a time travel paradox

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Paradox-Free Time Travel Is Theoretically Possible, Researchers Say

Matthew S. Schwartz

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered. Timothy A. Clary/AFP via Getty Images hide caption

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered.

"The past is obdurate," Stephen King wrote in his book about a man who goes back in time to prevent the Kennedy assassination. "It doesn't want to be changed."

Turns out, King might have been on to something.

Countless science fiction tales have explored the paradox of what would happen if you went back in time and did something in the past that endangered the future. Perhaps one of the most famous pop culture examples is in Back to the Future , when Marty McFly goes back in time and accidentally stops his parents from meeting, putting his own existence in jeopardy.

But maybe McFly wasn't in much danger after all. According a new paper from researchers at the University of Queensland, even if time travel were possible, the paradox couldn't actually exist.

Researchers ran the numbers and determined that even if you made a change in the past, the timeline would essentially self-correct, ensuring that whatever happened to send you back in time would still happen.

"Say you traveled in time in an attempt to stop COVID-19's patient zero from being exposed to the virus," University of Queensland scientist Fabio Costa told the university's news service .

"However, if you stopped that individual from becoming infected, that would eliminate the motivation for you to go back and stop the pandemic in the first place," said Costa, who co-authored the paper with honors undergraduate student Germain Tobar.

"This is a paradox — an inconsistency that often leads people to think that time travel cannot occur in our universe."

A variation is known as the "grandfather paradox" — in which a time traveler kills their own grandfather, in the process preventing the time traveler's birth.

The logical paradox has given researchers a headache, in part because according to Einstein's theory of general relativity, "closed timelike curves" are possible, theoretically allowing an observer to travel back in time and interact with their past self — potentially endangering their own existence.

But these researchers say that such a paradox wouldn't necessarily exist, because events would adjust themselves.

Take the coronavirus patient zero example. "You might try and stop patient zero from becoming infected, but in doing so, you would catch the virus and become patient zero, or someone else would," Tobar told the university's news service.

In other words, a time traveler could make changes, but the original outcome would still find a way to happen — maybe not the same way it happened in the first timeline but close enough so that the time traveler would still exist and would still be motivated to go back in time.

"No matter what you did, the salient events would just recalibrate around you," Tobar said.

The paper, "Reversible dynamics with closed time-like curves and freedom of choice," was published last week in the peer-reviewed journal Classical and Quantum Gravity . The findings seem consistent with another time travel study published this summer in the peer-reviewed journal Physical Review Letters. That study found that changes made in the past won't drastically alter the future.

Bestselling science fiction author Blake Crouch, who has written extensively about time travel, said the new study seems to support what certain time travel tropes have posited all along.

"The universe is deterministic and attempts to alter Past Event X are destined to be the forces which bring Past Event X into being," Crouch told NPR via email. "So the future can affect the past. Or maybe time is just an illusion. But I guess it's cool that the math checks out."

• time travel
• grandfather paradox

The Time-Travel Paradoxes

What happens if a time traveler kills his or her grandfather? What is a time loop? How do you stop a time machine from just appearing somewhere in space, millions of kilometers from home? And is there such a thing as free will?

Congratulations! You have a time machine! You can pop over to see the dinosaurs, be in London for the Beatles’ rooftop concert, hear Jesus deliver his Sermon on the Mount, save the books of the Library of Alexandria, or kill Hitler. Past and future are in your hands. All you have to do is step inside and press the red button.

Wait! Don’t do it!

Seriously, if you value your lives, if you want to protect the fabric of reality – run for the hills! Physics and logical paradoxes will be your undoing. From the grandfather paradox to laws of classic mechanics, we have prepared a comprehensive guide to the hazards of time travel. Beware the dangers that lie ahead.

 The Grandfather Paradox

Want to change reality? First think carefully about your grandparents’ contribution to your lives.

The grandfather paradox basically describes the following situation: For some reason or another, you have decided to go back in time and kill your grandfather in his youth. Yeah, sure, of course you love him – but this is a scientific experiment; you don’t have a choice. So your grandmother will never give birth to your parent – and therefore you will never be born, which means that you cannot kill your grandfather. Oh boy! This is quite a contradiction!

The extended version of the paradox touches upon practically every single change that our hypothetical time traveler will make in the past. In a chaotic reality, there is no telling what the consequences of each step will be on the reality you came from. Just as a butterfly flapping its wings in the Amazon could cause a tornado in Texas, there is no way of predicting what one wrong move on your part might do to all of history, let alone a drastic move like killing someone.

There is a possible solution to this paradox – but it cancels out free will: Our time traveler can only do what has already been done. So don’t worry – everything you did in the past has already happened, so it’s impossible for you to kill grandpa, or create any sort of a contradiction in any other way. Another solution is that the time traveler's actions led to a splitting of the universe into two universes – one in which the time traveler was born, and the other in which he murdered his grandfather and was not born.

Information passage from the future to the past causes a similar paradox. Let’s say someone from the future who has my best interests in mind tries to warn me that a grand piano is about to fall on my head in the street, or that I have a type of cancer that is curable if it’s discovered early enough. Because of this warning, I could take steps to prevent the event – but then, there is no reason to send back the information from the future that saves my life. Another contradiction!

Marty finds himself in hot water with the grandfather paradox, from ‘Back to the Future’ 1985

Let’s now assume the information is different: A richer future me builds a time machine to let the late-90s me know that I should buy stock of a small company called “Google”, so that I can make a fortune. If I have free will, that means I can refuse. But future me knows I already did it. Do I have a choice but to do what I ask of myself?

 The Time Loop

In the book All You Zombies by science fiction writer Robert A. Heinlein the Hero is sent back in time in order to impregnate a young woman who is later revealed to be him, following a sex change operation. The offspring of this coupling is the young man himself, who will meet himself at a younger age and take him back to the past to impregnate you know whom.

Confused? This is just one extreme example of a time loop – a situation where a past event is the cause of an event at another time and also the result of it. A simpler example could be a time traveler giving the young William Shakespeare a copy of the complete works of Shakespeare so that he can copy them. If that happens, then who is the genius author of Macbeth?

This phenomenon is also known as the Bootstrap Paradox , based on another story by Heinlein, who likened it to a person trying to pull himself up by his bootstraps (a phrase which, in turn, comes from the classic book The Surprising Adventures of Baron Munchausen). The word ‘paradox’ here is a bit misleading, since there is no contradiction in the loop – it exists in a sequence of events and feeds itself. The only contradiction is in the order of things that we are acquainted with, where cause leads to effect and nothing further, and there is meaning to the question “how did it all begin?”

 Terminator 2 (1991). The shapeshifting android (Arnold Schwarzenegger) destroys himself in order to break the time loop in which his mere presence in the present enabled his production in the future

Time travelers – where have all they gone?

In 1950, over lunch physicist Enrico Fermi famously asked: “If there is intelligent extraterrestrial life in the Universe – then where are they?” indicating that we have never met aliens or came across evidence of their existence, such as radio signals which would be proof of a technological society.  We could pose that same question about time travelers: “If time travel is possible, where are all the time travelers?”

The question, known as the Fermi Paradox, is an important one. After all, if it were possible to travel through time, would we not have bumped into a bunch of observers from the future at critical junctures in history? It is unlikely to assume that they all managed to perfectly disguise themselves, without making any errors in the design of the clothes they wore, their accents, their vocabulary, etc. Another option is that time travel is possible, but it is used with the utmost care and tight control, due to all the dangers we discuss here.

 On June 28, 2009, physicist Stephen Hawking carried out a scientific experiment which was meant to answer this question once and for all. He brought snacks, balloons and champagne and hosted a secret party for time travelers only – but sent out the invitations only on the next day. If no one showed up, he argued, that would be proof that time travel to the past is not possible. The invitees failed to arrive. “I sat and waited for a while, but nobody came,” he reported at the Seattle Science Festival in 2012.

Multiple time travelers also undermine the possibility of a fixed and consistent timeline, assuming that the past can indeed be changed. Imagine, for example, a nail-biting derby between the top clubs, Hapoel Jericho and Maccabi Jericho. Originally Maccabi won, so a Hapoel fan traveled back in time and managed to lead to his team’s victory. Maccabi fans would not give up and did the same. Soon, the whole stadium is filled with time travelers and paradoxes.

 One way or round trip?

When considering travel, it is always continuous – from point A to point B, through all the points in between. Time travel should supposedly be the same: travelers get into their machine, push the button, and go from time A to time B, through all the times in between. But there’s a catch, if we are only travelling through time, then to the casual observer, the time machine continuously exists in the same space between the points in time. The result is that our journey is one-way and the time travelers will stay stuck in the future or the past because the machine itself will block the time-path back. And that is before we even start wondering how to even build this thing in the first place if it already exists in the place where we want to build it.

If that’s the case, then there’s no choice but to assume that there is some way to jump from time to time or place to place and materialize at the destination. How will our machine “know” to jump to an empty area, and to avoid materializing into a wall or a living creature unlucky enough to occupy that same spot? The passengers will undoubtedly need effective navigation and observation equipment to prevent unfortunate accidents at the point of entry.

 Advanced time travel

In addition to the problems that time travel poses for anyone trying to keep the notion of  cause and effect in order, time travelers may also face – or already have faced – other challenges from physics, even classical physics.

One issue you have to consider during time travel, and which science fiction writers usually prefer to ignore for convenience sake, is the question of arrival at the specified time destination and what would happen to us there.

It is usually assumed, with no good reason, that if someone is travelling through time, he or she will land in the same place, but at a different time – past or future. But this is where we hit a snag: the Earth rotates around the sun at a speed of 110,000 kph, and the Solar System itself is moving in its trajectory around the galaxy at a speed of 750,000 kph. If we time-travel for even a few seconds and stay in the same coordinates of space, we will probably find ourselves floating in outer space and perhaps even manage a quick glance around before we die. Our time machine will have to take into account this movement of the heavenly bodies and place us at exactly the right spot in space.

This alone may be resolved, since time travel, in any case, takes place between two points in the four-dimensional space-time continuum. According to the theory of general relativity, the theoretical foundation for time travel, space and time are a single physical entity, known as space-time. This entity can be bent and distorted – in fact gravity itself is an external manifestation of space-time distortion.

The Time Lord ,Doctor Who explains what “time” is exactly (Doctor Who, Season 3, Chapter 10: Blink).

Time travel would be possible if we could create a closed space-time loop, or if we could go from one point to another through a shortcut called a “Wormhole”. This would, in any case, not be just moving from one point in time to another, but would also include moving through space. Thus, from the outset, the journey is not only in time, but necessarily includes a destination point in space, which we will need to pre-program on our machine, of course .

In practice, the situation is more complicated – especially if we want to go into the distant past or distant future. The speed of the celestial bodies, and even the Earth’s shape and the structure of the continents, the seas, and mountains on the face of the Earth, change over the years. And because even a tiny deviation in our knowledge of the past can land us in the core of the Earth, in outer space or somewhere else that immediately reduces life expectancy to zero – time travel becomes a Russian roulette.

 How to travel in time and stay alive

 Let’s assume we coped with this problem and managed to get to the exact point in space-time that can sustain life. Careful – we’re not there yet; we still have to deal with momentum.

Momentum is a conserved quantity, which basically represents the potential of a body to continue moving at the speed and direction in which it is already travelling. If we were to jump out of a moving car (heaven forbid!), conservation of momentum is what would cause us to roll on the ground and probably get injured (in the best-case scenario). And so, if we leap in time – say, a month back – and land at the exact same point on Earth – we would discover, much to our dismay, that even if we started motionless in relation to the ground, now the ground underneath us is moving quickly at one angle or another towards us . Thus, even if we were lucky enough not to crash immediately on impact, we’re likely to hit some obstacle. And if by some miracle we were to survive, we would quickly find ourselves burning up in the atmosphere or gasping for air in space – because we have far exceeded the escape velocity from Earth.

A possible solution to this problem is to plan our landing point ahead, so that the ground speed will be equal in size and direction to our exit speed, but this places many constraints on our journey. We could always leap into space, where there are hardly any moving objects to be bumped into, and only then land again at our point of destination on Earth.

Having said all that, this problem arises chiefly when we assume that time hopping is immediate – that we disappear from one point in time and immediately appear in another, without losing mass, energy, or momentum. But since a “realistic” journey in time is not instantaneous, rather it involves travelling along space-time, it is no different from other types of journeys. This being the case, we can hope that we could adjust our speed to the desired value and direction prior to landing, just like a spacecraft slowing down before landing on a planet.

We should also keep in mind that thankfully, we will have access to a powerful technology that would enable us to cope with such problems: Time-travel technology itself. For example, we might decide to send thousands of tiny probes ahead of us, each to a slightly different point in space-time. Some of them, maybe even most, will be destroyed for one of the reasons already mentioned. The others will wait patiently until the present and then feed their programmed coordinates into the time machine. Thus by definition, the destination entered will be safe for us, except, perhaps for the annoying probe shower hitting the travellers. For the travellers themselves, the entire process will be immediate.

Time Travelling Grammar

Finally, we come to the question: How do you actually talk about time travel? The three tenses – past, present, and future – are insufficient to discuss a future event that happened some time in the past with someone who is in the present, which is another’s past and yet another’s future. And what is the correct grammatical tense to use when we talk about an alternative future that would have been created after we killed our grandfather? Or how do we express the future-past tense (or past-future, or past-future-past?), when we get stuck in a time loop where what will happen leads to what had already taken place, and so on? And of course the biggest question that Hebrew editors and translators have faced for years – is there really such a thing as present continuous?

It’s complicated.

Arguing about tenses and a time machine, The Big Bang Theory, Season 8, Episode 5, 2014

In his book, The Restaurant at the End of the Universe, science fiction writer Douglas Adams suggests to his readers to consult (by Dr. Dan Streetmentioner) Time Traveler's Handbook of 1001 Tense Formations (by Dr. Dan Streetmentioner) to find the answers to these questions. That’s all very well, but, Adams tells us, “most readers get as far as the Future Semi-Conditionally Modified Subinverted Plagal Past Subjunctive Intentional before giving up; and in fact in later editions of the book all pages beyond this point have been left blank to save on printing costs.”

If, despite all of the above, you’re still intent on travelling back to Mount Sinai or the Apollo 11 moon landing – let us then wish you bon voyage, and please give our regards to Neil Armstrong!

Time Travel Paradoxes

• First Online: 11 May 2018

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• S. V. Krasnikov 28  

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 193))

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It seems appropriate now to turn attention to the most controversial issue related to the time machines—the time travel paradoxes. On the one hand, paradoxes seem to be something inherent to time machines (their main attribute, perhaps). On the other hand, the (supposed) paradoxicalness of time travel is traditionally the main objection against it and a good pretext for dismissing causality violating spacetimes from consideration. Recall, however, that in studying physics one meets a lot of ‘paradoxes’ (Ehrenfest’s, Gibbs’, Olbers’, etc.). Today they are just interesting and instructive toy problems. Our aim in this chapter is to examine the ‘temporal paradoxes’ and to reduce them to the same status. In particular, we are going to show that they do not increase the tension between the relativistic concept of spacetime and ‘the simple notion of free will’ (S. W. Hawking and G. F. R. Ellis (1973). The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge) [76]. As a by-product, we shall reveal, in the end of the chapter, a curious relation between the geometry of a spacetime and its matter content.

... Loads of them ended up killing their past or future selves by mistake! Hermiona in [158]

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Excerpts from: Time Travel and Time Machines

Physics and Our Intuitive Outlook on Time

Three facets of time-reversal symmetry

The term was coined by Tadasana.

In fact, this assumption is not that extravagant. I am not aware of a single strong argument against it. Note, in particular, that the apparent lack of contramotes in the everyday life and in astronomical observations is not an argument: the contramotes must be practically invisible to us comotes. Indeed, they almost do not radiate light. Instead, a contramote star, say, absorbs a powerful flux of photons emitted (for some mysterious reason) towards the star by other bodies.

For a collection of such pseudo paradoxes see [138].

