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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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Can we time travel? A theoretical physicist provides some answers

Emeritus professor, Physics, Carleton University

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Time travel makes regular appearances in popular culture, with innumerable time travel storylines in movies, television and literature. But it is a surprisingly old idea: one can argue that the Greek tragedy Oedipus Rex , written by Sophocles over 2,500 years ago, is the first time travel story .

But is time travel in fact possible? Given the popularity of the concept, this is a legitimate question. As a theoretical physicist, I find that there are several possible answers to this question, not all of which are contradictory.

The simplest answer is that time travel cannot be possible because if it was, we would already be doing it. One can argue that it is forbidden by the laws of physics, like the second law of thermodynamics or relativity . There are also technical challenges: it might be possible but would involve vast amounts of energy.

There is also the matter of time-travel paradoxes; we can — hypothetically — resolve these if free will is an illusion, if many worlds exist or if the past can only be witnessed but not experienced. Perhaps time travel is impossible simply because time must flow in a linear manner and we have no control over it, or perhaps time is an illusion and time travel is irrelevant.

Laws of physics

Since Albert Einstein’s theory of relativity — which describes the nature of time, space and gravity — is our most profound theory of time, we would like to think that time travel is forbidden by relativity. Unfortunately, one of his colleagues from the Institute for Advanced Study, Kurt Gödel, invented a universe in which time travel was not just possible, but the past and future were inextricably tangled.

We can actually design time machines , but most of these (in principle) successful proposals require negative energy , or negative mass, which does not seem to exist in our universe. If you drop a tennis ball of negative mass, it will fall upwards. This argument is rather unsatisfactory, since it explains why we cannot time travel in practice only by involving another idea — that of negative energy or mass — that we do not really understand.

Mathematical physicist Frank Tipler conceptualized a time machine that does not involve negative mass, but requires more energy than exists in the universe .

Time travel also violates the second law of thermodynamics , which states that entropy or randomness must always increase. Time can only move in one direction — in other words, you cannot unscramble an egg. More specifically, by travelling into the past we are going from now (a high entropy state) into the past, which must have lower entropy.

This argument originated with the English cosmologist Arthur Eddington , and is at best incomplete. Perhaps it stops you travelling into the past, but it says nothing about time travel into the future. In practice, it is just as hard for me to travel to next Thursday as it is to travel to last Thursday.

Resolving paradoxes

There is no doubt that if we could time travel freely, we run into the paradoxes. The best known is the “ grandfather paradox ”: one could hypothetically use a time machine to travel to the past and murder their grandfather before their father’s conception, thereby eliminating the possibility of their own birth. Logically, you cannot both exist and not exist.

Read more: Time travel could be possible, but only with parallel timelines

Kurt Vonnegut’s anti-war novel Slaughterhouse-Five , published in 1969, describes how to evade the grandfather paradox. If free will simply does not exist, it is not possible to kill one’s grandfather in the past, since he was not killed in the past. The novel’s protagonist, Billy Pilgrim, can only travel to other points on his world line (the timeline he exists in), but not to any other point in space-time, so he could not even contemplate killing his grandfather.

The universe in Slaughterhouse-Five is consistent with everything we know. The second law of thermodynamics works perfectly well within it and there is no conflict with relativity. But it is inconsistent with some things we believe in, like free will — you can observe the past, like watching a movie, but you cannot interfere with the actions of people in it.

Could we allow for actual modifications of the past, so that we could go back and murder our grandfather — or Hitler ? There are several multiverse theories that suppose that there are many timelines for different universes. This is also an old idea: in Charles Dickens’ A Christmas Carol , Ebeneezer Scrooge experiences two alternative timelines, one of which leads to a shameful death and the other to happiness.

Time is a river

Roman emperor Marcus Aurelius wrote that:

“ Time is like a river made up of the events which happen , and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.”

We can imagine that time does flow past every point in the universe, like a river around a rock. But it is difficult to make the idea precise. A flow is a rate of change — the flow of a river is the amount of water that passes a specific length in a given time. Hence if time is a flow, it is at the rate of one second per second, which is not a very useful insight.

Theoretical physicist Stephen Hawking suggested that a “ chronology protection conjecture ” must exist, an as-yet-unknown physical principle that forbids time travel. Hawking’s concept originates from the idea that we cannot know what goes on inside a black hole, because we cannot get information out of it. But this argument is redundant: we cannot time travel because we cannot time travel!

Researchers are investigating a more fundamental theory, where time and space “emerge” from something else. This is referred to as quantum gravity , but unfortunately it does not exist yet.

So is time travel possible? Probably not, but we don’t know for sure!

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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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How Time Travel Works

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Time Travel Paradoxes

As we mentioned before, the concept of traveling into the past becomes a bit murky the second causality rears its head. Cause comes before effect, at least in this universe, which manages to muck up even the best-laid time traveling plans.

For starters, if you traveled back in time 200 years, you'd emerge in a time before you were born. Think about that for a second. In the flow of time, the effect (you) would exist before the cause (your birth).

To better understand what we're dealing with here, consider the famous grandfather paradox . You're a time-traveling assassin, and your target just happens to be your own grandfather. So you pop through the nearest wormhole and walk up to a spry 18-year-old version of your father's father. You raise your laser blaster , but just what happens when you pull the trigger?

Think about it. You haven't been born yet. Neither has your father. If you kill your own grandfather in the past, he'll never have a son. That son will never have you, and you'll never happen to take that job as a time-traveling assassin. You wouldn't exist to pull the trigger, thus negating the entire string of events. We call this an inconsistent causal loop .

On the other hand, we have to consider the idea of a consistent causal loop . While equally thought-provoking, this theoretical model of time travel is paradox free. According to physicist Paul Davies, such a loop might play out like this: A math professor travels into the future and steals a groundbreaking math theorem. The professor then gives the theorem to a promising student. Then, that promising student grows up to be the very person from whom the professor stole the theorem to begin with.

Then there's the post-selected model of time travel, which involves distorted probability close to any paradoxical situation [source: Sanders]. What does this mean? Well, put yourself in the shoes of the time-traveling assassin again. This time travel model would make your grandfather virtually death proof. You can pull the trigger, but the laser will malfunction. Perhaps a bird will poop at just the right moment, but some quantum fluctuation will occur to prevent a paradoxical situation from taking place.

But then there's another possibility: The quantum theory that the future or past you travel into might just be a parallel universe . Think of it as a separate sandbox: You can build or destroy all the castles you want in it, but it doesn't affect your home sandbox in the slightest. So if the past you travel into exists in a separate timeline, killing your grandfather in cold blood is no big whoop. Of course, this might mean that every time jaunt would land you in a new parallel universe and you might never return to your original sandbox.

Confused yet? Welcome to the world of time travel.

Explore the links below for even more mind-blowing cosmology.

Related Articles

• How Time Works
• How Special Relativity Works
• What is relativity?
• Is Time Travel Possible?
• How Black Holes Work
• How would time travel affect life as we know it?

