Yes, Time Travel is not very Atomic Rocket. The main reason this page is here is because if you start messing around with faster than light travel, you are going to be ambushed by time travel sooner or later. The second reason is that some theories about time travel grow out the fourth dimension. Thirdly one of my favorite atomic rocket science fiction authors when I was growing up was Lady Andre Norton, and she has written some pretty sharp time travel novels.

This is actually such a huge topic that it deserves its own website. But somebody else can do that.

Suggested reading:

Physicists hold exceedingly tight on to the concept of Causality. Probably because without it the entire structure of physics crumbles into flaming ruin and you can't get any work done. Causality mandates that causes must precede effects, or it becomes impossible to make predictions. That is why it is called Causality. Science in general and physics in particular is all about making predictions. Step two of the Scientific Method, in fact.

So physicist hate time travel with every existential fiber of their being because time travel can kill causality like a sledge hammer splattering a roach.

You see, time travel raises the possibility of creating a dreaded temporal paradox. And all it takes is one of those bad boys to blow causality right out of the freaking water.

The only thing that makes physicists more angry is the fact that there appears to be nothing in physics that forbids time travel. And believe me, they have been looking real hard. More technically: the distinction between cause and effect is not made at the most fundamental level within the field of physics.

World-class physicist Dr. Stephen Hawking was so annoyed at this he formulated his Chronology protection conjecture, which holds that the laws of physics are such as to prevent time travel on all but submicroscopic scales. Why? Because time travel really cheeses him off, that's why.

As previously mentioned, this is also the reason physicists are so hostile to faster-than-light travel. As it turns out, Einstein's relativity shows that FTL travel and Time travel are two different terms for the exact same thing.

If a present-day scientist were confronted with a real time machine, he would certainly say that the machine had to be run by the rules of magic. His argument would go like this: Science is based on logic. Anything that produces logical paradoxes is not science. Time machines produce logical paradoxes. Therefore, if time machines exist, they must use magic, not science.

From Indistinguishable From Magic by Robert Forward (1995)

Time Travel Theory

To travel back in time, you need to follow what is called a Closed Timelike Curve. This is a world-line that is a loop, instead the more conventional wavy vertical line. Ordinary world line start at a given point in space and time, then moves into the future like all normal things do. A closed timelike curve eventually doubles back, traveling into the past until it connects with itself.

In his The Theory and Practice of Time Travel, Larry Niven notes that the basic motivation for inventing time travel is the same motivation of a child fervently wishing that something had never happened. If they could only change history so that they had hit the baseball to the left instead of the right it wouldn't have broken Old Man Smith's picture window. It is the desire for a "do-over" in other words.

So the primary motive for time travel is to change the past. Unsurprisingly that is also its prime danger.

All of the current "real" time machines visualized as thought-experiments by real physicists share a common flaw: you cannot use a time machine to travel to a point in time prior to the construction of that particular time machine.

Notice that this time machine, like all the other time machines that are allowed by the Einstein Theory of Gravity, can only take you backward in time to the moment that the machine is turned on, and forward in time to the moment that the machine is turned off. Einstein's laws do not allow a future time-machine maker to go back into time to tell himself how to make the machine. Thus, at least one of the possible time-machine paradoxes is avoided.

From Indistinguishable From Magic by Robert Forward (1995)

Larry Niven also makes a good case that in a given universe where time travel is possible, and using it to change the past is possible, the end result is that time travel will not be invented in that universe. Why? As he puts it:

Niven's Law

GIVEN: That the universe of discourse permits both time travel and the changing of the past.
THEN: A time machine will not be invented in that universe.

For, if a time machine is invented in that universe, somebody will change the past of that universe. There is just too much future subsequent to the invention of a time machine: too many people with too many good motives for meddling with too many events occurring in too much of the past.

If we assume that there is no historical inertia, no Conservation of Events, then each change makes a whole new universe. Every trip into the past means that all the dice have to be thrown over again. Every least change changes all the history books, until by chance and endless change we reach a universe where there is no time machine invented, ever, by any species.

Then that universe would not change.

(ed note: in other words, the universe will keep changing until it reaches a stable state, and that state is one where no time machines are invented)

Now assume that there is an inertia to history; that the past tends to remain unchanged; that probabilities change to protect the fabric of events. What is the simplest change in history that will protect the past from interference?

Right. No time machines!


