## 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.

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## Notes

- 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.

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

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### Keywords

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