how many time travel paradoxes are there

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5 Bizarre Paradoxes Of Time Travel Explained

December 20, 2014 James Miller Astronomy Lists , Time Travel 58

There is nothing in Einstein’s theories of relativity to rule out time travel , although the very notion of traveling to the past violates one of the most fundamental premises of physics, that of causality. With the laws of cause and effect out the window, there naturally arises a number of inconsistencies associated with time travel, and listed here are some of those paradoxes which have given both scientists and time travel movie buffs alike more than a few sleepless nights over the years.

Types of Temporal Paradoxes

The time travel paradoxes that follow fall into two broad categories:

1) Closed Causal Loops , such as the Predestination Paradox and the Bootstrap Paradox, which involve a self-existing time loop in which cause and effect run in a repeating circle, but is also internally consistent with the timeline’s history.

2) Consistency Paradoxes , such as the Grandfather Paradox and other similar variants such as The Hitler paradox, and Polchinski’s Paradox, which generate a number of timeline inconsistencies related to the possibility of altering the past.

1: Predestination Paradox

A Predestination Paradox occurs when the actions of a person traveling back in time become part of past events, and may ultimately cause the event he is trying to prevent to take place. The result is a ‘temporal causality loop’ in which Event 1 in the past influences Event 2 in the future (time travel to the past) which then causes Event 1 to occur.

This circular loop of events ensures that history is not altered by the time traveler, and that any attempts to stop something from happening in the past will simply lead to the cause itself, instead of stopping it. Predestination paradoxes suggest that things are always destined to turn out the same way and that whatever has happened must happen.

Sound complicated? Imagine that your lover dies in a hit-and-run car accident, and you travel back in time to save her from her fate, only to find that on your way to the accident you are the one who accidentally runs her over. Your attempt to change the past has therefore resulted in a predestination paradox. One way of dealing with this type of paradox is to assume that the version of events you have experienced are already built into a self-consistent version of reality, and that by trying to alter the past you will only end up fulfilling your role in creating an event in history, not altering it.

– Cinema Treatment

In The Time Machine (2002) movie, for instance, Dr. Alexander Hartdegen witnesses his fiancee being killed by a mugger, leading him to build a time machine to travel back in time to save her from her fate. His subsequent attempts to save her fail, though, leading him to conclude that “I could come back a thousand times… and see her die a thousand ways.” After then traveling centuries into the future to see if a solution has been found to the temporal problem, Hartdegen is told by the Über-Morlock:

“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 .”

• Movies : Examples of predestination paradoxes in the movies include 12 Monkeys (1995), TimeCrimes (2007), The Time Traveler’s Wife (2009), and Predestination (2014).
• Books : An example of a predestination paradox in a book is Phoebe Fortune and the Pre-destination Paradox by M.S. Crook.

2: Bootstrap Paradox

A Bootstrap Paradox is a type of paradox in which an object, person, or piece of information sent back in time results in an infinite loop where the object has no discernible origin, and exists without ever being created. It is also known as an Ontological Paradox, as ontology is a branch of philosophy concerned with the nature of being or existence.

– Information : George Lucas traveling back in time and giving himself the scripts for the Star War movies which he then goes on to direct and gain great fame for would create a bootstrap paradox involving information, as the scripts have no true point of creation or origin.

– Person : A bootstrap paradox involving a person could be, say, a 20-year-old male time traveler who goes back 21 years, meets a woman, has an affair, and returns home three months later without knowing the woman was pregnant. Her child grows up to be the 20-year-old time traveler, who travels back 21 years through time, meets a woman, and so on. American science fiction writer Robert Heinlein wrote a strange short story involving a sexual paradox in his 1959 classic “All You Zombies.”

These ontological paradoxes imply that the future, present, and past are not defined, thus giving scientists an obvious problem on how to then pinpoint the “origin” of anything, a word customarily referring to the past, but now rendered meaningless. Further questions arise as to how the object/data was created, and by whom. Nevertheless, Einstein’s field equations allow for the possibility of closed time loops, with Kip Thorne the first theoretical physicist to recognize traversable wormholes and backward time travel as being theoretically possible under certain conditions.

• Movies : Examples of bootstrap paradoxes in the movies include Somewhere in Time (1980), Bill and Ted’s Excellent Adventure (1989), the Terminator movies, and Time Lapse (2014). The Netflix series Dark (2017-19) also features a book called ‘A Journey Through Time’ which presents another classic example of a bootstrap paradox.
• Books : Examples of bootstrap paradoxes in books include Michael Moorcock’s ‘Behold The Man’, Tim Powers’ The Anubis Gates, and Heinlein’s “By His Bootstraps”

3: Grandfather Paradox

The Grandfather Paradox concerns ‘self-inconsistent solutions’ to a timeline’s history caused by traveling back in time. For example, if you traveled to the past and killed your grandfather, you would never have been born and would not have been able to travel to the past – a paradox.

Let’s say you did decide to kill your grandfather because he created a dynasty that ruined the world. You figure if you knock him off before he meets your grandmother then the whole family line (including you) will vanish and the world will be a better place. According to theoretical physicists, the situation could play out as follows:

– Timeline protection hypothesis: You pop back in time, walk up to him, and point a revolver at his head. You pull the trigger but the gun fails to fire. Click! Click! Click! The bullets in the chamber have dents in the firing caps. You point the gun elsewhere and pull the trigger. Bang! Point it at your grandfather.. Click! Click! Click! So you try another method to kill him, but that only leads to scars that in later life he attributed to the world’s worst mugger. You can do many things as long as they’re not fatal until you are chased off by a policeman.

– Multiple universes hypothesis: You pop back in time, walk up to him, and point a revolver at his head. You pull the trigger and Boom! The deed is done. You return to the “present,” but you never existed here. Everything about you has been erased, including your family, friends, home, possessions, bank account, and history. You’ve entered a timeline where you never existed. Scientists entertain the possibility that you have now created an alternate timeline or entered a parallel universe.

• Movies : Example of the Grandfather Paradox in movies include Back to the Future (1985), Back to the Future Part II (1989), and Back to the Future Part III (1990).
• Books : Example of the Grandfather Paradox in books include Dr. Quantum in the Grandfather Paradox by Fred Alan Wolf , The Grandfather Paradox by Steven Burgauer, and Future Times Three (1944) by René Barjavel, the very first treatment of a grandfather paradox in a novel.

4: Let’s Kill Hitler Paradox

Similar to the Grandfather Paradox which paradoxically prevents your own birth, the Killing Hitler paradox erases your own reason for going back in time to kill him. Furthermore, while killing Grandpa might have a limited “butterfly effect,” killing Hitler would have far-reaching consequences for everyone in the world, even if only for the fact you studied him in school.

The paradox itself arises from the idea that if you were successful, then there would be no reason to time travel in the first place. If you killed Hitler then none of his actions would trickle down through history and cause you to want to make the attempt.

• Movies/Shows : By far the best treatment for this notion occurred in a Twilight Zone episode called Cradle of Darkness which sums up the difficulties involved in trying to change history, with another being an episode of Dr Who called ‘Let’s Kill Hitler’.
• Books : Examples of the Let’s Kill Hitler Paradox in books include How to Kill Hitler: A Guide For Time Travelers by Andrew Stanek, and the graphic novel I Killed Adolf Hitler by Jason.

5: Polchinski’s Paradox

American theoretical physicist Joseph Polchinski proposed a time paradox scenario in which a billiard ball enters a wormhole, and emerges out the other end in the past just in time to collide with its younger version and stop it from going into the wormhole in the first place.

Polchinski’s paradox is taken seriously by physicists, as there is nothing in Einstein’s General Relativity to rule out the possibility of time travel, closed time-like curves (CTCs), or tunnels through space-time. Furthermore, it has the advantage of being based upon the laws of motion, without having to refer to the indeterministic concept of free will, and so presents a better research method for scientists to think about the paradox. When Joseph Polchinski proposed the paradox, he had Novikov’s Self-Consistency Principle in mind, which basically states that while time travel is possible, time paradoxes are forbidden.

However, a number of solutions have been formulated to avoid the inconsistencies Polchinski suggested, which essentially involves the billiard ball delivering a blow that changes its younger version’s course, but not enough to stop it from entering the wormhole. This solution is related to the ‘timeline-protection hypothesis’ which states that a probability distortion would occur in order to prevent a paradox from happening. This also helps explain why if you tried to time travel and murder your grandfather, something will always happen to make that impossible, thus preserving a consistent version of history.

• Books:  Paradoxes of Time Travel by Ryan Wasserman is a wide-ranging exploration of time and time travel, including Polchinski’s Paradox.

Are Self-Fulfilling Prophecies Paradoxes?

A self-fulfilling prophecy is only a causality loop when the prophecy is truly known to happen and events in the future cause effects in the past, otherwise the phenomenon is not so much a paradox as a case of cause and effect.  Say,  for instance, an authority figure states that something is inevitable, proper, and true, convincing everyone through persuasive style. People, completely convinced through rhetoric, begin to behave as if the prediction were already true, and consequently bring it about through their actions. This might be seen best by an example where someone convincingly states:

“High-speed Magnetic Levitation Trains will dominate as the best form of transportation from the 21st Century onward.”

