Causal Loops

You are a desperate college student trying to write your last term paper before you graduate. All you want to do is finish and celebrate your accomplishment. Suddenly, someone walks into the cubicle you have been working in all day. This person looks very much like you, but you do not have time to figure out what is going on, because she immediately hands you a USB drive and tells you to turn in the term paper that is on the drive. “Then you can go celebrate,” she says smiling as she quickly leaves, entering the nearby stairwell. You get up and go after her, but she seems to have vanished.

You go back to your cubicle, open up the paper’s file, read it over quickly, and, squashing your ethical dilemma, submit the paper. A few days later, you receive your grade: an A+. You go search for the mysterious stranger who gave you this wonderful paper and end up back at the stairwell. You walk in, pondering where the stranger could have gone. After a moment or two, you leave. As you walk by that cubicle you had toiled so long in, you see yourself! Then it dawns on you. You must give yourself the paper. Thankfully, you had put the USB drive in your pocket and, being an exceptionally unkempt college student, you are still wearing the same pants. You walk into the cubicle, hand yourself the USB drive, give yourself some instructions, and promptly flee back to the stairwell, which transports you back to your present.

How can you give yourself a paper that no one wrote? How can your turning in the paper cause you to turn in the paper?

What are Causal Loops?

A causal loop is a sequence of events e1, … , en. Each event in the loop is one of the causes of the next event. The last event en is one of the causes of the first event e1. If a causal loop has no external (outside-the-loop) causes or effects, then that causal loop is a closed causal loop; otherwise, it is an open causal loop. Think of closed causal loops as causally isolated. Think of open causal loops as causally embedded.

loopdefiimagesfinal

 

Figure 1: Two Kinds of Causal Loops

Are Causal Loops Impossible?

The idea of causal loops is sometimes seen as hosting an inherent paradox. The reasons for this concern vary.

Reason 1: There needs to be an uncaused first cause for every event. Loops can appear ex nihilo (from nothing) seemingly without an ultimate first cause. This concern can be resolved by comparing causal loops with more ordinary causal chains. A causal chain is a sequence of events with each event causing the next event in the sequence (a causal loop is a special sort of causal chain). Some causal chains do not loop; they consist of events in a sequence, each event temporally followed by and causing the next event in the sequence. There is nothing in this definition requiring that the chain have a beginning (or an end). Therefore, it seems a little odd to insist that events in a causal loop require an ultimate first cause. The only difference between causal chains and causal loops is that when following the causation along a causal loop one eventually ends up back where one started.

Furthermore, causal loops do, and even seem bound to have a first cause. Every causal loop will have at least one event that occurs earlier than all the rest. This event may not always turn out to be some ultimate first cause that explains the entire loop. Still, since the initial event occurs before the other events in the timeline, it is the first. It is the earliest event in the loop.

Reason 2: The threat of circular explanation. Consider earlier-than, a relationship that is often associated with an event causing another event. Traditionally, this relation is taken to be transitive; that is, if event a is earlier than event b and event b is earlier than event c, then event a is earlier than event c. However, if one takes causation to be transitive, and one applies it regarding a causal loop, matters break down. If causation is transitive, then every event in a causal loop is a cause of itself. In a three-event causal loop, causation works great for saying that event a causes b and b causes c and c causes a, but, unless causation is not transitive, these three facts lead to the conclusion that a caused a. Certainly, that a caused a is not an informative explanation of a. Fortunately, even if causation is transitive, we need not assume that the corresponding explanatory connections are preserved. Ulrich Meyer (2012, 261) holds that combining all the local explanations leads to a weaker explanation.

Reason 3: Every sequence of events must have a sufficient explanation of why the sequence occurs the way that it does. If every event sequence must have a sufficient explanation, the potential problem is illuminated by considering jinn. A jinni is an object that is a part of the events in the loop and appears to come into existence from nowhere. An excellent example of a jinni is Swann’s necklace from Timerider: The Adventure of Lyle Swann.

In this 1982 film, Swann is accidentally sent back in time and meets a woman named Claire, who eventually seduces him. After a series of spectacular events, the people who accidentally transported him back in time rescue Swann. Just before he is saved, Claire snatches the necklace that was handed down to Swann from his great-great-grandmother who stole it from his great-great-grandfather. The necklace is a jinni because Swann receives the necklace from his great-great-grandmother who stole the same necklace from him years earlier. As you might have guessed, Claire is Swann’s great-great-grandma. Swann himself is his own great-great-grandpa.

The problem posed by the necklace, and most jinn, is the source of their existence. How can a physical object like a necklace just exist? Who designed the necklace? What explains why it is a necklace rather than, say, a bracelet? There must be some explanation for why the necklace is the way it is. Right?

