space time travel paradox

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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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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."

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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. Paradoxes Lost?

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

• Supplement: Remarks and Limitations on the Toy Models

Modern physics strips away many aspects of the manifest image of time. Time as it appears in the equations of classical mechanics has no need for a distinguished present moment, for example. Relativity theory leads to even sharper contrasts. It replaces absolute simultaneity, according to which it is possible to unambiguously determine the time order of distant events, with relative simultaneity: extending an “instant of time” throughout space is not unique, but depends on the state of motion of an observer. More dramatically, in general relativity the mathematical properties of time (or better, of spacetime)—its topology and geometry—depend upon how matter is arranged rather than being fixed once and for all. So physics can be, and indeed has to be, formulated without treating time as a universal, fixed background structure. Since general relativity represents gravity through spacetime geometry, the allowed geometries must be as varied as the ways in which matter can be arranged. Alongside geometrical models used to describe the solar system, black holes, and much else, the scope of variation extends to include some exotic structures unlike anything astrophysicists have observed. In particular, there are spacetime geometries with curves that loop back on themselves: closed timelike curves (CTCs), which describe the possible trajectory of an observer who returns exactly back to their earlier state—without any funny business, such as going faster than the speed of light. These geometries satisfy the relevant physical laws, the equations of general relativity, and in that sense time travel is physically possible.

Yet circular time generates paradoxes, familiar from science fiction stories featuring time travel: [ 1 ]

• Consistency: Kurt plans to murder his own grandfather Adolph, by traveling along a CTC to an appropriate moment in the past. He is an able marksman, and waits until he has a clear shot at grandpa. Normally he would not miss. Yet if he succeeds, there is no way that he will then exist to plan and carry out the mission. Kurt pulls the trigger: what can happen?
• Underdetermination: Suppose that Kurt first travels back in order to give his earlier self a copy of How to Build a Time Machine. This is the same book that allows him to build a time machine, which he then carries with him on his journey to the past. Who wrote the book?
• Easy Knowledge: A fan of classical music enhances their computer with a circuit that exploits a CTC. This machine efficiently solves problems at a higher level of computational complexity than conventional computers, leading (among other things) to finding the smallest circuits that can generate Bach’s oeuvre—and to compose new pieces in the same style. Such easy knowledge is at odds with our understanding of our epistemic predicament. (This third paradox has not drawn as much attention.)

The first two paradoxes were once routinely taken to show that solutions with CTCs should be rejected—with charges varying from violating logic, to being “physically unreasonable”, to undermining the notion of free will. Closer analysis of the paradoxes has largely reversed this consensus. Physicists have discovered many solutions with CTCs and have explored their properties in pursuing foundational questions, such as whether physics is compatible with the idea of objective temporal passage (starting with Gödel 1949). Philosophers have also used time travel scenarios to probe questions about, among other things, causation, modality, free will, and identity (see, e.g., Earman 1972 and Lewis’s seminal 1976 paper).

We begin below with Consistency , turning to the other paradoxes in later sections. A standard, stone-walling response is to insist that the past cannot be changed, as a matter of logic, even by a time traveler (e.g., Gödel 1949, Clarke 1977, Horwich 1987). Adolph cannot both die and survive, as a matter of logic, so any scheme to alter the past must fail. In many of the best time travel fictions, the actions of a time traveler are constrained in novel and unexpected ways. Attempts to change the past fail, and they fail, often tragically, in just such a way that they set the stage for the time traveler’s self-defeating journey. The first question is whether there is an analog of the consistent story when it comes to physics in the presence of CTCs. As we will see, there is a remarkable general argument establishing the existence of consistent solutions. Yet a second question persists: why can’t time-traveling Kurt kill his own grandfather? Doesn’t the necessity of failures to change the past put unusual and unexpected constraints on time travelers, or objects that move along CTCs? The same argument shows that there are in fact no constraints imposed by the existence of CTCs, in some cases. After discussing this line of argument, we will turn to the palatability and further implications of such constraints if they are required, and then turn to the implications of quantum mechanics.

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 chronological 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. In other words, 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 & Young 1961: 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. There is also an intermediate state of the dial, after it interacts with its older self and before it is put into the time machine. We can represent the initial or intermediate states of the dial, before it goes into the time machine, as an angle \(x\) in the horizontal plane and the final state of the dial, after it comes out of the time machine, as an angle \(y\) in the vertical plane. All possible \(\langle x,y\rangle\) pairs can thus be visualized as a torus with each \(x\) value picking out a vertical circular cross-section and each \(y\) picking out a point on that cross-section. See figure 1 .

