The Double-Occupancy Problem

Imagine that our friend Tim is sitting in a time machine ready to travel back in time. At time 12:00:00 p.m., Tim pushes the activation button. At 12:00:01 p.m., Tim and his time machine are nowhere to be seen; they have embarked on a journey to the past. On the surface, there appears to be no problem; science-fiction characters time travel all the time. But Tim’s machine travels continuously backwards through time without changing its location. At 11:59:59 AM, Tim is in his time machine and he is about to push the button to go back in time. Yet, at this moment, Tim is also traveling back in time. There must be “two” time machines attempting to occupy exactly the same spatial location, one which is traveling forwards in time and the other traveling backwards in time.

But how can this be? How can separate, material objects come to occupy the same place at the same time? Imagine that you are sitting at a table and you accidentally drop your pencil beneath the table. The best course of action when retrieving your pencil would be to bend down to the left or the right of the table, since both your hand and the table top are solid. Any attempt to reach your hand through the table would result in bumping your hand and, if you tried hard enough, a nasty bruise. This is the essence of The Double-Occupancy Problem as it is discussed by Robin Le Poidevin (2005), Phil Dowe (2000, pp. 445-446), and others.1

Does it make a difference that the time machine traveling forward in time is the same one traveling back in time? In other words, does it matter that there aren’t really two machines? It may seem so because it can appear that there is only one material object trying to occupy one spatial location at 11:59:59 p.m. But given the constant nature of sub-atomic motion, not to mention the motion of the larger parts of the machine (like that button Tim pushed at noon), the parts of the forwards and the backwards traveling machines don’t occupy exactly the same spatial locations; parts that have moved are bound to overlap with other parts, so a portion of the forwards-traveling machine will have to share space with a different portion of the backwards-traveling machine.

There are many ways to approach The Double-Occupancy Problem. For example, we could consider whether double occupancy is really impossible; maybe there could be ghost-like material objects that pass through each other but not other material things. For another example, we could look to physics. The sorts of time travel thought to be consistent with the theory of general relativity, like worm-hole time travel, are a form of continuous backwards time travel that does not encounter double occupancy. Very roughly, there is no turnaround moment or event to worry about; locally the traveler only travels forwards in time (Dowe, pp. 447-448; Le Poidevin, p. 350).

We will go about things in a more direct way. To answer The Double-Occupancy Problem, building on the work of Dowe and Le Poidevin, we will describe a case of continuous backwards time travel that does not have distinct material objects simultaneously sharing a single spatial location even though there is a turnaround point. We will do so in a way that makes the time travel recognizable as a sort to be found in consistent, well-thought-out science fiction. The goal of this conceptual exercise is to gain a better understanding of some of our most basic common-sense assumptions about time and matter. In this regard, we will encounter something unexpected along the way: a reason why endurantists should believe in the possibility of temporal parts.

Discontinuous Time Travel

What if, at 12:00:00 p.m., Tim discontinuously time travels to two hours earlier: 10:00:00 a.m.? There could be a double-occupancy issue. If Tim arrives at the same spatial location he departed from (and if the time machine was not moved from 10:00:00 a.m. up to 12:00:00 p.m.), then the matter of the time machine from the future would find itself in the same place as that of the machine from the past. Nevertheless, double occupancy can be easily avoided if discontinuous time travel is an option merely by having the arrival be at a time and place where the forwards-traveling time machine (and anything else) is not present. In Back to the Future, Marty McFly and the DeLorean don’t have to worry about colliding with themselves when they pop into the 1950s; they discontinuously “jump” from the 1980s to a time before the DeLorean had been built or Marty had been born.

Though discontinuous time travel sidesteps double occupancy, it does invite an issue concerning identity over time. The issue concerns Marty’s jump to the past. Up until his departure, Marty’s existence can be (continuously) traced back to his birth—that is, with each passing instant, starting at the time of his birth, his spatial location changes continuously. But, when the Delorean pops into the 1950s, there is no longer any way to trace Marty’s existence to Marty’s birth. Marty seems to just appear and eventually to just disappear. Philosophers who think that spatial continuity through time is crucial to a person’s identity will contend that the person in the 1950s who looks exactly like Marty is not Marty; the person that emerges back then is not the one that departed in the 1980s.

Movement in Time and Space

Dowe (p. 446) suggests that if the time-traveling object is put into motion, then double occupancy is not an issue. What might such a process look like in real time?

In this first animation, an arrow (as shown in red) is traveling at a constant speed towards Point C. (When the arrow is time traveling backwards, it is shown in black.) As shown by the counter, the entire arrow begins time traveling backwards at (external) time t. That’s when the center of the arrow reaches C. From the perspective of the arrow, it continues traveling with the same velocity through space even as it time travels. However, external observers like us see what looks like two arrows—one arrow moving tip-first towards C from the left and one arrow moving towards C tail-first from the right.