From The Theory and Practice of Time Travel by Larry Niven (1971)

Mr. Niven also points out that time travel violates other laws of physics besides causality.

If your TARDIS travels from 3000 CE to 2000 CE, as far as the 2000 CE dwellers are concerned the TARDIS appears out of nowhere. Which is a drastic violation of the law of conservation of mass. The laws of the universe will be sternly unimpressed if you feebly protest that an equivalent tonnage of matter disappears a thousand years later. The laws will point out that for ten centuries there was an extra TARDIS around.

Sending a signal back in time is just as bad, only this time you are violating the law of conservation of energy. Which is to be expected since matter and energy are pretty much two aspects of the same thing.

And to top it off, physical time travel violates any law of motion, since motion always relates to time. Which has a ripple effect on conservation of momentum, statements about kinetic energy, plus any and all laws of gravity.

Admittedly these violated laws are of lesser importance than causality, but they are still nothing to sneeze at.

Time Travel Models

There are arguably five main theoretical models of time travel. There is no particular evidence for any of them, they are all theoretical.

No Timeline
AKA Presentism, Nowism. There is no past and future. The only thing that exists is the present movement. Obviously in this model time travel is impossible, thus it is immune to temporal paradox.
Fixed Timeline
AKA Fixed History, Determinism Eternalism, Block Time, Block Universe, Static interpretation of time, Four-dimensionalism. History cannot be changed by time travel (either past or future). Time travel is possible but all journeys have always been part of the timeline. There is no free will. Obviously immune to temporal paradox.
Self-Healing Timeline
AKA Elastic, Historical Inertia, Rubber-Band History, Plastic Time With High Resistance. This is in between Fixed Timeline and Dynamic Timeline. History can be changed by time travel, but due to inertia it tries to get back on its original track. Changes tend to be smoothed out. If a time traveler assassinated Hitler, some other similar leader would arise and World War 2 would happen anyway. This is sometimes called the "Law of Conservation of Reality." Danger of temporal paradox. The Novikov self-consistency principle may or may not apply (former is paradox-proof self-healing timeline).
Dynamic Timeline
AKA Overriding, Malleable, Plastic Time. History can be easily changed by time travel. This ranges on a spectrum between Temporal Balancing Act (where it is possible to make controlled changes and un-changes) and Temporal Chaos Theory {Chaotic Time} (where a traveler's mere presence causes zillions of uncontrolled changes per second, see Butterfly Effect). Danger of temporal paradox. The Novikov self-consistency principle may or may not apply (former is paradox-proof dynamic timeline).
Multiple Timelines
AKA Multiverse, Alternate Time lines, Parallel Universes, Cross-time, Branching Timelines. In the physics version, each time a time traveler changes history, the time line branches into the changed history line and the unchanged history line. In the literary version, each time any decision is made by any creature in the entire universe, the time line branches into one branch per possible outcome. Immune to temporal paradox

TV Tropes covers this at some length in its Temporal Mutability entry (aka The Sliding Scale Of How Easy It Is For Time Travelers To Change The Past, And Why).

No Timeline

AKA Presentism. There is no past and future. The only thing that exists is the present movement. Obviously in this model time travel is impossible, thus it is immune to temporal paradox.

How dull. In Clifford Simak's classic fix-up novel City, there is no room in the past or future for any past or future. That area is filled with parallel dimension worlds (the Cobbly worlds).

One world and then another, running like a chain. One world treading on the heels of another world that plodded just ahead. One world's tomorrow; another world's today. And yesterday is tomorrow and tomorrow is the past.

Except, there wasn't any past. No past, that was, except the figment of remembrance that flitted like a night-winged thing in the shadow of ones mind. No past that one could reach. No pictures painted on the wall of time. No film that one could run backward and see what once had been.

Joshua got up and shook himself, sat down and scratched a flea. Ichabod sat stiffly at the table, metal fingers tapping.

"It checks," the robot said. "There's nothing we can do about it. The factors check. We can't travel in the past."

"No," said Joshua.

"But," said Ichabod, "we know where the cobblies are."

"Yes," said Joshua, "we know where the cobblies are. And maybe we can reach them. Now we know the road to take."

One road was open, but another road was closed. Not closed, of course, for it had never been. For there wasn't any past, there never had been any, there wasn't room for one. Where there should have been a past there was another world.