Figure 1 [An extended description of figure 1 is in the supplement.]

Suppose that the dial starts at angle \(i\) which picks out vertical circle \(I\) on the torus. 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, circle \(I\) during this development will deform into some loop \(L\) on the torus. Loop \(L\) thus represents all possible intermediate angles \(x\) 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 \(\langle i,i\rangle\). Obviously any continuous deformation of \(I\) must still intersect \(C\) somewhere. So \(L\) must intersect \(C\) somewhere, say at \(\langle j,j\rangle\). 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.

Figure 2 [An extended description of figure 2 is in the supplement.]

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 \(\langle x,y\rangle\) 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 \(\langle x,y\rangle\) 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 \(\langle x,y\rangle\) 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? The arguments above depend on a well-known fixed point theorem (due to Schauder) that guarantees the existence of a fixed point for compact, convex state spaces. We do not know what subsequent extensions of this result imply regarding fixed points for a wider variety of systems, or whether there are other general results along these lines. (See Kutach 2003 for more on this issue.)

A further interesting question is whether this line of argument is sufficient to resolve Consistency (see also Dowe 2007). When they apply, these results establish the existence of a solution, such as the shade of uniform gray in the first example. But physicists routinely demand more than merely the existence of a solution, namely that solutions to the equations are stable—such that “small” changes of the initial state lead to “small” changes of the resulting trajectory. (Clarifying the two senses of “small” in this statement requires further work, specifying the relevant topology.) Stability in this sense underwrites the possibility of applying equations to real systems given our inability to fix initial states with indefinite precision. (See Fletcher 2020 for further discussion.) The fixed point theorems guarantee that for an initial state \(S_1\) there is a solution, but this solution may not be “close” to the solution for a nearby initial state, \(S'\). We are not aware of any proofs that the solutions guaranteed to exist by the fixed point theorems are also stable in this sense.

Time travel has recently been discussed quite extensively in the context of general relativity. General relativity places few constraints on the global structure of space and time. This flexibility leads to a possibility first described in print by Hermann Weyl:

Every world-point is the origin of the double-cone of the active future and the passive past [i.e., the two lobes of the light cone]. Whereas in the special theory of relativity these two portions are separated by an intervening region, it is certainly possible in the present case [i.e., general relativity] for the cone of the active future to overlap with that of the passive past; so that, in principle, it is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body), although it has a timelike direction at every point, to return to the neighborhood of a point which it has already once passed through. (Weyl 1918/1920 [1952: 274])

A time-like curve is simply a space-time trajectory such that the speed of light is never equaled or exceeded along this trajectory. Time-like curves represent possible trajectories of ordinary objects. In general relativity a curve that is everywhere timelike locally can nonetheless loop back on itself, forming a CTC. Weyl makes the point vividly in terms of the light cones: along such a curve, the future lobe of the light cone (the “active future”) intersects the past lobe of the light cone (the “passive past”). Traveling 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. CTCs can even arise when the spacetime is topologically \(\mathbb{R}^4\), due to the “tilting” of light cones produced by rotating matter (as in Gödel 1949’s spacetime).

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—there may be insurmountable practical obstacles. In Gödel’s spacetime, it is the case that there are CTCs passing through every point in the spacetime. Yet these CTCs are not geodesics, so traversing them requires acceleration. Calculations of the minimal fuel required to travel along the appropriate curve should discourage any would-be time travelers (Malament 1984, 1985; Manchak 2011). But more generally CTCs may be confined to smaller regions; 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 turn to 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 inextendable 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

• every point in the manifold can be connected by a timelike curve to \(S\), and
• 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 inextendable timelike curve which exists to the past of \(S\) intersects \(S\) exactly once, but (if there is time travel in the future) not every inextendable 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. [ 2 ] 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. [An extended description of figure 3 is in the supplement.]

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, analogous to the unwritten book: 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. [An extended description of figure 4 is in the supplement.]

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. [An extended description of figure 5 is in the supplement.]

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”. [An extended description of figure 6 is in the supplement.]

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. This is just like the case of the unwritten book: the book is never written, so nothing determines what fills its pages.

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 \(S\), or many possible books. 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. [An extended description of figure 7 is in the supplement.]

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. [ 3 ] 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 \(3W_{ad}\) and its value at node \(e\) will be \(3W_{ae} + 7W_{be}\). 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. [An extended description of figure 8 is in the supplement.]

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:

Solving these equations for \(z\) yields

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 \(vW_{vx}W_{xz} = -wW_{wy}W_{yz}\). 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.

