Time Travel Without Causal Loops by Bradley Monton University of Colorado at Boulder March 15, 2007

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Time Travel Without Causal Loops

by Bradley Monton

University of Colorado at Boulder

March 15, 2007

1. My thesis is that time travel can occur without causal loops. Specifically, I will show that, assuming backwards time travel (‘time travel’ for short) is logically possible, it is logically possible to have a universe where time travel occurs and yet where no causal loops occur.

Who would think otherwise? D. H. Mellor (1998: 131) is one. He argues against the possibility of time travel by arguing against the possibility of causal loops; he explains that his argument works by “ruling out the causal … loops that cyclical time and backward time travel need.” Jan Faye (2005) is another; he simply asserts that “time travel involves a causal loop”. David Lewis (1976, 74) famously expresses uncertainty regarding the matter: “Perhaps there must be loops if there is reversal; I am not sure.” I will show that there is no need for uncertainty. I’ll show this both for time travel where the world line of the time traveller is discontinuous, stopping at some time t2 and restarting at some earlier time t1, and for time travel where the world line of the time traveller forms a continuous zigzag shape, heading forward in time and then backward and then forward again (where the “and then” is from the standpoint of the time traveller’s personal time).

2. Imagine a possible (and clearly non-actual) world that consists of exactly three types of particles, A-particles, B-particles, and C-particles (all point particles). All interactions that take place between different particles happen via contact. The structure of spacetime of this universe is a standard Newtonian-style one – if time travel occurs in this universe, it occurs because of the worldlines that particles follow, not because of any sort of curved spacetime structure (such as wormholes). The universe is divided into two spatial regions, Region I and Region II. The border itself is part of Region II – Region I includes all points to the left of the border, while Region II includes the border and all points to the right of it. A-particles exist only in Region I, B-particles exist only in Region II, while C-particles exist in both regions. The C-particles can travel between the two regions unimpeded, and don’t interact with particles of the other types – they are put in simply to forestall any worries that Regions I and II are best considered as different universes. A force field prevents B-particles from crossing into Region I. A-particles can cross into Region II, but the moment they enter Region II, they turn into B-particles. Otherwise, the A-particles and B-particles don’t interact at all. Finally, suppose that there are no causal loops in this world.

Assuming jump time travel is possible in general, I will now show that jump time travel is possible in the possible world described above. An A-particle can disappear at time t2 in Region I, and can reappear at earlier time t1 in Region II, as a B-particle. This is time travel, analogous to the time traveller entering the time machine in Colorado in 2007 and instantaneously (from the standpoint of the time traveller’s personal time) emerging in Hawaii in 1987. But note that, given the nature of this possible world, this time travel does not produce causal loops. Once the time-travelling particle is in Region II, it is a B-particle, and hence cannot influence anything going on in Region I. Thus, I have provided a scenario where time travel has occurred without the existence of causal loops.

One might object: how could we know that the particle that appeared in Region II at t1 is the same particle that disappeared in Region I at t2? Wouldn’t it be more reasonable to postulate that the particle that appeared in Region II is a wholly new particle? Perhaps, but there are various ways to modify the story to make the persistence claim more plausible. For example, we can postulate that each particle has an unchanging unique property, and that the particle that appears in Region II has the same property that the particle that disappears in Region I has. But one might further object: how do we know that the particle that appeared in Region II is a later version of the particle that disappeared in Region I (from the standpoint of the particle’s personal time)? Here again, we can postulate properties to make the persistence claim more plausible. Specifically, we can postulate that each particle has a property whose value is zero at the moment of particle creation, and whose value continuously increases, analogously to a counter that keeps track of the age of the particle. The particle that disappears in Region I at t2 and the particle that appears in Region II at t1 will have the same value for that property.

3. Perhaps one believes in the possibility of time travel, but not in the possibility of jump time travel. Perhaps one believes that worldlines for particles have to be continuous. I will now show that such time travel is also possible in the world under consideration.