This example does not avoid double occupancy. Putting the arrow in motion throughout its journey is not enough. For instance, at external time t–15, there is an overlap problem involving the back portion and the front portion of the arrow. For another instance, immediately before time t, the forwards-traveling arrow and the backwards-traveling arrow occupy nearly all of the same spatial locations. Double occupancy still exists.

Dowe had something else in mind. Instead of letting the entire object begin to time travel at a specific time, the idea is to let it start time traveling bit by bit at a specific point in time and space. So, for this illustration, beginning at time t, as the tip of the arrow reaches C, only the tip of the arrow will begin to travel back in time. As each further bit of the arrow reaches C, that bit begins time traveling too. C acts as a time-travel field or portal causing what passes through it to begin traveling continuously into the past.

This animation shows the arrow moving from the left toward C. As the arrow crosses C, starting at time t, bit by bit, the object gradually begins to time travel. From the perspective of any bit of the arrow, that bit continues traveling with the same speed and direction as it crosses C. We, however, see something a little surprising. As outside observers we see the arrow moving tip-first towards C from the left and what looks like a second arrow moving tip-first (not tail-first) towards C from the right. We see arrows aimed at each other, moving together, seeming to annihilating each other as they hit C. Notice that there is never any overlap of the arrows. In this case, there is no double occupancy.

The arrow to the right of C is reversed in this case because the tip of the arrow to the left of C starts backwards time traveling first, at time t, and the other bits of this arrow start time traveling later. The other bits will, however, get back to time t. When they get then, they will be farther away from C than the tip of the arrow. The farther the bit of the arrow is from C on the left, the farther away it will be from C on the right.

The Cheshire Cat

Le Poidevin argues that although Dowe’s method escapes The Double-Occupancy Problem, it gives rise to another worry: The Cheshire Cat Problem. If an onlooker were to watch the arrow only while it is to the left of C, all he would see is the arrow disappearing into nothing as it hits C, just like Lewis Carroll’s famous disappearing cat. At any time after t, there are parts of the arrow “missing.” As we get closer to time t+30, only the tail of the arrow is there. Le Poidevin asks, “[H]ow can something continue to exist if there are (external) times when only a spatial part of it does so?” (p. 345). If less than half of the arrow exists at a time, is it true that the arrow exists at that time? What if only a molecule of the arrow exists? At some moment in time, some moment well before the final bits of fletching are gone, the arrow is no longer in existence. The Cheshire Cat Problem concerns the possibility of discontinuous existence. It is similar to the worry about whether Marty McFly ever makes it back to the 1950s.

A natural enough thought is that the so-called second arrow we see headed toward C from the right is the original arrow as it travels backwards in time. If so, it is easy to formulate a discontinuity concern. Consider time t. The arrow seems to be tip to tip with itself. So, somehow, the arrow on the left has to get turned around over to the right-hand side of C. Does it do so in a continuous manner? Apparently not: Looking at what goes on before time t will not help establish the continuous route as the left-side arrow and the right-side arrow are only further away from each other. So, the only hope is to consider what goes on after time t. The problem is that, at some moment after time t, the arrow will not exist and so will not have a spatial location at all. So, there is no continuous function from (external) times to the arrow’s spatial location(s) that links up the arrow as it is located to the left of C at time t with how it is located to the right of C at time t. The arrow seems to discontinuously flip over C. How does the arrow get from one side of C to the other?

That’s not the only puzzle, though. Indeed, it is not The Cheshire Cat Problem. For the sake of argument, drop the tempting assumption that the arrow moving toward C from the right is the original, once-forwards-traveling arrow. Also, focus on the original arrow just as its tip contacts C at time t. That’s when the arrow begins to bit by bit edge its way into the past, and as a result, before time t, to the right of point C, the arrow is spread out through time. To the right side of C before t, only a bit of the arrow exists at any external time, not the arrow itself.

…, before t, when the time machine is, bit by bit, edging its way into the past, are we willing to say that the machine exists when only a small spatial part of it does? Suppose the answer to these questions is no: the existence of only a small spatial part of the machine is not sufficient for the existence of the machine. Then we would have to say that in external time at least, the object has a discontinuous existence… (Le Poidevin, 346).

That is The Cheshire Cat Problem. When the arrow stops time traveling, when it is no longer spread out in time, it will come back into existence. Or will it? How could it if at the intermediate external times it did not exist?

Consider the following animation. It maps the personal time of each bit of the arrow to each bit’s spacetime location — to a spatial location, external time pair — at that personal time. It is personal time that is being counted in the lower right-hand corner.