This third point leads to a peculiar sort of indeterminism, illustrated by the case of the unwritten book: 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 2, 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. (See the supplement on

Remarks and Limitations on the Toy Models .

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 nor 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 more significant problem is underdetermination: the story can be consistently completed in many different ways.

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.

Figure 9 [An extended description of figure 9 is in the supplement.]

The threat of paradox in this case arises in the following form. Consider the initial trajectory of a ball as it approaches the time travel region. For some initial trajectories, the ball does not undergo a collision before reaching mouth 1, but upon exiting mouth 2 it will collide with its earlier self. This leads to a contradiction if the collision is strong enough to knock the ball off its trajectory and deflect it from entering 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? [ 4 ]

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, Morris, 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 & Morris 1991; Thorne 1994; Earman 1995; Earman, Smeenk, & Wüthrich 2009; and Dowe 2007.)

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 extendability of local solutions to global ones, and some simple models in which one does not have such extendability, 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, Morris, et al. 1990; Novikov 1992; Earman 1995; and Earman, Smeenk, & Wüthrich 2009). What are we to think of all of this?

The toy models above all treat billiard balls, fields, and other objects propagating through a background spacetime with CTCs. Even if we can show that a consistent solution exists, there is a further question: what kind of matter and dynamics could generate CTCs to begin with? There are various solutions of Einstein’s equations with CTCs, but how do these exotic spacetimes relate to the models actually used in describing the world? In other words, what positive reasons might we have to take CTCs seriously as a feature of the actual universe, rather than an exotic possibility of primarily mathematical interest?

We should distinguish two different kinds of “possibility” that we might have in mind in posing such questions (following Stein 1970). First, we can consider a solution as a candidate cosmological model, describing the (large-scale gravitational degrees of freedom of the) entire universe. The case for ruling out spacetimes with CTCs as potential cosmological models strikes us as, surprisingly, fairly weak. Physicists used to simply rule out solutions with CTCs as unreasonable by fiat, due to the threat of paradoxes, which we have dismantled above. But it is also challenging to make an observational case. Observations tell us very little about global features, such as the existence of CTCs, because signals can only reach an observer from a limited region of spacetime, called the past light cone. Our past light cone—and indeed the collection of all the past light cones for possible observers in a given spacetime—can be embedded in spacetimes with quite different global features (Malament 1977, Manchak 2009). This undercuts the possibility of using observations to constrain global topology, including (among other things) ruling out the existence of CTCs.

Yet the case in favor of taking cosmological models with CTCs seriously is also not particularly strong. Some solutions used to describe black holes, which are clearly relevant in a variety of astrophysical contexts, include CTCs. But the question of whether the CTCs themselves play an essential representational role is subtle: the CTCs arise in the maximal extensions of these solutions, and can plausibly be regarded as extraneous to successful applications. Furthermore, many of the known solutions with CTCs have symmetries, raising the possibility that CTCs are not a stable or robust feature. Slight departures from symmetry may lead to a solution without CTCs, suggesting that the CTCs may be an artifact of an idealized model.

The second sense of possibility regards whether “reasonable” initial conditions can be shown to lead to, or not to lead to, the formation of CTCs. As with the toy models above, suppose that we have a partial Cauchy surface \(S\), such that all the temporal funny business lies to the future. Rather than simply assuming that there is a region with CTCs to the future, we can ask instead whether it is possible to create CTCs by manipulating matter in the initial, well-behaved region—that is, whether it is possible to build a time machine. Several physicists have pursued “chronology protection theorems” aiming to show that the dynamics of general relativity (or some other aspects of physics) rules this out, and to clarify why this is the case. The proof of such a theorem would justify neglecting solutions with CTCs as a source of insight into the nature of time in the actual world. But as of yet there are several partial results that do not fully settle the question. One further intriguing possibility is that even if general relativity by itself does protect chronology, it may not be possible to formulate a sensible theory describing matter and fields in solutions with CTCs. (See SEP entry on Time Machines; Smeenk and Wüthrich 2011 for more.)

There is a different question regarding the limitations of these toy models. The toy models and related examples show that there are consistent solutions for simple systems in the presence of CTCs. As usual we have made the analysis tractable by building toy models, selecting only a few dynamical degrees of freedom and tracking their evolution. But there is a large gap between the systems we have described and the time travel stories they evoke, with Kurt traveling along a CTC with murderous intentions. In particular, many features of the manifest image of time are tied to the thermodynamical properties of macroscopic systems. Rovelli (unpublished) considers a extremely simple system to illustrate the problem: can a clock move along a CTC? A clock consists of something in periodic motion, such as a pendulum bob, and something that counts the oscillations, such as an escapement mechanism. The escapement mechanism cannot work without friction; this requires dissipation and increasing entropy. For a clock that counts oscillations as it moves along a time-like trajectory, the entropy must be a monotonically increasing function. But that is obviously incompatible with the clock returning to precisely the same state at some future time as it completes a loop. The point generalizes, obviously, to imply that anything like a human, with memory and agency, cannot move along a CTC.