We speak of the existence of the note and not of its appearance , because being a typical Cauchy demon, see Sect.  3 in Chap. 2, the note has always existed, without ever having come into being.

In fact, they often are too complex even when consist of billiard balls, see [39, 127].

As is known, ‘...either a tail is there or it isn’t there. You can’t make a mistake about it...’ [128]. The same is true for evolutions. So, we shall not speak of ‘self-inconsistent evolution’ or ‘trajectories with zero multiplicity’.

For a technical description see Example  74 in Chap. 1.

For a less trivial one see [65].

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Krasnikov, S.V. (2018). Time Travel Paradoxes. In: Back-in-Time and Faster-than-Light Travel in General Relativity. Fundamental Theories of Physics, vol 193. Springer, Cham. https://doi.org/10.1007/978-3-319-72754-7_6

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Paradoxes of Time Travel

Ryan Wasserman, Paradoxes of Time Travel , Oxford University Press, 2018, 240pp., $60.00, ISBN 9780198793335.

Reviewed by John W. Carroll, North Carolina State

Wasserman's book fills a gap in the academic literature on time travel. The gap was hidden among the journal articles on time travel written by physicists for physicists, the popular books on time travel by physicists for the curious folk, the books on the history of time travel in science fiction intended for a range of scholarly audiences, and the journal articles on time travel written for and by metaphysicians and philosophers of science. There are metaphysics books on time that give some attention to time travel, but, as far as I know, this is the first book length work devoted to the topic of time travel by a metaphysician homed in on the most important metaphysical issues. Wasserman addresses these issues while still managing to include pertinent scientific discussion and enjoyable time-travel snippets from science fiction. The book is well organized and is suitable for good undergraduate metaphysics students, for philosophy graduate students, and for professional philosophers. It reads like a sophisticated and excellent textbook even though it includes many novel ideas.

The research Wasserman has done is impressive. It reminds the reader that time travel as a topic of metaphysics did not start with David Lewis (1976). Wasserman (p. 2 n 4) identifies Walter B. Pitkin's 1914 journal article as (probably) the first academic discussion of time travel. The article includes a description of what has come to be called the double-occupancy problem, a puzzle about spatial location and time machines that trace a continuous path through space. The same note also includes a lovely passage, which anticipates paradoxes about changing the past, from Enrique Gaspar's 1887 book:

We may unwrap time but we don't know how to nullify it. If today is a consequence of yesterday and we are living examples of the present, we cannot unless we destroy ourselves, wipe out a cause of which we are the actual effects.

These are just two of the many useful bits of Wasserman's research.

Chapter 1 usefully introduces examples of time travel and some examples one might think would involve time travel, but do not (e.g., changing time zones). There is good discussion of Lewis's definition of time travel as a discrepancy between personal and external time, including a brief passage (p. 13) from a previously unpublished letter from Lewis to Jonathan Bennett on whether freezing and thawing is time travel. I had often wonder what Lewis would have said; now I know what he did say!

Chapter 2 dives into temporal paradoxes deriving from discussions of the status of tense and the ontology of time (presentism vs. eternalism vs. growing block vs. . . . ). Here, Wasserman also includes the double-occupancy problem as a problem for eternalism -- though it is not clear that it is only a problem for eternalism. Then he turns to the question of the compatibility of presentism and time travel, the compatibility of time travel and a version of growing block that accepts that there are no future-tensed truths, and finally to a section on relativity and time travel. The section on relativity is solid and seems to me to pull the rug out from under some earlier discussions. For example, Lewis's definition of time travel is shown not to work. It also becomes clear that presentism and the growing block are consistent with both time-dilation-style forward time travel and traveling-in-a-curved-spacetime "backwards" time travel.

Chapters 3 and 4 cover the granddaddies of all the time-travel paradoxes: the freedom paradoxes that include the grandfather paradox, the possibility of changing the past, and the prospects of such changes given models of branching time, models that invoke parallel worlds, and hyper time models. Chapter 4 gets serious about Lewis's treatment of the grandfather paradox and Kadri Vihvelin's treatment of the autoinfanticide paradox (about which I will have more to say).

Chapter 4 also includes discussion of "mechanical" paradoxes that, as stated, do not require modal premises about what something can and cannot do, and no notion of freedom or free will. (See Earman's bilking argument on p. 139 and the Polchinski paradox on p. 141.) Wasserman introduces modality to these paradoxes, but I would have liked them to be addressed on their own terms. As I see it, these paradoxes are introduced to show that backwards time travel or backwards causation in a certain situation validly lead to a contradiction. On their own terms, for these arguments to be valid, the premises of the arguments themselves must be inconsistent. How can one make trouble for backwards time travel if the argument is thus bound to be unsound?

Chapter 5 takes on the paradoxes generated by causal loops or more generally backwards causation including bilking arguments, the boot-strapping paradox (based on a presumption that self-causation is impossible), and the ex nihilo paradox with causal loops and object loops (i.e., jinn) that seem to have no cause or explanation.

Chapter 6 deals with paradoxes that arise from considerations regarding identity, with a focus on the self-visitation paradox from both perdurantist and endurantist perspectives. I was surprised to learn that Wasserman had defended an endurantist-friendly property compatibilism -- similar to my own -- to resolve the self-visitation paradox. I was then delighted to find out that he cleverly extends this sort of compatibilism to the time-travel-free problem of change (i.e., the so-called, temporary-intrinsics argument).

The outstanding scientific issue regarding backwards time travel is whether it is physically possible. There is no question that forwards time travel is actual, or even whether it is ubiquitous. There is also not much question that backwards time travel is consistent with general relativity. Still, we await more scientific progress before we will know whether backwards time travel really is consistent with the actual laws of nature. In the meantime, there is still much to be said about Lewis's treatment of the grandfather paradox and Vihvelin's stated challenge to that treatment in terms of the autoinfanticide paradox.

I will start by being somewhat critical of Lewis's approach. For his part (pp. 108-114), Wasserman does a terrific job of laying out Lewis's position as a metatheoretic discussion of the context sensitivity of 'can' and 'can't'. My concern is that not enough attention is given to the 'can' and 'can't' sentences that turn out true on the semantics. The semantics works only by a contextual restriction of possible worlds based on relevant facts -- the modal base -- associated with a conversational context. In meager contexts, false 'can' sentences will turn out true too easily. For example, suppose two people are having a conversation about Roger. Maybe all the two know about Roger is his name and that he is moving into the neighborhood. So, the proposition that Roger doesn't play the piano is not in the modal base. So, according to Lewis's semantics applied to 'can', 'Roger can play the piano' is true in this context. That seems wrong. This would be an unwarranted assertion for either of the participants in the conversation to make. Notice it is also true relative to the same meager context that Roger can play the harpsichord, the sousaphone, and the nyatiti. Quite a musician that Roger! [1]

Interestingly, though this problem arises for 'can', it does not arise for other "possibility" modals. For example, notice that, with the meager context described above, there is a big difference regarding the assertability of 'Roger could play the piano' and of 'Roger can play the piano'. Similarly, there is also no serious issue with regard to 'Roger might play the piano'. 'Could' and 'might' add tentativeness to the assertion that seems called for. There also seems to be no problem for the semantics insofar as it applies to 'is possible'. 'It is possible that Roger plays the piano' rings true relative to the context. But 'Roger can play the piano'? That shouldn't turn out true, especially if Roger is physically or psychologically unsuited for piano playing.

This issue has been frustrating for me, but Wasserman's book has me leaning toward the idea that what is needed is a contextual semantics that includes a distinguishing conditional treatment of 'can' of the sort Wasserman suggests:

(P1**) Necessarily, if someone would fail to do something no matter what she tried, then she cannot do it (p. 122).

This is a suggestion made by Wasserman on behalf of Vihvelin. I find (P1**) as a promising place to start in terms of the conditional treatment.

Speaking of Vihvelin, her thesis is "that no time traveler can kill the baby that in fact is her younger self, given what we ordinarily mean by 'can'" (1996, pp. 316-317). Vihvelin cites Paul Horwich as a defender of a can-kill solution, what she calls the standard reply :

The standard reply . . . goes something like this: Of course the time traveler . . . will not kill the baby who is her younger self . . . But that doesn't mean she can't . (Vihvelin 1996, p. 315)

Vihvelin's doing so is appropriate given what Horwich says about Charles attending the Battle of Hastings: "From the fact that someone did not do something it does not follow that he was not free to do it" (1975, 435). In contrast, it strikes me as odd that Vihvelin (1996, p. 329, fn. 1) also attributes the standard reply to Lewis. I presume that she does so based on some comments by Lewis. He says, "By any ordinary standards of ability , Tim can kill Grandfather," (1976, p. 150, my emphasis) and especially "what, in an ordinary sense , I can do" (1976, p. 151, my emphasis). So, admittedly, Vihvelin fairly highlights an aspect of Lewis's view as holding that, in the ordinary sense of 'can', Tim can kill Gramps. And I can see how this is a useful presentation of Lewis's position for her argumentative purposes.

Nevertheless, I take Lewis's talk of ordinary standards or an ordinary sense to just be a way to identify the ordinary contexts that arise with uses of 'can' in day-to-day dealings, where the possibility of time travel is not even on the table. Simple stuff like:

Hey, can you reach the pencil that fell on the floor?

Sure I can; here it is.

More importantly, we have to keep in mind that the basic semantics only has consequences about the truth of 'can' sentences once a modal base is in place. To me, the fact that Baby Suzy grows up to be Suzy is exactly the kind of fact that we do not ordinarily hold fixed. Lewis's commitment to the semantics does not make him either a can-kill guy or a can't-kill guy.

What is the upshot of this? There is a bit of underappreciation of Lewis's approach in Wasserman's discussion of Vihvelin's views. The pinching case on p. 119 provides a way to make the point. Consider:

(a) If Suzy were to try to kill Baby Suzy, then she would fail.

(b) If Suzy were to try to pinch Baby Suzy, then she would fail.

According to Wasserman, Vihvelin thinks that even in ordinary contexts (a) and (b) come apart (p. 119, note 32) -- (a) is true and (b) is false. As I see it, a natural context for (a) includes the fact that Baby Suzy grows up normally to be Suzy. That is a supposition that is crucial to the description of the scenario and so is likely to be part of the modal base. No canonical story or suppositions are tied to (b), though Vihvelin stipulates that Suzy travels back in time in both cases. We are not, however, told a story of Baby Suzy living a pinch-free life all the way to adulthood. We are not told whether Suzy decided go back in time because Baby Suzy deserved a pinch for some past transgression. My point is that the stories affect the context. So, with parallel background stories, (a) and (b) need not come apart.

I am not sure whether Wasserman was speaking for himself or for Vihvelin when he says about (a) and (b), "Self-defeating acts are paradoxical in a way other past-altering acts are not" (p. 120). Either way, I disagree. Lewis gives a more general way to resolve the past-alteration paradoxes that is not obviously in any serious conflict with Vihvelin's many utterances that turn out true relative to the contexts in which she asserts them. Wasserman also says, "The only disagreement between Lewis and Vihvelin is over whether Suzy's killing Baby Suzy is compatible with the kinds of facts we normally take as relevant in determining what someone can do" (p. 117). That is an odd thing for him to say. Lewis sketches a semantic theory that provides a framework for the truth conditions of 'can' and 'can't' sentences. He is not in disagreement with Vihvelin. For Lewis, there is one specification of truth conditions for 'can' that gives rise to both 'can kill' and 'can't kill' sentences turning out true relative to different contexts. Indeed, it is tempting to think that Vihvelin takes the fact that Baby Suzy grows up to be Adult Suzy as part of the modal base of the contexts from which she asserts the compelling 'can't-kill' sentences.

That all said, Wasserman's book is a significant contribution. There are those of us who focus a good chunk of our research on the paradoxes of time travel for their intrinsic interest, and especially because they are fun to teach. That is contribution enough for me. But, ultimately, from this somewhat esoteric, fun puzzle solving, we also learn more about the rest of metaphysics. The traditional issues of metaphysics: identity-over-time, freedom and determinism, causation, time and space, counterfactuals, personhood, mereology, and so on, all take on a new look when framed by the questions of whether time travel is possible and what time travel is or would be like. Wasserman's book is a wonderful source that spotlights these connections between the paradoxes of time travel and more traditional metaphysical issues.

Cargile, J., 1996. "Some Comments on Fatalism" The Philosophical Quarterly 46, No. 182 January 1996, 1-11.

Gaspar, E., 1887/2012. The Time-Ship: A Chronological Journey . Wesleyan University Press.

Horwich, P., 1975. "On Some Alleged Paradoxes of Time Travel" The Journal of Philosophy 72, 432-444.

Lewis, D., 1976 "The Paradoxes of Time Travel" American Philosophical Quarterly 13, 145-152.

Pitkin, W., 1914. "Time and Pure Activity" Journal of Philosophy, Psychology and Scientific Methods 11, 521-526.

Vihvelin, K., 1996. "What a Time Traveler Cannot Do" Philosophical Studies 81, 315-330.

[1] This criticism was first presented to me by Natalja Deng in the question-and-answer period for a presentation at the 2014 Philosophy of Time Society Conference. Later on, I found a parallel challenge in work by James Cargile (1996, 10-11) about Lewis's iconic, 'The ape can't speak Finnish, but I can'.

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Time Travel & the Predestination Paradox Explained

May 16, 2017 James Miller Time Travel 3

A Predestination Paradox refers to a phenomenon in which a person traveling back in time becomes part of past events, and may even have caused the initial event that caused that person to travel back in time in the first place. In this theoretical paradox of time travel, history is presented as being unalterable and predestined, with any attempts to change past events merely resulting in that event being fulfilled.

Science fiction has provided fertile ground for exploring this paradox of time travel , and over the years has provided much entertainment in the form of countless books and movies on the subject, some of which are mentioned in this article.

Etymology of Predestination Paradox

Origin of the term ‘predestination’.

The word ‘predestination’ derives from the Greek word “proorizo” with “pro” meaning “before” and the verb “orizo” meaning to “determine”. It has been in use since classical times, with the Greek physician Hippocrates (460-370 BC) using it to describe an intended result following the administration of medication. It is mentioned four times in the Bible, or more specifically in the Epistles of Paul, and over time in theology has come to represent God having immutably determined all events throughout eternity that will come to pass.

Origin of the term ‘Predestination Paradox’

The concept of a predestination paradox has been explored by scientific writers in the past, most notably by Robert A. Heinlein in his short stories entitled “By His Bootstraps” (1941) and “All You Zombies” (1959). However, it was the Star Trek franchise that coined the phrase “Predestination Paradox” in a 1996 episode of Star Trek: Deep Space Nine episode titled “Trials and Tribble-ations”.

The Deep Space Nine episode Trials and Tribble-ations was a homage to Star Trek the Original Series, and involves agents from Starfleet’s Department of Temporal Investigations visiting DS-9. The Department are there to determine whether the timeline has been corrupted after Captain Sisko took the USS Defiant back in time 105 years to save Captain James T. Kirk from being assassinated. The expression Predestination Paradox is used twice throughout the show. The first time is by two time agents who are questioning Captain Sisko while trying to establish his motive for traveling back in time :

LUCSLY: “So you’re not contending it was a predestination paradox?” DULMUR: “A time loop. That you were meant to go back into the past?”

In the second instance, Doctor Bashir worries that after being invited on a date by a woman bearing his great-grandmother’s name, Watley, he could be destined to fall in love with her and become his own great-grandfather, who no one had ever met. As a worried Bashir then ponders: “If I don’t meet with her tomorrow, I may never be born.”

What type of paradox is the Predestination Paradox?

Time travel paradoxes are generally categorized into either:

1) Closed Causal Loops: When an action resulting from time travel to the past ensures the fulfillment of a cause. Examples include the Bootstrap Paradox and Predestination Paradox.

2) Consistency Paradoxes: When an action resulting from time travel to the past stops the cause from ever happening. Examples include the Grandfather Paradox , Hitler Paradox, and Polchinski’s Paradox.

