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Which Worldlines Represent Possible Particle Histories?

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Based on three common interpretive commitments in general relativity, I raise a conceptual problem for the usual identification, in that theory, of timelike curves as those that represent the possible histories of (test) particles in spacetime. This problem affords at least three different solutions, depending on different representational and ontological assumptions one makes about the nature of (test) particles, fields, and their modal structure. While I advocate for a cautious pluralism regarding these options, I also suggest that re-interpreting (test) particles as field processes offers the most promising route for natural integration with the physics of material phenomena, including quantum theory.

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  1. I have set aside the energy-momentum tensor \(T_{ab}\); according to Einstein’s field equation, it is determined according to \(T_{ab} = (c^4/8\pi G)(R_{ab}- \frac{1}{2}Rg_{ab})\), where c is the speed of light, G is Newton’s constant, and \(R_{ab}\) and R are, respectively, the Ricci and scalar curvatures associated with the Levi-Civita connection compatible with \(g_{ab}\).

  2. Forbes [17] is almost an exception: he takes the temporal order of a (non-relativistic) world to be modally constructed from actual events and their counterparts in “branching” counterpart worlds—ones who have initial segments of events that are perfect counterparts (in the sense of Lewisian counterpart theory [16, 29]). For Forbes [17], therefore, events are still fundamental world-bound existents, while times are modal constructions therefrom.

  3. Malament assumes that \(\gamma \) is just a regular smooth curve, but notes later that this might be relaxed for particles undergoing collisions, which is why I have introduced piecewise smooth curves and curve segments; this restriction indeed makes no substantive difference in understanding the following interpretive principles. I have also omitted Malament’s footnotes and footnote symbols from the quotation to streamline the presentation.

  4. The role of regularity is to exclude such changes effected by buffering two non-co-oriented curve segments with a trivial (constant) curve, whose tangent vector vanishes. If one wanted to insist on the non-empty representational capacities of non-regular curves, then one could exchange regularity for a stronger and more complicated analog of orientation, but I shall not pursue such an approach here.

  5. There is some surprising variation in the definition of CTCs in the literature. O’Neill [35, p. 192] focuses on curve segments \({\bar{\gamma }}: [a,b] \rightarrow M\) that satisfy \({\bar{\gamma }}(a)={\bar{\gamma }}(b)\) and are such that at these two points, their tangent vectors are non-zero and positively proportional. This rules out non-smooth CTCs. Earman [12, p. 165] assumes that spacetime is temporally orientable and takes a CTC to be a smooth, future-directed timelike curve (segment) from some \(p \in M\) to itself. This rules out CTCs from existing in non-temporally orientable spacetimes. Perhaps there are sound arguments for either of these restrictions, but for present purposes they are not needed.

  6. Indeed, CTCs appear in a wide variety of general relativistic models used to describe parts of our universe, such as the interior of Kerr black holes, and (more speculatively) possibly certain wormhole solutions [12, pp. 168–169].

  7. Cf. the definition of bifurcate curves of the first kind in Hajicek [23, p. 158].

  8. I use the terms “right-” and “left-branching” rather than “future-” and “past-branching” because there is no need for the curve’s tangent vectors to be co-oriented with the temporal orientation of the spacetime into which they map, or even for that spacetime to have a temporal orientation at all.

  9. In addition, given any coordinate system containing the branching, at least one cannot be described as an analytic function in those coordinates.

  10. I emphasize that these must be “free” light rays because of course the presence of refractive media can make a difference—cf. the long quotation by Malament in Sect. 1.2.

  11. Whitehead, Russell, and Mayr are all concerned to construct simple events (i.e., those without parts) as, essentially, direct limits of series of nested processes. Their motivation comes, in part, from the empirical inaccessibility of such simple events. But for present purposes these differing motivations can be set aside.

  12. A further option suggested for quantum mechanics [44] and quantum field theory [27] is the trope-bundle ontology—see Rossanese [39] for discussion. Because tropes are particularized properties, rather than instances of universals, they seem to fit roughly with those who take events to be more fundamental than matter, at least for the present purposes.