Like two dogs walking in one another's tracks. One dog steps out and another dog steps in. Like along, endless row of ball-bearings running down a groove, almost touching, but not quite. Like the links of an endless chain running on a wheel with a billion billion sprockets.

From City by Clifford Simak (1952)

Fixed Timeline

AKA Fixed History, Determinism Eternalism, Block Time, Block Universe, Static interpretation of time, Four-dimensionalism. History cannot be changed by time travel (either past or future). Time travel is possible but all journeys have always been part of the timeline. There is no free will. Obviously immune to temporal paradox.

An easy fix to save Causality from an assassination attempt by a temporal paradox is to assert that it is impossible to use time travel to change history (aka the Post-selected model). Which is a big help, but no fun at all for science fiction authors (for one it kills off the do-over wish fulfilment aspect). And it also means all events are predetermined and you do not have any free will. Awkward, that.

If taken to extremes, it makes time travel very dangerous for the traveler. If they traveled into the past, raindrops would tear through their body like bullets because the traveler (and their body) could not alter the history of the raindrop's passage to the ground.

A fixed timeline is related to the concept of Block Time or the Block Universe. If the block extends from the past up to but stopping at the present, this is the Growing block universe. If the block extend from the past to the present and on to the future, this Eternalism.

What is Block Time? Here is an explanation from The Fourth Dimension: A Guided Tour of the Higher Universes by Rudy Rucker. Imagine that the fourth dimension is time (not the fourth spacial dimension). Since trying to visualize the fourth dimension will melt your brain, as an analogy people visualize two dimensional people living in a two dimensional space ("Flatland") and the third dimension is the flatlander's time dimension. A two-dimensional plane is the flatland universe, and the people are squares, triangles, and other geometric shapes sliding around.

Say Mr. Square is waiting somewhere, and around noon his father the triangle comes close for a second, then moves away. The block universe would look like Figure 1. The lower plane is 11:59, the middle plane is 12:00 and the upper plane is 12:01. Since Mr. Square is not moving his track through the time dimension (the third dimension in Flatland's case) is vertical. His father's track curves at angles because he is moving. In a way, it is like somebody made an overhead movie of flatland, took the film (ask your parents if you never hear of movie film), cut in into frames, and stacked them sequentially, as in Figure 2.

A larger view of Flatland's block universe would look like figure 3.

Note that from the view of a higher dimensional being (such as three-dimensional us), the block universe is static. This is why it works so well with the fixed timeline concept.

There is a temptation to visualize animating the block universe diagram, by illuminating successive slices to represent "now" (Figures 4 and 5). But this is incorrect, Dr. David Park calls this the "fallacy of the animated Minkowski diagram".

Author Richard Garfinkle wrote a science fiction novel All of an Instant, which is actually set around a block universe. It is just that entities who live outside of the block universe can ... alter it.

Once a time machine exists, then decisions made at one point on the time-line of an individual can affect not only the future of that individual, but also the past of that individual. Time machines will also raise philosophical questions. If you in the future had sent a message back in time to yourself in the past, then does the "you" in the future have any free will? For you know that you must and will send the message at the proper time in the future.

Yet, "free will" has always been limited by the laws of nature. For instance, if your past self has made the decision to jump off a bridge, your future self is bound by that decision. When time machines exist, your future decisions, in the same way, can bind your past self to the consequences of that "yet-to-be-made" decision.

From INDISTINGUISHABLE FROM MAGIC by Robert Forward (1995)

The oldest of all, going back to Greek times. Philosophers call it fatalism or determinism. A fatalist believes that everything that happens is predetermined to the end of time; that any attempt to change the predetermined future is fated, is a part of the predetermined future itself.

To a fatalist, the future looks exactly like the traditional picture of the past. Both are rigid, inflexible. The introduction of time travel would not alter the picture at all, for any attempt on the part of a time traveler to change the past has already been made, and is a part of the past.

Fatalism has been the basis for many a tale of a frantic time traveler caught in a web of circumstance such that every move he makes acts to bring about just the calamity he is trying to avert. The standard plot sketch is reminiscent of Oedipus Rex; when well done it has the same flavor of man heroically battling Fate and losing.

Notice how fatalism solves the Grandfather Paradox.

You can't kill your grandfather, because you didn't. You'll kill the wrong man if you try it; or your gun won't fire.

Fatalism ruins the wish-fulfillment aspect of time travel. Anything that averts the Grandfather Paradox will do that. The Grandfather Paradox is the wish-fulfillment aspect. Make it didn't happen.