Consistency Paradoxes vs. Causal Loops

To highlight the difference further, consistency paradoxes like the Grandfather Paradox create timeline inconsistencies caused by actually being able to change the past, including killing your own grandfather, thereby preventing your own existence. This would result in an inconsistent and altered version of a past event.

A Predestination Paradox, on the other hand, results in an internally consistent version of history, albeit involving an event that appears to predate the time traveler’s initial decision to travel to the past.  A  chrononaut visiting the past to prevent someone from being killed may still ultimately fail, but they are still able to use their time machine to return to their own present and continue living their lives in a linear fashion.

What is a Time Loop?

Time loops , on the other hand, are a favorite trope of time travel movies in which a person becomes stuck in a certain period of time after which the loop resets and they must repeat the time cycle endlessly. It is unclear whether these loops would be possible in our universe.

Predestination Paradoxes involving Objects

In Predestination (2014) , an intersex temporal agent who has undergone sexual reassignment surgery travels back in time to save his younger female self from falling in love and becoming pregnant by a mysterious male lover, who then disappears, completely ruining her life. Upon meeting his younger, female self, the time traveler subsequently falls in love and impregnates her, thus becoming the very stranger who caused all the heartache he traveled back in time to prevent.

As well as an example of a predestination paradox, the act of self-creation in which the time traveler is his own mother and father is an example of a bootstrap paradox, or a self-created entity (object, data, person) with no discernible point of origin.

A simpler predestination example involves a person traveling back in time to prevent a fire that broke out at a famous museum a century earlier resulting in the destruction of many valuable pieces of art, only to accidentally cause a kerosene lamp to fall, therefore creating the very fire that later motivated them to travel back in the first place. Likewise, a person traveling back in time to save a loved one from suffering a tragic death will be unable to save them from their fate as the event has already been determined.

– Movie Examples

In the 2002 remake of The Time Machine , the scientist Alex Hartdegen witnesses his girlfriend Emma being killed by a mugger looking to steal her engagement ring, after which Hartdegen devotes his life to building a time machine in order to change the past. Once completed, subsequent attempts to interfere with time sees Emma die under different circumstances, including being trampled by a horse, leading him to conclude that “I could come back a thousand times… and see her die a thousand ways.”

He then travels to the future to see whether scientists have discovered a solution on how to change the past, and during a conversation with the Über-Morlock in the distant future is told:

“You built your time machine because of Emma’s death. If she had lived, it would never have existed, so how could you use your machine to go back and save her? You are the inescapable result of your tragedy, just as I am the inescapable result of you.”

Other examples of predestination paradox movies involving physical time travel include the Terminator franchise (1984-2015), Back to the Future (1985), Bill and Ted’s Excellent Adventure (1989), Kate and Leopold (2001), Harry Potter and the Prisoner of Azkaban (2004), Timecrimes (2007), Looper (2012), and Interstellar (2014).

Finally, the movie 12 Monkeys (1995) also presents a worthy example, with the main protagonist James Cole traveling back thirty years in time to investigate a deadly plague that decimated humanity in 1996. During his investigation, he experiences flashbacks to when he was a boy and witnessed a man being shot at an airport, only at the end of the film becoming the very same man he witnessed being killed, while a younger version of himself in 1996 watches on from the airport.

Predestination Paradoxes involving Information

Instead of a person traveling back in time another type of predestination paradox involves information being sent from the future and causing a person to fulfill his part in an event yet to happen. Once again, any attempt to change either the past or future is doomed to ultimately fail.

Say, for instance, one day a man receives information from the future that he was fated to die from a heart attack. He subsequently takes up an active exercise regime in order to avoid his predestined fate but eventually ends up overexerting himself and dying from the very heart attack he set out to prevent. In another example, a person receives future information that they will die by drowning in the future, and so decides never to step foot off dry land. A decade later, her car falls off a collapsing bridge and she drowns in the river, having never learned to swim.

In both these examples, information from the future interacts with past events to form a causality loop, with both cause and effect running in a continuous circle. It is the fact that the information received from the future was truly known to occur that makes them examples of predestination paradoxes, though. Otherwise, it would just be a case of past events causing future actions.

– Literature and Movie Examples

The classic Greek tragedy Oedipus Rex (429 BC) includes a force beyond science component, as even the god Apollo warned King Laius about the supernatural curse placed on his family by King Pelops of Pisa.

The story centers around King Laius, Queen Jocasta and their son Oedipus, whom the oracle at Delphi prophecies will grow up to kill his father and marry his mother, thus bringing disaster on the city of Thebes. King Laius then leaves the infant on a mountainside side to die , which is subsequently found and raised by King Polybus and Queen Merope. After growing up, Oedipus learns of the prophecy and so leaves home to protect his adopted parents, but on his journey quarrels and kills a stranger (Laius), and after later saving the kingless Thebes from a monstrous Sphinx marries the king’s widow, Jocasta, thereby inadvertently fulfilling the prophecy.

In Star Wars Episode III: Revenge of the Sith (2005), Anakin Skywalker sees a premonition of the death of his wife Padmé Amidala while giving birth to Luke and Leia, leading him to turn to the Dark Side in an attempt to save his wife, ultimately causing her to lose the will to live, and die in childbirth.

Possible Solutions to the Predestination Paradox

In Predestination Paradox movies, the protagonist is usually depicted as helpless to change their fate either through a lack of free will, ignorance, or an external force seemingly controlling their actions and circumstances. This tallies with ‘Novikov’s self-consistency principle’ which asserts that a time traveler is constrained to only creating a consistent version of history. In other words, there must be zero probability of creating a time paradox.

According to another solution called the ‘ timeline-protection hypothesis ’, any attempts to change the timeline would result in a probability distortion being created to protect the timeline. Furthermore, a highly improbable event may occur in order to prevent a paradoxical, impossible event from taking place. The force which subsequently interferes with any attempts to alter past events may involve physical laws, fate, or even an improbable event.

A further possibility explored in sci-fi stories is that the time traveler is actually a willing participant in ensuring a paradox is maintained, such as in Predestination (2014), a movie inspired by the book All You Zombies (1959). In both instances, the time traveler not impregnating his younger transgender self would have resulted in him never having been born at all and therefore ceasing to exist.

History Must be Preserved

According to the Predestination Paradox, history is pre-written and anything interacting with past events will only be able to act in a consistent way that enables the already established past events to be preserved. One last classic example highlights this point nicely.

A person builds a time machine to prevent a loved one from being killed by a hit-and-run driver. After traveling to the past and driving to the scene of the crime, they accidentally run over their loved one and cause the very tragedy that they sought to prevent. They then flee the scene of the crime and return to their present and continue with their life knowing that history is pre-written, and that you cannot change an event in the past that has already taken place.

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2 Temporal Paradoxes

• Published: November 2017
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Chapter 2 surveys the various theories of time and explores their consequences for the possibility of time travel. Section 1 introduces the traditional debates over tense and distinguishes between three different views of temporal ontology: eternalism, presentism, and the growing block theory. Section 2 discusses eternalism and the double-occupancy paradox. Section 3 focuses on presentism and various versions of the “no destination” objection. Section 4 looks at the growing block theory and the worry that time travel would allow for future indeterminacy to creep back into the past. Finally, sections 5 and 6 look at the special and general theories of relativity and consider their implications for our understanding of time travel.

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April 26, 2023

Is Time Travel Possible?

The laws of physics allow time travel. So why haven’t people become chronological hoppers?

By Sarah Scoles

yuanyuan yan/Getty Images

In the movies, time travelers typically step inside a machine and—poof—disappear. They then reappear instantaneously among cowboys, knights or dinosaurs. What these films show is basically time teleportation .

Scientists don’t think this conception is likely in the real world, but they also don’t relegate time travel to the crackpot realm. In fact, the laws of physics might allow chronological hopping, but the devil is in the details.

Time traveling to the near future is easy: you’re doing it right now at a rate of one second per second, and physicists say that rate can change. According to Einstein’s special theory of relativity, time’s flow depends on how fast you’re moving. The quicker you travel, the slower seconds pass. And according to Einstein’s general theory of relativity , gravity also affects clocks: the more forceful the gravity nearby, the slower time goes.

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“Near massive bodies—near the surface of neutron stars or even at the surface of the Earth, although it’s a tiny effect—time runs slower than it does far away,” says Dave Goldberg, a cosmologist at Drexel University.

If a person were to hang out near the edge of a black hole , where gravity is prodigious, Goldberg says, only a few hours might pass for them while 1,000 years went by for someone on Earth. If the person who was near the black hole returned to this planet, they would have effectively traveled to the future. “That is a real effect,” he says. “That is completely uncontroversial.”

Going backward in time gets thorny, though (thornier than getting ripped to shreds inside a black hole). Scientists have come up with a few ways it might be possible, and they have been aware of time travel paradoxes in general relativity for decades. Fabio Costa, a physicist at the Nordic Institute for Theoretical Physics, notes that an early solution with time travel began with a scenario written in the 1920s. That idea involved massive long cylinder that spun fast in the manner of straw rolled between your palms and that twisted spacetime along with it. The understanding that this object could act as a time machine allowing one to travel to the past only happened in the 1970s, a few decades after scientists had discovered a phenomenon called “closed timelike curves.”

“A closed timelike curve describes the trajectory of a hypothetical observer that, while always traveling forward in time from their own perspective, at some point finds themselves at the same place and time where they started, creating a loop,” Costa says. “This is possible in a region of spacetime that, warped by gravity, loops into itself.”

“Einstein read [about closed timelike curves] and was very disturbed by this idea,” he adds. The phenomenon nevertheless spurred later research.

Science began to take time travel seriously in the 1980s. In 1990, for instance, Russian physicist Igor Novikov and American physicist Kip Thorne collaborated on a research paper about closed time-like curves. “They started to study not only how one could try to build a time machine but also how it would work,” Costa says.

Just as importantly, though, they investigated the problems with time travel. What if, for instance, you tossed a billiard ball into a time machine, and it traveled to the past and then collided with its past self in a way that meant its present self could never enter the time machine? “That looks like a paradox,” Costa says.

Since the 1990s, he says, there’s been on-and-off interest in the topic yet no big breakthrough. The field isn’t very active today, in part because every proposed model of a time machine has problems. “It has some attractive features, possibly some potential, but then when one starts to sort of unravel the details, there ends up being some kind of a roadblock,” says Gaurav Khanna of the University of Rhode Island.

For instance, most time travel models require negative mass —and hence negative energy because, as Albert Einstein revealed when he discovered E = mc 2 , mass and energy are one and the same. In theory, at least, just as an electric charge can be positive or negative, so can mass—though no one’s ever found an example of negative mass. Why does time travel depend on such exotic matter? In many cases, it is needed to hold open a wormhole—a tunnel in spacetime predicted by general relativity that connects one point in the cosmos to another.

Without negative mass, gravity would cause this tunnel to collapse. “You can think of it as counteracting the positive mass or energy that wants to traverse the wormhole,” Goldberg says.

Khanna and Goldberg concur that it’s unlikely matter with negative mass even exists, although Khanna notes that some quantum phenomena show promise, for instance, for negative energy on very small scales. But that would be “nowhere close to the scale that would be needed” for a realistic time machine, he says.

These challenges explain why Khanna initially discouraged Caroline Mallary, then his graduate student at the University of Massachusetts Dartmouth, from doing a time travel project. Mallary and Khanna went forward anyway and came up with a theoretical time machine that didn’t require negative mass. In its simplistic form, Mallary’s idea involves two parallel cars, each made of regular matter. If you leave one parked and zoom the other with extreme acceleration, a closed timelike curve will form between them.

Easy, right? But while Mallary’s model gets rid of the need for negative matter, it adds another hurdle: it requires infinite density inside the cars for them to affect spacetime in a way that would be useful for time travel. Infinite density can be found inside a black hole, where gravity is so intense that it squishes matter into a mind-bogglingly small space called a singularity. In the model, each of the cars needs to contain such a singularity. “One of the reasons that there's not a lot of active research on this sort of thing is because of these constraints,” Mallary says.

Other researchers have created models of time travel that involve a wormhole, or a tunnel in spacetime from one point in the cosmos to another. “It's sort of a shortcut through the universe,” Goldberg says. Imagine accelerating one end of the wormhole to near the speed of light and then sending it back to where it came from. “Those two sides are no longer synced,” he says. “One is in the past; one is in the future.” Walk between them, and you’re time traveling.

You could accomplish something similar by moving one end of the wormhole near a big gravitational field—such as a black hole—while keeping the other end near a smaller gravitational force. In that way, time would slow down on the big gravity side, essentially allowing a particle or some other chunk of mass to reside in the past relative to the other side of the wormhole.

Making a wormhole requires pesky negative mass and energy, however. A wormhole created from normal mass would collapse because of gravity. “Most designs tend to have some similar sorts of issues,” Goldberg says. They’re theoretically possible, but there’s currently no feasible way to make them, kind of like a good-tasting pizza with no calories.

And maybe the problem is not just that we don’t know how to make time travel machines but also that it’s not possible to do so except on microscopic scales—a belief held by the late physicist Stephen Hawking. He proposed the chronology protection conjecture: The universe doesn’t allow time travel because it doesn’t allow alterations to the past. “It seems there is a chronology protection agency, which prevents the appearance of closed timelike curves and so makes the universe safe for historians,” Hawking wrote in a 1992 paper in Physical Review D .

Part of his reasoning involved the paradoxes time travel would create such as the aforementioned situation with a billiard ball and its more famous counterpart, the grandfather paradox : If you go back in time and kill your grandfather before he has children, you can’t be born, and therefore you can’t time travel, and therefore you couldn’t have killed your grandfather. And yet there you are.

Those complications are what interests Massachusetts Institute of Technology philosopher Agustin Rayo, however, because the paradoxes don’t just call causality and chronology into question. They also make free will seem suspect. If physics says you can go back in time, then why can’t you kill your grandfather? “What stops you?” he says. Are you not free?

Rayo suspects that time travel is consistent with free will, though. “What’s past is past,” he says. “So if, in fact, my grandfather survived long enough to have children, traveling back in time isn’t going to change that. Why will I fail if I try? I don’t know because I don’t have enough information about the past. What I do know is that I’ll fail somehow.”

If you went to kill your grandfather, in other words, you’d perhaps slip on a banana en route or miss the bus. “It's not like you would find some special force compelling you not to do it,” Costa says. “You would fail to do it for perfectly mundane reasons.”

In 2020 Costa worked with Germain Tobar, then his undergraduate student at the University of Queensland in Australia, on the math that would underlie a similar idea: that time travel is possible without paradoxes and with freedom of choice.

Goldberg agrees with them in a way. “I definitely fall into the category of [thinking that] if there is time travel, it will be constructed in such a way that it produces one self-consistent view of history,” he says. “Because that seems to be the way that all the rest of our physical laws are constructed.”

No one knows what the future of time travel to the past will hold. And so far, no time travelers have come to tell us about it.

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Classic Time Travel Paradoxes (And How To Avoid Them)

[Movie still from  Time Machine , Warner Bros. and Dreamworks]

Editor’s Note: We’re bringing back one of our most loved posts because hey, time travel is always a relevant topic of discussion. Originally published 11/30/12.

Author’s Note: I assume that some day, this article will serve as an invaluable guide and warning for our time traveling ancestors-to-be (who will of course be unable to read books and learn these lessons for themselves, either because [a] all the books will have been burned, or [b] kids will have stopped reading books entirely, because grumble grumble, god damn kids, when I was your age, video games, blah blah, detriment to society, buncha hooligans, kids these days, no respect, etc). In the meantime, just enjoy it for all of its delightfully entertaining/convoluted/paradoxical pleasures.

As anyone who’s anyone who’s read any time travel story ever could easily tell you, time travel is a tricky subject. Temporal paradoxes might seem simple and straightforward at the start (no they don’t), but they always devolve quite quickly (linear time-wise) into some sort of trippy, philosophically complicated, timey-wimey conundrum that makes even the most convoluted middle school relationship make sense by comparison. Come to think of it, maybe the reason that all those cool kids in middle school suffer from impossibly complicated and melodramatic romances to begin with is because they’re all too “cool” to read time travel stories in the first place, which would obviously teach them the benefits of temporally linear dating, if nothing else.