Jet travel, relying on diminishing fuel supplies, will be reserved for ocean crossing, and local flights will be a thing of the past. People now start planning on building networks of high-speed trains that run on electricity. Infrastructure gears up to supply the needed parts and the prediction becomes true not because it was truly inevitable (though it is a smart idea), but because people behaved as if it were true.

It even works on a smaller scale – the scale of individuals. The basic methodology for all those “self-help” books out in the world is that if you modify your thinking that you are successful (money, career, dating, etc.), then with the strengthening of that belief you start to behave like a successful person. People begin to notice and start to treat you like a successful person; it is a reinforcement/feedback loop and you actually become successful by behaving as if you were.

Are Time Paradoxes Inevitable?

The Butterfly Effect is a reference to Chaos Theory where seemingly trivial changes can have huge cascade reactions over long periods of time. Consequently, the Timeline corruption hypothesis states that time paradoxes are an unavoidable consequence of time travel, and even insignificant changes may be enough to alter history completely.

In one story, a paleontologist, with the help of a time travel device, travels back to the Jurassic Period to get photographs of Stegosaurus, Brachiosaurus, Ceratosaurus, and Allosaurus amongst other dinosaurs. He knows he can’t take samples so he just takes magnificent pictures from the fixed platform that is positioned precisely to not change anything about the environment. His assistant is about to pick a long blade of grass, but he stops him and explains how nothing must change because of their presence. They finish what they are doing and return to the present, but everything is gone. They reappear in a wild world with no humans and no signs that they ever existed. They fall to the floor of their platform, the only man-made thing in the whole world, and lament “Why? We didn’t change anything!” And there on the heel of the scientist’s shoe is a crushed butterfly.

The Butterfly Effect is also a movie, starring Ashton Kutcher as Evan Treborn and Amy Smart as Kayleigh Miller, where a troubled man has had blackouts during his youth, later explained by him traveling back into his own past and taking charge of his younger body briefly. The movie explores the issue of changing the timeline and how unintended consequences can propagate.

Scientists eager to avoid the paradoxes presented by time travel have come up with a number of ingenious ways in which to present a more consistent version of reality, some of which have been touched upon here,  including:

• The Solution: time travel is impossible because of the very paradox it creates.
• Self-healing hypothesis: successfully altering events in the past will set off another set of events which will cause the present to remain the same.
• The Multiverse or “many-worlds” hypothesis: an alternate parallel universe or timeline is created each time an event is altered in the past.
• Erased timeline hypothesis : a person traveling to the past would exist in the new timeline, but have their own timeline erased.

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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!

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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.

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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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Physical Review D

Covering particles, fields, gravitation, and cosmology.

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Time travel paradoxes and multiple histories

Jacob hauser and barak shoshany, phys. rev. d 102 , 064062 – published 24 september 2020.

• Citing Articles (2)
• INTRODUCTION
• KRASNIKOV’S PARADOX MODEL
• GENERALIZING THE MODEL
• THE CASE OF UNLIMITED HISTORIES
• THE CASE OF FINITE CYCLIC HISTORIES
• ANALYSIS OF OUR MODEL
• SUMMARY AND FUTURE PLANS
• ACKNOWLEDGMENTS

If time travel is possible, it seems to inevitably lead to paradoxes. These include consistency paradoxes, such as the famous grandfather paradox, and bootstrap paradoxes, where something is created out of nothing. One proposed class of resolutions to these paradoxes allows for multiple histories (or timelines) such that any changes to the past occur in a new history, independent of the one where the time traveler originated. We introduce a simple mathematical model for a spacetime with a time machine and suggest two possible multiple-histories models, making use of branching spacetimes and covering spaces, respectively. We use these models to construct novel and concrete examples of multiple-histories resolutions to time travel paradoxes, and we explore questions such as whether one can ever come back to a previously visited history and whether a finite or infinite number of histories is required. Interestingly, we find that the histories may be finite and cyclic under certain assumptions, in a way which extends the Novikov self-consistency conjecture to multiple histories and exhibits hybrid behavior combining the two. Investigating these cyclic histories, we rigorously determine how many histories are needed to fully resolve time travel paradoxes for particular laws of physics. Finally, we discuss how observers may experimentally distinguish between multiple histories and the Hawking and Novikov conjectures.

• Received 10 January 2020
• Accepted 21 August 2020

DOI: https://doi.org/10.1103/PhysRevD.102.064062

© 2020 American Physical Society

Physics Subject Headings (PhySH)

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Authors & Affiliations

• 1 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
• 2 Pomona College, 333 North College Way, Claremont, California 91711, USA
• 3 Department of Physics, Brock University, 1812 Sir Isaac Brock Way, St. Catharines, Ontario L2S 3A1, Canada
• * [email protected]
• † [email protected];

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Vol. 102, Iss. 6 — 15 September 2020

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In the DP space, the line ( 1 , x ) is associated with ( − 1 , x ) for − 1 < x < 1 in Minkowski space. This is a simplified model for a wormhole time machine [ 2 ]. After traversing the wormhole, the particle emerges at an earlier value of t and travels in the same direction in x .

In the TDP space, ( 1 , x ) is instead associated with ( − 1 , − x ) for − 1 < x < 1 . After emerging from the wormhole, the particle will travel in the opposite direction in x .

The causality-violating region J 0 ( M ) for the TDP space M is contained between the two associated lines in x . The gray spacelike line indicates a choice of a reasonable surface on which to define initial conditions. We also see particles of two different colors, blue and green, emerging from the right and left; the meaning of these colors is explained in Sec.  2b .

The four possible distinct vertices for particle collisions in Krasnikov’s model. Time is the vertical axis, so the particles always come from the bottom. Note how each blue particle changes into a green particle, and vice versa, in every collision.

An illustration of the consistency and bootstrap paradoxes in Krasnikov’s model. The blue and green lines represent the two possible particle colors, as above. The gray lines indicate a particle which cannot be assigned a consistent color.

(a) Given the identification between colors and elements of Z C , this single general vertex captures all four vertices of Fig.  4 for C = 2 , as well as those for any other values of C . For illustration, the four colors in the figure—blue, green, orange, and magenta—represent any of the C possible colors for the case C ≥ 4 . (b) This vertex is the result of reversing time and parity and conjugating color with respect to the vertex in (a). Since each particle still leaves with a color one greater than it starts with, the result is a valid vertex. In fact, performing C T or P transformations independently also yields a valid vertex. In this example, we took blue = 0 , orange = 1 , green = 2 , magenta = 3 , c = 0 , c ′ = 2 , and C = 4 in both (a) and (b).

In the branching model, when the blue particle enters the time machine at h = 1 , it comes out twisted (since we are in a TDP space) at h = 2 . The new history has an identical copy of the initial blue particle, but this time it encounters itself (or more precisely, its copy from h = 1 ) and the two particles change their colors. A green particle then enters the time machine and continues to h = 3 , and so on. Thus, we have avoided both consistency and bootstrap paradoxes.

Unlike the branching model, the covering space model has no unique first history. Therefore, we depict two consecutive histories k and k + 1 . Without loss of generality, a green particle emerges from the time machine in history k , where it collides with the incoming blue particle; here we are using the color convention of Fig.  6 . Both particles increase their colors as in Fig.  6 : blue = 0 to orange = 1 and green = 2 to magenta = 3 . In history k + 1 , the same process occurs with a magenta particle emerging from the time machine instead of a green particle, and the magenta particle increases its color to blue = 4 (mod 4). Since there is a countably infinite number of time machines, the particle traversing the time machines never completes a CCC, nor does any copy of the incoming blue particle. Thus, we have again avoided both consistency and bootstrap paradoxes.

Since m is a point along the associated wormhole line, it appears twice in our representation of the TDP space—once at t = − 1 and once at t = + 1 . Therefore, our ball U around m is actually U = U + ∪ U − , the union of balls around each wormhole mouth. It is always possible to select such a ball which does not intersect a singularity: if m is a distance ϵ > 0 away from a singularity, then the ball can be chosen to have radius ϵ / 2 .

In our extension of the TDP space, wormhole points are now associated between adjacent histories. As a result, the ball around the point m k + 1 (the point overlapping histories k and k + 1 , which projects down to m under the map p ) is equal to U k + ∪ U k + 1 − . The preimage p − 1 ( U ) = ⋃ k ( U k + ∪ U k + 1 − ) is composed of a countably infinite number of such balls, each of which is homeomorphic to U + ∪ U − from Fig.  9 .

When C = 2 , the consistency paradox can be solved with two cyclic histories. The blue particle entering the time machine in h = 1 comes out of the time machine in h = 2 , and the green particle entering the time machine in h = 2 comes out of the time machine back in h = 1 . Since we interpret the vertices as elastic collisions, we now have a bootstrap paradox: the particle traveling along the CCC only exists within the CCC itself. We will discuss how to resolve this in Sec.  5b . Unlike in the scenario of Fig.  7 , here there is no first history where nothing has come out of the time machine yet (in fact, in Fig.  7 the past exit of the time machine does not even exist for h = 1 ).