Suppose that an A-particle heads toward the boundary between Regions I and II, and at the very moment it reaches the boundary it simultaneously turns into a B-particle and starts heading backwards in time. Further, suppose that the initial conditions of this possible world ensure that this is the only A-particle that ever reaches the boundary. Suppose that while it is travelling backwards in time, its worldline follows the boundary, thus ensuring that it can’t interact with B-particles from Region II (since they can’t reach the boundary) nor other A-particles that have turned into B-particles (since no other A-particle makes it to the boundary). At the moment that it stops travelling backwards in time, it moves from the boundary into Region II. Here again, we have a case of time travel without causal loops. The time-travelling particle is causally isolated from all the other particles in the universe while it is travelling backwards in time. Once it stops travelling backwards, the region where it started time travel is causally inaccessible, thus ensuring no causal loop is produced.

4. One might think that my argument is too convoluted. Is there an easy way to show that time travel is possible without causal loops?1 Richard Hanley (2004: 130) thinks so.2 He claims that the following time travel story is one where time travel occurs, but no causal loops occur. Leaving out the details, the basic story is that Max is born and grows up in Sydney, and then in 2000 time travels to New York in 1900. He does not father any children, and generally has little impact in New York before dying. Hanley concludes: ‘this is a story in which reverse causation obtains. But are any of the [New York] events causes of any of the [Sydney events]? This is doubtful.’

Hanley is wrong about this. In a universe governed by Newtonian physics, at least, there are events associated with Max’s visit to New York that do causally influence events associated with Max’s pre-departure life in Sydney. (It’s hard to say whether this would be the case according to the true fundamental theory of physics, because we don’t have any completely worked-out candidates for such a theory.) It is difficult to pick out exactly what events these are, without more detail to the story, but we can be confident that such events exist. For example, Max’s arrival in New York exerted gravitational influence on some nearby particles, and this caused some of the particles to change locations. In 2000, some particles in New York – either the original affected ones or ones causally related to the original ones – were still in slightly different locations than they would have been had Max not arrived. (This holds because of the bideterminism of Newtonian physics – setting aside the exotic failures of Newtonian determinism such as described by Norton 2003: 8-9.) It follows that in 2000 these particles in New York exerted a gravitational influence on the particles that composed Max’s body in Sydney in a slightly different way than they would have had Max not arrived in New York. Hence, some of the particles in Max’s body in 2000 would have been in slightly different locations had Max not arrived in New York in 1900. Thus, there is a causal loop: the particular configuration of the particles in Max’s body just before he shows up in New York in 1900 is causally affected by Max’s showing up in New York in 1900.3 I conclude that it is not easy to show that time travel is possible without causal loops – a more careful argument (such as the one I’ve presented above) is required.4

Faye, J. 2005. Backward causation. The Stanford Encyclopedia of Philosophy (Fall 2005 Edition), ed. E. N. Zalta. Available at http://plato.stanford.edu/archives/fall2005/entries/causation-backwards/
Hanley, R. 2004. No end in sight: causal loops in philosophy, physics, and fiction. Synthese 141: 123-52.
Lewis, D. 1976. The paradoxes of time travel. American Philosophical Quarterly 13: 145-52. Reprinted in his Philosophical Papers Volume II. Oxford: Oxford University Press.
Mellor, D. H. 1998. Real Time II. London: Routledge.
Norton, J. D. 2003. Causation as folk science. Philosophers’ Imprint 3. Available at http://www.philosophersimprint.org/003004
Riggs, P. J. 1991. A critique of Mellor’s argument against ‘backwards’ causation. British Journal for the Philosophy of Science 42: 75-86.
Tooley, M. 2002. Backwards causation and the Stalnaker-Lewis approach to counterfactuals. Analysis 62: 191-97.

1 The reader who recalls Tooley 2002 may wonder whether Tooley has already shown that time travel is possible without causal loops. But in fact, Tooley doesn’t show this. Tooley presents a situation where backwards causation occurs without causal loops, but this backwards causation is in no way backwards time travel. Instead, the backwards causation occurs via the following sort of causal law: if location x has properties P and Q at time t, then that state of affairs causes a related location x – x to have P and lack Q at the earlier time t – t. As the reader can see, nothing is travelling backwards in time here.

2 And for a similar argument to Hanley’s, see Riggs 1991: 78-9.

3 Note that I am following Hanley in thinking that there is a close connection between causation and counterfactual dependence – but I take it that the same conclusion would be reached whatever plausible analysis of causation is endorsed.

4 Thanks to Chris Heathwood and the students in my Fall 2006 time travel class for helpful comments and discussion.

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