I’m looking at you, River Song.

For the most part, any paradox related to time travel can generally be resolved or avoided by the Novikov self-consistency principle, which essentially asserts that for any scenario in which a paradox might arise, the probability of that event actually occurring is zero — or, to quote from LOST, “whatever happened, happened,” meaning that no matter what anyone does, they can’t actually create a paradox, because the laws of quantum physics will self-correct to avoid such a situation. Still, I’m wary of such a loose explanation for things, and so below, I’ve compiled a list of a few of the more popular time travel paradoxes — and what to do to avoid them.

ONTOLOGICAL PARADOX : Also known as the “Bootstraps Paradox,” an ontological paradox arises when a person or object is sent through time and recovered by another person, whose actions then lead to the original person or object back to the time from when it came in the first place, thus creating an endless loop with no discernible point of origin. Thus, the original person or object is essentially “pulling itself up by its own bootstraps,” hence the nickname (thanks in no small part to the Robert Heinlein story “By His Bootstraps”).

Example : The Terminator films are a prime and popular example of the Ontological Paradox. In the future, a Terminator is sent back in time to kill the mother of resistance leader John Connor before he is born. While the original T-800 is ultimately destroyed, the leftover pieces are found by scientists who use the technological to…develop and create Skynet, and the Terminator-series robots. Skynet would have never been created if Skynet hadn’t taken over the world and then sent a Terminator back in time to get destroyed and ultimately lead to the creation of Skynet. Trippy, right?

There’s also the fact that Future John Connor sends his buddy Kyle Reese back in time to protect his mother from the T-800, only Kyle ends up totally bangin’ John’s mom (dude high five! I mean, not cool, man) and impregnates her with his buddy John Connor. So to top it all off, if John hadn’t sent his friend back in time, his friend would never have had sex with John’s mom, and John would never have been born (meaning that Kyle Reese is either the best or worst friend, ever).

How to Avoid : No one’s really sure if a real-life ontological paradox would lead to some massive hemorrhaging of spacetime, or if the closed loop is kind of automatically self-corrected since it all works itself out evenly in the end anyway. Still, better to avoid these kind of complicated situations, and the best way to do that would simply be to stop taking candy from strangers — “candy” in this case being mysterious or alien artifacts with questionable origins, possibly given to you by mysterious people who may or may not come from the future. See? Maybe all those warnings that your Mom gave you when you were a little kid still mean something today. Or maybe all along she was just trying to prevent you from sending your friends back in time to sleep with her. Or perhaps encourage it…

PREDESTINATION PARADOX : The predestination paradox is similar to the ontological paradox in that the Cause leads to an Effect which then leads back to the initial Cause. The basic tenant of the predestination paradox is similar to that of a self-fulfilling prophecy: the motivation for the time traveler to travel in time is ultimately realized to have been the time traveler’s fault, due to his or her decision to time travel in the first place, or else otherwise unavoidable. Stories involving predestination paradoxes often involve a heavy sense of irony — the time traveler might go back in time in order to change something, for example, but his or her actions inadvertently lead to the exact situation that inspired the time traveler to have gone back and changed things. Thus, nothing ultimately changes. Determinism is a bleak friend.

Example : In Twelve Monkeys, James Cole is sent back in time to prevent a mysterious disaster involving the “Army of the Twelve Monkeys.” His wild rantings in the past about the terrible future from which he came are overheard by Jeffrey Goines, a mental patient who is remembered in the future as the leader of Army of the Twelve Monkeys. Ultimately, Cole’s efforts to prevent his future from happening inspire the actions that lead to his future coming to be. And in a cruel twist of irony, James Cole’s childhood memory of a man in a airport being shot and falling into the arms of a beautiful blonde — the memory that haunts him for the rest of his life — turns out that the guy who was shot was actually him, in the future, dooming young James Cole to grow up and repeat the cycle all over again.

How to Avoid : This one’s tricky, because philosophically, it’s all about free will (or lack thereof). So in fact, by trying to teach you to how to avoid falling victim to the tenants of the predestination paradox, I’m probably going to inspire you to go back in time and create the French film La jetée, which in turn inspires Terry Gilliam to make Twelve Monkeys, which in turn inspires me to use it as an example in this article, et cetera et cetera. Basically we’re all screwed, unless we avoid time travel and time travelers all together. Even a many worlds theory/alternate timeline thing can’t prevent this, because your actions wouldn’t even create a divergent timeline — they would just result in your present situation. So, sorry dude, nothing you can do is going to change anything. Again, unless you don’t do anything at all, although that still doesn’t guarantee anything.

GRANDFATHER PARADOX : This one perfectly demonstrates the aforementioned Novikov self-consistency principle. The basic idea is that, no matter how hard you try, you can’t go back in time and kill your grandfather, because if you did, your mother or father would never have been born, which means that you would never have been born, which means you couldn’t have gone back in time and killed your grandfather, which means that you didn’t go back in time and kill your grandfather, because you can’t go back in time and kill your grandfather, because if you did, you wouldn’t be born, which you obviously have already been born because if you were never born then you couldn’t have gone back in time and tried (and failed) to kill your grandfather in the first place.

That’s just a simple and straightforward summary though. You know, in Layman’s terms.

Basically, the Grandfather paradox conveys the idea of a self-correcting universe and/or fixed points in time. Even if you were able to go back in time and, I don’t know, shoot your Grandpa in the head before he ever meets your Grandma (jeez, you must really hate that guy, huh?), your Grandfather would turn out to be an early sperm donor or something, who would still manage even posthumously to impregnate your Grandmother, because you would have to exist in order to have shot him in the head in the first place. So you might be able to fudge a few temporal details here and there, but no matter what you do, the end result stays the same.

Example : Let’s just say that when you’re LOST on a magical tropical island somewhere in the Pacific Ocean (ish?) and you end up skipping through time and decide to try to kill that evil guy while he’s still a kid and/or stop a nuclear bomb you’ve so affectionately nicknamed “The Jughead” from exploding and causing all kinds of electromagnetic problems and inconsistencies on your already-mystical island home, the best that’s going to happen is you get some kind of weird Hindu sideways limbo reality that works as a parallel narrative to the entire last season of your television show. Oh, and that little kid you shot still turns out to be pretty evil, and it’s all your fault.

How to Avoid : Uhh, don’t try to kill your grandfather in the past before the birth of your father? Take that as a metaphor all you’d like.

HITLER’S MURDER PARADOX : This is similar to the Grandfather Paradox, in that the time traveller goes back in time to change something significant that has already happened. Unlike the Grandfather Paradox (which we assume would self-correct despite our best efforts), the change that one wishes to affect in the Hitler’s Murder Paradox is one that is more technically feasible — as in not intrinsically paradoxical — but still ultimately problematic.

The name comes from the idea that one could theoretically go back in time and kill Adolf Hitler before the Holocaust happened, thus preventing the systematic annihilation of some six million Jews and other minorities. Which, ya know, all sounds good and well, except that it tends to lead to some kind of downward spiraling domino effect with plenty of other consequences that the well-intentioned time traveler probably didn’t consider, and which ultimately might lead to a worse situation than that which the time traveler had hoped to prevent.

Example : This kind of stuff is rampant in comic books, especially X-Men, but the best example of it was the early 90s Age of Apocalypse storyline, in which Professor Xavier’s schizophrenic mutant son, Legion, decides to make daddy proud by helping his dream of mutant-human co-existence come true. Legion concludes that the best way to do this is to go back in time and kill Magneto before he becomes, ya know, Magneto. The only problem is, Magneto and Xavier were like totally BFF back then, so Xavier ends up taking the bullet for Magneto and dies (so yes, Legion does technically end up killing his own father, but that’s not the point).

As a result of there being no Charles Xavier, the psycho evil Darwinist uber-mutant Apocalypse ends up taking over the world before Magneto’s team of X-Men (named in honor of his deceased friend) are able to stop him, which leads to all kinds of crazy situations like evil Hank McCoy aka Dark Beast, who works alongside the evil versions of Cyclops and Havok, or a Sabretooth who is actually a pretty likeable superhero and a member of the X-Men. Oh, also, Magneto and Rogue totally have the sex, and humans are being systematically slaughtered in concentration camps by Apocalypse and his cronies. So basically, in his attempt to kill a perceived “Hitler” in the form of Magneto, Legion caused a real and even more twisted Holocaust to happen. WHOOPS.

How to Avoid : In addition to the whole alternate-reality-that-is-ironically-worse-than-the-world-as-it-used-to-be problem, there’s also the moral compromise of killing an innocent child, even though you know that child is going to grow up to become pretty much the worst (greatest?) mass murderer in history. The best way to avoid it is simply and sadly to accept that you cannot change the past and shouldn’t even try. That is, unless you’re smart enough to have eliminated any possibility of negative domino effect resulting out of your actions.

For example, if you went back in time and eliminated M. Night Shyamalan shortly before the release of Signs, there would be nothing but positive results; the world would mourn the tragic and mysterious loss of a gifted young filmmaker taken before his time, we would all be so blinded by the shock of his death that we’d be able to ignore how bad the aliens looked in that movie (and the fact that seeing them at all was completely unnecessary), and the rest of us wouldn’t have been forced to endure such awful schlock as The Happening or Lady in the Water. See? That way everyone wins!

BUTTERFLY EFFECT : Similar to the cascading domino effect of the Hitler’s Murder Paradox, but on a different level. Whereas killing Hitler would obviously be a landmark event with quite a significant historical impact, something like, say, accidentally stepping on a bug in the past probably wouldn’t have as big of an effect, right?

Have you even been paying attention? Of course it will! That’s the whole point of a time travel paradox! Just like the way that a butterfly flapping its wings in Brazil can affect a weather system in Texas, one tiny change in the past can lead to all kinds of Rube Goldbergian complications that can subtly — or seriously — affect the present. The term “Butterfly Effect” is actually derived from “A Sound of Thunder,” a short story by Ray Bradbury, in which a character accidentally steps on a butterfly in prehistoric times and causes catastrophic changes in the future from which he came.

Example : In Orpheus With Clay Feet by Philip K. Dick, the main character, Jesse Slade, enlists in the services of a time travel tourism agency, who set him up with a trip that allows him to go back in time and act as a muse for some significant historical figure. Slade chooses to go back and inspire his favorite science fiction writer Jack Dowland (which was also Dick’s pen name). Unfortunately, in his efforts to inspire Dowland’s monumental science fiction work, Slade directly reveals to Dowland that he is a time traveler hoping to inspire his work. Dowland takes this as an insulting ruse, and as a result, never becomes the great science fiction writer that he is meant to be. He does, however, publish a single science short story, under the pen name Philip K. Dick: a story called Orpheus With Clay Feet, about a time traveler that goes back in time to inspire his favorite science fiction writer, a man named Jack Dowland.

How to Avoid : Watch your step

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Time Travel and Modern Physics

Time travel has been a staple of science fiction. With the advent of general relativity it has been entertained by serious physicists. But, especially in the philosophy literature, there have been arguments that time travel is inherently paradoxical. The most famous paradox is the grandfather paradox: you travel back in time and kill your grandfather, thereby preventing your own existence. To avoid inconsistency some circumstance will have to occur which makes you fail in this attempt to kill your grandfather. Doesn't this require some implausible constraint on otherwise unrelated circumstances? We examine such worries in the context of modern physics.

1. A Botched Suicide

2. why do time travel suicides get botched, 3. topology and constraints, 4. the general possibility of time travel in general relativity, 5. two toy models, 6. remarks and limitations on the toy models, 7. slightly more realistic models of time travel, 8. even if there are constraints, so what, 9. quantum mechanics to the rescue, 10. conclusions, other internet resources, related entries.

You are very depressed. You are suicidally depressed. You have a gun. But you do not quite have the courage to point the gun at yourself and kill yourself in this way. If only someone else would kill you, that would be a good thing. But you can't really ask someone to kill you. That wouldn't be fair. You decide that if you remain this depressed and you find a time machine, you will travel back in time to just about now, and kill your earlier self. That would be good. In that way you even would get rid of the depressing time you will spend between now and when you would get into that time machine. You start to muse about the coherence of this idea, when something amazing happens. Out of nowhere you suddenly see someone coming towards you with a gun pointed at you. In fact he looks very much like you, except that he is bleeding badly from his left eye, and can barely stand up straight. You are at peace. You look straight at him, calmly. He shoots. You feel a searing pain in your left eye. Your mind is in chaos, you stagger around and accidentally enter a strange looking cubicle. You drift off into unconsciousness. After a while, you can not tell how long, you drift back into consciousness and stagger out of the cubicle. You see someone in the distance looking at you calmly and fixedly. You realize that it is your younger self. He looks straight at you. You are in terrible pain. You have to end this, you have to kill him, really kill him once and for all. You shoot him, but your eyesight is so bad that your aim is off. You do not kill him, you merely damage his left eye. He staggers off. You fall to the ground in agony, and decide to study the paradoxes of time travel more seriously.

The standard worry about time travel is that it allows one to go back and kill one's younger self and thereby create paradox. More generally it allows for people or objects to travel back in time and to cause events in the past that are inconsistent with what in fact happened. (See e.g., Gödel 1949, Earman 1972, Malament 1985a&b, Horwich 1987.) A stone-walling response to this worry is that by logic indeed inconsistent events can not both happen. Thus in fact all such schemes to create paradox are logically bound to fail. So what's the worry?

Well, one worry is the question as to why such schemes always fail. Doesn't the necessity of such failures put prima facie unusual and unexpected constraints on the actions of people, or objects, that have traveled in time? Don't we have good reason to believe that there are no such constraints (in our world) and thus that there is no time travel (in our world)? We will later return to the issue of the palatability of such constraints, but first we want to discuss an argument that no constraints are imposed by time travel.

Wheeler and Feynman (1949) were the first to claim that the fact that nature is continuous could be used to argue that causal influences from later events to earlier events, as are made possible by time travel, will not lead to paradox without the need for any constraints. Maudlin (1990) showed how to make their argument precise and more general, and argued that nonetheless it was not completely general.

Imagine the following set-up. We start off having a camera with a black and white film ready to take a picture of whatever comes out of the time machine. An object, in fact a developed film, comes out of the time machine. We photograph it, and develop the film. The developed film is subsequently put in the time machine, and set to come out of the time machine at the time the picture is taken. This surely will create a paradox: the developed film will have the opposite distribution of black, white, and shades of gray, from the object that comes out of the time machine. For developed black and white films (i.e. negatives) have the opposite shades of gray from the objects they are pictures of. But since the object that comes out of the time machine is the developed film itself it we surely have a paradox.

However, it does not take much thought to realize that there is no paradox here. What will happen is that a uniformly gray picture will emerge, which produces a developed film that has exactly the same uniform shade of gray. No matter what the sensitivity of the film is, as long as the dependence of the brightness of the developed film depends in a continuous manner on the brightness of the object being photographed, there will be a shade of gray that, when photographed, will produce exactly the same shade of gray on the developed film. This is the essence of Wheeler and Feynman's idea. Let us first be a bit more precise and then a bit more general.

For simplicity let us suppose that the film is always a uniform shade of gray (i.e. at any time the shade of gray does not vary by location on the film). The possible shades of gray of the film can then be represented by the (real) numbers from 0, representing pure black, to 1, representing pure white.

Let us now distinguish various stages in the chronogical order of the life of the film. In stage S 1 the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage S 2 of the life of the film, it has been developed and is about to enter the time machine. Stage S 3 occurs just after it exits the time machine and just before it is photographed. Stage S 4 occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage S 1 in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages S 1 and S 2 . During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. I.e., the shade of gray that the film acquires at stage S 2 depends on the shade of gray it has at stage S 3 . The influence of the shade of gray of the film at stage S 3 , on the shade of gray of the film at stage S 2 , can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function f from [0,1] to [0,1] must intersect the line x = y somewhere, and thus there must be at least one point x such that f ( x )= x . Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox.