More Great Links

• NOVA Online: Time Travel
• Into the Universe with Stephen Hawking
• Cleland, Andrew. Personal interview. April 2010.
• Davies, Paul. "How to Build a Time Machine." Penguin. March 25, 2003.
• Davies, Paul. Personal interview. April 2010.
• Franknoi, Andrew. "Light as a Cosmic Time Machine." PBS: Seeing in the Dark. March 2008. (March 1, 2011)http://www.pbs.org/seeinginthedark/astronomy-topics/light-as-a-cosmic-time-machine.html
• Hawking, Stephen. "How to build a time machine." Mail Online. May 3, 2010. (March 1, 2011)http://www.dailymail.co.uk/home/moslive/article-1269288/STEPHEN-HAWKING-How-build-time-machine.html
• "Into the Universe with Stephen Hawking." Discovery Channel.
• Kaku, Michio. "Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos." Anchor. Feb. 14, 2006.
• "Kerr Black Holes and time travel." NASA. Dec. 8, 2008. (March 1, 2011)http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/041130a.html
• Sanders, Laura. "Physicists Tame Time Travel by Forbidding You to Kill Your Grandfather." WIRED. July 20, 2010. (Mach 1, 2011)http://www.wired.com/wiredscience/2010/07/time-travel/

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2 Temporal Paradoxes

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

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What Is the Grandfather Paradox of Time Travel?

For nearly 100 years, the famous grandfather paradox of time travel has fascinated both sci-fi fans and scientists. has anyone ever come up with a solution for it.

If you went back in time and accidentally killed one of your ancestors, what would happen to you?

When it comes to hypotheticals about time travel, this is one of the most popular and compelling questions of all, well, time. And it isn’t just a plot device that science fiction writers and filmmakers use to drive a story. The so-called Grandfather Paradox has been weighed by physicists and philosophers alike for nearly a century.

What Is the Grandfather Paradox?

Broadly speaking, the Grandfather Paradox is a colloquial term for just about any causality paradox . Causality, as you may remember from school, is the relationship between cause and effect. While not quite a law, causality is certainly fundamental to our understanding of basic physics, which depends in part upon the idea that cause must come before effect.

The paradox part of the Grandfather Paradox occurs when a time traveler creates a self-contradicting scenario where effect precedes cause. In this example, if you kill your grandfather before he has kids, one of your parents is never born. Consequently, you are never born. But if you’re never born, then you wouldn’t exist to go back and kill your grandfather in the first place, would you? Paradox!

Of course, the paradox doesn’t apply just to instances of grand-patricide, but to all hypothetical situations where a time traveler goes back to prevent something from occurring. If they’re successful, and the event never happens, the time traveler has no reason to go back and change something. So, they might try again and ... paradox! Whatever the traveler changes doesn’t have to be intentional either; accidentally interfering with a past event can also trigger a paradox.

Who Invented the Grandfather Paradox?

While the concept of going back in time and changing the past has existed almost since the dawn of science fiction, credit for naming the Grandfather Paradox often goes to early sci-fi magazine fans and writers.

The specific time travel trope of killing your grandfather (and the consequences that might ensue) is referenced several times in the 1920s letter pages of pulp magazines like the famous Amazing Stories . By the late 1920s and early ‘30s, now-obscure writers like Charles Cloukey , Nathaniel Schachner and Rene Barjavel were delving into the paradox long before it would be used in famous short stories by Ray Bradbury, or in popular films and TV shows ranging from Star Trek to Back to the Future to The Umbrella Academy . 

Has Anyone Solved the Grandfather Paradox?

As a pure logic problem, scientists and other thinkers have devoted a surprising amount of time to pondering the Grandfather Paradox over the past several decades. Physicists who study quantum mechanics, for example, have published some compelling but not easily explained papers that allow for a temporal paradox to exist without necessarily violating any laws in quantum physics.

Meanwhile, others have offered less scientific but more creative ideas to resolve the paradox. One of the more popular notions — a real have-your-cake-and-eat-it-too theory — is that you wouldn’t actually create a paradox or cancel out your own existence by killing your grandfather.

Parallel Universe

Instead, once your grandfather dies, you’d find yourself still very much alive, but in an alternate timeline. The future (or your past, depending on how you look at it) would remain unchanged in your prime timeline while the alternate timeline continues to unfold elsewhere in the multiverse. In fact, some theorists feel that this solution doesn’t really speak to time travel so much as it addresses traveling to parallel universes, in this case a place exactly like your own, right up until you left it.

Read More: Scientists Attempt to Map the Multiverse

Still, in the end, both possibilities — killing Grandpa and not deleting yourself from existence — could occur in this scenario. Paradox solved! (Although this solution raises a different thorny question: Once you’ve killed your grandfather, can you get back to your original timeline, or will you be stuck in the alternate version?)

Self Consistency Principle

Other theories maintain that the Grandfather Paradox just goes to show you that time travel isn’t possible at all. The past cannot be changed ; somehow, the universe would course-correct before you could ever have a chance to alter the past. In short, you might find that your time machine simply won’t work, the ultimate universal safety measure for preventing any paradox. A variation of this theory treats causality as an unbreakable physical law, like gravity. In other words, just as you couldn’t simply decide to float off the ground, so too would you be physically unable to do anything, accidentally or deliberately, that would create a time paradox.

Predestination Paradox

This brings us to the “what happened, happened” school of thought, which basically says that, even if time machines were real, you’re kidding yourself to think you can change things in the past. This theory relies less on causality and more on the idea that everything is simply predetermined . Your "going back in time to make a change" — or what you think is a change — was always supposed to happen.

Read More: A Major Time Travel Perk May Be Technically Impossible

For example, let’s say you do go back in time and kill your grandfather. Plot twist: It might turn out that he was never your grandfather after all! In a cosmic case of mistaken identity, some other guy could be revealed as your actual grandfather and you would still exist in the future, as you were always meant to. (But then Grandma might have some explaining to do.)

An even wilder variation on this theme “solves” the paradox by introducing another paradox. Here, the idea is that if you did manage to kill your grandfather, you would end up taking his place and become, yes, your own grandpa. Admittedly, scenarios like this tend to play out more in the minds of fiction writers than they do in logicians and theorists.   

Is Time Travel Possible?

Don’t laugh! This question isn’t as far-fetched as you might think. Are time machines real? No. Will they ever be? Probably not. As the late Stephen Hawking and others have noted, if such devices were ever going to be real, we’d likely have seen tourists from the future by now. But that doesn’t mean that time travel itself is impossible, at least theoretically.