A collision of p particles from the left and q particles from the right.

A single history’s causality-violating region can be partitioned into three zones, each of which contains a group collision of particles.

Here, a reflected version of the h = 2 causality-violating region is stacked on top of the h = 1 causality-violating region. These two regions lie in different spaces, as indicated by the separate coordinate axes. However, this representation makes it easy to see how particles evolve over multiple histories, and what the consistency constraints are: that particles on the last wormhole surface match those on the first one.

In this illustration, with C = 2 and H = 2 , one particle is solid and the other is dashed. The illustration demonstrates an interpretation in which the particles do not collide; instead, they pass through each other. This allows us to avoid a bootstrap paradox. However, the same vertices in Fig.  4 still apply.

One of the two consistent solutions obtained by sending particles toward the causality-violating region from both sides.

The second of the two consistent solutions obtained by sending particles toward the causality-violating region from both sides. Note that the initial conditions and final outcomes are the same as in Fig.  16 —two blue particles coming in and two green particles coming out—but the evolution inside the causality-violating region is different. Thus, evolution in this region cannot be predicted.

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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.

Hello, curious kids! Do you have a question you’d like an expert to answer? Ask an adult to send your question to [email protected] . Please tell us your name, age and the city where you live.

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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• Special Relativity
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• Stephen Hawking
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Philosophers, Physicists, and the Time Travel Paradoxes

• First Online: 25 December 2016

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• Paul J. Nahin 15  

Part of the book series: Science and Fiction ((SCIFICT))

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More than 30 years ago Quentin Smith, a philosopher who believes in a finite length to the past, wrote a refutation to those who believe in an infinite past and, while that paper has nothing to do with the paradoxes of time travel, in the course of presenting his reasoning he included the following curious passage:

“He felt the intellectual desperation of any honest philosopher. He knew that he had about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.”

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A time traveler admits (to himself) how perplexed he is by paradoxes in Robert Heinlein’s classic tale “By His Bootstraps,” Astounding Science Fiction , October 1941.

Excerpt from a conversation between two paradox-puzzled time travelers in Larry Niven’s story “Bird in the Hand,” The Magazine of Fantasy & Science Fiction , October 1970.

Q. Smith, “Kant and the Beginning of the World,” New Scholasticism , Summer 1985, pp. 339–346.

L. Niven, “Rotating Cylinders and the Possibility of Global Causality Violation,” Analog Science Fiction , August 1977. Niven took this title from a physics paper with that title, authored by Tipler, that had appeared three years earlier in Physical Review D (April 15, 1974, pp. 2203–2206).

L. Sprague de Camp, “A Gun for Dinosaur,” Galaxy Science Fiction , March 1956.

This is a statement of the belief that the past cannot be changed, an idea we will examine later in this chapter.

Look back at note 93 in Chap. 1 .

This is what is called a bilking paradox , and such paradoxes will be discussed later in this chapter. Brown gave this story a lot of thought. At one point in the tale one of the colleagues, puzzled by how the inventor will be able to place the cube into the time machine at three if it has already vanished from his hand and appeared in the machine, asks “How can you place it there, then?” Replies the inventor, “It will, as my hand approaches, vanish from the [machine] and appear in my hand to be placed there.”

L. Dwyer, “Time Travel and Some Alleged Logical Asymmetries Between Past and Future,” Canadian Journal of Philosophy , March 1978, pp. 15–38.

M. Jameson, “Dead End,” Thrilling Wonder Stories , March 1941.

E. Binder, “The Time Cheaters,” Thrilling Wonder Stories , March 1940.

V. Grigoriev, “Vanya,” in Last Door to Aiya (M. Ginsburg, editor), S. G. Phillips 1968.

M. A. Abramowicz and J. P. Lasota, “On Traveling Round Without Feeling It and Uncurving Curves,” American Journal of Physics , October 1986, pp. 936–939.

F. J. Bridge, “Via the Time Accelerator,” Amazing Stories , January 1931.

This story idea (the letter was signed only with the initials T.J.D.) may well have been the inspiration for R. Rocklynne, “Time Wants a Skeleton,” Astounding Science Fiction , June 1941. Not all fans agreed with T.J.D. A few years later, for example, a teenager named P. Schuyler Miller (1912–1974), who would author several time travel classics himself, wrote a letter to the editor of Astounding Stories (June 1931) stating “there is nothing in physics … to prevent yourself from going into the past … and shaking hands with yourself or killing yourself.” That did, however, provoke the following harsh reply from another, more skeptical reader (in the December 1933 issue): “P. S. Miller once wrote that time traveling is not incompatible with any laws of physics … ‘he don’t know from nothing.’”

C. B. White, “The Lost Continent,” Amazing Stories , July 1927.

F. Flagg, “The Machine Man of Ardathia,” Amazing Stories , November 1927.

C. Cloukey, “Paradox,” Amazing Stories Quarterly , Summer 1929.

H. F. Kirkham, “The Time Oscillator,” Science Wonder Stories , December 1929.

By this time Gernsback had lost control of Amazing , and Science Wonder was his come-back as a publisher of pulp ‘scientifiction.’

E. L. Rementer, “The Time Deflector,” Amazing Stories , December 1929. Gernsback may well have been the editor, before he lost Amazing , who bought this story, and the magazine’s new management simply used what remained in inventory.

F. Flagg, “An Adventure in Time,” Science Wonder Stories , April 1930.

Look back at note 26 in “Some First Words.”

Despite these words, Gernsback apparently hadn’t given up entirely on the ‘invisibility of time travelers’ view, as he had only a few months earlier published another such tale: R. A. Palmer, “The Time Ray of Jandra,” Wonder Stories , June 1930. In this story (one either silly or hilarious, take your pick) a time traveler moves into the future by means of a ‘time ray.’ Unfortunately, the ray works differently on the various chemical elements, and not at all on either hydrogen or oxygen. Thus the time traveler—or at least much of him—and his machine do vanish into the future, but left behind are “several gallons of water spilled on the floor.” (The human body is about 60 % H 2 O.)

For example, Nicholson’s item (2) is a precise plot outline for L. Raphael, “The Man Who Saw Through Time,” Fantastic Adventures , September 1941, and a version of item (4) is in Robert Heinlein’s famous “All You Zombies—,“ Magazine of Fantasy & Science Fiction , March 1959.

One cannot, however, fault the imaginative powers of James Nicholson (1916–1972). He eventually became President of American International Films, the company that made such science fiction ‘classics’ as Attack of the Crab Monsters (1957), the 1963 X (The Man with the X-Ray Eyes) , and The Time Travelers (1964).

The opening line to F. M. Busby, “A Gun for Grandfather,” Future Science Fiction , September 1957.

Jiri Benovsky, “Endurance and Time Travel,” Kriterion — Journal of Philosophy , 2011, pp. 65–72.

R. P. McArthur and M. P. Slattery, “Peter Damian and Undoing the Past,” Philosophical Studies , February 1974, pp. 137–141; P. Remnant, “Peter Damian: Could God Change the Past?” Canadian Journal of Philosophy , June 1978, pp. 259–268; R. Gaskin, “Peter Damian on Divine Power and the Contingency of the Past,” The British Journal for the History of Philosophy , September 1997, pp. 229–247.

This work is in the form of a letter to his friend Desiderius (who later became Pope Victor III), in which Damian rebutted Desiderius’ defense of St. Jerome’s claim that “while God can do all things, he cannot cause a virgin to be restored after she has fallen.” Desiderius thought the reason God could not restore virgins is that he does not want to, to which Damian replied that this meant God is unable to do whatever he does not want to do, but this meant that God would then be less powerful than men, who are able to do things they don’t want to do (such as go without food for a month). This is a good example of the dangers involved when getting into debates with theologians.

The Argentinian writer Jorge Luis Borges (1899–1986) was so inspired by Damian’s view that the past could be changed that he wrote a short story based on it (see “The Other Death,” originally published in The New Yorker , November 2, 1968) and put a character in it named after Damian.

E. Stump and N. Kretzmann, “Eternity,” Journal of Philosophy , August 1981, pp. 429–458.

P. Geach, God and the Soul , Routledge & Kegan Paul 1969.

A summary of those opinions can be found in G. Brown, “Praying About the Past,” Philosophical Quarterly , January 1985, pp. 83–86. Debate continues on the retroactive prayer into the 21st century: see, for example, K. Timpe, “Prayers for the Past,” September 2005, pp. 305–322, and T. J. Mawson, “Praying for Known Outcomes,” March 2007, pp. 71–87, both in Religious Studies .

M. Dummett, “Bringing About the Past,” Philosophical Review , July 1964, pp. 338–359.

Reprinted in the posthumously published Grumbles from the Grave (edited by Heinlein’s widow, Virginia Heinlein), Del Rey 1990.