Self-Healing Timeline

AKA Elastic, Historical Inertia, Rubber-Band History, Plastic Time With High Resistance. This is in between Fixed Timeline and Dynamic Timeline. History can be changed by time travel, but due to inertia it tries to get back on its original track. Changes tend to be smoothed out. If a time traveler assassinated Hitler, some other similar leader would arise and World War 2 would happen anyway. This is sometimes called the "Law of Conservation of Reality." Danger of temporal paradox. The Novikov self-consistency principle may or may not apply (former is paradox-proof self-healing timeline).

At one end of the spectrum we have the free-will fun kind of time travel (Dynamic Timeline), where history can be altered but causality is toast. At the other end we have the deterministic no-free-will no-fun kind of time travel (Fixed Timeline), where history is unalterable but causality is safe and secure.

In between is a sort of compromise, where historical events have "inertia" or "hysteresis". A history conservation law so to speak. So if you go back in time and kill Hitler, some other German charismatic malcontent will take his place and World War 2 will occur on schedule anyway. The time traveler will really have to work at it to overcome the inertia leading to the war, just killing Hitler is not enough.

Between the deterministic and free will modes of time travel lies a kind of compromise position:

We assume a kind of inertia, or hysteresis effect, or special conservation law for time travel. The past resists change. Breaks in time tend to heal. Kill Charlemagne and someone will take his place, conquer his empire, mate with his wives, breed sons very like his. Changes will be minor and local.

Fritz Leiber used Conservation of Events to good effect in the Change War stories. In TRY AND CHANGE THE PAST, his protagonist went to enormous lengths to prevent a bullet from smashing through a man's head.

He was sincere. It was his own head. In the end he succeeded—and watched a bullet-sized meteorite smash into his alter-self's forehead.

Probabilities change to protect history. This is the safest form of time travel in that respect. But one does have to remember that the odds have changed.

Try to save {insert major historical figure here} with a submachine gun, and the gun will positively jam.

But if you did succeed in killing your own six-year-old grandfather, you would stand a good chance of taking his place. Conservation of Events requires someone to take his place; and everyone else is busy filling his own role. Except you, an extraneous figure from another time. Now Conservation of Events acts to protect you in your new role!

Besides, you're already carrying the old man's genes.


Dynamic Timeline

AKA Overriding, Malleable, Plastic Time. History can be easily changed by time travel. This ranges on a spectrum between Temporal Balancing Act (where it is possible to make controlled changes and un-changes) and Temporal Chaos Theory {Chaotic Time} (where a traveler's mere presence causes zillions of uncontrolled changes per second, see Butterfly Effect). Danger of temporal paradox. The Novikov self-consistency principle may or may not apply (former is paradox-proof dynamic timeline).

The way to get the most fun out of time travel is to accept it for what it is. Give up relativity and the conservation laws. Allow changes in the past and present and future, reversals in the order of cause and effect, effects whose cause never happens...

Fatalistic time travel also allows these causative loops, but they are always simple, closed 1oops with no missing parts. The appearance of a time machine somewhere always implies its disappearance somewhere—and somewhen else. But with this new, free will kind of time travel…

We assume that there is only one reality, one past and one future; but that it can be changed at will via the time machine. Cause and effect may loop toward the past; and sometimes a loop is pinched off, to vanish from the time stream. The traveler who kills his six-year-old grandfather eliminates the cause of himself, but he and his time machine remain-until someone else changes the past even further back.

RocketCat sez

After Robert Heinlein died, Jack Williamson took the mantle of "The Dean of Science Fiction". They sure got that right! Actually, Jack's 1931 novel Birth of a New Republic probably inspired Heinlein to write The Moon is a Harsh Mistress.

Williamson's early stuff was pure space opera, but even there he got the science right (he invented the word "Seetee" for antimatter). Williamson was quite aware of Heisenberg's uncertainty principle and used it accurately in quite a few novels. I was impressed with how Heisenberg could be applied to ghostly astral monsters in his Darker Than You Think, but applying it to time travel was sheer genius.

Somebody in Hollywood is missing a bet. Movies based on Williamson's The Legion of Space, After World's End, or The Legion of Time would be box-office gold. The Three Musketeers paired with Shakespeare's Falstaff with ray-guns, spaceships, cosmic horror, and a goddesss-like leading lady armed with the ultimate weapon; what's not to like? It would certainly show Star Wars how real space opera is done!