Let us now be more general and allow color photography. One can represent each possible color of an object (of uniform color) by the proportions of blue, green and red that make up that color. (This is why television screens can produce all possible colors.) Thus one can represent all possible colors of an object by three points on three orthogonal lines x , y and z , that is to say, by a point in a three-dimensional cube. This cube is also known as the ‘Cartesian product’ of the three line segments. Now, one can also show that any continuous map from such a cube to itself must have at least one fixed point. So color photography can not be used to create time travel paradoxes either!

Even more generally, consider some system P which, as in the above example, has the following life. It starts in some state S 1 , it interacts with an object that comes out of a time machine (which happens to be its older self), it travels back in time, it interacts with some object (which happens to be its younger self), and finally it grows old and dies. Let us assume that the set of possible states of P can be represented by a Cartesian product of n closed intervals of the reals, i.e., let us assume that the topology of the state-space of P is isomorphic to a finite Cartesian product of closed intervals of the reals. Let us further assume that the development of P in time, and the dependence of that development on the state of objects that it interacts with, is continuous. Then, by a well-known fixed point theorem in topology (see e.g., Hocking and Young 1961, p 273), no matter what the nature of the interaction is, and no matter what the initial state of the object is, there will be at least one state S 3 of the older system (as it emerges from the time travel machine) that will influence the initial state S 1 of the younger system (when it encounters the older system) so that, as the younger system becomes older, it develops exactly into state S 3 . Thus without imposing any constraints on the initial state S 1 of the system P , we have shown that there will always be perfectly ordinary, non-paradoxical, solutions, in which everything that happens, happens according to the usual laws of development. Of course, there is looped causation, hence presumably also looped explanation, but what do you expect if there is looped time?

Unfortunately, for the fan of time travel, a little reflection suggests that there are systems for which the needed fixed point theorem does not hold. Imagine, for instance, that we have a dial that can only rotate in a plane. We are going to put the dial in the time machine. Indeed we have decided that if we see the later stage of the dial come out of the time machine set at angle x , then we will set the dial to x +90, and throw it into the time machine. Now it seems we have a paradox, since the mapping that consists of a rotation of all points in a circular state-space by 90 degrees does not have a fixed point. And why wouldn't some state-spaces have the topology of a circle?

However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage S 2 must be a continuous function of the state of the dial at stage S 3 . But, the state of the dial at stage S 2 is arrived at by taking the state of the dial at stage S 1 , and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage S 2 , should be a continuous function of the cause, namely the state of the dial at stage S 3 . It is additionally the case that path taken to get there, the way the dial is rotated between stages S 1 and S 2 must be a continuous function of the state at stage S 3 . And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case.

Forget time travel for the moment. Suppose that you and I each have a watch with a single dial neither of which is running. My watch is set at 12. You are going to announce what your watch is set at. My task is going to be to adjust my watch to yours no matter what announcement you make. And my actions should have a continuous (single valued) dependence on the time that you announce. Surprisingly, this is not possible! For instance, suppose that if you announce “12”, then I achieve that setting on my watch by doing nothing. Now imagine slowly and continuously increasing the announced times, starting at 12. By continuity, I must achieve each of those settings by rotating my dial to the right. If at some point I switch and achieve the announced goal by a rotation of my dial to the left, I will have introduced a discontinuity in my actions, a discontinuity in the actions that I take as a function of the announced angle. So I will be forced, by continuity, to achieve every announcement by rotating the dial to the right. But, this rotation to the right will have to be abruptly discontinued as the announcements grow larger and I eventually approach 12 again, since I achieved 12 by not rotating the dial at all. So, there will be a discontinuity at 12 at the latest. In general, continuity of my actions as a function of announced times can not be maintained throughout if I am to be able to replicate all possible settings. Another way to see the problem is that one can similarly reason that, as one starts with 12, and imagines continuously making the announced times earlier, one will be forced, by continuity, to achieve the announced times by rotating the dial to the left. But the conclusions drawn from the assumption of continuous increases and the assumption of continuous decreases are inconsistent. So we have an inconsistency following from the assumption of continuity and the assumption that I always manage to set my watch to your watch. So, a dial developing according to a continuous dynamics from a given initial state, can not be set up so as to react to a second dial, with which it interacts, in such a way that it is guaranteed to always end up set at the same angle as the second dial. Similarly, it can not be set up so that it is guaranteed to always end up set at 90 degrees to the setting of the second dial. All of this has nothing to do with time travel. However, the impossibility of such set ups is what prevents us from enacting the rotation by 90 degrees that would create paradox in the time travel setting.

Let us now give the positive result that with such dials there will always be fixed point solutions, as long as the dynamics is continuous. Let us call the state of the dial before it interacts with its older self the initial state of the dial. And let us call the state of the dial after it emerges from the time machine the final state of the dial. We can represent the possible initial and final states of the dial by the angles x and y that the dial can point at initially and finally. The set of possible initial plus final states thus forms a torus. (See figure 1.)

Suppose that the dial starts at angle I . The initial angle I that the dial is at before it encounters its older self, and the set of all possible final angles that the dial can have when it emerges from the time machine is represented by the circle I on the torus (see figure 1). Given any possible angle of the emerging dial the dial initially at angle I will develop to some other angle. One can picture this development by rotating each point on I in the horizontal direction by the relevant amount. Since the rotation has to depend continuously on the angle of the emerging dial, ring I during this development will deform into some loop L on the torus. Loop L thus represents the angle x that the dial is at when it is thrown into the time machine, given that it started at angle I and then encountered a dial (its older self) which was at angle y when it emerged from the time machine. We therefore have consistency if x = y for some x and y on loop L . Now, let loop C be the loop which consists of all the points on the torus for which x = y . Ring I intersects C at point < i , i >. Obviously any continuous deformation of I must still intersect C somewhere. So L must intersect C somewhere, say at < j , j >. But that means that no matter how the development of the dial starting at I depends on the angle of the emerging dial, there will be some angle for the emerging dial such that the dial will develop exactly into that angle (by the time it enters the time machine) under the influence of that emerging dial. This is so no matter what angle one starts with, and no matter how the development depends on the angle of the emerging dial. Thus even for a circular state-space there are no constraints needed other than continuity.

Unfortunately there are state-spaces that escape even this argument. Consider for instance a pointer that can be set to all values between 0 and 1, where 0 and 1 are not possible values. That is, suppose that we have a state-space that is isomorphic to an open set of real numbers. Now suppose that we have a machine that sets the pointer to half the value that the pointer is set at when it emerges from the time machine.

Suppose the pointer starts at value I . As before we can represent the combination of this initial position and all possible final positions by the line I . Under the influence of the pointer coming out of the time machine the pointer value will develop to a value that equals half the value of the final value that it encountered. We can represent this development as the continuous deformation of line I into line L , which is indicated by the arrows in Figure 2. This development is fully continuous. Points < x , y > on line I represent the initial position x = I of the (young) pointer, and the position y of the older pointer as it emerges from the time machine. Points < x , y > on line L represent the position x that the younger pointer should develop into, given that it encountered the older pointer emerging from the time machine set at position y . Since the pointer is designed to develop to half the value of the pointer that it encounters, the line L corresponds to x = 1 / 2 y . We have consistency if there is some point such that it develops into that point, if it encounters that point. Thus, we have consistency if there is some point < x , y > on line L such that x = y . However, there is no such point: lines L and C do not intersect. Thus there is no consistent solution, despite the fact that the dynamics is fully continuous.

Of course if 0 were a possible value L and C would intersect at 0. This is surprising and strange: adding one point to the set of possible values of a quantity here makes the difference between paradox and peace. One might be tempted to just add the extra point to the state-space in order to avoid problems. After all, one might say, surely no measurements could ever tell us whether the set of possible values includes that exact point or not. Unfortunately there can be good theoretical reasons for supposing that some quantity has a state-space that is open: the set of all possible speeds of massive objects in special relativity surely is an open set, since it includes all speeds up to, but not including, the speed of light. Quantities that have possible values that are not bounded also lead to counter examples to the presented fixed point argument. And it is not obvious to us why one should exclude such possibilities. So the argument that no constraints are needed is not fully general.

An interesting question of course is: exactly for which state-spaces must there be such fixed points. We do not know the general answer. (But see Kutach 2003 for more on this issue.)

Time travel has recently been discussed quite extensively in the context of general relativity. Time travel can occur in general relativistic models in which one has closed time-like curves (CTC's). A time like curve is simply a space-time trajectory such that the speed of light is never equalled or exceeded along this trajectory. Time-like curves thus represent the possible trajectories of ordinary objects. If there were time-like curves which were closed (formed a loop), then travelling along such a curve one would never exceed the speed of light, and yet after a certain amount of (proper) time one would return to a point in space-time that one previously visited. Or, by staying close to such a CTC, one could come arbitrarily close to a point in space-time that one previously visited. General relativity, in a straightforward sense, allows time travel: there appear to be many space-times compatible with the fundamental equations of General Relativity in which there are CTC's. Space-time, for instance, could have a Minkowski metric everywhere, and yet have CTC's everywhere by having the temporal dimension (topologically) rolled up as a circle. Or, one can have wormhole connections between different parts of space-time which allow one to enter ‘mouth A ’ of such a wormhole connection, travel through the wormhole, exit the wormhole at ‘mouth B ’ and re-enter ‘mouth A ’ again. Or, one can have space-times which topologically are R4, and yet have CTC's due to the ‘tilting’ of light cones (Gödel space-times, Taub-NUT space-times, etc.)

General relativity thus appears to provide ample opportunity for time travel. Note that just because there are CTC's in a space-time, this does not mean that one can get from any point in the space-time to any other point by following some future directed timelike curve. In many space-times in which there are CTC's such CTC's do not occur all over space-time. Some parts of space-time can have CTC's while other parts do not. Let us call the part of a space-time that has CTC's the “time travel region" of that space-time, while calling the rest of that space-time the "normal region". More precisely, the “time travel region" consists of all the space-time points p such that there exists a (non-zero length) timelike curve that starts at p and returns to p . Now let us start examining space-times with CTC's a bit more closely for potential problems.

In order to get a feeling for the sorts of implications that closed timelike curves can have, it may be useful to consider two simple models. In space-times with closed timelike curves the traditional initial value problem cannot be framed in the usual way. For it presupposes the existence of Cauchy surfaces, and if there are CTCs then no Cauchy surface exists. (A Cauchy surface is a spacelike surface such that every inextendible timelike curve crosses it exactly once. One normally specifies initial conditions by giving the conditions on such a surface.) Nonetheless, if the topological complexities of the manifold are appropriately localized, we can come quite close. Let us call an edgeless spacelike surface S a quasi-Cauchy surface if it divides the rest of the manifold into two parts such that a) every point in the manifold can be connected by a timelike curve to S , and b) any timelike curve which connects a point in one region to a point in the other region intersects S exactly once. It is obvious that a quasi-Cauchy surface must entirely inhabit the normal region of the space-time; if any point p of S is in the time travel region, then any timelike curve which intersects p can be extended to a timelike curve which intersects S near p again. In extreme cases of time travel, a model may have no normal region at all (e.g., Minkowski space-time rolled up like a cylinder in a time-like direction), in which case our usual notions of temporal precedence will not apply. But temporal anomalies like wormholes (and time machines) can be sufficiently localized to permit the existence of quasi-Cauchy surfaces.

Given a timelike orientation, a quasi-Cauchy surface unproblematically divides the manifold into its past (i.e., all points that can be reached by past-directed timelike curves from S ) and its future (ditto mutatis mutandis ). If the whole past of S is in the normal region of the manifold, then S is a partial Cauchy surface : every inextendible timelike curve which exists to the past of S intersects S exactly once, but (if there is time travel in the future) not every inextendible timelike curve which exists to the future of S intersects S . Now we can ask a particularly clear question: consider a manifold which contains a time travel region, but also has a partial Cauchy surface S , such that all of the temporal funny business is to the future of S . If all you could see were S and its past, you would not know that the space-time had any time travel at all. The question is: are there any constraints on the sort of data which can be put on S and continued to a global solution of the dynamics which are different from the constraints (if any) on the data which can be put on a Cauchy surface in a simply connected manifold and continued to a global solution? If there is time travel to our future, might we we able to tell this now, because of some implied oddity in the arrangement of present things?

It is not at all surprising that there might be constraints on the data which can be put on a locally space-like surface which passes through the time travel region: after all, we never think we can freely specify what happens on a space-like surface and on another such surface to its future, but in this case the surface at issue lies to its own future. But if there were particular constraints for data on a partial Cauchy surface then we would apparently need to have to rule out some sorts of otherwise acceptable states on S if there is to be time travel to the future of S . We then might be able to establish that there will be no time travel in the future by simple inspection of the present state of the universe. As we will see, there is reason to suspect that such constraints on the partial Cauchy surface are non-generic. But we are getting ahead of ourselves: first let's consider the effect of time travel on a very simple dynamics.

The simplest possible example is the Newtonian theory of perfectly elastic collisions among equally massive particles in one spatial dimension. The space-time is two-dimensional, so we can represent it initially as the Euclidean plane, and the dynamics is completely specified by two conditions. When particles are traveling freely, their world lines are straight lines in the space-time, and when two particles collide, they exchange momenta, so the collision looks like an ‘ X ’ in space-time, with each particle changing its momentum at the impact. [ 1 ] The dynamics is purely local, in that one can check that a set of world-lines constitutes a model of the dynamics by checking that the dynamics is obeyed in every arbitrarily small region. It is also trivial to generate solutions from arbitrary initial data if there are no CTCs: given the initial positions and momenta of a set of particles, one simply draws a straight line from each particle in the appropriate direction and continues it indefinitely. Once all the lines are drawn, the worldline of each particle can be traced from collision to collision. The boundary value problem for this dynamics is obviously well-posed: any set of data at an instant yields a unique global solution, constructed by the method sketched above.

What happens if we change the topology of the space-time by hand to produce CTCs? The simplest way to do this is depicted in figure 3: we cut and paste the space-time so it is no longer simply connected by identifying the line L − with the line L +. Particles “going in” to L + from below “emerge” from L − , and particles “going in” to L − from below “emerge” from L +.

Figure 3: Inserting CTCs by Cut and Paste

How is the boundary-value problem changed by this alteration in the space-time? Before the cut and paste, we can put arbitrary data on the simultaneity slice S and continue it to a unique solution. After the change in topology, S is no longer a Cauchy surface, since a CTC will never intersect it, but it is a partial Cauchy surface. So we can ask two questions. First, can arbitrary data on S always be continued to a global solution? Second, is that solution unique? If the answer to the first question is no , then we have a backward-temporal constraint: the existence of the region with CTCs places constraints on what can happen on S even though that region lies completely to the future of S . If the answer to the second question is no , then we have an odd sort of indeterminism: the complete physical state on S does not determine the physical state in the future, even though the local dynamics is perfectly deterministic and even though there is no other past edge to the space-time region in S 's future (i.e., there is nowhere else for boundary values to come from which could influence the state of the region).

In this case the answer to the first question is yes and to the second is no : there are no constraints on the data which can be put on S , but those data are always consistent with an infinitude of different global solutions. The easy way to see that there always is a solution is to construct the minimal solution in the following way. Start drawing straight lines from S as required by the initial data. If a line hits L − from the bottom, just continue it coming out of the top of L + in the appropriate place, and if a line hits L + from the bottom, continue it emerging from L − at the appropriate place. Figure 4 represents the minimal solution for a single particle which enters the time-travel region from the left:

Figure 4: The Minimal Solution

The particle ‘travels back in time’ three times. It is obvious that this minimal solution is a global solution, since the particle always travels inertially.

But the same initial state on S is also consistent with other global solutions. The new requirement imposed by the topology is just that the data going into L + from the bottom match the data coming out of L − from the top, and the data going into L - from the bottom match the data coming out of L + from the top. So we can add any number of vertical lines connecting L - and L + to a solution and still have a solution. For example, adding a few such lines to the minimal solution yields:

Figure 5: A Non-Minimal Solution

The particle now collides with itself twice: first before it reaches L + for the first time, and again shortly before it exits the CTC region. From the particle's point of view, it is traveling to the right at a constant speed until it hits an older version of itself and comes to rest. It remains at rest until it is hit from the right by a younger version of itself, and then continues moving off, and the same process repeats later. It is clear that this is a global model of the dynamics, and that any number of distinct models could be generating by varying the number and placement of vertical lines.