See Jan Pinkerton, “Backward Time Travel, Alternate Universes, and Edward Everett Hale,” Extrapolation , Summer 1979, pp. 168–175. The time machine in Mitchell’s story is more fantasy than anything else. It is simply stated that if the clock runs backward, then it travels backward in time.

Wells’ failure to use paradox in his famous novel surprises most modern readers and, in fact, one of the first reviewers specifically criticized him for this lapse. See the 1895 review of The Time Machine that appeared in Pall Mall Magazine , by Israel Zangwill, reprinted in Parrinder’s book (note 1 in the Introduction).

T. Powers, The Anubis Gates , Ace 1983, a work with equal shares of physics and magic.

C. L. Moore, “Tryst in Time,” Astounding Stories , December 1936.

See, for example, P. J. Riggs, “The Principal Paradox of Time Travel,” Ratio , April 1997, pp. 48–64. The ‘principal paradox’ is that time travel is inherently contradictory because it permits the possibility of traveling to an earlier time to prevent the trip. The grandfather paradox is a special case of this. For more discussion, see T. Chambers, “Time Travel: How Not to Defuse the Principal Paradox,” Ratio , September 1999, pp. 296–301.

R. Bradbury, “A Sound of Thunder,” Collier’s , June 1952.

R. Silverberg, “The Assassin,” Imaginative Tales , July 1957.

See, for example, the novel by C. L. Harness, Krono , Franklin Watts 1988.

A. Porges, “The Rescuer,” Analog Science Fiction , July 1962.

P. S. Miller, “Status Quondam,” New Tales of Space and Time , November 1951. This is the Miller I mentioned back in note 15.

N. Schachner, “Ancestral Voices,” Astounding Stories , December 1933.

N. Schachner, “The Time Imposter,” Astounding Stories , March 1934.

C. Dye, “Time Goes to Now,” Science Fiction Quarterly , May 1953.

The ‘paradox’ is that, assuming you do arrive in the past with a working gun, why can’t you kill your grandfather? After all, you must fail in that quest because otherwise you wouldn’t be there from the future to even try. But why must you fail? It is, of course, not actually necessary to try to kill your grandfather to run into this paradoxical situation—just go back in time to any moment in the past and try to kill yourself! You won’t succeed (if the past is unchangeable), but why not ? (To argue ‘because the past is unchangeable’ is to beg the question. We need more insight than that.)

M. Reynolds and F. Brown, “Dark Interlude,” Galaxy Science Fiction , January 1951.

C. South, “The Time Mirror,” Amazing Stories , December 1942.

F. Brown, “First Time Machine,” Honeymoon in Hell , Bantam 1958.

J. W. Meiland, “A Two-Dimensional Passage Model of Time for Time Travel,” Philosophical Studies , November 1974, pp. 153–173. Science fiction had already considered time travel suicide in, for example, K. Neville, “Mission,” Fantasy and Science Fiction , April 1953.

F. Arntzenius, “Causal Paradoxes in Special Relativity,” British Journal for the Philosophy of Science , June 1990, pp. 223–243.

P. Worth, “Typewriter from the Future,” Amazing Stories ,” February 1950. See also note 106 in Chap. 1 .

For example, in J. H. Schmidt, “Newcomb’s Paradox Realized with Backward Causation,” British Journal for the Philosophy of Science , March 1998, pp. 67–87, we read that “there are as yet no generally accepted solutions” to the grandfather paradox.

S. Gorovitz, “Leaving the Past Alone,” Philosophical Review , July 1964, pp. 360–371.

F. M. Busby, “A Gun for Grandfather,” Future Science Fiction , Fall 1957.

David Horacek, “Time Travel in Indeterministic Worlds,” The Monist , July 2005, pp. 423–436.

S. Krasnikov, “Time Travel Paradox,” Physical Review D , February 14, 2002, pp. 064013–1 to -8.

R. G. Swinburne, “Affecting the Past,” Philosophical Quarterly , October 1966, pp. 341–347.

“The Red Queen’s Race,” Astounding Science Fiction , January 1949.

O. Saari, “The Time Bender,” Astounding Stories , August 1937.

This story describes something a bit more than ‘simply’ affecting the past; it has a causal loop in it. The time traveling historian makes his trip because of an event in the past that his trip causes. Such paradoxes will be the subject of the next section.

L. Sprague de Camp, “Aristotle and the Gun,” Astounding Science Fiction , February 1958. Asimov and de Camp were close friends, and their two stories with similar premises are clearly the result of a bit of friendly rivalry.

P. Geach, “Some Problems About Time,” in Studies in the Philosophy of Thought and Action (P. F. Strawson, editor), Oxford 1968.

M. Sinclair, “Where Their Fire Is Not Quenched,” in After the Darkness Falls (B. Karloff, editor), World Publishing 1946.

D. Beason, “Ben Franklin’s Laser,” Analog , December 1990.

Like just about everything concerning time travel, however, not all think this. For example, the great German physicist Max Planck (1858–1947), the 1918 Nobel physics laureate, said (in 1922): “Physics hence is inclined to view the principle of least action more as a formal and accidental curiosity than as a pillar of physical knowledge.” Still, he did also declare that he thought it unlikely “the dominance of such a simple law could be a mere accident.” Quoted from Marc Lange, “Conservation Laws in Scientific Explanations: Constraints or Coincidences,” Philosophy of Science , July 2011, pp. 333–352.

See A. Carlini, et al., “Time Machines: The Principle of Self-Consistency as a Consequence of the Principle of Minimal Action,” October 1995, pp. 557–580, and “Time Machines and the Principle of Self-Consistency as a Consequence of the Principle of Stationary Action (II): The Cauchy Problem for a Self-Interacting Relativistic Particle,” October 1996, pp. 445–479, both in International Journal of Modern Physics D .

I. D. Novikov, “An Analysis of the Operation of a Time Machine,” Soviet Physics JETP , March 1989, pp. 439–443.

R. D. Driver, “Can the Future Influence the Present?” Physical Review D , February 15, 1979, pp. 1098–1107.

R. F. Young, “The Dandelion Girl,” The Saturday Evening Post , April 1, 1961. See also note 49 in Chap. 2 .

R. Wilson, “The Message,” Astounding Stories , March 1942.

G. C. Edmondson, “The Misfit,” Fantasy and Science Fiction , February 1959.

M. Weisinger, “Thompson’s Time Traveling Theory,’ Amazing Stories , March 1944.

“Typewriter from the Future”: see note 106 in Chap. 1 .

Theodore Sider, “Time Travel, Coincidences, and Counterfactuals,” Philosophical Studies , August 2002, pp. 115–138.

See, for example, David King, “Time Travel and Self-Consistency: Implications for Determinism and the Human Condition,” Ratio , September 1999, pp. 271–278.

Nicholas J. J. Smith, “Bananas Enough for Time Travel?” British Journal for the Philosophy of Science , September 1997, pp. 363–389.

This does present us with the curious (although non-paradoxical) situation that the time traveler will find, upon his appearing in the future, the date he will write (in his personal future) when he returns to just after he left on his time trip.

A science fiction suggestion that in certain situations (particularly causal loops), might actually be good advice! From L. Sprague de Camp’s “The Best-Laid Scheme,” Astounding Science Fiction , February 1941.

P. Bolton, “The Time Hoaxes,” Amazing Stories , August 1931.

J. Blish, “Weapon Out of Time,” Science Fiction Quarterly , Spring 1941.

A. B. Chandler, “The Tides of Time,” Fantastic Adventures , June 1948.

W. Sheldon, “A Bit of Forever,” Super Science Stories , July 1950.

M. Leinster, “The Gadget Had a Ghost,” Thrilling Wonder Stories , June 1952.

D. I. Massor, “A Two-Timer,” New Worlds SF , February 1966.

H. Harrison, The Technicolor Time Machine , Doubleday 1967.

C. Simak, “The Birch Clump Cylinder,” Stellar 1 (J. del Rey, editor), Ballantine 1974.

D. Knight, “The Man Who Went Back,” Amazing Stories , November 1985. This same idea was used earlier in the story “Compounded Interest,” ( Magazine of Fantasy and Science Fiction , August 1956) by Mack Reynolds, in which the inventor of a time machine has the money to build his gadget because he uses it to go back into the past where he deposits a small sum, which then grows (through the ‘magic’ of compound interest) into the cash he needs to fund his time machine.

G. Fulmer, “Understanding Time Travel,” Southwestern Journal of Philosophy , Spring 1980, pp. 151–156.

G. Fulmer, “Time Travel, Determinism, and Fatalism,” Philosophical Speculations in Science Fiction and Fantasy , Spring 1981, pp. 41–48.

M. MacBeath, “Communication and Time Reversal,” Synthese , July 1983, pp. 27–46.

H. Reichenbach, The Direction of Time , University of California Press 1956, p. 37.

J. L. Friedman et al., “Cauchy Problem in Spacetimes with Closed Timelike Curves,” Physical Review D , September 15, 1990, pp. 1915–1930. Another physicist, however, has flatly rejected this need for the Principle, calling it redundant: see D. Deutsch, “Quantum Mechanics Near Closed Timelike Lines,” Physical Review D , November 15, 1991, pp. 3197–3217.