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 \(\boldsymbol{E}\) on an initial surface to be related to the (simultaneous) charge density distribution \(\varrho\) by the equation \(\varrho = \text{div}(\boldsymbol{E})\). (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 \(\varrho = \text{div}(\boldsymbol{E})\) 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 \(\varrho = \text{div}(\boldsymbol{E})\). The constraints imposed by \(\varrho = \text{div}(\boldsymbol{E})\) on the state on a space-like hypersurface are:

• 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),
• quite independent of the global space-time structure,
• quite independent of how the space-like surface in question is embedded in a given space-time, and
• 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 & Horowitz 1979: 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, & Wüthrich 2009). 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 in section 1. 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. (Recall that the changes only effect test particles.) 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? [ 5 ]

An alternative approach to time travel (initiated by Deutsch 1991) abstracts away from the idealized toy models described above. [ 6 ] This computational approach considers instead the evolution of bits (simple physical systems with two discrete states) through a network of interactions, which can be represented by a circuit diagram with gates corresponding to the interactions. Motivated by the possibility of CTCs, Deutsch proposed adding a new kind of channel that connects the output of a given gate back to its input —in essence, a backwards-time step. More concretely, given a gate that takes \(n\) bits as input, we can imagine taking some number \(i \lt n\) of these bits through a channel that loops back and then do double-duty as inputs. Consistency requires that the state of these \(i\) bits is the same for output and input. (We will consider an illustration of this kind of system in the next section.) Working through examples of circuit diagrams with a CTC channel leads to similar treatments of Consistency and Underdetermination as the discussion above (see, e.g., Wallace 2012: § 10.6). But the approach offers two new insights (both originally due to Deutsch): the Easy Knowledge paradox, and a particularly clear extension to time travel in quantum mechanics.

A computer equipped with a CTC channel can exploit the need to find consistent evolution to solve remarkably hard problems. (This is quite different than the first idea that comes to mind to enhance computational power: namely to just devote more time to a computation, and then send the result back on the CTC to an earlier state.) The gate in a circuit incorporating a CTC implements a function from the input bits to the output bits, under the constraint that the output and input match the i bits going through the CTC channel. This requires, in effect, finding the fixed point of the relevant function. Given the generality of the model, there are few limits on the functions that could be implemented on the CTC circuit. Nature has to solve a hard computational problem just to ensure consistent evolution. This can then be extended to other complex computational problems—leading, more precisely, to solutions of NP -complete problems in polynomial time (see Aaronson 2013: Chapter 20 for an overview and further references). The limits imposed by computational complexity are an essential part of our epistemic situation, and computers with CTCs would radically change this.

We now turn to the application of the computational approach to the quantum physics of time travel (see Deutsch 1991; Deutsch & Lockwood 1994). By contrast with the earlier discussions of constraints in classical systems, they claim to show that time travel never imposes any constraints on the pre-time travel state of quantum systems. The essence of this account is as follows. [ 7 ]

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 10 ).

Figure 10 [An extended description of figure 10 is in the supplement.]

Deutsch assumes that the set of possible states of this system are the mixed states, i.e., are represented by the density matrices over the Hilbert space of that system. Deutsch then shows that for any initial state \(S_1\), any unitary interaction between the older and younger self, and any unitary development during time travel, there is a consistent solution, i.e., there is at least one pair of states \(S_2\) and \(S_3\) such that when \(S_1\) interacts with \(S_3\) it will change to state \(S_2\) and \(S_2\) will then develop into \(S_3\). The states \(S_2, S_3\) and \(S_4\) will typically be not be pure states, i.e., will be non-trivial mixed states, even if \(S_1\) is pure. In order to understand how this leads to interpretational problems let us give an example. Consider a system that has a two dimensional Hilbert space with as a basis the states \(\vc{+}\) and \(\vc{-}\). Let us suppose that when state \(\vc{+}\) of the young system encounters state \(\vc{+}\) of the older system, they interact and the young system develops into state \(\vc{-}\) and the old system remains in state \(\vc{+}\). In obvious notation:

Similarly, suppose that:

Let us furthermore assume that there is no development of the state of the system during time travel, i.e., that \(\vc{+}_2\) develops into \(\vc{+}_3\), and that \(\vc{-}_2\) develops into \(\vc{-}_3\).