Knowing the data on S , then, gives us only incomplete information about how things will go for the particle. We know that the particle will enter the CTC region, and will reach L +, we know that it will be the only particle in the universe, we know exactly where and with what speed it will exit the CTC region. But we cannot determine how many collisions the particle will undergo (if any), nor how long (in proper time) it will stay in the CTC region. If the particle were a clock, we could not predict what time it would indicate when exiting the region. Furthermore, the dynamics gives us no handle on what to think of the various possibilities: there are no probabilities assigned to the various distinct possible outcomes.

Changing the topology has changed the mathematics of the situation in two ways, which tend to pull in opposite directions. On the one hand, S is no longer a Cauchy surface, so it is perhaps not surprising that data on S do not suffice to fix a unique global solution. But on the other hand, there is an added constraint: data “coming out” of L − must exactly match data “going in” to L +, even though what comes out of L − helps to determine what goes into L +. This added consistency constraint tends to cut down on solutions, although in this case the additional constraint is more than outweighed by the freedom to consider various sorts of data on L +/ L -.

The fact that the extra freedom outweighs the extra constraint also points up one unexpected way that the supposed paradoxes of time travel may be overcome. Let's try to set up a paradoxical situation using the little closed time loop above. If we send a single particle into the loop from the left and do nothing else, we know exactly where it will exit the right side of the time travel region. Now suppose we station someone at the other side of the region with the following charge: if the particle should come out on the right side, the person is to do something to prevent the particle from going in on the left in the first place. In fact, this is quite easy to do: if we send a particle in from the right, it seems that it can exit on the left and deflect the incoming left-hand particle.

Carrying on our reflection in this way, we further realize that if the particle comes out on the right, we might as well send it back in order to deflect itself from entering in the first place. So all we really need to do is the following: set up a perfectly reflecting particle mirror on the right-hand side of the time travel region, and launch the particle from the left so that— if nothing interferes with it —it will just barely hit L +. Our paradox is now apparently complete. If, on the one hand, nothing interferes with the particle it will enter the time-travel region on the left, exit on the right, be reflected from the mirror, re-enter from the right, and come out on the left to prevent itself from ever entering. So if it enters, it gets deflected and never enters. On the other hand, if it never enters then nothing goes in on the left, so nothing comes out on the right, so nothing is reflected back, and there is nothing to deflect it from entering. So if it doesn't enter, then there is nothing to deflect it and it enters. If it enters, then it is deflected and doesn't enter; if it doesn't enter then there is nothing to deflect it and it enters: paradox complete.

But at least one solution to the supposed paradox is easy to construct: just follow the recipe for constructing the minimal solution, continuing the initial trajectory of the particle (reflecting it the mirror in the obvious way) and then read of the number and trajectories of the particles from the resulting diagram. We get the result of figure 6:

Figure 6: Resolving the “Paradox”

As we can see, the particle approaching from the left never reaches L +: it is deflected first by a particle which emerges from L -. But it is not deflected by itself , as the paradox suggests, it is deflected by another particle. Indeed, there are now four particles in the diagram: the original particle and three particles which are confined to closed time-like curves. It is not the leftmost particle which is reflected by the mirror, nor even the particle which deflects the leftmost particle; it is another particle altogether.

The paradox gets it traction from an incorrect presupposition: if there is only one particle in the world at S then there is only one particle which could participate in an interaction in the time travel region: the single particle would have to interact with its earlier (or later) self. But there is no telling what might come out of L − : the only requirement is that whatever comes out must match what goes in at L +. So if you go to the trouble of constructing a working time machine, you should be prepared for a different kind of disappointment when you attempt to go back and kill yourself: you may be prevented from entering the machine in the first place by some completely unpredictable entity which emerges from it. And once again a peculiar sort of indeterminism appears: if there are many self-consistent things which could prevent you from entering, there is no telling which is even likely to materialize.

So when the freedom to put data on L − outweighs the constraint that the same data go into L +, instead of paradox we get an embarrassment of riches: many solution consistent with the data on S . To see a case where the constraint “outweighs” the freedom, we need to construct a very particular, and frankly artificial, dynamics and topology. Consider the space of all linear dynamics for a scalar field on a lattice. (The lattice can be though of as a simple discrete space-time.) We will depict the space-time lattice as a directed graph. There is to be a scalar field defined at every node of the graph, whose value at a given node depends linearly on the values of the field at nodes which have arrows which lead to it. Each edge of the graph can be assigned a weighting factor which determines how much the field at the input node contributes to the field at the output node. If we name the nodes by the letters a , b , c , etc., and the edges by their endpoints in the obvious way, then we can label the weighting factors by the edges they are associated with in an equally obvious way.

Suppose that the graph of the space-time lattice is acyclic , as in figure 7. (A graph is Acyclic if one can not travel in the direction of the arrows and go in a loop.)

Figure 7: An Acyclic Lattice

It is easy to regard a set of nodes as the analog of a Cauchy surface, e.g., the set { a , b , c }, and it is obvious if arbitrary data are put on those nodes the data will generate a unique solution in the future. [ 2 ] If the value of the field at node a is 3 and at node b is 7, then its value at node d will be 3 W ad and its value at node e will be 3 W ae + 7 W be . By varying the weighting factors we can adjust the dynamics, but in an acyclic graph the future evolution of the field will always be unique.

Let us now again artificially alter the topology of the lattice to admit CTCs, so that the graph now is cyclic. One of the simplest such graphs is depicted in figure 8: there are now paths which lead from z back to itself, e.g., z to y to z .

Figure 8: Time Travel on a Lattice

Can we now put arbitrary data on v and w , and continue that data to a global solution? Will the solution be unique?

In the generic case, there will be a solution and the solution will be unique. The equations for the value of the field at x , y , and z are:

x = v W vx + z W zx y = w W wy + z W zy z = x W xz + y W yz .

Solving these equations for z yields

z = ( v W vx + z W zx ) W xz + ( w W wy + z W zy ) W yz , or z = ( v W vx W xz + w W wy W yz )/ (1 − W zx W xz − W zy W yz ),

which gives a unique value for z in the generic case. But looking at the space of all possible dynamics for this lattice (i.e., the space of all possible weighting factors), we find a singularity in the case where 1−W zx W xz − W zy W yz = 0. If we choose weighting factors in just this way, then arbitrary data at v and w cannot be continued to a global solution. Indeed, if the scalar field is everywhere non-negative, then this particular choice of dynamics puts ironclad constraints on the value of the field at v and w : the field there must be zero (assuming W vx and W wy to be non-zero), and similarly all nodes in their past must have field value zero. If the field can take negative values, then the values at v and w must be so chosen that v W vx W xz = − w W wy W yz . In either case, the field values at v and w are severely constrained by the existence of the CTC region even though these nodes lie completely to the past of that region. It is this sort of constraint which we find to be unlike anything which appears in standard physics.

Our toy models suggest three things. The first is that it may be impossible to prove in complete generality that arbitrary data on a partial Cauchy surface can always be continued to a global solution: our artificial case provides an example where it cannot. The second is that such odd constraints are not likely to be generic: we had to delicately fine-tune the dynamics to get a problem. The third is that the opposite problem, namely data on a partial Cauchy surface being consistent with many different global solutions, is likely to be generic: we did not have to do any fine-tuning to get this result. And this leads to a peculiar sort of indeterminism: the entire state on S does not determine what will happen in the future even though the local dynamics is deterministic and there are no other “edges” to space-time from which data could influence the result. What happens in the time travel region is constrained but not determined by what happens on S , and the dynamics does not even supply any probabilities for the various possibilities. The example of the photographic negative discussed in section 3, then, seems likely to be unusual, for in that case there is a unique fixed point for the dynamics, and the set-up plus the dynamical laws determine the outcome. In the generic case one would rather expect multiple fixed points, with no room for anything to influence, even probabilistically, which would be realized.

It is ironic that time travel should lead generically not to contradictions or to constraints (in the normal region) but to underdetermination of what happens in the time travel region by what happens everywhere else (an underdetermination tied neither to a probabilistic dynamics or to a free edge to space-time). The traditional objection to time travel is that it leads to contradictions: there is no consistent way to complete an arbitrarily constructed story about how the time traveler intends to act. Instead, though, it appears that the problem is underdetermination: the story can be consistently completed in many different ways.

The two toys models presented above have the virtue of being mathematically tractable, but they involve certain simplifications and potential problems that lead to trouble if one tries to make them more complicated. Working through these difficulties will help highlight the conditions we have made use of.

Consider a slight modification of the first simple model proposed to us by Adam Elga. Let the particles have an electric charge , which produces forces according to Coulomb’s law. Then set up a situation like that depicted in figure 9:

Figure 9: Set-up for Elga's Paradox

The dotted line indicates the path the particle will follow if no forces act upon it. The point labeled P is the left edge of the time-travel region; the two labels are a reminder that the point at the bottom and the point at the top are one and the same.

Elga's paradox is as follows: if no force acts on the particle, then it will enter the time-travel region. But if it enters the time travel region, and hence reappears along the bottom edge, then its later self will interact electrically with its earlier self, and the earlier self will be deflected away from the time-travel region. It is easy to set up the case so that the deflection will be enough to keep the particle from ever entering the time-travel region in the first place. (For instance, let the momentum of the incoming particle towards the time travel region be very small. The mere existence of an identically charged particle inside the time travel region will then be sufficient to deflect the incoming particle so that it never reaches L + .) But, of course, if the particle never enters the region at all, then it will not be there to deflect itself….

One might suspect that some complicated collection of charged particles in the time-travel-region can save the day, as it did with our mirror-reflection problem above. But (unless there are infinitely many such particles) this can't work, as conservation of particle number and linear momentum show. Suppose that some finite collection of particles emerges from L - and supplies the repulsive electric force needed to deflect the incoming particle. Then exactly the same collection of particles must be “absorbed” at L + . So at all times after L + , the only particle there is in the world is the incoming particle, which has now been deflected away from its original trajectory.

The deflection, though, means that the linear momentum of the particle has changed from what is was before L - . But that is impossible, by conservation of linear momementum. No matter how the incoming particle interacts with particles in the time-travel region, or how those particle interact with each other, total linear momentum is conserved by the interaction. And whatever net linear momentum the time-travelling particles have when they emerge from L - , that much linear momentum most be absorbed at L + . So the momentum of the incoming particle can't be changed by the interaction: the particle can't have been deflected. (One could imagine trying to create a sort of “S” curve in the trajectory of the incoming particle, first bending to the left and then to the right, which leaves its final momentum equal to its initial momentum, but moving it over in space so it misses L + . However, if the force at issue is repulsive, then the bending back to the right can't be done. In the mirror example above, the path of the incoming particle can be changed without violating the conservation of momentum because at the end of the process momentum has been transferred to the mirror.)

How does Elga's example escape our analysis? Why can't a contintuity principle guarantee the existence of a solution here?

The continuity assumption breaks down because of two features of the example: the concentration of the electric charge on a point particle, and the way we have treated (or, more accurately, failed to treat) the point P , the edge of L + (and L - ). We have assumed that a point particle either hits L + , and then emerges from L - , or else it misses L + and sails on into the region of space-time above it. This means that the charge on the incoming particle only has two possibilities: either it is transported whole back in time or it completely avoids time travel altogether. Let's see how it alters the situation to imagine the charge itself to be continuous divisible.

Suppose that, instead of being concentrated at a point, the incoming object is a little stick, with electric charge distributed even across it (figure 10).

Figure 10: Elga's Paradox with a Charged Bar

Once again, we set things up so that if there are no forces on the bar, it will be completely absorbed at L + . But we now postulate that if the bar should hit the point P , it will fracture: part of it (the part that hits L+ ) will be sent back in time and the rest will continue on above L + . So continuity of a sort is restored: now we have not just the possibility of the whole charge being sent back or nothing, we have the continuum degrees of charge in between.

It is not hard to see that the restoration of continuity restores the existence of a consistent solution. If no charge is sent back through time, then the bar is not deflected and all of it hits L + (and hence is sent back through time). If all the charge is sent back through time, then is incoming bar is deflected to an extent that it misses L + completely, and so no charge is sent back. But if just the right amount of charge is sent back through time, then the bar will be only partially deflected, deflected so that it hits the edge point P , and is split into a bit that goes back and a bit that does not, with the bit that goes back being just the right amount of charge to produce just that deflection (figure 11).

Figure 11: Solution to Elga's Paradox with a Charged Bar

Our problem about conservation of momentum is also solved: piece of the bar that does not time travel has lower momentum to the right at the end than it had initially, but the piece that does time travel has a higher momentum (due to the Coulomb forces), and everything balances out.

Is it cheating to model the charged particle as a bar that can fracture? What if we insist that the particle is truly a point particle, and hence that its time travel is an all-or-nothing affair?

In that case, we now have to worry about a question we have not yet confronted: what happens if our point particle hits exactly at the point P on the diagram? Does it time-travel or not? Confronting this question requires us to face up to a feature of the rather cheap way we implemented time travel in our toy models by cut-and-paste. The way we rejiggered the space-time structure had a rather severe consequence: the resulting space-time is no longer a manifold : the topological structure at the point P is different from the topological structure elsewhere. Mathematical physicists simply don't deal with such structures: the usual procedure is to eliminate the offending point from the space-time and thus restore the manifold structure. In this case, that would leave a bare singularity at point P , an open edge to space-time into which anything could disappear and out of which, for all the physics tells us, anything could emerge.

In particular, if we insist that our particle is a point particle, then if its trajectory should happen to intersect P it will simply disappear. What could cause the extremely fortuitous result that the trajectory strikes precisely at P ? The emergence of some other charged particle, with just the right charge and trajectory, from P (on L - ). And we are no longer bound by any conservation laws: the bare singularity can both swallow and produce whatever mass or change or momentum we like. So if we insist on point particles, then we have to take account of the singularity, and that again saves the day.

Consideration of these (slightly more complicated) toy models does not replace the proving of theorems, of course. But they do serve to illustrate the sorts of consideration that necessarily come into play when trying to spell out the physics of time travel in all detail. Let us now discuss some results regarding some slightly more realistic models that have been discussed in the physics literature.

Echeverria, Klinkhammer and Thorne (1991) considered the case of 3-dimensional single hard spherical ball that can go through a single time travel wormhole so as to collide with its younger self.

The threat of paradox in this case arises in the following form. There are initial trajectories (starting in the non-time travel region of space-time) for the ball such that if such a trajectory is continued (into the time travel region), assuming that the ball does not undergo a collision prior to entering mouth 1 of the wormhole, it will exit mouth 2 so as to collide with its earlier self prior to its entry into mouth 1 in such a way as to prevent its earlier self from entering mouth 1. Thus it seems that the ball will enter mouth 1 if and only if it does not enter mouth 1. Of course, the Wheeler-Feynman strategy is to look for a ‘glancing blow’ solution: a collision which will produce exactly the (small) deviation in trajectory of the earlier ball that produces exactly that collision. Are there always such solutions? [ 3 ]

Echeverria, Klinkhammer & Thorne found a large class of initial trajectories that have consistent ‘glancing blow’ continuations, and found none that do not (but their search was not completely general). They did not produce a rigorous proof that every initial trajectory has a consistent continuation, but suggested that it is very plausible that every initial trajectory has a consistent continuation. That is to say, they have made it very plausible that, in the billiard ball wormhole case, the time travel structure of such a wormhole space-time does not result in constraints on states on spacelike surfaces in the non-time travel region.

In fact, as one might expect from our discussion in the previous section, they found the opposite problem from that of inconsistency: they found underdetermination. For a large class of initial trajectories there are multiple different consistent ‘glancing blow’ continuations of that trajectory (many of which involve multiple wormhole traversals). For example, if one initially has a ball that is traveling on a trajectory aimed straight between the two mouths, then one obvious solution is that the ball passes between the two mouths and never time travels. But another solution is that the younger ball gets knocked into mouth 1 exactly so as to come out of mouth 2 and produce that collision. Echeverria et al. do not note the possibility (which we pointed out in the previous section) of the existence of additional balls in the time travel region. We conjecture (but have no proof) that for every initial trajectory of A there are some, and generically many, multiple ball continuations.