S. Mines, “Find the Sculptor,” Thrilling Wonder Stories , Spring 1946.

R. M. Farley, “The Man Who Met Himself,” Top-Notch Magazine , August 1935 ( Top-Notch was an adventure pulp published between 1910 and 1937).

The reason for this line in the story is that earlier the question of “Where did the time machine come from originally ?” was raised. The answer: “There was never any ‘original.’ … There is no round-and-round circle of events, no repetition. Merely one closed cycle.” This is, in fact, the modern view of causal loops, expressed in a 1935 (!) science fiction story.

See note 99 in Chap. 1 , and the related discussion there.

One philosopher calls this bit of dialog “unhelpful,” while ignoring the fact that it appeared in a science fiction pulp magazine and not a scholarly journal, and was clearly meant to dazzle teenage boys (see note 39 and related discussion in “Some First Words”) with the concept of a causal loop, rather than to break new ground in metaphysical thought. See Richard Hanley, “No End in Sight: Causal Loops in Philosophy, Physics and Fiction,” Synthese , July 2004, pp. 123–152.

G. Nerlich, “Can Time Be Finite?” Pacific Philosophical Quarterly , July 1981, pp. 227–239.

P. S. Miller, “As Never Was,” Astounding Science Fiction , January 1944. This is the same Miller who appears in note 15 (and see note 46, too).

It is not sufficient to say that perhaps she polished the watch. Polishing would remove material from the watch, which means she gives him a watch different from the one he gives her in the past.

Bradley Monton, “Time Travel Without Causal Loops,” The Philosophical Quarterly , January 2009, pp. 54–67.

Professor Hanley (note 105) says the answer to such questions is “straightforward”: the information comes “from itself.” I think the issue is rather deeper than that.

See D. Franson, “Package Deal,” in Microcosmic Tales , Taplinger 1980. The British philosopher J. R. Lucas had a similar scenario in mind when he wrote, in his book A Treatise on Time and Space (Methuen 1973, p. 50), “It is very important, not only for reasons of modesty, that I should not be able to use a Time Machine to go into a public library and read my own biography.” Robert Heinlein didn’t agree with Lucas: in his 1956 novel The Door Into Summer the protagonist, an inventor, travels thirty years into the future, where he reads some patent disclosures for inventions that he doesn’t remember, even though they are in his name. He then returns to his own time and promptly files the patents!

M. R. Levin, “Swords’ Points,” Analysis , March 1980, pp. 69–70.

P. Anderson and G. Dickson, “Trespass,” Fantastic Story Quarterly , Spring 1950.

Why would God do such a thing? Apparently ‘just to have some fun with geologists and biologists,’ as creationists call such ancient fossils ‘sports of nature.’

G. Klein, “Party Line,” The Best from the Rest of the World (D. A. Wolheim, editor), Doubleday 1976 (story originally published in France in 1973).

J. Finney, “The Love Letter,” The Saturday Evening Post , August 1959.

T. N. Scortia, “When You Hear the Tone,” Galaxy Science Fiction , January 1971. See also L. Padgett, “Line to Tomorrow,” Astounding Science Fiction , November 1945.

M. Leinster, “Sam, This Is You,” Galaxy Science Fiction , May 1955. This story was later broadcast as an episode on the “X-Minus One” radio drama program. See also F. A. Reeds, “Forever Is Not So Long,” Astounding Science Fiction , May 1942.

L. Del Rey, “Fools’ Errand,” Science Fiction Quarterly , November 1951.

N. Schachner, “Lost in the Dimensions,” Astounding Stories , November 1937.

‘Time traveling’ by crossing time zones is an idea that one can trace at least as far back as to Edgar Allen Poe’s 1841 short story “Three Sundays in a Week.”

From Robert Forward’s 1992 novel Timemaster .

J. Harrison, “Jocasta’s Crime,” Analysis , March 1979, p. 65.

M. MacBeath, “Who Was Dr. Who’s Father?” Synthese , June 1982, pp. 397–430.

W. Godfrey-Smith, “Traveling in Time,” Analysis , March 1980, pp. 72–73. This false claim had already been raised by a physicist (L. S. Schulman, “Tachyon Paradoxes ,” American Journal of Physics , May 1971, pp. 481–484), and even earlier by a science fiction writer (P. Anderson, “Time Patrol,” Magazine of Fantasy and Science Fiction , May 1955).

Phil Dowe, “The Coincidences of Time Travel,” Philosophy of Science , July 2003, pp. 574–589. See also J. Berkovich, “On Chance in Causal Loops,” Mind , January 2001, pp. 1–23, and P. Dowe, “Causal Loops and Independence of Causal Facts,” Philosophy of Science , September 2001, pp. 89–97.

R. Dee (this is not the ‘Dee’ of Harrison’s story!), “The Poundstone Paradox,” Magazine of Fantasy and Science Fiction , May 1954.

C. L. Harness, “Child By Chronos,” Magazine of Fantasy and Science Fiction , June 1953.

R. Heinlein, “All You Zombies—,” Magazine of Fantasy and Science Fiction , March 1959.

S. Lem, “The Time-Travel Story and Related Matters of SF Structuring,” Science Fiction Studies , Spring 1974, pp. 143–154.

G. Benford, “Down the River Road,” After the King: Stories in Honor of J. R. Tolkien (C. Tolkien and M. Greenberg, editors), Tor 1991.

M. Gardner, “Mathematical Games,” Scientific American , March 1979.

See, for example, Jack Williamson’s 1938 novel The Legion of Time , and C. L. Moore’s, “Tryst in Time,” Astounding Stories , December 1936. L. Sprague de Camp (1907–2000), too, was an early pioneer in the exploration of the MWI idea in science fiction long before Everett. In his 1941 novel Lest Darkness Fall , for example, he uses the analogy of a tree (the “main time line”) that is always sprouting new branches.

N. Bond, “Parallel in Time,” Thrilling Wonder Stories , June 1940. See also S. N. Faber, “Trans Dimensional Imports,” Isaac Asimov’s Science Fiction Magazine , August 1980.

W. Sell, “Other Tracks,” Astounding Science Fiction , October 1938.

Just like Marty McFly does at the end of the 1985 film Back to the Future . The movie is fun, but pulp science fiction did it first.

M. Leinster, “Sidewise in Time,” Astounding Stories , June 1934. Splitting universes with multiple time tracks and time loops became quite popular after Leinster’s and Sell’s stories; you can find the basic idea repeated yet again in Alfred Bester’s “The Probable Man,” Astounding Science Fiction , July 1941, for example, in which each new journey into the past causes the future to fan out into an infinity of new time tracks.

L. Watt-Evans, “The Drifter,” Amazing Stories , October 1991.

In Jack Haldeman’s 1990 novel The Hemingway Hoax we read that “there is not just one [parallel] universe, but actually uncountable zillions of them.”

In “Quantum Mechanics for Cosmologists,” Speakable and Unspeakable in Quantum Mechanics , Cambridge University Press 1987.

J. R. Pierce, “Mr. Kinkaid’s Pasts,” Magazine of Fantasy & Science Fiction , August 1953.

A. Everett, “Time Travel Paradoxes, Path Integrals, and the Many Worlds Interpretation of Quantum Mechanics,” Physical Review D , June 25, 2004, pp. 124023–1:124023–14.

For a modern philosophical argument specifically rebutting Deutsch’s enthusiasm for the MWI, see Theodore Sider, “A New Grandfather Paradox?” Philosophy and Phenomenological Research , March 1997, pp. 139–144.

See note 34 in Chap. 3 .

R. A. Healy, “How Many Worlds?” Nous , November 1984, pp. 591–616.

B. S. DeWitt, “Quantum Mechanics and Reality,’ in The Many-Worlds Interpretation of Quantum Mechanics (B. S. DeWitt and N. Graham, editors), Princeton 1973.

B. Shaw, “What Time Do You Call This?” Amazing Science Fiction , September 1971. When a bank robber in one world tries to make his escape into the other world, he literally runs into ‘himself’ trying to escape after robbing the ‘same’ bank in the parallel world!

D. S. Goldwirth et al. , “Quantum Propagator for a Nonrelativistic Particle in the Vicinity of a Time Machine,” Physical Review D , April 15, 1994, pp. 3951–3957. See, too, the earlier D. S. Goldwirth et al. , “The Breakdown of Quantum Mechanics in the Presence of Time Machines,” General Relativity and Gravitation , January 1993, pp. 7–13.

D. R. Daniels, “The Branches of Time,” Wonder Stories , August 1935.

These sad, resigned words were written when the author, David R. Daniels (1915–1936), was just twenty years old. A year later he committed suicide.

A. Shimony, “Events and Processes in the Quantum World,” Quantum Concepts in Space and Time (R. Penrose and C. J. Isham, editors), Oxford University Press 1986.