(ed note: Jack Williamson's time travel theory is a unique combination of multiple timelines and fixed timelines joined with Heisenberg's uncertainty principle. With the fallacy of the animated Minkowski diagram thrown in, just for fun.

In the novel there are two contradictory future time-lines. Only one can exist, so they are battling each other for survival. Basically each is trying to increase the probability of their timeline at the expense of their opponent. There is the Utopian future of Jonbar, with their ethereal virginal princess Lethonee, and the hideous nightmare of Gyronchi, with the evil but savagely beautiful Sorainya quote "clad in a gleaming crimson tunic of woven mail that swelled with her womanly curves" unquote. Scarlet woman indeed.

Nice usage of C. S. Lewis' "male heart haunted by a terrestrial and an infernal Venus" motif from The Screwtape Letters. You can see this in the Tarot major arcana card "The Lovers" in some decks with Paris of Troy standing between Helen and the goddess Aphrodite. Or a man torn between the virgin and the temptress, hand-in-hand with the virgin but simultaneously clutching at the temptress' skirt. Maybe even the barest hint of the Madonna–whore complex. But I digress.

Anyway, our hero Dennis Lanning is the only one who can thwart the sinister designs of Sorainya. He is visited by both ladies and draw into the conflict. He also discovers that the time war started due to the meddling of his old college roommate Wilmont McLan. Wilmont creates a time-viewer, which unbeknown to him damages the probability of Jonbar's existence. Sorainya beguiles Wilmont into creating a time machine so he can travel to meet her. Whereupon she captures him, torturing him for ten years while her scientists study the time machine. Finally Lethonee manages to engineer Wilmont's escape. He then recruits Dennis and other soldiers from different times for the fight to defeat Sorainya)

     (Wilmont McLan wrote) "To an external observer, gifted with four-dimensional senses, our quadraxial universe must appear complete, fixed, and forever unchanging. The sweep of time is no more than the hand of a subjective watch; it is no more than the intangible ray of consciousness, illuminating human experience. In any absolute sense, the events of yesterday and tomorrow are alike eternal, immutable as the structure of space itself."

(ed note: but in his latter book Wilmont McLan realizes that Heisenberg's uncertainty principle makes the future quite changeable.)

     "I'll try to tell you, Denny." Her (Lethonee) face was illuminated, like a shrine, by the shimmer of the jewel in her hands. "The world is a long corridor, from the beginning of existence to the end. Events are groups in a sculptured frieze that runs endlessly along the walls. And time is a lantern carried steadily through the hall, to illuminate the groups one by one. It is the light of awareness, the subjective reality of consciousness.
     "Again and again the corridor branches, for it is the museum of all that is possible. The bearer of the lantern may take one turning, or another. And always, many halls that might have been illuminated with reality are left forever in the dark.
     "My world of Jonbar is one such possible way. It leads through splendid halls, bright vistas that have no limit Gyronchi is another. But it is a barren track, through narrowing, ugly passages, that comes to a dead and useless end."
     The wide solemn eye of Lethonee looked at him, over the slumberous flame of the jewel. Lanning tensed and caught his breath, as if a light cold hand, from nowhere, had touched his shoulder.
     "You, Denny Lanning," she went on, "are destined, for a little time, to carry the lantern. Yours is the choice of reality. Neither I nor Sorainya can come to you, bodily—unless perhaps at the moment of your death. But, through a partial mastery of time, we can each call to you, begging you to carry the lamp into our different halls. Denny—" The silver voice caught with emotion.
     "Denny, think well before you choose. For your choice will bring life to one possible world. And it will leave another in the darkness, never to be born."

     "Don't forget, Sorainya," he (Denny) muttered. "I saw the shark."
     She tossed back her head, and her hair fell like a yellow torrent across the colored cushions. And the lure of her smile set a pain to throbbing in his throat.
     "The shark would have killed you, Denny. But you should know that death alone can bring you to me—and to the strong new life the gyrane gives. For our lives were cast far apart in the stream of time. And not all the power of the gyrane can lift you out of the time-stream, living—for then the whole current must be deflected. But the stream has little grasp upon a few dead pounds of clay. I can carry that to Glarath, to be returned to life."