  1. Arnol’d, V.I.: Ordinary Differential Equations, 3rd edn. Springer, Berlin (1992). Trans. Roger Cooke

    MATH  Google Scholar 

  2. Auyang, S.Y.: How Is Quantum Field Theory Possible?. Oxford University Press, New York (1995)

    Google Scholar 

  3. Auyang, S.Y.: Spacetime as a fundamental and inalienable structure of fields. Stud. Hist. Philos. Mod. Phys. 32(2), 205–215 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baker, D.: Against field interpretations of quantum field theory. Br. J. Philos. Sci. 60(3), 585–609 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Belot, G.: Understanding electromagnetism. Br. J.Philos. Sci. 49(4), 531–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Binkoski, J.: Geometry, fields, and spacetime. Br. J. Philo. Sci. (2018).

  7. Casati, R., Varzi, A.: Events. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, winter 2015 edn. Metaphysics Research Lab, Stanford University, Stanford (2015)

    Google Scholar 

  8. Dieks, D.: Space and time in particle and field physics. Stud. Hist. Philos. Mod. Phys. 32(2), 217–241 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dieks, D.: Events and covariance in the interpretation of quantum field theory. In: Kuhlmann, M., Lyre, H., Wayne, A. (eds.) Ontological Aspects of Quantum Field Theory, pp. 215–234. World Scientific, Singapore (2002)

    Chapter  Google Scholar 

  10. Dixon, W.G.: Dynamics of extended bodies in general relativity. I. momentum and angular momentum. Proc. R. Soc. Lond. A 314(1519), 499–527 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  11. Dorato, M.: Events and the ontology of quantum mechanics. Topoi 34, 369–378 (2015)

    Article  MathSciNet  Google Scholar 

  12. Earman, J.: Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press, Oxford (1995)

    Google Scholar 

  13. Einstein, A.: Relativity: The Special and General Theory. Crown Publishers, New York (1961)

    MATH  Google Scholar 

  14. Fletcher, S.C.: Indeterminism, gravitation, and spacetime theory. In: Hofer-Szabó, G., Wroński, L. (eds.) Making it Formally Explicit: Probability, Causality and Indeterminism, pp. 179–191. Springer International Publishing, Cham (2017)

    Chapter  Google Scholar 

  15. Fletcher, S.C.: On representational capacities, with an application to general relativity. Found. Phys. (2018).

  16. Forbes, G.: The Metaphysics of Modality. Oxford University Press, Oxford (1985)

    Google Scholar 

  17. Forbes, G.: Time, events, and modality. In: Le Poidevin, R. (ed.) The Philosophy of Time, pp. 80–95. Oxford University Press, Oxford (1993)

    Google Scholar 

  18. Fraser, D.: The fate of ‘particles’ in quantum field theories with interactions. Stud. Hist. Philos. Mod. Phys. 39(4), 841–859 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gallois, A.: Identity over time. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, winter 2016 edn. Metaphysics Research Lab, Stanford University, Stanford (2016)

  20. Geroch, R.: General Relativity from A to B. University of Chicago Press, Chicago (1978)

    Google Scholar 

  21. Gilmore, C.: Location and mereology. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, fall 2018 edn. Metaphysics Research Lab Stanford University, Stanford (2018)

    Google Scholar 

  22. Haag, R.: On the sharpness of localization of individual events in space and time. Found. Phys. 43, 1295–1313 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Hajicek, P.: Bifurcate space-times. J. Math. Phys. 12(1), 157–160 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  25. Hobson, A.: There are no particles, there are only fields. Am. J. Phys. 81(3), 211–223 (2012)

    Article  ADS  Google Scholar 

  26. Knox, E.: Newtonian spacetime structure in light of the equivalence principle. Br. J. Philos. Sci. 65(4), 863–880 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kuhlmann, M.: The ultimate constituents of the material world. Ontos Verlag, Frankfurt (2010)

