possible locally). I argue that shifting from 4D space-time to 5D space-time-actuality does not change the situation with respect to the Autonomy Principle, since the shift does not allow us to have a coincidence-free local dynamical theory.

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  1. 1.

    Lockwood (2005: p. 172 and p. 327). See also Deutsch and Lockwood (1994: p. 71).

  2. 2.

    Arntzenius and Maudlin (2002) argue that this is false. I think their arguments are convincing, but in this paper I will grant Lockwood and Deutsch their point and pursue another line of criticism. In general, there are several plausible objections to AUTONOMY: for instance, it seems to be false even outside time travel scenarios, given that many theories encompass not only dynamical evolution equations, but also constraint equations on initial data. However, for the purpose of the present paper, I will set aside those worth discussing issues. Thanks to an anonymous referee who helped me clarify this point.

  3. 3.

    Throughout the paper I am setting aside relativistic considerations and I am always speaking as if a reference system of temporal and spatial coordinates has been fixed. This simplification is just for ease of exposition. Also, I am considering only backwards time travel. Although forward time travel scenarios raise interesting questions too, I am not touching upon them here.

  4. 4.

    Lockwood (2005: pp. 169–72).

  5. 5.

    Harrison (1971), Thom (1975), Lewis (1976), and Riggs (1997) suggest solutions that entail that such events will happen in travel scenarios, and they all stress that they are compatible with ordinary physics.

  6. 6.

    The argument is criticised by Sider (2002). A version of it is defended by Vihvelin (1996).

  7. 7.

    In criticising their position, Dowe (2007) rightly points out that what prevents the paradoxical outcome to happen does not need to be “already there” (as in the case of a meteorite that is following its trajectory), since it may arrive from the future as well (as if out of the tunnel arrived something that blocks or destroys the train). See also Dowe (2003).

  8. 8.

    A ‘coincidence-free’ local dynamical theory is one on which an explanation that is coincidence-free in the sense introduced above can be based. Thanks to anonymous referees for useful comments on the relation between the issue of “coincidences” and that of the compatibility between TIME-TRAVEL and AUTONOMY (again, only the latter is my main focus), and for having suggested the more general point on the requirement of having a coincidence-free local dynamical theory.

  9. 9.

    The two dimensional model that Lewis briefly discusses is the one in (17, Meiland 1973), which is fundamentally B-theoretic (i.e. tenseless) in spirit. More recently, A-theoretic (i.e. tensed) two-dimensional models of time have been discussed in the context of time travel by van Inwagen (2009), Hudson and Wasserman (2009), and Bernstein (2014).

  10. 10.

    UNITY is vague, given that what the “temporal dimension” is may depend on the background theory that we assume. However, I take the basic idea to be clear enough: whatever counts as the (closest approximation of the) temporal dimension in the background theory is unique. For instance, UNITY is meant to be compatible with special and general relativity: even if relativistic space-times have no frame-independent temporal dimension, it is neither the case that the temporal coordinate duplicates within each frame of reference, nor that the “negative” coordinate of the four-dimensional manifold is not unique.

  11. 11.

    An analogously “purified” version of the grandfather paradox is in Earman (1972), who arrives at conclusions that are different from Lewis’s, although similar in spirit. See also Sider (1997).

  12. 12.

    As an anonymous referee rightly noticed, Lockwood’s proposal is roughly that of a branching time structure. However, I will stick to the original formulation in terms of the dimensionality of the manifold, rather than rephrase it in terms of its topological structure. Firstly, I wish to be as faithful as possible to Lockwood’s text. Secondly, a five dimensional manifold is per se incompatible with UNITY, given that the temporal dimension is multiplied (or split) into multiple temporal dimension in the different points of actuality, but I am unsure about how to rephrase such incompatibility in terms of a branching time structure. Besides, I am conscious that there may be deeper problems concerning the the possibility of formulating a mathematical and physical model of such a spacetime structure, regardless of the basic language from which we start (as also the same anonymous referee notices). However, for the purpose of the present paper I grant to Lockwood the general viability of the proposal.

  13. 13.

    The idea of a 5D space-time-actuality manifold is close to that of a multiverse. Some maintains that in a multiverse backwards time travel would actually be “universe-hopping”, namely a kind of space travel through universes. See Hewlett (1994), Abruzzese (2001), Richmond (2000b), and Effingham (2012). One might expect such worries to carry over to the 5D case. For my purposes, however, this does not matter: for my purposes a difficulty arises insofar as someone can leave a point of actuality at a temporal coordinate \(t'\) and “then” reach a different point of actuality at a preceding temporal coordinate t.

  14. 14.

    Of course, in so far as it is an empirical issue whether we inhabit a 5D manifold or not, dialectical considerations can be overridden by experimentally corroborated theories.

  15. 15.

    Besides, there could be ways of concatenating events at different actuality points together, so that even if an object doesn’t arrive in the past of the same actuality from which it departed, some counterpart of it is guaranteed to do so. Thanks to an anonymous referee for suggesting this further scenario.

  16. 16.

    For helpful comments and discussions thanks to John W. Carroll, Giuseppe Spolaore, Cody Gilmore, Claudio Calosi, Achille Varzi, Sara Bernstein, Samuele Iaquinto, Cristian Mariani, Dave Ingram and Nick Young. Thanks to the projects CUP G45F15000050007 (Regione Lombardia and Fondazione Cariplo) and CUP G45C16000000001 (University of Milan) for financial supports.


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Torrengo, G. Time travel and coincidence-free local dynamical theories. Synthese 197, 4835–4846 (2020). https://doi.org/10.1007/s11229-017-1433-9

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  • Time travel
  • Wormoholes
  • Multiverse