J. MacCreigh, “A Hitch in Time,” Thrilling Wonder Stories , June 1947.

J. Blish, “Beep,” Galaxy Science Fiction , February 1954.

I. Asimov, “Fair Exchange?” Asimov’s Science Fiction Adventure Magazine , Fall 1978.

M. F. Flynn, “The Forest of Time,” Analog Science Fiction , June 1987.

Lord Dunsany, “Lost,” The Fourth Book of Jorkens , Arkham House 1948.

Q. Smith, “A New Topology of Temporal and Atemporal Permanence,” Nous , June 1989, pp. 307–330.

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Nahin, P.J. (2017). Philosophers, Physicists, and the Time Travel Paradoxes. In: Time Machine Tales. Science and Fiction. Springer, Cham. https://doi.org/10.1007/978-3-319-48864-6_4

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Paradoxes of Time Travel

Professor of Philosophy

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Paradoxes of Time Travel is a comprehensive study of the philosophical issues raised by the possibility of time travel. The book begins, in Chapter 1, by explaining the concept of time travel and clarifying the central question to be addressed: Is time travel compatible with the laws of metaphysics and, in particular, the laws concerning time, freedom, causation, and identity? Chapter 2 then explores the various temporal paradoxes, including the double-occupancy problem, the no-destination argument, and the famous twin paradox of special relativity. Chapters 3 and 4 focus on the paradoxes of freedom, including various versions of the grandfather paradox. Chapter 5 covers causal paradoxes, including the bootstrapping paradox, the problems of backward causation, and the various puzzles raised by causal loops. Chapter 6 then concludes by looking at various paradoxes of identity. This includes a discussion of different theories of change and persistence, and an exploration of the various puzzles raised by self-visitation.

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Time travel: Is it possible?

Science says time travel is possible, but probably not in the way you're thinking.

Albert Einstein's theory

• General relativity and GPS
• Wormhole travel
• Alternate theories

Science fiction

Is time travel possible? Short answer: Yes, and you're doing it right now — hurtling into the future at the impressive rate of one second per second. 

You're pretty much always moving through time at the same speed, whether you're watching paint dry or wishing you had more hours to visit with a friend from out of town. 

But this isn't the kind of time travel that's captivated countless science fiction writers, or spurred a genre so extensive that Wikipedia lists over 400 titles in the category "Movies about Time Travel." In franchises like " Doctor Who ," " Star Trek ," and "Back to the Future" characters climb into some wild vehicle to blast into the past or spin into the future. Once the characters have traveled through time, they grapple with what happens if you change the past or present based on information from the future (which is where time travel stories intersect with the idea of parallel universes or alternate timelines). 

Related: The best sci-fi time machines ever

Although many people are fascinated by the idea of changing the past or seeing the future before it's due, no person has ever demonstrated the kind of back-and-forth time travel seen in science fiction or proposed a method of sending a person through significant periods of time that wouldn't destroy them on the way. And, as physicist Stephen Hawking pointed out in his book " Black Holes and Baby Universes" (Bantam, 1994), "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."

Science does support some amount of time-bending, though. For example, physicist Albert Einstein 's theory of special relativity proposes that time is an illusion that moves relative to an observer. An observer traveling near the speed of light will experience time, with all its aftereffects (boredom, aging, etc.) much more slowly than an observer at rest. That's why astronaut Scott Kelly aged ever so slightly less over the course of a year in orbit than his twin brother who stayed here on Earth. 

Related: Controversially, physicist argues that time is real

There are other scientific theories about time travel, including some weird physics that arise around wormholes , black holes and string theory . For the most part, though, time travel remains the domain of an ever-growing array of science fiction books, movies, television shows, comics, video games and more. 

Einstein developed his theory of special relativity in 1905. Along with his later expansion, the theory of general relativity , it has become one of the foundational tenets of modern physics. Special relativity describes the relationship between space and time for objects moving at constant speeds in a straight line. 

The short version of the theory is deceptively simple. First, all things are measured in relation to something else — that is to say, there is no "absolute" frame of reference. Second, the speed of light is constant. It stays the same no matter what, and no matter where it's measured from. And third, nothing can go faster than the speed of light.

From those simple tenets unfolds actual, real-life time travel. An observer traveling at high velocity will experience time at a slower rate than an observer who isn't speeding through space. 

While we don't accelerate humans to near-light-speed, we do send them swinging around the planet at 17,500 mph (28,160 km/h) aboard the International Space Station . Astronaut Scott Kelly was born after his twin brother, and fellow astronaut, Mark Kelly . Scott Kelly spent 520 days in orbit, while Mark logged 54 days in space. The difference in the speed at which they experienced time over the course of their lifetimes has actually widened the age gap between the two men.

"So, where[as] I used to be just 6 minutes older, now I am 6 minutes and 5 milliseconds older," Mark Kelly said in a panel discussion on July 12, 2020, Space.com previously reported . "Now I've got that over his head."

General relativity and GPS time travel

The difference that low earth orbit makes in an astronaut's life span may be negligible — better suited for jokes among siblings than actual life extension or visiting the distant future — but the dilation in time between people on Earth and GPS satellites flying through space does make a difference. 

Read more: Can we stop time?

The Global Positioning System , or GPS, helps us know exactly where we are by communicating with a network of a few dozen satellites positioned in a high Earth orbit. The satellites circle the planet from 12,500 miles (20,100 kilometers) away, moving at 8,700 mph (14,000 km/h). 

According to special relativity, the faster an object moves relative to another object, the slower that first object experiences time. For GPS satellites with atomic clocks, this effect cuts 7 microseconds, or 7 millionths of a second, off each day, according to the American Physical Society publication Physics Central .  

Read more: Could Star Trek's faster-than-light warp drive actually work?

Then, according to general relativity, clocks closer to the center of a large gravitational mass like Earth tick more slowly than those farther away. So, because the GPS satellites are much farther from the center of Earth compared to clocks on the surface, Physics Central added, that adds another 45 microseconds onto the GPS satellite clocks each day. Combined with the negative 7 microseconds from the special relativity calculation, the net result is an added 38 microseconds. 

This means that in order to maintain the accuracy needed to pinpoint your car or phone — or, since the system is run by the U.S. Department of Defense, a military drone — engineers must account for an extra 38 microseconds in each satellite's day. The atomic clocks onboard don’t tick over to the next day until they have run 38 microseconds longer than comparable clocks on Earth.

Given those numbers, it would take more than seven years for the atomic clock in a GPS satellite to un-sync itself from an Earth clock by more than a blink of an eye. (We did the math: If you estimate a blink to last at least 100,000 microseconds, as the Harvard Database of Useful Biological Numbers does, it would take thousands of days for those 38 microsecond shifts to add up.) 

This kind of time travel may seem as negligible as the Kelly brothers' age gap, but given the hyper-accuracy of modern GPS technology, it actually does matter. If it can communicate with the satellites whizzing overhead, your phone can nail down your location in space and time with incredible accuracy. 

Can wormholes take us back in time?

General relativity might also provide scenarios that could allow travelers to go back in time, according to NASA . But the physical reality of those time-travel methods is no piece of cake. 

Wormholes are theoretical "tunnels" through the fabric of space-time that could connect different moments or locations in reality to others. Also known as Einstein-Rosen bridges or white holes, as opposed to black holes, speculation about wormholes abounds. But despite taking up a lot of space (or space-time) in science fiction, no wormholes of any kind have been identified in real life. 

Related: Best time travel movies

"The whole thing is very hypothetical at this point," Stephen Hsu, a professor of theoretical physics at the University of Oregon, told Space.com sister site Live Science . "No one thinks we're going to find a wormhole anytime soon."

Primordial wormholes are predicted to be just 10^-34 inches (10^-33 centimeters) at the tunnel's "mouth". Previously, they were expected to be too unstable for anything to be able to travel through them. However, a study claims that this is not the case, Live Science reported . 

The theory, which suggests that wormholes could work as viable space-time shortcuts, was described by physicist Pascal Koiran. As part of the study, Koiran used the Eddington-Finkelstein metric, as opposed to the Schwarzschild metric which has been used in the majority of previous analyses.

In the past, the path of a particle could not be traced through a hypothetical wormhole. However, using the Eddington-Finkelstein metric, the physicist was able to achieve just that.

Koiran's paper was described in October 2021, in the preprint database arXiv , before being published in the Journal of Modern Physics D.

Alternate time travel theories

While Einstein's theories appear to make time travel difficult, some researchers have proposed other solutions that could allow jumps back and forth in time. These alternate theories share one major flaw: As far as scientists can tell, there's no way a person could survive the kind of gravitational pulling and pushing that each solution requires.

Infinite cylinder theory

Astronomer Frank Tipler proposed a mechanism (sometimes known as a Tipler Cylinder ) where one could take matter that is 10 times the sun's mass, then roll it into a very long, but very dense cylinder. The Anderson Institute , a time travel research organization, described the cylinder as "a black hole that has passed through a spaghetti factory."