Friedman et al. 1990 examined the case of source free non-self-interacting scalar fields traveling through such a time travel wormhole and found that no constraints on initial conditions in the non-time travel region are imposed by the existence of such time travel wormholes. In general there appear to be no known counter examples to the claim that in ‘somewhat realistic’ time-travel space-times with a partial Cauchy surface there are no constraints imposed on the state on such a partial Cauchy surface by the existence of CTC's. (See e.g., Friedman and Morris 1991, Thorne 1994, and Earman 1995; in the Other Internet Resources, see Earman, Smeenk, and Wüthrich 2003.)

How about the issue of constraints in the time travel region T ? Prima facie , constraints in such a region would not appear to be surprising. But one might still expect that there should be no constraints on states on a spacelike surface, provided one keeps the surface ‘small enough’. In the physics literature the following question has been asked: for any point p in T , and any space-like surface S that includes p is there a neighborhood E of p in S such that any solution on E can be extended to a solution on the whole space-time? With respect to this question, there are some simple models in which one has this kind of extendibility of local solutions to global ones, and some simple models in which one does not have such extendibility, with no clear general pattern. The technical mathematical problems are amplified by the more conceptual problem of what it might mean to say that one could create a situation which forces the creation of closed timelike curves. (See e.g. Yurtsever 1990, Friedman et al. 1990, Novikov 1992, Earman 1995 and Earman, Smeenk and Wüthrich 2009; in the Other Internet Resources, see Earman, Smeenk and Wüthrich 2003). What are we to think of all of this?

Since it is not obvious that one can rid oneself of all constraints in realistic models, let us examine the argument that time travel is implausible, and we should think it unlikely to exist in our world, in so far as it implies such constraints. The argument goes something like the following. In order to satisfy such constraints one needs some pre-established divine harmony between the global (time travel) structure of space-time and the distribution of particles and fields on space-like surfaces in it. But it is not plausible that the actual world, or any world even remotely like ours, is constructed with divine harmony as part of the plan. In fact, one might argue, we have empirical evidence that conditions in any spatial region can vary quite arbitrarily. So we have evidence that such constraints, whatever they are, do not in fact exist in our world. So we have evidence that there are no closed time-like lines in our world or one remotely like it. We will now examine this argument in more detail by presenting four possible responses, with counterresponses, to this argument.

Response 1. There is nothing implausible or new about such constraints. For instance, if the universe is spatially closed, there has to be enough matter to produce the needed curvature, and this puts constraints on the matter distribution on a space-like hypersurface. Thus global space-time structure can quite unproblematically constrain matter distributions on space-like hypersurfaces in it. Moreover we have no realistic idea what these constraints look like, so we hardly can be said to have evidence that they do not obtain.

Counterresponse 1. Of course there are constraining relations between the global structure of space-time and the matter in it. The Einstein equations relate curvature of the manifold to the matter distribution in it. But what is so strange and implausible about the constraints imposed by the existence of closed time-like curves is that these constraints in essence have nothing to do with the Einstein equations. When investigating such constraints one typically treats the particles and/or field in question as test particles and/or fields in a given space-time, i.e., they are assumed not to affect the metric of space-time in any way. In typical space-times without closed time-like curves this means that one has, in essence, complete freedom of matter distribution on a space-like hypersurface. (See response 2 for some more discussion of this issue). The constraints imposed by the possibility of time travel have a quite different origin and are implausible. In the ordinary case there is a causal interaction between matter and space-time that results in relations between global structure of space-time and the matter distribution in it. In the time travel case there is no such causal story to be told: there simply has to be some pre-established harmony between the global space-time structure and the matter distribution on some space-like surfaces. This is implausible.

Response 2. Constraints upon matter distributions are nothing new. For instance, Maxwell's equations constrain electric fields E on an initial surface to be related to the (simultaneous) charge density distribution ρ by the equation ρ = div( E ). (If we assume that the E field is generated solely by the charge distribution, this conditions amounts to requiring that the E field at any point in space simply be the one generated by the charge distribution according to Coulomb's inverse square law of electrostatics.) This is not implausible divine harmony. Such constraints can hold as a matter of physical law. Moreover, if we had inferred from the apparent free variation of conditions on spatial regions that there could be no such constraints we would have mistakenly inferred that ρ = div( E ) could not be a law of nature.

Counterresponse 2. The constraints imposed by the existence of closed time-like lines are of quite a different character from the constraint imposed by ρ = div( E ). The constraints imposed by ρ = div( E ) on the state on a space-like hypersurface are: (i) local constraints (i.e., to check whether the constraint holds in a region you just need to see whether it holds at each point in the region), (ii) quite independent of the global space-time structure, (iii) quite independent of how the space-like surface in question is embedded in a given space-time, and (iv) very simply and generally stateable. On the other hand, the consistency constraints imposed by the existence of closed time-like curves (i) are not local, (ii) are dependent on the global structure of space-time, (iii) depend on the location of the space-like surface in question in a given space-time, and (iv) appear not to be simply stateable other than as the demand that the state on that space-like surface embedded in such and such a way in a given space-time, do not lead to inconsistency. On some views of laws (e.g., David Lewis' view) this plausibly implies that such constraints, even if they hold, could not possibly be laws. But even if one does not accept such a view of laws, one could claim that the bizarre features of such constraints imply that it is implausible that such constraints hold in our world or in any world remotely like ours.

Response 3. It would be strange if there are constraints in the non-time travel region. It is not strange if there are constraints in the time travel region. They should be explained in terms of the strange, self-interactive, character of time travel regions. In this region there are time-like trajectories from points to themselves. Thus the state at such a point, in such a region, will, in a sense, interact with itself. It is a well-known fact that systems that interact with themselves will develop into an equilibrium state, if there is such an equilibrium state, or else will develop towards some singularity. Normally, of course, self-interaction isn't true instantaneous self-interaction, but consists of a feed-back mechanism that takes time. But in time travel regions something like true instantaneous self-interaction occurs. This explains why constraints on states occur in such time travel regions: the states ‘ab initio’ have to be ‘equilibrium states’. Indeed in a way this also provides some picture of why indeterminism occurs in time travel regions: at the onset of self-interaction states can fork into different equi-possible equilibrium states.

Counterresponse 3. This is explanation by woolly analogy. It all goes to show that time travel leads to such bizarre consequences that it is unlikely that it occurs in a world remotely like ours.

Response 4. All of the previous discussion completely misses the point. So far we have been taking the space-time structure as given, and asked the question whether a given time travel space-time structure imposes constraints on states on (parts of) space-like surfaces. However, space-time and matter interact. Suppose that one is in a space-time with closed time-like lines, such that certain counterfactual distributions of matter on some neighborhood of a point p are ruled out if one holds that space-time structure fixed. One might then ask “Why does the actual state near p in fact satisfy these constraints? By what divine luck or plan is this local state compatible with the global space-time structure? What if conditions near p had been slightly different?” And one might take it that the lack of normal answers to these questions indicates that it is very implausible that our world, or any remotely like it, is such a time travel universe. However the proper response to these question is the following. There are no constraints in any significant sense. If they hold they hold as a matter of accidental fact, not of law. There is no more explanation of them possible than there is of any contingent fact. Had conditions in a neighborhood of p been otherwise, the global structure of space-time would have been different. So what? The only question relevant to the issue of constraints is whether an arbitrary state on an arbitrary spatial surface S can always be embedded into a space-time such that that state on S consistently extends to a solution on the entire space-time.

But we know the answer to that question. A well-known theorem in general relativity says the following: any initial data set on a three dimensional manifold S with positive definite metric has a unique embedding into a maximal space-time in which S is a Cauchy surface (see e.g., Geroch and Horowitz 1979, p. 284 for more detail), i.e., there is a unique largest space-time which has S as a Cauchy surface and contains a consistent evolution of the initial value data on S . Now since S is a Cauchy surface this space-time does not have closed time like curves. But it may have extensions (in which S is not a Cauchy surface) which include closed timelike curves, indeed it may be that any maximal extension of it would include closed timelike curves. (This appears to be the case for extensions of states on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, and Wüthrich 2003 in the Other Internet Resources). But these extensions, of course, will be consistent. So properly speaking, there are no constraints on states on space-like surfaces. Nonetheless the space-time in which these are embedded may or may not include closed time-like curves.

Counterresponse 4. This, in essence, is the stonewalling answer which we indicated at the beginning of section 2. However, whether or not you call the constraints imposed by a given space-time on distributions of matter on certain space-like surfaces ‘genuine constraints’, whether or not they can be considered lawlike, and whether or not they need to be explained, the existence of such constraints can still be used to argue that time travel worlds are so bizarre that it is implausible that our world or any world remotely like ours is a time travel world.

Suppose that one is in a time travel world. Suppose that given the global space-time structure of this world, there are constraints imposed upon, say, the state of motion of a ball on some space-like surface when it is treated as a test particle, i.e., when it is assumed that the ball does not affect the metric properties of the space-time it is in. (There is lots of other matter that, via the Einstein equation, corresponds exactly to the curvature that there is everywhere in this time travel worlds.) Now a real ball of course does have some effect on the metric of the space-time it is in. But let us consider a ball that is so small that its effect on the metric is negligible. Presumably it will still be the case that certain states of this ball on that space-like surface are not compatible with the global time travel structure of this universe.

This means that the actual distribution of matter on such a space-like surface can be extended into a space-time with closed time-like lines, but that certain counterfactual distributions of matter on this space-like surface can not be extended into the same space-time. But note that the changes made in the matter distribution (when going from the actual to the counterfactual distribution) do not in any non-negligible way affect the metric properties of the space-time. Thus the reason why the global time travel properties of the counterfactual space-time have to be significantly different from the actual space-time is not that there are problems with metric singularities or alterations in the metric that force significant global changes when we go to the counterfactual matter distribution. The reason that the counterfactual space-time has to be different is that in the counterfactual world the ball's initial state of motion starting on the space-like surface, could not ‘meet up’ in a consistent way with its earlier self (could not be consistently extended) if we were to let the global structure of the counterfactual space-time be the same as that of the actual space-time. Now, it is not bizarre or implausible that there is a counterfactual dependence of manifold structure, even of its topology, on matter distributions on spacelike surfaces. For instance, certain matter distributions may lead to singularities, others may not. We may indeed in some sense have causal power over the topology of the space-time we live in. But this power normally comes via the Einstein equations. But it is bizarre to think that there could be a counterfactual dependence of global space-time structure on the arrangement of certain tiny bits of matter on some space-like surface, where changes in that arrangement by assumption do not affect the metric anywhere in space-time in any significant way . It is implausible that we live in such a world, or that a world even remotely like ours is like that.

Let us illustrate this argument in a different way by assuming that wormhole time travel imposes constraints upon the states of people prior to such time travel, where the people have so little mass/energy that they have negligible effect, via the Einstein equation, on the local metric properties of space-time. Do you think it more plausible that we live in a world where wormhole time travel occurs but it only occurs when people's states are such that these local states happen to combine with time travel in such a way that nobody ever succeeds in killing their younger self, or do you think it more plausible that we are not in a wormhole time travel world? [ 4 ]

There has been a particularly clear treatment of time travel in the context of quantum mechanics by David Deutsch (see Deutsch 1991, and Deutsch and Lockwood 1994) in which it is claimed that quantum mechanical considerations show that time travel never imposes any constraints on the pre-time travel state of systems. The essence of this account is as follows.

A quantum system starts in state S 1, interacts with its older self, after the interaction is in state S 2 , time travels while developing into state S 3 , then interacts with its younger self, and ends in state S 4 (see figure 13).

1 3 develops into 2 4 .

Similarly, suppose that:

1 3 develops into 2 4 , 1 3 develops into 2 4 , and 1 3 develops into 2 4 .

This clarification of why Deutsch needs his mixtures does however indicate a serious worry about the simplifications that are part of Deutsch's account. After the interaction the old and young system will (typically) be in an entangled state. Although for purposes of a measurement on one of the two systems one can say that this system is in a mixed state, one can not represent the full state of the two systems by specifying the mixed state of each separate part, as there are correlations between observables of the two systems that are not represented by these two mixed states, but are represented in the joint entangled state. But if there really is an entangled state of the old and young systems directly after the interaction, how is one to represent the subsequent development of this entangled state? Will the state of the younger system remain entangled with the state of the older system as the younger system time travels and the older system moves on into the future? On what space-like surfaces are we to imagine this total entangled state to be? At this point it becomes clear that there is no obvious and simple way to extend elementary non-relativistic quantum mechanics to space-times with closed time-like curves. There have been more sophisticated approaches than Deutsch's to time travel, using technical machinery from quantum field theory and differentiable manifolds (see e.g., Friedman et al 1991, Earman, Smeenk, and Wüthrich 2003 in the Other Internet Resources, and references therein). But out of such approaches no results anywhere near as clear and interesting as Deutsch's have been forthcoming.

How does Deutsch avoid these complications? Deutsch assumes a mixed state S 3 of the older system prior to the interaction with the younger system. He lets it interact with an arbitrary pure state S 1 younger system. After this interaction there is an entangled state S ′ of the two systems. Deutsch computes the mixed state S 2 of the younger system which is implied by this entangled state S ′. His demand for consistency then is just that this mixed state S 2 develops into the mixed state S 3 . Now it is not at all clear that this is a legitimate way to simplify the problem of time travel in quantum mechanics. But even if we grant him this simplification there is a problem: how are we to understand these mixtures?

Now whatever one thinks of the merits of many worlds interpretations, and of this understanding of it applied to mixtures, in the end one does not obtain genuine time travel in Deutsch's account. The systems in question travel from one time in one world to another time in another world, but no system travels to an earlier time in the same world. (This is so at least in the normal sense of the word ‘world,’ the sense that one means when, for instance, one says “there was, and will be, only one Elvis Presley in this world.”) Thus, even if it were a reasonable view, it is not quite as interesting as it may have initially seemed.

What remains of the killing-your-earlier-self paradox in general relativistic time travel worlds is the fact that in some cases the states on edgeless spacelike surfaces are ‘overconstrained’, so that one has less than the usual freedom in specifying conditions on such a surface, given the time-travel structure, and in some cases such states are ‘underconstrained’, so that states on edgeless space-like surfaces do not determine what happens elsewhere in the way that they usually do, given the time travel structure. There can also be mixtures of those two types of cases. The extent to which states are overconstrained and/or underconstrained in realistic models is as yet unclear, though it would be very surprising if neither obtained. The extant literature has primarily focused on the problem of overconstraint, since that, often, either is regarded as a metaphysical obstacle to the possibility time travel, or as an epistemological obstacle to the plausibility of time travel in our world. While it is true that our world would be quite different from the way we normally think it is if states were overconstrained, underconstraint seems at least as bizarre as overconstraint. Nonetheless, neither directly rules out the possibility of time travel.

If time travel entailed contradictions then the issue would be settled. And indeed, most of the stories employing time travel in popular culture are logically incoherent: one cannot “change” the past to be different from what it was, since the past (like the present and the future) only occurs once. But if the only requirement demanded is logical coherence, then it seems all too easy. A clever author can devise a coherent time-travel scenario in which everything happens just once and in a consistent way. This is just too cheap: logical coherence is a very weak condition, and many things we take to be metaphysically impossible are logically coherent. For example, it involves no logical contradiction to suppose that water is not molecular, but if both chemistry and Kripke are right it is a metaphysical impossibility. We have been interested not in logical possibility but in physical possibility. But even so, our conditions have been relatively weak: we have asked only whether time-travel is consistent with the universal validity of certain fundamental physical laws and with the notion that the physical state on a surface prior to the time travel region be unconstrained. It is perfectly possible that the physical laws obey this condition, but still that time travel is not metaphysically possible because of the nature of time itself. Consider an analogy. Aristotle believed that water is homoiomerous and infinitely divisible: any bit of water could be subdivided, in principle, into smaller bits of water. Aristotle's view contains no logical contradiction. It was certainly consistent with Aristotle's conception of water that it be homoiomerous, so this was, for him, a conceptual possibility. But if chemistry is right, Aristotle was wrong both about what water is like and what is possible for it. It can't be infinitely divided, even though no logical or conceptual analysis would reveal that.