See the diagonal arrow heading diagonally upward and to the right? Arguably, this is the arrow, before t, to the right of C, heading bit by bit into the past and having only a small spatial part existing at any of these times. Before t, on the right-hand side of C, the arrow is spread out through space and time. On the right-hand side of line C, before time t, there is no external time shown at which the arrow exists. The Cheshire Cat Problem then becomes the problem of how the backwards traveling arrow could survive going out of existence through this stretch of external time.2

What of the arrow to the right of C in Animation 2? Is it the original arrow we see to the left of C? We think that this is a pretty good puzzle in and of itself. Notice that this arrow at each external time before t has the same basic parts as the original arrow, arranged in a nicely arrowish way, in the same way that they were arranged in the original arrow. There are even causal connections between all the parts of the arrow to the right of C and the arrow to the left of C. But, there are some strange goings-on that lead us to conclude that the arrow to the right of C in Animation 2 is not the original arrow that we see to the left of C. The parts of the arrow to the right of C at any external time are all different ages, they have all experienced different amounts of personal time—the closer a point is to the tail, the older it is. Furthermore, if the arrow we see to the right of C in Animation 2 is the original, then it wholly exists at a single time, but we find the thought that the original arrow is spread out in space and time to the right of C before t to be very intuitive. Finally, if the arrow we see to the right of C in Animation 2 is not the original, we can sidestep one discontinuity concern: There is no worry about how the arrow gets flipped from one side of C to the other at time t. It doesn’t. At time t, on the right-hand side of C, the arrow doesn’t exist. At time t and at all times prior to t, at most only a tiny bit of the original arrow exists.

There is a neat consequence of adopting the position that the time traveling arrow is along a temporal diagonal to the right of Point C: The time-traveling arrow has temporal parts. Indeed, to make it from time t, to a time when it is no longer time traveling, it perdures. It persists from t to that other time by having temporal parts. These temporal parts are not similar to the temporal parts perdurantists like David Lewis have tried to persuade us all of. They are not arrow stages; they are not short-lived, arrow-shaped things. They are not the equivalent of the person stages you and I are supposed to have making up the lifetime long spacetime worms that we are. No, these are little bits of the arrow that exist at different times. They are true temporal parts of the arrow. As far as we can tell, these are temporal parts that even an endurantist can love. Maybe this is what a perduring material object is really like.

Continuity Relative to Personal Time

We do favor the position that the original arrow seen to the left of C in all of the animations does not exist at any external time to the right of C; only a minimal part of it does. We do so for the reasons just given and because we believe that it permits nice answers to The Cheshire Cat Problem.

Remember what that problem is: If the arrow doesn’t exist to the right of C at any external time before t, if only a tiny part of it does, then how could the arrow survive its backwards time travel trip? If the arrow were to stop time traveling, then how could it come back into existence? If it did, it would seem to have a discontinuous existence. Proponents of a continuity requirement on identity over time should be concerned.

One answer invokes personal time, but we have to be very careful. Invoking personal time with respect to continuity can easily trivialize a continuity constraint on identity over time. As illustration, consider Marty once again. Relative to his personal time, his trip to the 1950s is continuous. He ends up in the same spatial location from which he left. He goes precisely to where the strip-mall parking lot of his departure would later be built. He continuously changes spatial locations as his personal time passes. Unfortunately, as we have seen, it is exactly this kind of time travel that makes friends of continuity constraints suspicious; they doubt that Marty survives such a trip even given that there is continuity relative to his personal time.

Our first answer to The Cheshire Cat Problem appeals to personal time, but as part of a slightly different sort of continuity. Instead of focusing on spatial location, we should focus on spatial and temporal location. So, instead of spatial continuity in relation to personal time, we should consider spatiotemporal continuity in relation to personal time. For the arrow example, the relevant function is from personal times of the arrow to the spatial-plus-temporal location of the arrow at the particular personal time. Marty doesn’t have this sort of continuity. There is a big temporal jump in Marty’s trip, one from the 1980s to the 1950s though there is no corresponding spatial jump. With the arrow, there are no jumps at all. As Animation 3 illustrates, if the continuity under consideration is spacetime locations in relation to the personal time of the arrow, then it’s smooth sailing.

In presenting this solution, we are making an assumption that Le Poidevin would question. It is that it makes sense to talk of the personal time of the entire arrow in a case like this in which the arrow gradually time travels. He is right to be concerned; we doubt that David Lewis’s official definition of personal time (1976, pp. 146 and 149) works here, because it is defined in terms of traditional perdurantist temporal parts, i.e., person stages. The temporal parts of the arrow as it time travels are only tiny bits of an arrow, not arrow stages.