There are explanations for stages of the necklace’s existence. The necklace has causes. Swann’s receiving the necklace from his grandmother is a cause of him taking it with him back in time. The necklace’s going back in time is a cause of Claire being able to steal the necklace, and so on. In addition, one could argue that the universe and natural laws must have a specific structure in order for causal loops to exist. These laws would also be a source for useful explanations.

timerider-timeline

 Figure 2: Timerider Timeline

Some facts, though, seem bound to go unexplained, facts like that the necklace is a necklace and not a bracelet. In addition, why is there a causal loop rather than no causal loop? Does our inability to explain these facts show that there is something incoherent about causal loops? No; the problem with this reasoning is that similar issues arise regarding normal objects. You can see the causes of a chair because you can see the carpenter build the chair from wood, but what made the wood? Even more so, what made the atoms that compose the wood? One can keep asking these questions, but a fully sufficient and complete explanation may be all but impossible to advance in normal circumstances. There are many facts and objects for which we may never find a good explanation.

To take this analysis a step further consider the origin of the artistic design of the necklace. The necklace appeared to be a normal necklace, one that had been crafted with intent and artistry. This begs the question of where the artistry came from. Who’s (or what’s) skill and knowledge went in to creating this necklace? Storrs McCall (2010) says that there is no solution to this problem. Perhaps, some facts simply do not have explanations. Insisting that everything must have an explanation is unwarranted.

Does Time Travel Require Causal Loops?

The assertion that all time travel must include at least one causal loop is widely held. In his 2009 essay on this subject, Bradley Monton presents this position using a quote of D.H. Mellor. According to Monton, Mellor argues against the possibility of time travel by “‘ruling out the causal loops…that cyclical time and backwards time travel need’” (Monton, 2009, 55; Mellor 1998, 131). According to Mellor and others, when a time traveler goes back in time, his actions in the past will always have affects that influence the time traveler’s journey back into time.The assertion that all time travel must include at least one causal loop is widely held. In his 2009 essay on this subject, Bradley Monton presents this position using a quote of D.H. Mellor. According to Monton, Mellor argues against the possibility of time travel by “‘ruling out the causal loops…that cyclical time and backwards time travel need’” (Monton, 2009, 55; Mellor 1998, 131). According to Mellor and others, when a time traveler goes back in time, his actions in the past will always have affects that influence the time traveler’s journey back into time.

Here is one simple example of this phenomenon: In 2020, Jim builds a time machine from some plans he found in his attic and decides to travel back to see his younger self. So Jim travels back to 1990 and finds his younger self. Jim gives his younger self the plans for the time machine. His younger self finds the idea of time travel preposterous and so stuffs the plans in his attic. Then in 2020 Jim finds those plans and builds his time machine.

Clearly, Jim being able to go on his time-travel adventure is dependent on Jim going on the adventure in the first place. Jim’s actions or even simply his presence on arrival could have somehow affected his younger self in a way that leads to his trip back in time. More to the point, it seems that any trip to the past would in some way interact with people, objects, or particles that will eventually move from the past to the future, meaning that all time travel results in a causal loop. The potential for a change that in some way causes a chain of events that influences the trip to the past that produced the original change appears to be genuine.

However, just having the potential to cause an event does not require that the event occur. Monton claims to have come up with a hypothetical situation that involves time travel without causal loops. He describes a universe that is split into two regions that only contain A, B, and C particles. (See Figure 1.) Region 1 contains only A and C particles and the area up to the border of the two regions. Region 2 contains only B and C particles and the area up to and including the same border. A force field keeps B particles from crossing over the border into Region 1. The C particles never interact with either A or B particles and move freely between Regions 1 and 2. When an A particle crosses onto the border, the particle is turned into a B particle immediately.

Now consider the following scenario: An A particle is moving towards the boundary. At the same instance that the particle reaches the boundary it is transformed into a B particle and also begins traveling back in time. While traveling in time the A particle follows the boundary, keeping the A particle from interacting with any of the other particles. Based on the initial conditions of this example, this A particle is the only one that can reach the boundary. As soon as the particle ceases to time travel the A particle moves into Region 2 (Monton 2009, 60).

Montontwo_regionsginal

Figure 3: Monton’s Example

It would seem that the A particle is able to time travel without interacting with any of the particles in either region, including its younger self. While it is time traveling to the past and after it has stopped time traveling, the boundary prevents this particle from interacting with its departure to the past. This means that particle A has successfully time traveled without entering into a causal loop, since the A particle’s time traveling could not have caused this very particle to have initiated its time-travel journey.

In general, especially in more realistic situations, in situations with a physics more like our own, the concept of time travel to the past without causal loops occurring in some form seems unlikely. An extremely careful and specific description is needed in order to generate Monton’s clever scenario.

Do Causal Loops Require Time Travel? 

A causal loop will always contain backwards causation simply because at some point one of the events in the loop has to be a cause of an earlier event. However, this does not prove that all causal loops will include time travel. Does backwards causation always involve time travel?