After spinning this black hole spaghetti a few billion revolutions per minute, a spaceship nearby — following a very precise spiral around the cylinder — could travel backward in time on a "closed, time-like curve," according to the Anderson Institute. 

The major problem is that in order for the Tipler Cylinder to become reality, the cylinder would need to be infinitely long or be made of some unknown kind of matter. At least for the foreseeable future, endless interstellar pasta is beyond our reach.

Time donuts

Theoretical physicist Amos Ori at the Technion-Israel Institute of Technology in Haifa, Israel, proposed a model for a time machine made out of curved space-time — a donut-shaped vacuum surrounded by a sphere of normal matter.

"The machine is space-time itself," Ori told Live Science . "If we were to create an area with a warp like this in space that would enable time lines to close on themselves, it might enable future generations to return to visit our time."

Amos Ori is a theoretical physicist at the Technion-Israel Institute of Technology in Haifa, Israel. His research interests and publications span the fields of general relativity, black holes, gravitational waves and closed time lines.

There are a few caveats to Ori's time machine. First, visitors to the past wouldn't be able to travel to times earlier than the invention and construction of the time donut. Second, and more importantly, the invention and construction of this machine would depend on our ability to manipulate gravitational fields at will — a feat that may be theoretically possible but is certainly beyond our immediate reach.

Time travel has long occupied a significant place in fiction. Since as early as the "Mahabharata," an ancient Sanskrit epic poem compiled around 400 B.C., humans have dreamed of warping time, Lisa Yaszek, a professor of science fiction studies at the Georgia Institute of Technology in Atlanta, told Live Science .  

Every work of time-travel fiction creates its own version of space-time, glossing over one or more scientific hurdles and paradoxes to achieve its plot requirements. 

Some make a nod to research and physics, like " Interstellar ," a 2014 film directed by Christopher Nolan. In the movie, a character played by Matthew McConaughey spends a few hours on a planet orbiting a supermassive black hole, but because of time dilation, observers on Earth experience those hours as a matter of decades. 

Others take a more whimsical approach, like the "Doctor Who" television series. The series features the Doctor, an extraterrestrial "Time Lord" who travels in a spaceship resembling a blue British police box. "People assume," the Doctor explained in the show, "that time is a strict progression from cause to effect, but actually from a non-linear, non-subjective viewpoint, it's more like a big ball of wibbly-wobbly, timey-wimey stuff." 

Long-standing franchises like the "Star Trek" movies and television series, as well as comic universes like DC and Marvel Comics, revisit the idea of time travel over and over. 

Related: Marvel movies in order: chronological & release order

Here is an incomplete (and deeply subjective) list of some influential or notable works of time travel fiction:

Books about time travel:

• Rip Van Winkle (Cornelius S. Van Winkle, 1819) by Washington Irving
• A Christmas Carol (Chapman & Hall, 1843) by Charles Dickens
• The Time Machine (William Heinemann, 1895) by H. G. Wells
• A Connecticut Yankee in King Arthur's Court (Charles L. Webster and Co., 1889) by Mark Twain
• The Restaurant at the End of the Universe (Pan Books, 1980) by Douglas Adams
• A Tale of Time City (Methuen, 1987) by Diana Wynn Jones
• The Outlander series (Delacorte Press, 1991-present) by Diana Gabaldon
• Harry Potter and the Prisoner of Azkaban (Bloomsbury/Scholastic, 1999) by J. K. Rowling
• Thief of Time (Doubleday, 2001) by Terry Pratchett
• The Time Traveler's Wife (MacAdam/Cage, 2003) by Audrey Niffenegger
• All You Need is Kill (Shueisha, 2004) by Hiroshi Sakurazaka

Movies about time travel:

• Planet of the Apes (1968)
• Superman (1978)
• Time Bandits (1981)
• The Terminator (1984)
• Back to the Future series (1985, 1989, 1990)
• Star Trek IV: The Voyage Home (1986)
• Bill & Ted's Excellent Adventure (1989)
• Groundhog Day (1993)
• Galaxy Quest (1999)
• The Butterfly Effect (2004)
• 13 Going on 30 (2004)
• The Lake House (2006)
• Meet the Robinsons (2007)
• Hot Tub Time Machine (2010)
• Midnight in Paris (2011)
• Looper (2012)
• X-Men: Days of Future Past (2014)
• Edge of Tomorrow (2014)
• Interstellar (2014)
• Doctor Strange (2016)
• A Wrinkle in Time (2018)
• The Last Sharknado: It's About Time (2018)
• Avengers: Endgame (2019)
• Tenet (2020)
• Palm Springs (2020)
• Zach Snyder's Justice League (2021)
• The Tomorrow War (2021)

Television about time travel:

• Doctor Who (1963-present)
• The Twilight Zone (1959-1964) (multiple episodes)
• Star Trek (multiple series, multiple episodes)
• Samurai Jack (2001-2004)
• Lost (2004-2010)
• Phil of the Future (2004-2006)
• Steins;Gate (2011)
• Outlander (2014-2023)
• Loki (2021-present)

Games about time travel:

• Chrono Trigger (1995)
• TimeSplitters (2000-2005)
• Kingdom Hearts (2002-2019)
• Prince of Persia: Sands of Time (2003)
• God of War II (2007)
• Ratchet and Clank Future: A Crack In Time (2009)
• Sly Cooper: Thieves in Time (2013)
• Dishonored 2 (2016)
• Titanfall 2 (2016)
• Outer Wilds (2019)

Additional resources

Explore physicist Peter Millington's thoughts about Stephen Hawking's time travel theories at The Conversation . Check out a kid-friendly explanation of real-world time travel from NASA's Space Place . For an overview of time travel in fiction and the collective consciousness, read " Time Travel: A History " (Pantheon, 2016) by James Gleik. 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

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Vicky Stein is a science writer based in California. She has a bachelor's degree in ecology and evolutionary biology from Dartmouth College and a graduate certificate in science writing from the University of California, Santa Cruz (2018). Afterwards, she worked as a news assistant for PBS NewsHour, and now works as a freelancer covering anything from asteroids to zebras. Follow her most recent work (and most recent pictures of nudibranchs) on Twitter. 

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Engineers Garage

Time Travel: Theories, Possibilities, and Paradoxes Explained

By Neha Rastogi February 16, 2017

Time Travel has been a matter of great interest for Science fiction since ages. Whether it’s the movies like Planet of the Apes (1968) or modern franchises like “Doctor Who” and “Star Trek” ; the concept is grabbing a lot of eyeballs. Not only movies and shows but even some mythological tales like Mahabharata and the Japanese story of Urashima Taro support the evidence that time travel exists. We often see stories where characters use time machines to jaunt through the years but the reality is far more complex and inexplicable.

Understanding the Concept of Time Travel

Time Travel is defined as the phenomenon of moving between different points in time through a hypothetical device called “Time Machine”. Despite being predominantly related to the field of philosophy and fiction, it’s somehow supported to a small extent by physics in conjunction with quantum mechanics. However, before getting into the argument of how real it is, let’s comprehend the fundamental meaning of time.

Basically, the whole idea of Time Travel is administered by the concept of time. Usually, people believe that time is constant but the famous Physicist Albert Einstein introduced the “Theory of Relativity” as per which, time is relative. In other words, time slows down or speeds up depending on how fast the observer moves relative to something else. According to him, a person traveling inside a spaceship at the speed of light would age much slower than his/her twin back at home.

Time is Relative

After Einstein’s Theory of Relativity, his teacher Herman Minkowski emphasized on space-time, a mathematical model that joins both space and time in a continuum. This implies that time and space cannot exist without each other. Space is a 3-dimensional arena consisting of length, width, and height. This is joined by Time with the fourth dimension called direction. So anything that happens in the universe takes place in this space-time continuum. Although this validates that space travelers are slightly younger than their twins when they return to earth, yet a huge leap in the past or future is not possible with the current technology.

Time Machines

It is believed that in order to travel back or forward in time, one would require a device called Time Machine . The research on such a device would involve bending space-time to such an extent that time lines turn back on themselves to form a loop, which is termed as “closed time-like curve.” Such an action demands the use of an exotic form of matter with “negative energy density” that has a unique property of moving in the opposite direction of the normal matter when pushed. Even if it exists, the quantity would be too small to construct a machine.

Pictorial Representation of Time Travel through closed time-like curve

However, some another research suggests that time machines can also be constructed by building a doughnut-shaped hole enveloped within a sphere of normal matter. Inside this doughnut-shaped hole filled with vacuum, gravitational force can be used to bend the space-time so as to form a closed time-like curve. After racing around inside this doughnut a traveler would be able to go back in time with each lap. But in reality, it’s quite complex because the gravitational fields have to be very strong and would demand precise manipulation.

Time Travel Approaches in Physics

After studying and researching about Time Travel, various physicists have come up with approaches that may support its possibility, at least theoretically. Let’s take a look at these concepts so as to understand how Time Travel could actually work someday.