     (Wilmont told Denny) "Mere probability is all we have left. And my first actual invention was a geodesic tracer, designed for probability analysis. It was a semi-mathematical instrument, essentially a refinement of the old harmonic analyzer. Tracing the possible world lines of material particles through time, it opened a window to futurity."
     The hoarse whisper paused, and old Wil McLan limped to the side of the dome. His scarred trembling hands lifted a black velvet cover from a rectangular block of some clear crystal mounted on the top of a metal cabinet.
     "Here is the chronoscope," he said. "A sort of window into time. It creates special fields, that bend radiation into the time-axis. We get a stereoscopic image in the crystal screen—there's a selective fluorescence to the beat frequencies projected from below."
     The old man snapped a switch, manipulated dials at the end of the crystal block. It lit with a cloudy green. The green cleared, and a low cry escaped Lanning's lips. Within the crystal, microscopically clear, he saw a new world in miniature.

     "It happened," the hoarse voiceless gasp went on, "that Gyronchi was the first future world, out of those possible, that the chronoscope revealed. Happened that I found Sorainya, splendid in her armor, fencing with one of her human ants.
     "You can see that she is—well, attractive. At first the range of the instrument was limited to her youth, where scenes of such barbarity are less frequent. Remember, Denny, I was thirty years younger when I first saw her, back in 1945. Her glorious beauty, the military pomp of her empire—I was swept away.
     "Neglecting all the other possible worlds, I followed her, for months—years. I didn't know, then, all the harm the temporal searchbeam was doing." His white head bowed; for a moment he was speechless. "But no process whatever can reveal the state of an electron without changing that state. The quanta of my scanning ray were absorbed by the atoms that refracted them. The result was an increase in the probability factor of Gyronchi—that is the root of all the tragedy."
     The scarred face made a grimace of pain.
     "The blame is mine. For, before I was aware of it, the absorption had cut down the probability of all other possible worlds, so that Gyronchi was the only one the limited power of my instrument could reach. That blinded me to the crime that I was doing.

     (Wilmont said) There is a terrific resistance to the displacement of any body in time. For the geodesies are anchored in the future, as well as in the past. The removal of a living person, which might warp all futurity, is impossible. And even to dislodge inert matter requires tremendous power.
     "Nothing less than atomic energy, I soon perceived, could even begin to overcome that resistance. I set out, therefore, with the searching ray of the chronoscope, to study the atomic science of the future. But there I met a curious difficulty…
     "For the instrument, which, after all, can only trace out probabilities, sometimes queerly blurred the fine detail of script or printing. Los Alamos and the Kremlin were equally open to the searching beam. I studied the works of many future scientists—of John Barr and Ivor Gyros and many more. But essential words always faded.
     "There is a law of sequence and progression, I found at last, operating along a fifth rather than the temporal dimension, which imposes inexorable limits. It is that progression which actually creates reality out of possibility. And it is that higher law which prohibits all. the trite absurdities met with in the old speculation about travel in time, such as the adventurer in time who returns to kill himself. The familiar logic of cause and effect is not abolished, but simply advanced to a higher dimension.

     "I went alone, (foolishly using his time machine to travel to Sorainya)" Wil McLan looked back to him, with hollow, haunted eyes. "For the Chronion, with all her millions of horsepower, could not have drawn a crew of sound men from their places in time. Even alone, I had difficulty. An overloaded field coil burned out. The laboratory caught fire, and I was badly injured. The very accident, however, so weakened my future geodesies that the time-drive could pull me out. At the very instant the burning building collapsed, we broke free into the tune stream."

     (after ten years in Sorainya torture chambers, a projection of Lethonee helps Wilmont escape) "With seconds to spare, I got aboard the Chronion, started the converters, and escaped into time. I returned to the early twentieth century (that point in time where the time-lines branch into Jonbar and Gyronchi). And then at last, guided by Lethonee down the fainter geodesies of her possible world, I came to Jonbar."
     "Jonbar—" Lanning interrupted again, with a quick gesture at the crystal block of the chronoscope. "Can we see Jonbar, in that? And—Lethonee?"
     Very gravely, Wil McLan shook his white, haggard head.
     "Presently, we shall try," he whispered. "But the probability factor of Jonbar has become so small that I can reach it only with the utmost power of the scanning beam, and then the images are very poor. For Jonbar is at the brink of doom."
     "But there is still one chance." A stern light flashed in his hollowed eyes. "Jonbar hasn't given up. It was Lethonee's father, an archeologist digging in the Rockies where my laboratory used to be, who found there the charred books and age-rusted mechanisms from which he rediscovered the secret of time.
     "He made the time crystal. With it, Lethonee soon discovered the menace born of my unwitting tampering with probability. And she brought me to Jonbar to aid the defense. That is why I have been gathering up you and your men, Denny."
     Lanning was staring at him, frowning.
     "I don't understand," he muttered. "What can we do?"
     "These two possible worlds, each armed with the secret of time, are fighting for survival." A fierce glint burned in the old man's eyes. "Either Jonbar or Gyronchi—either Lethonee or Sorainya—may exist. But not both. The battle is on, all along the front of time. The outcome will be fixed by that higher progression, in the fifth dimension."
     "But you can see the future," broke in Lanning. "Can't you tell?"
     "The chronoscope reveals no certainties," said McLan. "Only probabilities—which it changes even as it reveals them." His white head shook. "I know, though, that the balance of probability is far in favor of Sorainya."

