## Abstract

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.

This is a preview of subscription content, access via your institution.

## Notes

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 \(g_{ab}\).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.

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.

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.

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.

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].

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

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.

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

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.

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.

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.

## References

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

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

Auyang, S.Y.: Spacetime as a fundamental and inalienable structure of fields. Stud. Hist. Philos. Mod. Phys.

**32**(2), 205–215 (2001)Baker, D.: Against field interpretations of quantum field theory. Br. J. Philos. Sci.

**60**(3), 585–609 (2009)Belot, G.: Understanding electromagnetism. Br. J.Philos. Sci.

**49**(4), 531–555 (1998)Binkoski, J.: Geometry, fields, and spacetime. Br. J. Philo. Sci. (2018). https://doi.org/10.1093/bjps/axy002

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)

Dieks, D.: Space and time in particle and field physics. Stud. Hist. Philos. Mod. Phys.

**32**(2), 217–241 (2001)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)

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)Dorato, M.: Events and the ontology of quantum mechanics. Topoi

**34**, 369–378 (2015)Earman, J.: Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press, Oxford (1995)

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

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)

Fletcher, S.C.: On representational capacities, with an application to general relativity. Found. Phys. (2018). https://doi.org/10.1007/s10701-018-0208-6

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

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

Fraser, D.: The fate of ‘particles’ in quantum field theories with interactions. Stud. Hist. Philos. Mod. Phys.

**39**(4), 841–859 (2008)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)

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

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)

Haag, R.: On the sharpness of localization of individual events in space and time. Found. Phys.

**43**, 1295–1313 (2013)Hajicek, P.: Bifurcate space-times. J. Math. Phys.

**12**(1), 157–160 (1971)Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)

Hobson, A.: There are no particles, there are only fields. Am. J. Phys.

**81**(3), 211–223 (2012)Knox, E.: Newtonian spacetime structure in light of the equivalence principle. Br. J. Philos. Sci.

**65**(4), 863–880 (2014)Kuhlmann, M.: The ultimate constituents of the material world. Ontos Verlag, Frankfurt (2010)

Lazarovici, D.: Against fields. Eur. J. Philos. Sci.

**8**(2), 145–170 (2017)Lewis, D.: On the Plurality of Worlds. Basil Blackwell, Oxford (1986)

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

Mathisson, M.: Neue mechanik materieller systeme. Acta Phys. Pol.

**6**, 163–209 (1937)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)Meyer, U.: The Nature of Time. Clarendon Press, Oxford (2013)

Norton, J.D.: Approximation and idealization: why the difference matters. Philos. Sci.

**79**(2), 207–232 (2012)O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, San Diego (1983)

Papapetrou, A.: Spinning test-particles in general relativity. I. Proc. R. Soc. Lond. A

**209**(1097), 248–258 (1951)Romero, G.: On the ontology of spacetime: substantivalism, relationism, eternalism, and emergence. Found. Sci.

**22**(1), 141–159 (2017)Romero, G.E.: A formal ontological theory based on timeless events. Philosophia

**44**(2), 607–622 (2016)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)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)

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)

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

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

Simons, P.: Particulars in particular clothing: three trope theories of substance. Philos. Phenomenol. Res.

**54**(3), 553–575 (1994)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)

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

Westman, H., Sonego, S.: Events and observables in generally invariant spacetime theories. Found. Phys.

**38**(10), 908–915 (2008)Westman, H., Sonego, S.: Coordinates, observables and symmetry in relativity. Ann. Phys.

**324**(8), 1585–1611 (2009)Whitehead, A.N.: The Concept of Nature. Cambridge University Press, Cambridge (1920)

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

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Fletcher, S.C. Which Worldlines Represent Possible Particle Histories?.
*Found Phys* **50, **582–599 (2020). https://doi.org/10.1007/s10701-019-00260-4

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10701-019-00260-4

### Keywords

- General relativity
- Closed timelike curves
- Particle ontology
- Field ontology
- Event ontology
- Process ontology