Theory of Relativity

Einstein’s theory of relativity established that time and space are linked, and that the faster you travel through space, the slower you will experience time. This is known as time dilation and you’ve probably heard the concept explained via the famous example of two twins — one who takes a trip in a fast rocket ship, and one who remains on Earth. The closer the astronaut twin’s ship approaches the speed of light, the slower that twin will experience time. When that twin returns to Earth, they will be notably younger than the twin who remained on the planet. In essence, the astronaut twin will have traveled into the future. Of course, this would be a one-way trip, and humans can’t yet (or possibly ever) make a ship that approaches light-speed, but never mind — relativity still says this form of time travel is possible.

Wormhole Theory

Then, there’s the wormhole theory . Einstein’s theory of general relativity allows for the existence of wormholes. A sufficiently large source of mass and energy (such as a black hole, or a merging of two black holes) could create these structures, making a tear in the very fabric of the universe, connecting two different points of space-time. Traveling through such a wormhole would, for our purposes, be the equivalent of time travel. But again, humanity has no way to create such a phenomenon — even if we could find a naturally occurring wormhole stable enough to enter, it’s unlikely we’d ever be able to construct a vessel powerful enough to survive a trip through it.

So, unfortunately for us — but luckily for all of our grandfathers — true time travel remains a practical impossibility. Instead, humans will just have to make do with the same form of time travel we’ve always used — not to the past, but to the future, one second at a time.

Read More: Is There a Particle That Can Travel Back in Time?

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The Time-Travel Paradoxes

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

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

Wait! Don’t do it!

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

 The Grandfather Paradox

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

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

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

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

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

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

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

 The Time Loop

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

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

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

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

Time travelers – where have all they gone?

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

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

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

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

 One way or round trip?

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

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

 Advanced time travel

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

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

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

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

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

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

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

 How to travel in time and stay alive

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

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

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

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

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

Time Travelling Grammar

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

It’s complicated.

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

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

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

University of Notre Dame

Notre Dame Philosophical Reviews

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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'.

Time Travel Movies Rely on the ‘Bootstrap Paradox.’ It Could Explain Real-Life Destiny

It may not be as famous as the so-called “grandfather paradox,” but that doesn’t make the idea of an infinite causal loop any less troubling ... or fascinating .

Arriving back slightly before your time, you watch your friend as she teaches her younger self how to build the time machine in the first place. “But … ” you stammer, cautious in case your friend—or her younger counterpart—takes offense and travels back in time to wipe you from existence, “if you used the time machine to travel back in time to inform yourself how to build the time machine, isn’t that a paradox?” It is , because no one in this story ever actually sat down to figure out how to create a time machine. This is the bootstrap paradox, which details an infinite cause-and-effect loop that develops around information, an object, or even a person that can no longer be given a discernible point of origin in a time travel scenario.

This results in a “closed causal loop,” in which some event or object is a key player in its own origin and couldn’t exist without its later self. The idea is woven through the imaginations of science fiction writers, theoretical physicists, and philosophers, meaning its roots in science and pop culture may be irrevocably interwoven.

Though less famous than the grandfather paradox —in which a time traveler kills their grandfather before they have children, and can no longer exist to go back in time and perform grand patricide—the bootstrap paradox is still an idea that philosophers and physicists have been grappling with for almost 100 years.

“The bootstrap paradox in time travel occurs when a piece of information that has no right to exist, nonetheless exists,” Seth Lloyd, a professor of mechanical engineering at the Massachusetts Institute of Technology and a self-described “quantum mechanic,” tells Popular Mechanics . “To illustrate the bootstrap paradox, consider the version known as the ‘unproved theorem paradox.’”

Lloyd explains that in the unproved theorem paradox, a time traveler finds a beautiful and elegant proof of a theorem in a book, then goes back in time and shows the proof to a mathematician, who includes the proof in his book. “The book, of course, is the same book in which the time traveler found the proof in the future,” he adds. “So who proved the theorem? No one proved the theorem! Where did the proof come from? Nowhere!”

Tim Maudlin, a philosopher of science who investigates the metaphysical foundations of physics and logic, says there is a key difference between the bootstrap paradox and the grandfather paradox.

“The bootstrap paradox occurs in certain logically consistent time travel scenarios. Unlike the grandfather paradox, which is a self-contradictory and hence an impossible scenario, in a bootstrap story, everything fits together in a logically coherent way,” Maudlin tells Popular Mechanics. “Still, the paradox arises because certain things or information [are] ‘self-caused’ or ‘come from itself.’” That means, in a sense, that these bits of information appear in the story without ever having been created, just like the instructions for building your friend’s time machine.

Maudlin — who alongside Oxford University philosopher Frank Arntzenius wrote the Stanford Encyclopedia of Philosophy article “Time Travel and Modern Physics”— adds that since it is not a matter of self-contradiction or logical incoherence, logic alone cannot rule out these bootstrap paradox scenarios. This has helped make the concept a staple of time travel stories in popular culture.

The Origins of the Bootstrap Paradox

Physicists first began to pay serious attention to the concept of time travel in the early years of the 20th century, following the work of Albert Einstein .

As he formulated the theory of special relativity, published in 1905, Einstein united the concepts of space and time into a single four-dimensional entity — spacetime — and also imposed the rule that particles with mass would need infinite energy to accelerate to the speed of light . This led to speculation about hypothetical massless particles called “tachyons” that could exceed the speed of light, but as a consequence, would travel back through time.

.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;} “The paradox suggests the potential for ‘predestination’ to causality, or the idea that the past is dependent on the future.”

While this opened the door to physicists thinking about time travel, Einstein would violently tear that hypothetical door down ten years later when he formulated the theory of general relativity . Primarily a theory that describes the effect of mass on spacetime, general relativity implies that four-dimensional spacetime could be twisted into any shape. This means it could be possible to create a path that forms a loop passing back through time, returning to its original starting point — which would become known as a “closed time-like curve” or CTC.

The idea of time travel caused a serious threat to causality, the relationship between cause and effect, in that it could make an effect precede its own cause—a clear contradiction. The bootstrap paradox is more subtle than this (cause still precedes effect), but there is no starting point in the process. Instead, the paradox suggests the potential for “predestination” to causality, or the idea that the past is dependent on the future.

As the bootstrap paradox worked its way into physicists’ notebooks and blackboards, it was also making its mark on pop culture, with the two occasionally overlapping.

The Bootstrap Paradox in Pop Culture

The bootstrap paradox takes its name from an expression that dates back to 1834: to “pull oneself over a fence by one’s bootstraps,” which came to represent the idea of performing a ludicrous or impossible task. (In the startup world, it’s come to mark the idea of launching a company with your own money or whatever you can scrounge from friends and family—certainly a Herculean task.)

In popular culture, the paradox lent its name to the 1941 Robert A. Heinlein story, By His Bootstraps , likely one of the paradox’s earliest presentations to the public. In the story, a similar scenario plays out to the one we described about your time-traveling friend:

“A time travel machine is built by following the instructions in a book, which itself is sent back from the future. But how did the information get into the book? By being copied into a newly made notebook from the older version of itself,” Maudlin explains. “That is, no one ever sat down and figured out how to build a time machine, the information about how to do it is just there in a closed causal loop.”