Now, if the only possible states of the system were \(\vc{+}\) and \(\vc{-}\) (i.e., if there were no superpositions or mixtures of these states), then there is a constraint on initial states: initial state \(\vc{+}_1\) is impossible. For if \(\vc{+}_1\) interacts with \(\vc{+}_3\) then it will develop into \(\vc{-}_2\), which, during time travel, will develop into \(\vc{-}_3\), which inconsistent with the assumed state \(\vc{+}_3\). Similarly if \(\vc{+}_1\) interacts with \(\vc{-}_3\) it will develop into \(\vc{+}_2\), which will then develop into \(\vc{+}_3\) which is also inconsistent. Thus the system can not start in state \(\vc{+}_1\).

But, says Deutsch, in quantum mechanics such a system can also be in any mixture of the states \(\vc{+}\) and \(\vc{-}\). Suppose that the older system, prior to the interaction, is in a state \(S_3\) which is an equal mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\). Then the younger system during the interaction will develop into a mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), which will then develop into a mixture of 50% \(\vc{+}_3\) and 50% \(\vc{-}_3\), which is consistent! More generally Deutsch uses a fixed point theorem to show that no matter what the unitary development during interaction is, and no matter what the unitary development during time travel is, for any state \(S_1\) there is always a state \(S_3\) (which typically is not a pure state) which causes \(S_1\) to develop into a state \(S_2\) which develops into that state \(S_3\). Thus quantum mechanics comes to the rescue: it shows in all generality that no constraints on initial states are needed!

One might wonder why Deutsch appeals to mixed states: will superpositions of states \(\vc{+}\) and \(\vc{-}\) not suffice? Unfortunately such an idea does not work. Suppose again that the initial state is \(\vc{+}_1\). One might suggest that that if state \(S_3\) is

one will obtain a consistent development. For one might think that when initial state \(\vc{+}_1\) encounters the superposition

it will develop into superposition

and that this in turn will develop into

as desired. However this is not correct. For initial state \(\vc{+}_1\) when it encounters

will develop into the entangled state

In so far as one can speak of the state of the young system after this interaction, it is in the mixture of 50% \(\vc{+}_2\) and 50% \(\vc{-}_2\), not in the superposition

So Deutsch does need his recourse to mixed states.

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: we apparently need to characterize not just the entanglement between two systems, but entanglement relative to specific spacetime descriptions.

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?

If we take an ignorance interpretation of mixtures we run into trouble. For suppose that we assume that in each individual case each older system is either in state \(\vc{+}_3\) or in state \(\vc{-}_3\) prior to the interaction. Then we regain our paradox. Deutsch instead recommends the following, many worlds, picture of mixtures. Suppose we start with state \(\vc{+}_1\) in all worlds. In some of the many worlds the older system will be in the \(\vc{+}_3\) state, let us call them A -worlds, and in some worlds, B -worlds, it will be in the \(\vc{-}_3\) state. Thus in A -worlds after interaction we will have state \(\vc{-}_2\) , and in B -worlds we will have state \(\vc{+}_2\). During time travel the \(\vc{-}_2\) state will remain the same, i.e., turn into state \(\vc{-}_3\), but the systems in question will travel from A -worlds to B -worlds. Similarly the \(\vc{+}\) \(_2\) states will travel from the B -worlds to the A -worlds, thus preserving consistency.

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. (See Wallace 2012 for a more sympathetic treatment, that explores several further implications of accepting time travel in conjunction with the many worlds interpretation.)

We close by acknowledging that Deutsch’s starting point—the claim that this computational model captures the essential features of quantum systems in a spacetime with CTCs—has been the subject of some debate. Several physicists have pursued a quite different treatment of evolution of quantum systems through CTC’s, based on considering the “post-selected” state (see Lloyd et al. 2011). Their motivations for implementing the consistency condition in terms of the post-selected state reflects a different stance towards quantum foundations. A different line of argument aims to determine whether Deutsch’s treatment holds as an appropriate limiting case of a more rigorous treatment, such as quantum field theory in curved spacetimes. For example, Verch (2020) establishes several results challenging the assumption that Deutsch’s treatment is tied to the presence of CTC’s, or that it is compatible with the entanglement structure of quantum fields.