Sometimes no, and sometimes yes. In a universe where objects can only affect an object that exists at the same time it does, time travel would be necessary to affect objects at an earlier time. In order for an object to affect objects in a time different from the object’s own time, that object would have to travel to the different time and at that time affect the other object. However, if the universe allowed objects to affect each other from different points in time then time travel would no longer be necessary. Since backwards causation does not require time travel, then by extension neither do causal loops.

Causal Loops and Multi-Dimensional Time

The structure of time with branching of timelines—sometimes called multi-dimensional time—removes most of the interesting features of causal loops. (See the Multi-Dimensional Time topic page of our website.) In fact, it is sometimes introduced in order to keep causal loops out (Deutsch 1991, Deutsch and Lockwood 1994). With multi-dimensional time, time traveling causes timelines to split, so an event cannot cause an event along its own past branch. This “unwraps” the loops and all that is left is a series of split causal chains. One consequence of this is that, if multi-dimensional time were to be true, then the answer to the question of whether time travel to the past always involve a causal loop would be a resounding no. A time traveler creates branches instead of loops.

Causal Loops and Physics

To introduce some theoretical causal loops in the context of physics, let us consider the idea of a time-like curve. A time-like curve is an object’s path through space-time where the object persists locally forward in time with time-like connections between each interval. A causal loop occurs when an object’s time-like curve loops back on itself.

One way of introducing a causal loop is with the idea that the universe have a rolled up space-time (Gott 2001, 82-85). The best analogy for this idea is a cylinder where the dimensions that make up space are the axis of the cylinder. This structure allows an object’s time-like curve to loop around the cylinder and meet up with itself.

Image: Cylindrical Space-Time

Figure 4: Curved Space-Time

Wormhole-based time travel also allows for closed time-like curves, see the Relativity and Time Travel topic page of our website.

Physics poses some serious problems for the possibility of jinn. According to the second law of thermodynamics, entropy (or disorder) always increases with time. Consider the example of the necklace in Timerider. According to thermodynamics, in normal circumstances, the entropy of the necklace would increase from the moment that Claire steals the necklace to when the necklace is being passed down to Swann and until Swann travels back in time. Now most understandings of time travel do not alter the state of objects as they travel back in time. However, since the entropy of the necklace right before Swann goes back should have the same amount of entropy as when Swann arrives in the past, this would produce a contradiction. A contradiction arises because the necklace’s entropy just before Swann leaves both equals and is greater than the entropy when Swann arrives in the past. This contradiction means that in order for jinn to exist time travel models must in some way account for reducing the entropy for its return to the past (Gott 2001, 23).

A final interesting application of causal loops in physics is the hypothesis that, rather than originating from a big bang, the universe began as a space-time ‘doughnut’ from which the rest of the universe branched. The authors of this theory, J. Richard Gott and Li-Xin Li (Gott 2001, 186-199), formulated this theory based on an alternative solution to Einstein’s field equations. The space-time doughnut is essentially a causal loop with both closed and open paths around the loop. So, some paths through space-time exist as a loop, but there are others that branch off to make the rest of universe and its contents (cf., Meyer 2012, 259).

Conclusion

Going back to the opening example about that morally ambiguous term paper, let us take a closer look. As you may have guessed from the rest of this discussion, while we can analyze some aspects of this story successfully, many interesting questions are still open for discussion. For instance, did you thereby plagiarize when you submitted the paper? You never sat down and wrote the paper, but you also did not copy or even rely on anyone’s work! For another instance, are the ideas in the paper jinn? Is the digital information on the USB drive a jinni? Is the USB drive itself a jinni? Is the story consistent with thermodynamics? Nothing in the story suggests that entropy somehow fails to increase while the USB drive languished in your pants for a week. Despite these open questions, we have not encountered any inherent paradox associated with causal loops.

 

References and Further Reading

Arntzenius, Frank and Maudlin, Tim. “Time Travel and Modern Physics.” The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), <http://plato.stanford.edu/archives/win2013/entries/time-travel-phys/>.

Dear, William. (director). Timerider: The Adventures of Lynn Swann [Motion Picture]. USA: Zoomo Productions, 1982.

Deutsch, David. ‘Quantum Mechanics Near Closed Timelike Lines.’ Physical Review D 44 (1991): 3197-3217.

Deutsch, David, and Lockwood, Michael. ‘The Quantum Physics of Time Travel.’ Scientific American 270 (1990): 68-74.

Gott, J. Richard. Time Travel in Einstein’s Universe. Boston: Houghton-Mifflin, 2001.

McCall, Storrs. “An Insoluble Problem.” Analysis 70 (2010): 647-648.

Mellor, D. H. Real Time. London: Routledge, 1998.

Meyer, Ulrich. “Explaining Causal Loops.” Analysis 72 (2012): 259-264.
Monton, Bradley. “Time Travel without Causal Loops.” The Philosophical Quarterly 59 (2009): 54-67.

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