Since the publication of Heinlein’s tale, the bootstrap paradox has made its way into a wealth of science fiction tales, though it often goes unnamed. For instance, in the popular Doctor Who episode “Blink,” the time-traveler that David Tennett plays must combat a menace called the Weeping Angels by passing messages from the past—when he is trapped without his own method of time travel, the TARDIS—to the present.

At the end of the story, with evil defeated, a log of the messages is passed back to the timelord before he is trapped in the past. Therefore, the log of messages has no set origin; It cycles between the past and the future in a closed causal loop.

Meanwhile, the Terminator franchise features a wealth of potential time travel paradoxes across the series. For example, the main mission of the earliest film is for Arnold Schwarzenegger’s T-800 to prevent the birth of John Connor. Yet, if the Terminator kills his mother Sarah, and prevents the birth of John Conner before he can lead the rebellion, then he never existed as a threat to the Terminator controlling Skynet in the first place; That means they’d never need to send a Terminator back to kill him at all,eEssentially creating a version of the Grandfather paradox with more robots.

Artificial intelligence-based boogeyman Skynet is itself, however, an example of the bootstrap paradox. The AI system that would go on to rule Earth and subjugate mankind was created from the leftover parts of the T-800 Terminator that was sent back in time to kill Sarah Connor and prevent the birth of John Connor in the first place. Eventually, Skynet’s own killing machines, including the T-800, were retro-engineered from these parts, meaning the T-800 is the source of its own origin. Classic bootstrap!

These are just a few of the more infamous examples of the bootstrap paradox in fiction, and some of these tales have even helped inform scientific theory on the subject.

“As part of our efforts to construct a self-consistent quantum theory of time travel via closed timelike curves, I did extensive reading of time travel narratives and viewing of time travel films,” Lloyd explains. “All time travel paradoxes seem to be versions of either the grandfather paradox or of the bootstrap paradox.”

Escaping the Bootstrap Paradox

Hugh Everett’s Many-Worlds Interpretation of quantum mechanics suggests that for every quantum possibility, a distinct universe is created. This can provide a solution to apparent contradictions in time travel.

“It has been suggested that paradoxes such as the grandfather paradox could be avoided by Everett’s Many Worlds—or more accurately, ‘relative state’—interpretation of quantum mechanics ,” Lloyd says. “When the time traveler goes back and kills her grandfather, she is simply forcing a transition into a separate ‘world,’ or branch of the wave function, from the world in which she came.”

That means a time traveler hitching a ride on a closed time-like curve into the past actually causes the universe to diverge, and the world she arrives in is distinct from the one she left. Therefore, it is not her grandfather she kills in the past, but a version of him that will never go on to have a child or grandchild. Our time traveler’s birth, therefore, is not prevented.

So in the case of the bootstrap paradox, returning to our earlier scenario, our friend doesn’t pass the information for building a time machine to her earlier self, but a version of herself in the past in a separate world.

That means each backward leap passes the information to a new universe.

Even so, this doesn’t fully solve the Bootstrap paradox, as it doesn’t explain where this information came from in the first place; The instructions for the time travel machine still don’t have an origin point.

“Since no contradiction is involved, one can’t raise the same logical objection as in the grandfather paradox case,” Maudlin says. “But still, many people think that such ‘self-causation’ or lack of an originator of the information, makes the scenario impossible. And of course, if reverse causation is not possible, then this closed causal loop is not either.”

Maudlin does point out that the bootstrap paradox isn’t as intrinsically intractable as the grandfather paradox, however. In fact, it could be replete with a wealth of hitherto undiscovered exit routes.

“If one allows for closed causal loops, then in a way, the opposite thing happens as in the grandfather paradox case. With the grandfather paradox, there seems to be no logically self-consistent solution,” Maudlin concludes. “But in the bootstrapping cases, there are often several different solutions, with the closed loops taking different forms.

“In fact, there would not seem to be any explanation of why one, rather than another, of these solutions would be realized.”

Testing the Bootstrap Paradox

While Maudlin doesn’t believe that the bootstrap paradox could ever be experimentally tested, Lloyd disagrees; He has suggested an experiment to investigate the unproved theorem paradox version of the bootstrap paradox.

“We devised experiments to test the pre/retrodictions of our theory for both of the central paradoxes of time travel—the grandfather paradox and the bootstrap paradox—a simple experiment to test the pre/retrodictions of our theory for the unproved theorem paradox,” Lloyd says. “The test can be performed on a simple quantum computer . In the experiment, a bit of information—call it Bit A—is copied to a second bit (Bit B).

“Bit B is sent back into the past, where it turns out that B in the past is in fact the same bit that becomes A in the future. Rather than engendering a self-inconsistency, however, the bootstrap/unproved theorem experiment is entirely self-consistent.”

This means for Lloyd, perhaps the greatest paradox surrounding the bootstrap paradox is that it doesn’t have to be a paradox at all.

“The bootstrap paradox isn’t always paradoxical. In our proposed experiment, for example, we predict that the ‘unproved’ bit will turn out to be entirely random,” he says. “This makes sense, as at no point in the future or the past was there any bias introduced to make it anything other than random .”

For now, however, Lloyd doesn’t intend to attempt such an experiment. Instead, alongside his colleague Michele Reilly and artist Andrey Kezzyn, Lloyd expressed his interest in time travel in the form of the 2022 movie Steeplechase .

“The experimental demonstration of the grandfather paradox gave rise to so much publicity—not to say notoriety—however, that we decided to hold off on further time travel experiments for the foreseeable future,” the physicist explains.

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.  

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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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The Grandfather Paradox: Is Time Travel Possible?

T he concept of time travel has long captivated the human imagination, inspiring countless works of science fiction and fueling speculation about the possibility of journeying backward or forward in time. One of the most famous thought experiments in the realm of time travel is the Grandfather Paradox—a hypothetical scenario in which a time traveler could potentially alter the past in a way that prevents their own existence. In this article, we will delve into the intricacies of the Grandfather Paradox, examine the challenges it poses to the concept of time travel, and explore current scientific theories and speculations regarding the feasibility of traversing the fourth dimension.

• The Grandfather Paradox Explained:

The Grandfather Paradox is a classic paradoxical scenario that arises from the notion of time travel. It posits a hypothetical situation in which a person travels back in time and kills their own grandfather before their parent is conceived, thus preventing their own birth. This creates a logical contradiction—if the time traveler’s grandfather is never born, then the time traveler would never exist to travel back in time and commit the act of patricide.

• Paradoxes and Causality:

The Grandfather Paradox highlights the potential contradictions and complexities inherent in the concept of time travel, particularly when it comes to causality—the relationship between cause and effect. In the context of the paradox, the time traveler’s actions in the past have the potential to alter the course of history in ways that defy conventional notions of cause and effect, leading to logical inconsistencies and paradoxical outcomes.