Time Dilation

Time Dilation Explanation

An important aspect of Einstein’s relativity theory is the term “time dilation” , which is defined as the difference of elapsed time between two events as measured by observers who are either moving relative to each other or are situated at different locations from the gravitational mass. As per the theory, time dilation can be summarized as a phenomenon which occurs due to the difference in either gravity or relative velocity.

In special relativity the time dilation effect is reciprocal i.e. when two clocks are in motion with respect to each other, for both the observers, the other one will be time dilated or the other clock will move slower. However, in general relativity, an observer at the top of the tower will find the clock closer to the ground to be slower and the other observer would agree about the direction and magnitude of this difference.

Due to the concept of time dilation, the current human time travel record is held by Russian cosmonaut Sergei Krikalev . Owing to the high-speed (7.66 km/s) of ISS and the length of time spent in space, it is believed that the cosmonaut actually arrived 0.02 seconds in the future while returning to the earth.

Cosmic String

Diagram Depicting Cosmic Strings

In 1991 J Richard Gott gave the idea of Cosmic Strings , which are believed to be left over from the early cosmos. These are defined as string-like objects or narrow tubes of energy that are stretched across the entire length of the universe. Owing to the huge amount of mass and massive gravitational pull, it would allow objects attached to the Cosmic Strings to travel at the speed of light.

So if two strings are pulled close to each other or one of them is stretched near the black hole, it might warp space-time to such an extent that would lead to creating a closed time-like curve and hence leading to the possibility of time travel. Theoretically, the gravity generated by these two Cosmic strings would help in propelling a spaceship into the past.

However, coming to the reality, the loop of strings is required to contain half the mass-energy of an entire galaxy so as to travel one year back in time. This implies that powering a time machine would require splitting half the atoms present in the whole galaxy.

Black holes

Illustration of Kerr Hole

When stars (having a mass of more than four times our sun) reach their end of life and all their fuel is burned up, they collapse under the pressure of their own weight creating “Black Holes” . The boundary of a Black Hole, called Event Horizon , has such a strong gravitational pull that it doesn’t even allow light to pass through it. Since light travels at the fastest speed, everything else traveling through a black hole is also dragged back. Such a non-rotating black hole is named as Schwarzschild black hole .

However, traveling to a parallel universe is possible through a rotating black hole named Kerr Hole . It was proposed in 1963 by a mathematician named Roy Kerr . As per his theory, if dying stars collapse into a rotating ring of neutron stars, that would produce enough centrifugal force to prevent the formation of singularity.

Note: Singularity can be perceived as the point into which the black hole tapers much like an ice-cream cone. At this point, the laws of Physics cease to exist and all the matter is crushed beyond recognition.

Since there will be no singularity, it would be safe to pass through a black hole without being crushed and exit out of a “White Hole” . A white hole is believed to be the exhaust end of a black hole which pushes everything away from it. Hence we may travel into another time or even another universe.

Although Kerr Holes are just theoretical, if they exist then we may find our way to a one-way trip to the past or future. However, physicist Kip Thorne believes that such a black hole doesn’t exist and it would suck everything before someone even reaches the Singularity.

Diagrammatic Representation of Wormhole 

Wormholes, also known as Einstein-Rosen Bridges , are believed to be the most potential means for time travel. It could allow us to travel several light years from earth and in much less time as compared to the conventional space travel methods. The possibility of wormholes is based on Einstein’s theory of relativity which says that any mass curves space-time. The following example is used to explain this curvature.

If two persons are holding a bed sheet stretching it tight and a baseball is placed on the sheet, its weight will make it roll to the middle of the sheet creating a curve at that point. Now if a marble is placed on the sheet, it would travel towards the baseball because of the curve. Here space is depicted as a two-dimensional plane than the four dimensions that actually makes up space-time.

Now if this sheet is folded over leaving a space at the top and bottom, placing the baseball on the top would form a curvature. If an equal mass is placed at the bottom part at a point corresponding to the location of the baseball, the second mass would eventually meet with the baseball. Similarly, wormholes might develop.

In space, masses that place pressure on different parts of the universe combine together to form a tunnel. Theoretically, this tunnel joins two separate times and allows passage between them. However, it’s possible that certain unforeseen physical properties may prevent the occurrence of wormholes and even if they exist, these might be really unstable.

Possibly someday human may learn to capture, stabilize and enlarge these tunnels but according to Dr. Hawking, prolonging the life of a tunnel through folded space-time may lead to a radiation feedback loop destroying the time tunnel.

Time Travel Paradoxes

If we ever work out a theory for time travel, we would give way to certain complexities known as paradoxes. A paradox is something that contradicts itself. In other words, time travel is not believed to be a practical concept because of certain situations that are likely to arise as the after-effects. These are broadly classified as -:

1. Closed Casual Loops: The cause and effect run in a circle causing a loop and is also internally consistent with the timeline’s history.

Diagram depicting time loop

• Predestination Paradox

It is defined as a situation when a traveler going back in time causes the event which he is trying to prevent from happening. It implies that any attempt to stop any event from occurring in the past would simply lead to the cause itself. The paradox suggests that things are destined to turn out the way they have happened and anyone attempting to change the past would find himself trapped in the repeating loop of time. For example, if you travel in the past to prevent your lover from dying in a road accident, you will find out that you were the one who accidentally ran over her.

• Bootstrap Paradox

A bootstrap paradox, also known as an Ontological Paradox where an object, person, or piece of information sent back in time leads to an infinite loop where the object has no discernible origin and is believed to exist without ever being created. It implies that the past, present and future and not defined, thus making it complicated to pinpoint the origin of anything. It raises questions like how were the objects created and by whom.

2. Consistency Paradox: It generates a number of timeline inconsistencies related to the possibility of altering the past. It can be further divided into the following categories.

• The Grandfather Paradox

Grandfather Paradox

This paradox talks about a hypothetical situation where a person travels back in time and kills his paternal grandfather at the time when his grandfather didn’t even meet his grandmother. In such a situation, his father would never have been born and neither would the traveler himself. So if he was never born, how would he travel to the past to kill his grandfather?

The paradox also talks about auto-infanticide where a time traveler goes into the past to kill himself when he was an infant. Now if he killed himself when he was a kid, how would he exist in the future to come back in time? Some physicists say that you would be able to go back in time but you won’t be able to change it, while others suggest that you would be born in one universe but unborn in another universe.

• The Hitler Paradox

Similar to the grandfather paradox, the killing Hitler paradox erases the reason for which you would want to go into the past and kill Hitler. Moreover, killing grandfather might have a “butterfly effect” but killing Hitler would have a far-reaching impact on the History as it would change the whole course of events. If you were successful in killing Hitler, there’d be no reason that would make you want to go back in time and kill him.

This paradox has been explained very well in a Twilight Zone episode called “Cradle of Darkness” as well as an episode “Let’s Kill Her” from Dr. Who.

• Polchinski’s Paradox

American physicist Joseph Polchinski proposed a paradox where a billiard ball enters a wormhole and emerges out of the other end in the past just in time to collide with its younger version and prevents it from entering the wormhole in the first place. While proposing this scenario, Joseph had Novikov’s Self Consistency Principle in his mind which states that time travel is possible but time paradoxes are forbidden.

A number of solutions have been suggested to avoid these inconsistencies like the billiard ball will deliver a blow which changes the course of the younger version of the ball but it would not stop it from entering the wormhole. This also explains that if you go back in time to kill your grandfather then something or the other will happen to prevent you from making it happen thus preserving the consistency of the History.

Solutions for the Paradoxes

In order to come up with a solution for these above-mentioned paradoxes, scientists have proposed some explanations which are enlisted below

The Solution: Time Travel is impossible because of the paradoxes that it creates.

Self-Healing Hypothesis: If we succeed to change the events in the past, it will set off another set of events that will keep the present unchanged.

The Multiverse: Every time an event in the past is altered, an alternate parallel universe or timeline is created.

Erased Timeline Hypothesis: A person traveling to the past would exist in the new timeline but their own timeline would be erased.

Is Time Travel Possible?

Nobody seems to have a definite answer in support or against the existence of Time Travel. On one hand, Einstein suggested to traveling at the speed of light in order to jaunt through the future but this would mean an unimaginable amount of energy would be required. Moreover, the centrifugal force on the body would prove to be fatal. Although it has been observed that space travelers age a little slower as compared to their identical twin on earth but some believe that there is no definite answer to travel back in space.

Theoretical physicist Brian Greene of Columbia University says that “No one has given a definite proof that you can’t travel to the past. But every time we look at the proposals and detail it seems kind of clear that they’re right at the edge of the known laws of physics.” Besides, Prof. Hawking feels that “Today’s science fiction is tomorrow’s science fact.”

However, the paradoxes, especially the grandfather paradox, have imposed a big question mark on the possibility of Time Travel. Basically, with the present laws and knowledge of Physics, the human won’t be able to survive in the process of Time Travel. So, we need certain developments in the quantum theories till we are sure as to how the paradoxes can be solved.

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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

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.