What remains of the grandfather 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 topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Adlam, Emily, unpublished, “ Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling ”, 2022, arXiv: 2208.02721. doi:10.48550/ARXIV.2208.02721
• Rovelli, Carlo, unpublished, “ Can We Travel to the Past? Irreversible Physics along Closed Timelike Curves ”, arXiv: 1912.04702. doi:10.48550/ARXIV.1912.04702

causation: backward | determinism: causal | quantum mechanics | quantum mechanics: retrocausality | space and time: being and becoming in modern physics | time machines | time travel

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Is time travel possible? An astrophysicist explains

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.

Related: The speed of light, explained

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.

Time travel 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.

This article first appeared on the Conversation. You can read the original here .

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.

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

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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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• Stephen Hawking
• Curious Kids
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Is time travel possible? Why one scientist says we 'cannot ignore the possibility.'

A common theme in science-fiction media , time travel is captivating. It’s defined by the late philosopher David Lewis in his essay “The Paradoxes of Time Travel” as “[involving] a discrepancy between time and space time. Any traveler departs and then arrives at his destination; the time elapsed from departure to arrival … is the duration of the journey.”

Time travel is usually understood by most as going back to a bygone era or jumping forward to a point far in the future . But how much of the idea is based in reality? Is it possible to travel through time? 

Is time travel possible?

According to NASA, time travel is possible , just not in the way you might expect. Albert Einstein’s theory of relativity says time and motion are relative to each other, and nothing can go faster than the speed of light , which is 186,000 miles per second. Time travel happens through what’s called “time dilation.”

Time dilation , according to Live Science, is how one’s perception of time is different to another's, depending on their motion or where they are. Hence, time being relative. 

Learn more: Best travel insurance

Dr. Ana Alonso-Serrano, a postdoctoral researcher at the Max Planck Institute for Gravitational Physics in Germany, explained the possibility of time travel and how researchers test theories. 

Space and time are not absolute values, Alonso-Serrano said. And what makes this all more complex is that you are able to carve space-time .

“In the moment that you carve the space-time, you can play with that curvature to make the time come in a circle and make a time machine,” Alonso-Serrano told USA TODAY. 

She explained how, theoretically, time travel is possible. The mathematics behind creating curvature of space-time are solid, but trying to re-create the strict physical conditions needed to prove these theories can be challenging. 

“The tricky point of that is if you can find a physical, realistic, way to do it,” she said. 

Alonso-Serrano said wormholes and warp drives are tools that are used to create this curvature. The matter needed to achieve curving space-time via a wormhole is exotic matter , which hasn’t been done successfully. Researchers don’t even know if this type of matter exists, she said.

“It's something that we work on because it's theoretically possible, and because it's a very nice way to test our theory, to look for possible paradoxes,” Alonso-Serrano added.

“I could not say that nothing is possible, but I cannot ignore the possibility,” she said. 

She also mentioned the anecdote of  Stephen Hawking’s Champagne party for time travelers . Hawking had a GPS-specific location for the party. He didn’t send out invites until the party had already happened, so only people who could travel to the past would be able to attend. No one showed up, and Hawking referred to this event as "experimental evidence" that time travel wasn't possible.

What did Albert Einstein invent?: Discoveries that changed the world

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USA TODAY is exploring the questions you and others ask every day. From "How to watch the Marvel movies in order" to "Why is Pluto not a planet?" to "What to do if your dog eats weed?" – we're striving to find answers to the most common questions you ask every day. Head to our Just Curious section to see what else we can answer for you. 

The Time-Travel Paradoxes

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

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

Wait! Don’t do it!

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

 The Grandfather Paradox

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

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

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

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

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

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

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

 The Time Loop

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

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

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

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

Time travelers – where have all they gone?

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

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

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

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

 One way or round trip?

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

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

 Advanced time travel

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

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

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

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

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

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

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

 How to travel in time and stay alive

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

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

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

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

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

Time Travelling Grammar

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

It’s complicated.

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

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

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

Time travel: 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. 

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Ailsa is a staff writer for How It Works magazine, where she writes science, technology, space, history and environment features. Based in the U.K., she graduated from the University of Stirling with a BA (Hons) journalism degree. Previously, Ailsa has written for Cardiff Times magazine, Psychology Now and numerous science bookazines. 

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Paradoxes of Time Travel

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

Reviewed by John W. Carroll, North Carolina State

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sure I can; here it is.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Infamous ‘Grandfather Paradox’ Doesn’t Make Time Travel Impossible After All

It just means you can’t go back in time and kill your grandfather.

• Closed timelike curves, or paths through spacetime that lead to the past, allow time travel.
• An MIT experiment suggests any jaunt that would lead to a paradox in time travel is canceled preemptively.

It’s a classic science fiction trope: a time traveler journeys back in time and causes a change in history that has disastrous effects on the present or even threatens their very existence.