• Temporal Mechanics and Alternate Timelines:

One proposed solution to the Grandfather Paradox is the idea of alternate timelines or parallel universes. According to this interpretation, when the time traveler alters the past, they create a new timeline or branch of reality in which their actions have different consequences. In this scenario, the original timeline remains intact, unaffected by the time traveler’s interference, while the altered timeline diverges into a separate reality with its own set of events and outcomes.

• Closed Timelike Curves and Wormholes:

In theoretical physics, closed timelike curves (CTCs) are hypothetical paths through spacetime that loop back on themselves, allowing for the possibility of time travel. Wormholes, which are hypothetical tunnels or shortcuts through spacetime, have been proposed as potential mechanisms for traversing CTCs and facilitating time travel. While the existence of wormholes remains speculative, their theoretical properties raise intriguing questions about the nature of spacetime and the feasibility of time travel.

• Time Dilation and Relativity:

The theory of relativity, formulated by Albert Einstein in the early 20th century, provides a theoretical framework for understanding the nature of spacetime and the effects of gravity on the passage of time. According to the theory of special relativity, time dilation occurs when an object is in motion relative to an observer, causing time to pass more slowly for the moving object. This phenomenon has been experimentally confirmed and plays a crucial role in various applications of modern physics, including GPS satellites and particle accelerators.

• Quantum Mechanics and Entanglement:

Quantum mechanics, the branch of physics that describes the behavior of particles at the smallest scales, introduces additional complexities to the concept of time travel. Quantum entanglement, a phenomenon in which particles become correlated in such a way that the state of one particle instantaneously influences the state of another, raises questions about the nature of causality and the possibility of information transfer across spacetime. While quantum mechanics offers intriguing insights into the nature of reality, its implications for time travel remain speculative and subject to ongoing debate.

• The Arrow of Time and Thermodynamics:

The arrow of time, which refers to the directionality of time’s flow from past to future, is intimately linked to the second law of thermodynamics, which states that entropy, or disorder, tends to increase over time. The concept of time travel raises intriguing questions about the nature of causality and the relationship between the past, present, and future. While time travel remains a staple of science fiction, its feasibility in the real world is a topic of ongoing scientific inquiry and speculation.

• Philosophical Implications:

Beyond its scientific and theoretical implications, the concept of time travel raises profound philosophical questions about the nature of free will, determinism, and personal identity. The Grandfather Paradox, in particular, forces us to confront the possibility of altering the past and the implications of such actions for our own existence and the course of history. While time travel remains a tantalizing prospect, its potential consequences for the fabric of reality and the human experience are subjects of speculation and debate.

• Ethical Considerations:

The prospect of time travel also raises ethical considerations regarding the potential impact of altering the past on individuals, societies, and the course of history. Questions about the responsibility of time travelers, the preservation of historical integrity, and the consequences of unintended consequences loom large in discussions of time travel ethics. As we contemplate the feasibility of time travel, it is essential to consider the ethical implications of altering the past and the potential risks and benefits associated with such endeavors.

• The Search for Answers:

While the Grandfather Paradox and other paradoxes of time travel present formidable challenges to our understanding of the nature of reality, they also inspire curiosity, imagination, and inquiry. Scientists, philosophers, and science fiction authors continue to explore the mysteries of time travel, probing the boundaries of theoretical physics, metaphysics, and human imagination in search of answers to some of the most profound questions about existence, time, and the universe.

Conclusion:

The Grandfather Paradox and the concept of time travel represent a fascinating intersection of science, philosophy, and speculation. While the feasibility of time travel remains uncertain and the Grandfather Paradox poses formidable challenges to our understanding of causality and reality, the quest to unravel the mysteries of time continues to inspire curiosity and inquiry. Whether through theoretical physics, philosophical reflection, or imaginative storytelling, the exploration of time travel invites us to contemplate the nature of existence, the passage of time, and the boundless possibilities of the cosmos. As we grapple with the paradoxes and possibilities of time travel, we are reminded of the enduring human quest to unlock the secrets of the universe and journey into the unknown realms of spacetime.

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Mindfulness

How time perspective affects travel, do you live in the past, present, or future.

Posted April 15, 2024 | Reviewed by Michelle Quirk

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• Understanding your own time perspective can enhance your life experience.
• Our characteristic types are neither good nor bad, just different from one another.
• Children are present-oriented, while adults favor the future. Seniors tend to preserve the past.

As an enjoyable vacation winds down, some of us become impatient to get home and move on to the next thing. Maybe that’s you. But, instead, you might be someone who tries to preserve, or even expand, every remaining moment. In either case, you’ll attempt to lock these precious flashes into your memory bank with mental snapshots. But without any effort, and all too quickly, the present quickly fades into the past. How we experience time is relevant to travel. Understanding your own time perspective can enhance your experience.

Stanford University Professor Emeritus Phil Zimbardo, author of The Time Paradox , notes that we are all oriented to time in one of the following characteristic ways—past, present, or future. According to his profile, I am future-oriented. What might your style be? Let’s see.

Those of us in the future category are goal-driven, focused on the future consequences of our actions, and forward-looking in general. Then there are the present-hedonic folks, the pleasure seekers who enjoy things in real-time, with less concern about tomorrow. Folks who live in the present tend to be open to experiences they didn’t necessarily plan, and they don’t need to check it off their bucket list. If this style fits, you’re probably most content with the moment-to-moment flow of your travel.

Past-oriented people make up the remaining category. This might be you if you compare current experiences with memories of past events or situations. Past-oriented folks determine the value of travel, according to Zimbardo, by assigning a pleasure quotient to the comparison—better or worse and by how much? This style is more analytic and rational, and based less on emotional factors than is true for present-focused folks. Does this sound like you?

Our characteristic types are neither good nor bad—just different from one another. Future- and past-oriented travelers provide a logical, systematic understanding of where travel fits into human experience. These styles have great evolutionary value. Our distant ancestors, who chronicled the past and predicted the future, tended to be the shaman and storytellers of the tribe. Reviewing the past and predicting the future was critical to human survival.

Present-oriented people tend to have more fun in the moment, and every society needs this type of person to keep things from getting too serious. Savoring the present is an acquired skill and is worth the effort to cultivate! Also, by expanding the present-pleasant and then reviewing a trip in the past-positive, you can have both good feelings and pleasurable memories. Since, as Zimbardo’s research indicates, we have characteristic ways of perceiving time, maintaining a present focus may require some work—if this isn’t naturally how you see the world.

Zimbardo points to another dimension of time—one that is age-related. In general, children are present-oriented while adults favor the future. Seniors tend to preserve the past. As a future-focused senior, I'm aware of the need to put my foot on the brake and try to prolong the present—particularly the pleasing moments while vacationing. This takes some work.