    Book  Google Scholar 

  28. Lazarovici, D.: Against fields. Eur. J. Philos. Sci. 8(2), 145–170 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lewis, D.: On the Plurality of Worlds. Basil Blackwell, Oxford (1986)

    Google Scholar 

  30. Malament, D.B.: Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. University of Chicago Press, Chicago (2012)

    Book  MATH  Google Scholar 

  31. Mathisson, M.: Neue mechanik materieller systeme. Acta Phys. Pol. 6, 163–209 (1937)

    MATH  Google Scholar 

  32. Mayr, D.: A constructive-axiomatic approach to the Lie structure in general spacetime by the principle of approximative reproducibility. Found. Phys. 13(7), 731–743 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  33. Meyer, U.: The Nature of Time. Clarendon Press, Oxford (2013)

    Book  MATH  Google Scholar 

  34. Norton, J.D.: Approximation and idealization: why the difference matters. Philos. Sci. 79(2), 207–232 (2012)

    Article  MathSciNet  Google Scholar 

  35. O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, San Diego (1983)

    MATH  Google Scholar 

  36. Papapetrou, A.: Spinning test-particles in general relativity. I. Proc. R. Soc. Lond. A 209(1097), 248–258 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Romero, G.: On the ontology of spacetime: substantivalism, relationism, eternalism, and emergence. Found. Sci. 22(1), 141–159 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Romero, G.E.: A formal ontological theory based on timeless events. Philosophia 44(2), 607–622 (2016)

    Article  Google Scholar 

  39. Rossanese, E.: Trope ontology and algebraic quantum field theory: an evaluation of Kuhlmann’s proposal. Stud. Hist. Philos. Mod. Phys. 44(4), 417–423 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rovelli, C.: Halfway through the woods: contemporary research on space and time. In: Earman, J., Norton, J.D. (eds.) The Cosmos of Science: Essays of Exploration, pp. 180–223. University of Pittsburgh Press, Pittsburgh (1997)

    Google Scholar 

  41. Rovelli, C.: ‘Localization’ in quantum field theory: how much of QFT is compatible with what we know about space-time? In: Cao, T.Y. (ed.) Conceptual Foundations of Quantum Field Theory, pp. 207–232. Cambridge University Press, Cambridge (1999)

    Chapter  MATH  Google Scholar 

  42. Russell, B.: The Analysis of Matter. Kegan Paul, Trench, Trubner, London (1927)

    MATH  Google Scholar 

  43. Russell, B: Our Knowledge of the External World, 2nd edition. W. W. Norton, New York (1929/1914)

  44. Simons, P.: Particulars in particular clothing: three trope theories of substance. Philos. Phenomenol. Res. 54(3), 553–575 (1994)

    Article  MathSciNet  Google Scholar 

  45. Teller, P.: Space-time as a physical quantity. In: Kargon, R., Achinstein, P. (eds.) Kelvin’s Baltimore Lectures and Modern Theoretical Physics, pp. 425–448. MIT Press, Cambridge (1987)

    Google Scholar 

  46. Weatherall, J.O: Geometry and motion in general relativity. arXiv preprint arXiv:1810.09046 (2018)

  47. Westman, H., Sonego, S.: Events and observables in generally invariant spacetime theories. Found. Phys. 38(10), 908–915 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Westman, H., Sonego, S.: Coordinates, observables and symmetry in relativity. Ann. Phys. 324(8), 1585–1611 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Whitehead, A.N.: The Concept of Nature. Cambridge University Press, Cambridge (1920)

    MATH  Google Scholar 

  50. Whitehead, A.N.: Process and Reality. In: D.R. Griffin and D.W. Sherburne (eds) corrected edition. Free Press, New York (1978/1929)

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Fletcher, S.C. Which Worldlines Represent Possible Particle Histories?. Found Phys 50, 582–599 (2020).

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