If these changes jeopardize their ability to travel back through time in the first place, then surely the traveler can’t make that change to time, right? But then they can go back in time again, so, can make those changes again … and so forth.

🌀 Physics can be trippy. We’ll be your guide—join Pop Mech Pro.

That’s the essence of a trap called the “grandfather paradox,” an idea that has been used to great effect in books, films, and TV shows—from Ray Bradbury’s short story A Sound of Thunder to Futurama to Back to the Future . And as much fun as this concept is in science fiction, it’s also something that actual physicists and philosophers are intensely thinking about.

“The argument runs like this, if you could ‘go back in time’ then you could go back to a time before your grandfather had had any children and murder him,” Tim Maudlin, a philosopher of science who investigates the metaphysical foundations of physics and logic, explains to Popular Mechanics . “But if that happened, then one of your parents would not have been born, so you would not have been born, so there would be no you to go back in time. Contradiction.”

This problem arises from the risk time travel would present to one of the most preserved ideas in physics — causality , the idea that cause must proceed effect in all circumstances.

“The grandfather paradox is usually presented as a reductio ad absurdum, or a refutation of the proposition that time travel is possible,” Maudlin says. “So the hypothesis must be impossible because of the grandfather paradox; time travel  — or reverse causation — is not possible.”

Though he doesn’t ultimately think travel backward through time is possible, Maudlin thinks that the grandfather paradox shouldn’t prevent time travel in and of itself. Instead, the paradox just prevents what actions can be conducted on a trip through time.

“The grandfather paradox does not prove that you can’t go back in time, just that you can’t go back in time and kill your grandfather,” he says. “There would be nothing logically wrong with going back in time and, say, saying ‘Hello’ to your grandfather.”

Researchers from the Massachusetts Institute of Technology (MIT) have an idea of just how causality violation could be prevented.

Time Travel That Protects Granddad

Seth Lloyd, a professor of mechanical engineering at the Massachusetts Institute of Technology and a self-described “quantum mechanic,” has been conducting research for over a decade that suggests a way of going back in time and avoiding the grandfather paradox altogether.

This involves the physics of closed timelike curves (CTCs), paths through time and space that return to their starting point, which are allowed by general relativity — Albert Einstein’s theory of gravity and the effect mass has on space and time, or the single entity of spacetime.

.css-2l0eat{font-family:UnitedSans,UnitedSans-roboto,UnitedSans-local,Helvetica,Arial,Sans-serif;font-size:1.625rem;line-height:1.2;margin:0rem;padding:0.9rem 1rem 1rem;}@media(max-width: 48rem){.css-2l0eat{font-size:1.75rem;line-height:1;}}@media(min-width: 48rem){.css-2l0eat{font-size:1.875rem;line-height:1;}}@media(min-width: 64rem){.css-2l0eat{font-size:2.25rem;line-height:1;}}.css-2l0eat b,.css-2l0eat strong{font-family:inherit;font-weight:bold;}.css-2l0eat em,.css-2l0eat i{font-style:italic;font-family:inherit;} “If you follow a closed timelike curve in your spaceship, you can end up interacting with your former self.”

“A closed timelike curve is a path through spacetime that leads to the past,” Loyd tells Popular Mechanics . “If you follow a closed timelike curve in your spaceship, you can end up interacting with your former self. That is, closed timelike curves allow time travel.”

There are a few different types of CTC models, which Lloyd illustrates with examples from popular fiction.

“There are basically two different possible types of models for CTCs. In one — which we call, imaginatively, Type I — the time traveler can intervene to change the past as she remembers it, at which point she enters into a different quantum branch of the universe  — as in Back to the Future , Hot Tub Time Machine , and other time-travel narratives,” he explains. “In such Type I theories of time travel, it’s perfectly possible for the time traveler to kill her grandfather.”

In the other type of CTC model , which is predictably called Type II, time travel has to obey a principle of self-consistency. Sometimes called the Novikov self-consistency principle, or Niven’s Law of the conservation of history, this principle prevents causality violation by placing some events in order on the same CTC. This self-consistency would prevent our time-traveler from landing her machine on granddad, even if she wanted to. Some effect would always divert her course.

“In Type II theories, the time traveler cannot change the past, no matter how hard she tries,” Lloyd says. “Examples of Type II time travel narratives include Harry Potter and the Prisoner of Azkaban , and the Terry Gilliam film, Twelve Monkeys .”

Terminator Photons: Back in Time With a Mission to Kill

Lloyd and his team set about exploring a version of Type II CTCs that combine the concepts of quantum teleportation with post-selection — the factor in a computation that allows certain results to be accepted while others are rejected.