Similarly, even if all of our consistency conditions can be met, it does not follow that time travel is physically possible, only that some specific physical considerations cannot rule it out. The only serious proof of the possibility of time travel would be a demonstration of its actuality. For if we agree that there is no actual time travel in our universe, the supposition that there might have been involves postulating a substantial difference from actuality, a difference unlike in kind from anything we could know if firsthand. It is unclear to us exactly what the content of possible would be if one were to either maintain or deny the possibility of time travel in these circumstances, unless one merely meant that the possibility is not ruled out by some delineated set of constraints. As the example of Aristotle's theory of water shows, conceptual and logical “possibility” do not entail possibility in a full-blooded sense. What exactly such a full-blooded sense would be in case of time travel, and whether one could have reason to believe it to obtain, remain to us obscure.

• Deutsch, D. 1991. “Quantum mechanics near closed timelike curves,” Physical Review D , 44: 3197-3217.
• Deutsch, D. and Lockwood, M. 1994. “The quantum physics of time travel,” Scientific American , 270 (3): 68-74.
• Earman, J. 1972. “Implications of causal propagation outsider the null cone,” in Foundations of Space-Time Theory , Minnesota Studies in the Philosophy of Science , Vol VII, Earman, J., Glymour, C., and Stachel, J. (eds), pp. 94-108. Minneapolis: University of Minnesota Press.
• Earman, J. 1995. Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalities in Relativistic Spacetimes , New York: Oxford University Press.
• Earman, J., Smeenk, C., and Wüthrich, C. 2009.“Do the laws of physics forbid the operation of a time machine?,” Synthese , 169 (1): 91-124.
• Echeverria, F., Klinkhammer, G., and Thorne, K. 1991. “Billiard ball in wormhole spacetimes with closed timelike curves: classical theory,” Physical Review D , 44 (4): 1077-1099.
• Friedman, J. et al. 1990. “Cauchy problem in spacetimes with closed timelike lines,” Physical Review D , 42: 1915-1930.
• Friedman, J. and Morris, M. 1991. “The Cauchy problem for the scalar wave equation is well defined on a class of spacetimes with closed timelike curves,” Physical Review letters , 66: 401-404.
• Geroch, R. and Horowitz, G. 1979. “Global structures of spacetimes,” in General Relativity, an Einstein Centenary Survey , S. Hawking and W. Israel (eds.), Cambridge: Cambridge University Press.
• Gödel, K. 1949. “A remark about the relationship between relativity theory and idealistic philosophy,” in Albert Einstein: Philosopher-Scientist , P. Schilpp (ed.), La Salle: Open Court, pp. 557-562.
• Hocking, J., and Young, G. 1961. Topology , New York: Dover Publications.
• Horwich, P. 1987. “Time travel,” in Asymmetries in time , Cambridge, MA: MIT Press.
• Kutach, D. 2003. “Time travel and consistency constraint”, Philosophy of Science , 70: 1098-1113.
• Malament, D. 1985a. “’Time travel’ in the Gödel universe,” PSA 1984, 2: 91-100, P. Asquith and P. Kitcher (eds.), East Lansing, MI: Philosophy of Science Association.
• Malament, D. 1985b. “Minimal acceleration requirements for ‘time travel’ in Gödel spacetime,” Journal of Mathematical Physics , 26: 774-777.
• Maudlin, T. 1990. “Time Travel and topology,” PSA 1990, 1: 303-315, East Lansing, MI: Philosophy of Science Association.
• Novikov, I. 1992. “Time machine and self-consistent evolution in problems with self-interaction,” Physical Review D , 45: 1989-1994.
• Thorne, K. 1994. Black Holes and Time Warps, Einstein's Outrageous Legacy , London and New York: W.W. Norton.
• Wheeler, J. and Feynman, R. 1949. “Classical electrodynamics in terms of direct interparticle action,” Reviews of Modern Physics , 21: 425-434.
• Yurtsever, U. 1990. “Test fields on compact space-times,” Journal of Mathematical Physics , 31: 3064-3078.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up this entry topic at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Earman, J., Smeenk, C. and Wüthrich, C., 2003, “ Take a ride on a time machine ”, manuscript available at the PhilSci Archive, University of Pittsburgh.
• Time Travel in Flatland (Cal Tech Particle Theory Group)

determinism: causal | -->Gödel, Kurt: contributions to relativity theory --> | time machines | time travel

Acknowledgments

Thanks to Edward N. Zalta, who spotted that we incorrectly stated one of the consequences of Maxwell's equations as E = div(ρ) rather than as ρ = div( E ).

Copyright © 2009 by Frank Arntzenius Tim Maudlin < twm3 @ nyu . edu >

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Is time travel even possible? An astrophysicist explains the science behind the science fiction

Assistant Professor of Astronomy and Astrophysics, University of Maryland, Baltimore County

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Curious Kids is a series for children of all ages. If you have a question you’d like an expert to answer, send it to [email protected] .

Will it ever be possible for time travel to occur? – Alana C., age 12, Queens, New York

Have you ever dreamed of traveling through time, like characters do in science fiction movies? For centuries, the concept of time travel has captivated people’s imaginations. Time travel is the concept of moving between different points in time, just like you move between different places. In movies, you might have seen characters using special machines, magical devices or even hopping into a futuristic car to travel backward or forward in time.

But is this just a fun idea for movies, or could it really happen?

The question of whether time is reversible remains one of the biggest unresolved questions in science. If the universe follows the laws of thermodynamics , it may not be possible. The second law of thermodynamics states that things in the universe can either remain the same or become more disordered over time.

It’s a bit like saying you can’t unscramble eggs once they’ve been cooked. According to this law, the universe can never go back exactly to how it was before. Time can only go forward, like a one-way street.

Time is relative

However, physicist Albert Einstein’s theory of special relativity suggests that time passes at different rates for different people. Someone speeding along on a spaceship moving close to the speed of light – 671 million miles per hour! – will experience time slower than a person on Earth.

People have yet to build spaceships that can move at speeds anywhere near as fast as light, but astronauts who visit the International Space Station orbit around the Earth at speeds close to 17,500 mph. Astronaut Scott Kelly has spent 520 days at the International Space Station, and as a result has aged a little more slowly than his twin brother – and fellow astronaut – Mark Kelly. Scott used to be 6 minutes younger than his twin brother. Now, because Scott was traveling so much faster than Mark and for so many days, he is 6 minutes and 5 milliseconds younger .

Some scientists are exploring other ideas that could theoretically allow time travel. One concept involves wormholes , or hypothetical tunnels in space that could create shortcuts for journeys across the universe. If someone could build a wormhole and then figure out a way to move one end at close to the speed of light – like the hypothetical spaceship mentioned above – the moving end would age more slowly than the stationary end. Someone who entered the moving end and exited the wormhole through the stationary end would come out in their past.

However, wormholes remain theoretical: Scientists have yet to spot one. It also looks like it would be incredibly challenging to send humans through a wormhole space tunnel.

Paradoxes and failed dinner parties

There are also paradoxes associated with time travel. The famous “ grandfather paradox ” is a hypothetical problem that could arise if someone traveled back in time and accidentally prevented their grandparents from meeting. This would create a paradox where you were never born, which raises the question: How could you have traveled back in time in the first place? It’s a mind-boggling puzzle that adds to the mystery of time travel.

Famously, physicist Stephen Hawking tested the possibility of time travel by throwing a dinner party where invitations noting the date, time and coordinates were not sent out until after it had happened. His hope was that his invitation would be read by someone living in the future, who had capabilities to travel back in time. But no one showed up.

As he pointed out : “The best evidence we have that time travel is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future.”

Telescopes are time machines

Interestingly, astrophysicists armed with powerful telescopes possess a unique form of time travel. As they peer into the vast expanse of the cosmos, they gaze into the past universe. Light from all galaxies and stars takes time to travel, and these beams of light carry information from the distant past. When astrophysicists observe a star or a galaxy through a telescope, they are not seeing it as it is in the present, but as it existed when the light began its journey to Earth millions to billions of years ago.

NASA’s newest space telescope, the James Webb Space Telescope , is peering at galaxies that were formed at the very beginning of the Big Bang, about 13.7 billion years ago.

While we aren’t likely to have time machines like the ones in movies anytime soon, scientists are actively researching and exploring new ideas. But for now, we’ll have to enjoy the idea of time travel in our favorite books, movies and dreams.

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And since curiosity has no age limit – adults, let us know what you’re wondering, too. We won’t be able to answer every question, but we will do our best.

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10 best movies about time travel paradoxes.

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The One Problem Sci-Fi Time Travel Movies NEVER Resolve

Back to the future writer explains marty’s parents plot hole, 2024's the crow box office edges past global milestone after just 3 weeks in theaters.

• Time travel movies often contain paradoxes, creating confusion for viewers. Each film's unique rules and characters' reactions to those paradoxes shape the plot.
• Different types of paradoxes exist, such as bootstrap, predestination, and temporal paradoxes, which add depth and complexity to time travel movies.
• Despite the presence of paradoxes, time travel movies can still be entertaining and thought-provoking, providing great storytelling and exploration of love, fate, and the concept of free will.

Time travel can't exist without paradoxes, and neither can science fiction movies about time travel. The way time travel works in each individual film is the first thing a director needs to think through to make sure their movie is consistent and viewers aren't left scratching their heads afterward. Unfortunately, time travel is a tricky subject, and specific rules do not guarantee the absence of paradoxes. The only difference between all time travel movies is that some characters acknowledge the paradox and try to do something about it, and others just ignore its existence and proceed with their goal no matter what.

There are quite a few types of time paradoxes. For instance, a bootstrap paradox is about information or objects that seemingly have no starting point in their timeline; a predestination paradox centers on the cause of someone's time travel being of their own doing in the past; and a temporal paradox revolves around someone's actions in the past that remove the necessity to time travel in the first place. From Interstellar to About Time , time travel movies are riddled with paradoxes , sometimes for the better, providing a great story, and sometimes for the worse, confusing anyone who tries to follow the plot.

Almost every Sci-Fi time travel movie runs into the causal loop paradox, but not all films deal with the logic of time travel in the same way.

10 Interstellar

Cooper gives himself the idea of contacting murth, interstellar.

Christopher Nolan's movies are largely regarded as sci-fi masterpieces, and Interstellar is no exception. The movie's main mystery, the identity of the ghost, was based on a time paradox. At the beginning of the movie, a book fell out of a shelf on its own, and Interstellar 's surprising ending revealed that it was Cooper who made the book fall out via the Tesseract mechanism to send his past self a message . However, Cooper just did what he'd already seen happen, so the concept raises the question of who originally thought of sending the message in this way. Still, this mind-bending time loop worked against all odds.

9 The Terminator

Kyle reese is john connor's father.

James Cameron's epic sci-fi tale is a classic example of a predestination paradox. In The Terminator , Kyle Reese arrived from the future to stop the Terminator from killing Sarah Connor , the mother of his colleague John. Unknowingly, Kyle ended up fathering John when he developed a romantic relationship with Sarah. If the Terminator hadn't been sent to kill Sarah, and Kyle hadn't followed him, John wouldn't have been born, since his father wouldn't have traveled to the past and met his mother. The Terminator 's paradox ending was controversial, and yet the movie managed to make the story entertaining enough to look past it.

8 The Time Traveler's Wife

Henry & clare meet out of order.

The Time Traveler's Wife explores a beautiful notion that love can transcend any boundaries — apparently, that includes the boundaries of time. The movie didn't pretend to be a serious sci-fi title, but it was essentially based on a paradox, specifically, the incorrect order, in which Henry met Clare. Henry first saw Clare when he time traveled to 1991 , but she already knew him because she had met Henry when she was but a child. That is a confusing concept that raises two questions: when their first meeting took place and how they ended up together at all. The Time Traveler's Wife' s paradoxical love story is endearing nevertheless.

7 The Butterfly Effect

Evan causes his own blackouts, the butterfly effect.

The Butterfly Effect is one of the most mind-blowing time travel movies out there, partly because the rules are very specific, and yet they make no sense whatsoever. The movie featured quite a few time paradoxes, but the biggest one was probably the existence of Evan's blackouts. Young Evan experienced blackouts, caused by his adult self's time travel ; adult Evan had to travel to his past because he knew that he was supposed to cause these blackouts. It is unclear how blackouts appeared in the first place. This plot detail makes The Butterfly Effect 's understanding of time circular rather than linear, but the paradox is still there.

6 About Time

Tim prevents the car crash.

About Time 's central point was Tim going back to the past to prevent the car crash, as this resulted in the erasure of his daughter Posy from existence. Although the film is full of inconsistencies, this event in particular showcases the classic paradox of time travel movies — if the car crash motivated Tim to go back in time to prevent it , then in doing so, he erased the very reason for him to travel to the past. Still, Domhnall Gleeson and accidental time-travel expert Rachel McAdams make up a dynamic duo, and it is impossible not to feel for Tim's struggle to help everyone through his gift.

The Protagonist Founds Tenet

The ending of Tenet , Christopher Nolan's sci-fi follow-up to Interstellar , turned out to be even more confusing than that of its predecessor, and not just because of the inverted entropy concept. The entire plot wouldn't exist if the Protagonist hadn't founded the mysterious organization Tenet that helped him in the first place and led him to create it in the future. Tenet explores the notion of a person's future and past intertwining and being part of the same time loop, with no one able to tell what the original cause of the event was. The Protagonist's survival in the film depended solely on himself from the future, who apparently knew that his past self once needed saving.

4 Back to the Future

Marty mcfly has to bring his parents together, back to the future.

Back to the Future is the movie that started the time travel film craze in the first place, and it features one of the most well-known paradoxes in the genre. When Marty traveled to the past, he saved George's life by preventing a car accident, but in the aftermath of the events, he accidentally jeopardized his own existence and had to make his parents fall in love with each other all over again. However, since there was a possibility that Marty would never be born , he should have disintegrated right then and there in the 1950s before he had a chance to fix his own timeline.

Back to the Future co-writer Bob Gale explains one of the movie's perceived plot holes, concerning Marty's parents not recognizing him at the end.

3 Donnie Darko

The plane comes out of nowhere, donnie darko.

Donnie Darko is a stunning dark tale with a timeless message about a person forging their own fate. In the film, the protagonist sent the engine of the plane that would kill his mother and sister back in time and allowed it to fall on him to prevent the catastrophe. However, Donnie Darko 's timeline created a paradox in the fact that the plane existed in the first place. If the future had rewound and the plane had never started to crash, the engine couldn't have been there in the past to kill Donnie in his bedroom. The movie is incredibly thought-provoking in a way that more lighthearted time travel films never are.

2 12 Monkeys

James cole originates the virus.

12 Monkeys' post-apocalyptic nature paints its deterministic narrative in dark colors, adding to the eerie atmosphere of the story at hand. Bruce Willis' James Cole traveled to the past to prevent humanity's extinction , but his every action just led to the devastating virus scenario taking place in the end. 12 Monkeys' predestination paradox lied in the fact that if Cole hadn't planted the idea of the viral outbreak in the past, it wouldn't have happened at all. The protagonist's desperate attempts to stop the apocalypse explore the notion that there is no such thing as free will and that everything in life is already determined.

Aaron & Abe Create A Causal Loop

Primer is decidedly the time-travel movie that has the most rules on the subject and surprisingly follows them through with the help of extremely complicated tech jargon. Aaron and Abe discovered how to create a causal loop and use it to their own advantage. Unfortunately, their actions unraveled in a heap of consequences, and each attempt to fix the problem just made it a lot worse. Primer requires at least two or three watches to fully understand its core concepts and follow the characters' decisions with ease, but it is worth every minute of the time spent.