Regardless of the type that best explains you, here are some strategies to expand your time orientation:

• If you’re naturally drawn to the past or future , notice these tendencies and gently nudge yourself toward the present moment. When you catch yourself reminiscing about the last time you were in Paris, as you sit at an outdoor café savoring your steaming latte and munching on a croissant, remind yourself that the people you see strolling by are there right now—not last time or next time. The weather is uniquely now, not needing a contrast with a warmer or sunnier last visit. The present can be pleasant without any backward reference—or simply less.
• Future- oriented travelers tend to spend their present moments imagining future trips, which makes sense in planning life but can steal from the here and now. Recently, on a river cruise through Austria, I was struck by how much conversation I overheard about planning the next trip. Busily sharing these thoughts with fellow travelers, these vacationers sat by a large picture window as the ship sailed into a new city—totally missing the present moment, unnoticed outside of the window.
• Again, if future is your natural mode, keep that in mind as you travel. Learn to prolong the only moment that truly exists—this one that you anticipated for months or maybe years. The first step involves gently guiding your awareness back to the present. Practicing meditation even a few minutes a day will make this process easier.

This article is based on a chapter from my book: Inward Traveler: 51 Ways to Explore the World Mindfully, 2018.

Stanford University professor emeritus, Phil Zimbardo, authored The Time Paradox, Free Press, N.Y., 2008.

Francine Toder, Ph.D. , is an emeritus faculty member of California State University, Sacramento and is a clinical psychologist retired from private practice.

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Time travel, Babes With Blades style

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What is better than a time travel play? Not much. A rare species in itself, it can only be improved upon by something equally awesome—like centering women, impeccable stage combat, and a rock-solid plot that accounts for the bootstrap paradox. This is exactly what the Babes With Blades world premiere of The S Paradox pulled off on opening night at Factory Theater. Written by Jillian Leff (Joining Sword & Pen International Playwright Competition winner and Margaret W. Martin Award winner), and directed by Morgan Manasa, The S Paradox delivers thrill after thrill, tightly choreographed combat (created by fight director Samantha Kaufman), multiple dimensions, and snappy, character-driven dialogue that keeps you invested in the story.

The S Paradox Through 5/18: Thu-Sat 8 PM, Sun 3 PM; open captions 5/4-5/5 and 5/9-5/10, sensory-friendly performances 4/21 and 4/28, online streaming available 5/11-5/12 and 5/16-5/17; Factory Theater, 1623 W. Howard, babeswithblades.org , $28-$35

With 12 ensemble cast members, the small stage somehow never seems crowded—or empty—as characters zip in and out between scenes, creating and erasing alternative realities as they go. As Kayla Marie Klammer plays Sloane, the would-be cop in a utopian world, and Maureen Yasko plays her older, more wizened self, “S,” the situation quickly becomes a conundrum. Fortunately, their genius geek friend (with a huge crush on them) named Dez (her younger self played by Elisabeth Del Toro and older self by Sonja Lynn Mata) comes to the rescue, creating time travel in her angsty, people-pleasing way. Both versions of Dez brought comic relief and an easy-to-relate-to geek foil to Sloane’s earnest badass timecop persona.

Most impressively, true to its title, the play skitters on the edge of the theory of relativity, deftly explaining away paradoxes in the plot and timeline with humor and much more sound reasoning than most time travel fiction.

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Politics and Pugilists: A Review of “The S Paradox” at Babes with Blades Theatre Company

by Tristan Bruns | April 17, 2024

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Babes with Blades’ “The S Paradox,” with Kayla Marie Klammer and Maureen Yasko/Photo: Joe Mazza/Brave Lux

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Time travel is a tricky business. Altering events in the past can cause unexpected and paradoxical ripples that may result in dire consequences for the future. Television and film, from “The Terminator” to “The Simpsons,” have used this plot device to great effect; Babes with Blades bring time-traveling hijinks and sword-based combat to the stage in playwright Jillian Leff’s “The S Paradox,” running through May 18 at Factory Theater.

“The S Paradox” imagines a near future where capitalism has capitulated to a socialist government that has effectively ended wealth disparity and homelessness and has outlawed all guns. Police uphold the law using long swords combined with tasers, but this makes them no less deadly.

Babes with Blades are self-described as using “stage combat to tell stories that elevate the voices of underrepresented communities and dismantle the patriarchy.” True to their word, the combat in “The S Paradox” is a step above typical theatrical melee. The swords sound heavy as they clang against each other, and downward swings aimed at necks feel like they really could take a head off. Fast-paced fisticuffs produce loud smacks as flesh blocks flesh. These intricately choreographed bouts will surely have patriarchy quaking in their boots.

Babes with Blades’ “The S Paradox,” with (L to R) Jessica Pennachio, Tina-Kim Nyguen, Steve Peebles, Thomas Russell and Deanna Palmer/Photo: Joe Mazza/Brave Lux

Sloane (Kayla Marie Klammer) is an officer-in-training and outstanding swashbuckler, but is held back from full status due to her unwillingness to kill. Quirky weapons cache manager Dez (Elizabeth Del Toro) pulls strings to allow Sloane to retain her beloved blade, citing an ominous portent delivered by a future version of herself.

Meanwhile, a parallel story unfolds in the future as a mysterious figure known as S (Maureen Yasko) takes leap after leap back in time to prevent society from descending into a capitalist dystopia. Working with time-travel expert Older Dez (Luz Espinosa), S causes timeline alterations that coincide with previous scenes in the alternate storyline, explaining moments that were before ambiguous and creating genuine aha moments.

Even in a socialist utopia, there are shadowy forces in the works. William Hale (Steve Peebles), a well-dressed, slick-tongued representative of the watchdog intelligence agency, the CRC, approaches Sloane with an offer to lead his clandestine corporate guard, The Nameless, promising a salary above the government-mandated pay cap and forcing Sloane to choose between her socialist ideals and material gain.

Amidst sword fighting and comedic banter are deep conversations about political philosophy. Sloane’s girlfriend Ava (Cat Evans) is against all weaponry, as she remembers a time (like today) when anyone who wanted a gun could get one and all the trouble that caused. William is an archetype of a libertarian capitalist, placing the rights of the individual above the good of society. And, if socialism has alleviated all our problems, why is there still a need for a police force, anyway? Critiques of both the Left and Right create a middle ground for further philosophical investigations.

Politics and pugilists collide in “The S Paradox,” a fun time-travel adventure that provides commentary on society’s pressing issues. Political philosophy can be confusing, but when arguments are decided by the tip of the sword, I think you’ll get the point.

“The S Paradox” is presented by Babes with Blades Theatre Company through May 18 at The Factory Theater, 1623 West Howard. Performances are Thursdays through Saturdays at 8pm, Sundays at 3pm. Sensory friendly performances: April 21 & 28. Streaming: May 11, 12, 16, 17. Tickets are $28-$35 at babeswithblades.org .

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The journey from Moscow to Elektrostal by train is 32.44 mi and takes 2 hr 7 min. There are 71 connections per day, with the first departure at 12:15 AM and the last at 11:46 PM. It is possible to travel from Moscow to Elektrostal by train for as little as or as much as . The best price for this journey is .