“Quantum teleportation is a process in which a quantum system dematerializes here and then rematerializes somewhere else based on the counter-intuitive quantum phenomenon of entanglement [the idea that two or more particles can be linked in such a way that a change in one instantaneously causes a change in the other no matter how distant they are],” Lloyd says. “In the quantum theory of CTCs that we developed, travel through the closed timelike curve is closely related to teleportation .”

The quantum mechanic added that adding post-selection to quantum measurement makes the process deterministic rather than probabilistic and it effectively bans events that would prove to be paradoxical.

Lloyd set about testing this idea by developing an “in principle” time machine — a quantum simulation that effectively sends a photon a few billionths of a second backward in time to have it attempt to “kill” its previous self.

The results showed that the closer a photon got to doing something self-inconsistent, the more frequently the experiment failed. Lloyd’s results could hint that time travel might work in the same way — any jaunt that would lead to a paradox is canceled preemptively.

Could Quantum Physics Provide an Exit to the Grandfather Paradox?

Quantum physics might provide another out to the Grandfather Paradox. One particular interpretation of quantum mechanics — Hugh Everett’s Many World Interpretations — suggests that for every quantum possibility that exists, a separate and distinct world emerges.

Physicist David Deutsch , a pioneer in quantum computing , imagined the Many Worlds idea in the case of time travel. He envisioned a particle traveling along a CTC loop through time in a quantum superposition  — a phenomenon that exists in quantum physics that allows a system to exist in multiple, potentially contradictory, states at once.

To avoid paradoxes at the end of the journey and ensure a particle arrives back at its starting point the same as it was when it left, a world is created for each possible state. Let’s see how that would work for a human traveler in time if such a thing was possible.

Imagine a hypothetical time traveler ,  who we’ll call Susan , takes a CTC-based journey back through time to meet her grandfather as a child in 1963. Being hyper-literal and overprecise, she lands this time machine exactly where granddad was standing in Totter’s Lane scrapyard, London, squishing him dead. Susan waits to disappear from existence, but the Many Worlds interpretation of quantum physics may protect her.

This is because when Susan arrived in 1963, she created a world that is distinct from the world she left. In the world she left , let’s call it Earth 1 , her grandfather wasn’t squashed. He went on to have a granddaughter called Susan who once disappeared in a time machine . So, the child Susan landed on in the past isn’t her grandfather at all, just a version of him from an alternative world.

Traveling back to the future, Susan would find it different from the world she left—not because it’s been altered by her actions, but rather because this world, Earth 2, was created by her — it’s not the same world.

The Many Worlds Interpretation has a consequence for our time traveler; Everett insisted that one of the rules of his theorems was that worlds couldn’t interfere with each other. This means that our time traveler can’t get back to Earth 1.

If Susan attempts to travel back in time to 1963 to prevent the death of her grandfather, she creates a third world — Earth 3 — in which two time-travelers appeared in Totter’s Lane scrapyard in 1963. She travels forward again realizing she now can’t get back to Earth 1 or Earth 2.

Somewhere on Earth 1 and in that timeline, Susan’s wistful grandfather awaits her return, which will never come about.

Of course, the Grandfather Paradox isn’t the only argument against time travel. One very sensible question is: if time travel is possible, when are all the time travelers?

“For what it’s worth since we put forward the theory and performed the proof of principle experiment, many people have written to me claiming to be time travelers who are stuck in time and asking me if they can use our time machine to get back to their own time,” Lloyd says. “I advise them to wait until the bugs have been worked out.”

Robert Lea is a freelance science journalist focusing on space, astronomy, and physics. Rob’s articles have been published in Newsweek , Space , Live Science , Astronomy magazine and New Scientist . He lives in the North West of England with too many cats and comic books.  

.css-cuqpxl:before{padding-right:0.3125rem;content:'//';display:inline;} Physics .css-xtujxj:before{padding-left:0.3125rem;content:'//';display:inline;}

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Time Travel: Theories, Possibilities, and Paradoxes Explained

By Neha Rastogi

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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Mathematics

The mathematician who worked out how to time travel.

Mathematics suggested that time travel is physically possible – and Kurt Gödel proved it. Mathematician Karl Sigmund explains how the polymath did it

By Karl Sigmund

5 April 2024

Gödel proved that, mathematically speaking, time travel is physically possible

Quality Stock / Alamy

The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time for free  here .

There may be no better way to get truly lost in space-time than to travel to the past and fiddle around with causality. Polymath Kurt Gödel suggested that you could, for instance, land…

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