Still, much more so than Le Poidevin, we are optimistic that some closely related notion of personal time does make sense and can be applied here. For example, one might think of personal time more like the way physicists employ the notion of proper time. Or, one might think of personal time, as Lewis first informally introduces it (p. 146), as that which is measured by the traveler’s wristwatch (assuming that the wristwatch is functioning properly). All we really need to assume is that the time travel process is not speeding up or slowing down the passage of personal time, and so there is an increase of one second of personal time for each second (positive or negative) that passes in external time for all the bits of the arrow. This is tantamount to assuming that, whether time traveling or not, the would-be teeny tiny wristwatches are ticking in the same way for all the bits of the arrow. It is these assumptions that permit us to speak of the personal time of the entire arrow. Animation 3 makes perfectly good sense whether or not we can give some rigorous definition of personal time.

A Continuous Set of Spatiotemporal Locations

There is a second answer to The Cheshire Cat Problem that should be more to Le Poidevin’s liking. It takes seriously the idea that the time-traveling arrow is spread out through time, as did our first solution. But our second solution provides a continuity condition on identity over time that does not include an appeal to personal time.

As before, what we are looking for is a continuity constraint that distinguishes Marty McFly (a supposedly discontinuous time traveler) from the time-traveling arrow. We now suggest requiring that there be a continuous region of spacetime occupied by the person or object in question. The key to this solution is to be careful about the spacetime location of what time travels. We need to recognize again that an object that starts gradually to travel backwards in time will have locations in spacetime that are spread out along the temporal dimension. So, the idea is:

If x is a spacetime location of person/object p and y is also a spacetime location of p, then there is a continuous set of spacetime locations that includes x and y and only other spacetime locations of p.

Consider Marty’s spacetime location in the 1980s just before he time travels and the spacetime location when he supposedly arrives in the 1950s. There is no continuous set of Marty’s spacetime locations that includes these two locations. There is a continuous set of Marty’s spacetime locations starting from his birth until his departure, and there is a continuous set of spacetime locations of Marty’s supposed locations in the 1950s. As the story is told, however, none of Marty’s spacetime locations link these two sets up. With the arrow, the set of all its spacetime locations is a continuous set.

Regarding this second solution, plotting the arrow’s location in spacetime relative to personal time (as we do in Animation 3) turns out to be just a useful way to convince ourselves (and hopefully others) where in spacetime the arrow is as it time travels; personal time isn’t required for specifying the continuity condition.


Thus we have a perfectly germane and coherent example. It is essentially Dowe’s original example with some details filled in and supplemented with a discussion of how the time travel is continuous. The example is recognizable as a case of backwards time travel. It is a case not subject to The Double-Occupancy Problem, and the time travel involved is continuous time travel either in virtue of the arrow’s spatiotemporal continuity relative to personal time or in virtue of the continuity of its set of spacetime locations. In short, the example provides an answer to The Double-Occupancy Problem that is not subject to The Cheshire Cat Problem.

References and Further Reading

Bernstein, Sara “Nowhere Man: Time Travel and Spatial Location.” Midwest Studies in Philosophy 39 (2015): 158-168.

Carroll, John. “Self  Visitation, Traveler Time, and Compatible Properties.” Canadian Journal of Philosophy 41 (2011): 359-370.

Cook, Monte. “Tips for Time Travel.” Philosophers Look at Science Fiction, Nicholas Smith (ed.). Chicago: Nelson-Hall, 1982.

Dowe, Phil. “The Case for Time Travel.” Philosophy 75 (2000): 441-451.

Grey, William. “Troubles with Time Travel.” Philosophy 74 (1999): 55-70.

Harrison, Jonathan. “Dr. Who and the Philosophers or Time-Travel for Beginners.” Aristotelian Society, Supplementary Volume 45 (1971): 1-24.

Le Poidevin, Robin. “The Cheshire Cat Problem and Other Spatial Obstacles to Backwards Time Travel.” Monist 88 (2005): 336-352.

Lewis, David. “The Paradoxes of Time Travel.” American Philosophical Quarterly 13 (1976): 145-152.

Nahin, Paul. Time Machines. New York: Springer-Verlag, 1999.

For page credits, see the “Topics-Page Credits” page.

1. Also see Harrison (1971, 13-15), Cook (1982, 52), Nahin (1999, 23-25) and Grey (1999, 60-62).

2. We are simplifying a bit. The arrow is really located at a lot of spatiotemporal locations at each personal time. (See Carroll 2011.) What we are depicting in Animation 3 is a mapping from personal time of the tip of the arrow to a location of the tip and its co-parts relative to its location at that personal time, first focusing only on the spatiotemporal locations of the tip to the left of C, and finishing looking at the spatiotemporal locations of the tip to the right of C. We will continue making this simplification.