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Virail's search tool will provide you with the options you need when you want to go from Moscow to Elektrostal. All you need to do is enter the dates of your planned journey, and let us take care of everything else. Our engine does the hard work, searching through thousands of routes offered by our trusted travel partners to show you options for traveling by train, bus, plane, or carpool. You can filter the results to suit your needs. There are a number of filtering options, including price, one-way or round trip, departure or arrival time, duration of journey, or number of connections. Soon you'll find the best choice for your journey. When you're ready, Virail will transfer you to the provider's website to complete the booking. No matter where you're going, get there with Virail.

How can I find the cheapest train tickets to get from Moscow to Elektrostal?

Prices will vary when you travel from Moscow to Elektrostal. On average, though, you'll pay about for a train ticket. You can find train tickets for prices as low as , but it may require some flexibility with your travel plans. If you're looking for a low price, you may need to prepare to spend more time in transit. You can also often find cheaper train tickets at particular times of day, or on certain days of the week. Of course, ticket prices often change during the year, too; expect to pay more in peak season. For the lowest prices, it's usually best to make your reservation in advance. Be careful, though, as many providers do not offer refunds or exchanges on their cheapest train tickets. Unfortunately, no price was found for your trip from Moscow to Elektrostal. Selecting a new departure or arrival city, without dramatically changing your itinerary could help you find price results. Prices will vary when you travel from Moscow to Elektrostal. On average, though, you'll pay about for a train ticket. If you're looking for a low price, you may need to prepare to spend more time in transit. You can also often find cheaper train tickets at particular times of day, or on certain days of the week. Of course, ticket prices often change during the year, too; expect to pay more in peak season. For the lowest prices, it's usually best to make your reservation in advance. Be careful, though, as many providers do not offer refunds or exchanges on their cheapest train tickets.

How long does it take to get from Moscow to Elektrostal by train?

The journey between Moscow and Elektrostal by train is approximately 32.44 mi. It will take you more or less 2 hr 7 min to complete this journey. This average figure does not take into account any delays that might arise on your route in exceptional circumstances. If you are planning to make a connection or operating on a tight schedule, give yourself plenty of time. The distance between Moscow and Elektrostal is around 32.44 mi. Depending on the exact route and provider you travel with, your journey time can vary. On average, this journey will take approximately 2 hr 7 min. However, the fastest routes between Moscow and Elektrostal take 1 hr 3 min. If a fast journey is a priority for you when traveling, look out for express services that may get you there faster. Some flexibility may be necessary when booking. Often, these services only leave at particular times of day - or even on certain days of the week. You may also find a faster journey by taking an indirect route and connecting in another station along the way.

How many journeys from Moscow to Elektrostal are there every day?

On average, there are 71 daily departures from Moscow to Elektrostal. However, there may be more or less on different days. Providers' timetables can change on certain days of the week or public holidays, and many also vary at particular times of year. Some providers change their schedules during the summer season, for example. At very busy times, there may be up to departures each day. The providers that travel along this route include , and each operates according to their own specific schedules. As a traveler, you may prefer a direct journey, or you may not mind making changes and connections. If you have heavy suitcases, a direct journey could be best; otherwise, you might be able to save money and enjoy more flexibility by making a change along the way. Every day, there are an average of 18 departures from Moscow which travel directly to Elektrostal. There are 53 journeys with one change or more. Unfortunately, no connection was found for your trip from Moscow to Elektrostal. Selecting a new departure or arrival city, without dramatically changing your itinerary could help you find connections.

Book in advance and save

If you're looking for the best deal for your trip from Moscow to Elektrostal, booking train tickets in advance is a great way to save money, but keep in mind that advance tickets are usually not available until 3 months before your travel date.

Stay flexible with your travel time and explore off-peak journeys

Planning your trips around off-peak travel times not only means that you'll be able to avoid the crowds, but can also end up saving you money. Being flexible with your schedule and considering alternative routes or times will significantly impact the amount of money you spend on getting from Moscow to Elektrostal.

Always check special offers

Checking on the latest deals can help save a lot of money, making it worth taking the time to browse and compare prices. So make sure you get the best deal on your ticket and take advantage of special fares for children, youth and seniors as well as discounts for groups.

Unlock the potential of slower trains or connecting trains

If you're planning a trip with some flexible time, why not opt for the scenic route? Taking slower trains or connecting trains that make more stops may save you money on your ticket – definitely worth considering if it fits in your schedule.

Best time to book cheap train tickets from Moscow to Elektrostal

The cheapest Moscow - Elektrostal train tickets can be found for as low as $35.01 if you’re lucky, or $54.00 on average. The most expensive ticket can cost as much as $77.49.

Find the best day to travel to Elektrostal by train

When travelling to Elektrostal by train, if you want to avoid crowds you can check how frequently our customers are travelling in the next 30-days using the graph below. On average, the peak hours to travel are between 6:30am and 9am in the morning, or between 4pm and 7pm in the evening. Please keep this in mind when travelling to your point of departure as you may need some extra time to arrive, particularly in big cities!

Moscow to Elektrostal CO2 Emissions by Train

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40 facts about elektrostal.

Written by Lanette Mayes

Modified & Updated: 02 Mar 2024

Reviewed by Jessica Corbett

Elektrostal is a vibrant city located in the Moscow Oblast region of Russia. With a rich history, stunning architecture, and a thriving community, Elektrostal is a city that has much to offer. Whether you are a history buff, nature enthusiast, or simply curious about different cultures, Elektrostal is sure to captivate you.

This article will provide you with 40 fascinating facts about Elektrostal, giving you a better understanding of why this city is worth exploring. From its origins as an industrial hub to its modern-day charm, we will delve into the various aspects that make Elektrostal a unique and must-visit destination.

So, join us as we uncover the hidden treasures of Elektrostal and discover what makes this city a true gem in the heart of Russia.

Key Takeaways:

• Elektrostal, known as the “Motor City of Russia,” is a vibrant and growing city with a rich industrial history, offering diverse cultural experiences and a strong commitment to environmental sustainability.
• With its convenient location near Moscow, Elektrostal provides a picturesque landscape, vibrant nightlife, and a range of recreational activities, making it an ideal destination for residents and visitors alike.

Known as the “Motor City of Russia.”

Elektrostal, a city located in the Moscow Oblast region of Russia, earned the nickname “Motor City” due to its significant involvement in the automotive industry.

Home to the Elektrostal Metallurgical Plant.

Elektrostal is renowned for its metallurgical plant, which has been producing high-quality steel and alloys since its establishment in 1916.

Boasts a rich industrial heritage.

Elektrostal has a long history of industrial development, contributing to the growth and progress of the region.

Founded in 1916.

The city of Elektrostal was founded in 1916 as a result of the construction of the Elektrostal Metallurgical Plant.

Located approximately 50 kilometers east of Moscow.

Elektrostal is situated in close proximity to the Russian capital, making it easily accessible for both residents and visitors.

Known for its vibrant cultural scene.

Elektrostal is home to several cultural institutions, including museums, theaters, and art galleries that showcase the city’s rich artistic heritage.

A popular destination for nature lovers.

Surrounded by picturesque landscapes and forests, Elektrostal offers ample opportunities for outdoor activities such as hiking, camping, and birdwatching.

Hosts the annual Elektrostal City Day celebrations.

Every year, Elektrostal organizes festive events and activities to celebrate its founding, bringing together residents and visitors in a spirit of unity and joy.

Has a population of approximately 160,000 people.

Elektrostal is home to a diverse and vibrant community of around 160,000 residents, contributing to its dynamic atmosphere.

Boasts excellent education facilities.

The city is known for its well-established educational institutions, providing quality education to students of all ages.

A center for scientific research and innovation.

Elektrostal serves as an important hub for scientific research, particularly in the fields of metallurgy, materials science, and engineering.

Surrounded by picturesque lakes.

The city is blessed with numerous beautiful lakes, offering scenic views and recreational opportunities for locals and visitors alike.

Well-connected transportation system.

Elektrostal benefits from an efficient transportation network, including highways, railways, and public transportation options, ensuring convenient travel within and beyond the city.

Famous for its traditional Russian cuisine.

Food enthusiasts can indulge in authentic Russian dishes at numerous restaurants and cafes scattered throughout Elektrostal.

Home to notable architectural landmarks.

Elektrostal boasts impressive architecture, including the Church of the Transfiguration of the Lord and the Elektrostal Palace of Culture.

Offers a wide range of recreational facilities.

Residents and visitors can enjoy various recreational activities, such as sports complexes, swimming pools, and fitness centers, enhancing the overall quality of life.

Provides a high standard of healthcare.

Elektrostal is equipped with modern medical facilities, ensuring residents have access to quality healthcare services.

Home to the Elektrostal History Museum.

The Elektrostal History Museum showcases the city’s fascinating past through exhibitions and displays.

A hub for sports enthusiasts.

Elektrostal is passionate about sports, with numerous stadiums, arenas, and sports clubs offering opportunities for athletes and spectators.

Celebrates diverse cultural festivals.

Throughout the year, Elektrostal hosts a variety of cultural festivals, celebrating different ethnicities, traditions, and art forms.

Electric power played a significant role in its early development.

Elektrostal owes its name and initial growth to the establishment of electric power stations and the utilization of electricity in the industrial sector.

Boasts a thriving economy.

The city’s strong industrial base, coupled with its strategic location near Moscow, has contributed to Elektrostal’s prosperous economic status.

Houses the Elektrostal Drama Theater.

The Elektrostal Drama Theater is a cultural centerpiece, attracting theater enthusiasts from far and wide.

Popular destination for winter sports.

Elektrostal’s proximity to ski resorts and winter sport facilities makes it a favorite destination for skiing, snowboarding, and other winter activities.

Promotes environmental sustainability.

Elektrostal prioritizes environmental protection and sustainability, implementing initiatives to reduce pollution and preserve natural resources.

Home to renowned educational institutions.

Elektrostal is known for its prestigious schools and universities, offering a wide range of academic programs to students.

Committed to cultural preservation.

The city values its cultural heritage and takes active steps to preserve and promote traditional customs, crafts, and arts.

Hosts an annual International Film Festival.

The Elektrostal International Film Festival attracts filmmakers and cinema enthusiasts from around the world, showcasing a diverse range of films.

Encourages entrepreneurship and innovation.

Elektrostal supports aspiring entrepreneurs and fosters a culture of innovation, providing opportunities for startups and business development.

Offers a range of housing options.

Elektrostal provides diverse housing options, including apartments, houses, and residential complexes, catering to different lifestyles and budgets.

Home to notable sports teams.

Elektrostal is proud of its sports legacy, with several successful sports teams competing at regional and national levels.

Boasts a vibrant nightlife scene.

Residents and visitors can enjoy a lively nightlife in Elektrostal, with numerous bars, clubs, and entertainment venues.

Promotes cultural exchange and international relations.

Elektrostal actively engages in international partnerships, cultural exchanges, and diplomatic collaborations to foster global connections.

Surrounded by beautiful nature reserves.

Nearby nature reserves, such as the Barybino Forest and Luchinskoye Lake, offer opportunities for nature enthusiasts to explore and appreciate the region’s biodiversity.

Commemorates historical events.

The city pays tribute to significant historical events through memorials, monuments, and exhibitions, ensuring the preservation of collective memory.

Promotes sports and youth development.

Elektrostal invests in sports infrastructure and programs to encourage youth participation, health, and physical fitness.

Hosts annual cultural and artistic festivals.

Throughout the year, Elektrostal celebrates its cultural diversity through festivals dedicated to music, dance, art, and theater.

Provides a picturesque landscape for photography enthusiasts.

The city’s scenic beauty, architectural landmarks, and natural surroundings make it a paradise for photographers.

Connects to Moscow via a direct train line.

The convenient train connection between Elektrostal and Moscow makes commuting between the two cities effortless.

A city with a bright future.

Elektrostal continues to grow and develop, aiming to become a model city in terms of infrastructure, sustainability, and quality of life for its residents.

In conclusion, Elektrostal is a fascinating city with a rich history and a vibrant present. From its origins as a center of steel production to its modern-day status as a hub for education and industry, Elektrostal has plenty to offer both residents and visitors. With its beautiful parks, cultural attractions, and proximity to Moscow, there is no shortage of things to see and do in this dynamic city. Whether you’re interested in exploring its historical landmarks, enjoying outdoor activities, or immersing yourself in the local culture, Elektrostal has something for everyone. So, next time you find yourself in the Moscow region, don’t miss the opportunity to discover the hidden gems of Elektrostal.

Q: What is the population of Elektrostal?

A: As of the latest data, the population of Elektrostal is approximately XXXX.

Q: How far is Elektrostal from Moscow?

A: Elektrostal is located approximately XX kilometers away from Moscow.

Q: Are there any famous landmarks in Elektrostal?

A: Yes, Elektrostal is home to several notable landmarks, including XXXX and XXXX.

Q: What industries are prominent in Elektrostal?

A: Elektrostal is known for its steel production industry and is also a center for engineering and manufacturing.

Q: Are there any universities or educational institutions in Elektrostal?

A: Yes, Elektrostal is home to XXXX University and several other educational institutions.

Q: What are some popular outdoor activities in Elektrostal?

A: Elektrostal offers several outdoor activities, such as hiking, cycling, and picnicking in its beautiful parks.

Q: Is Elektrostal well-connected in terms of transportation?

A: Yes, Elektrostal has good transportation links, including trains and buses, making it easily accessible from nearby cities.

Q: Are there any annual events or festivals in Elektrostal?

A: Yes, Elektrostal hosts various events and festivals throughout the year, including XXXX and XXXX.

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