     "But we can help?" he demanded. "What is our part?"
     "No direct geodesies link Jonbar and Gyronchi," explained McLan. "Therefore they have no common reality. They are contradictory. They can explore each other's trains of probability. But there can be no physical contact, because the existence of each is a denial of the other. Their forces, therefore, can never come directly to grips.
     "Our contemporary world, however, joined by direct geodesies with all possible futurities, has a common existence with both Lethonee and Sorainya. That's how you get into the picture, Denny."
     "Huh?" Lanning leaned forward desperately. "They both talked of destiny. You can tell me what they meant?"
     "You are in the key position, Denny," breathed McLan. "Fate has made you the champion of Jonbar. Your triumph alone can save it. If you fail, it is lost."
     "And that's why they came to me?"
     "Sorainya has sought to cause your death." The old man nodded. "To carry you to Gyronchi, where your aid would insure her victory. And Lethonee took it upon herself to watch over you, until the moment we could pull you aboard the Chronion."
     "Death..." Lanning whispered the echo. "Then we are—dead?"
     "I came back to find you and a band of your contemporaries, to serve Jonbar. Since it is impossible to draw a sound, living man from his place in time—to do so might wrap the whole continuum—we had to wait until the moment when each of you was actually dead, to draw you aboard through the temporal field. Jonbar has provided a corps of surgeons, who were able to revive you immediately."

     Wil McLan had been collecting weapons. There were a dozen Mauser rifles, two dozen Luger pistols, four crated machine guns, several boxes of hand grenades, and a hundred thousand rounds of assorted ammunition, that all had come, along with a stock of food and a few medical supplies, from a sinking munitions ship.
     "The first precaution," McLan told him. "We located a torpedoed ship, when we first came back from Jonbar, to collect supplies and arms—and test our technique of recovery. Weapons from Jonbar, you see, wouldn't function against targets from Gyronchi."
     Since McLan's helpers from Jonbar would be unable to enter Gyronchi, Lanning detailed Clark, Barinin, and Willie Rand as a crew for the Chronion, and himself learned something of her navigation, as the time ship drove steadily down the geodesies of Jonbar. The hydrogen converter throbbed endlessly beneath the deck, but Wil McLan seemed disheartened with their progress.
     "The world we seek is now all but impossible," he rasped. "The full power of the field drives us forward very slowly. And at any instant the geodesies of Jonbar may break, for they are weak enough already, and leave us—nowhere!"

     "We are already doing all that can be done," he said. "The geodesies of Jonbar are like microscopic wires drawn out thinner and thinner by the attenuation of probability. If the tracer loses them, or if they snap, Jonbar is—lost!"

     (Our Heroes arrive at Jonbar and Lethonee) "Gyronchi has carried out some new attack," she (Lethonee) told him. "The dynon (the future race that springs from Jonbar) tried to bring a warning from the future, but they were cut off. Now the time crystal shows no future at all, beyond tonight. This is the last possible night for Jonbar. Unless—"
     Her haunted eyes clung desperately to Lanning's face.
     "Unless the tide of probability is changed."

     "How can you be—not real?" Lanning stood gazing at her quiet loveliness, framed against the terrace garden. "What's the difference between reality and—such a seeming as you are?"
     She hesitated, with a little frown of thought.
     "There is a flow from probability to certainty, along the fifth dimension," she explained. "Probabilities are infinite, but there is only one reality. Many conflicting futures are possible, but the past is simple and complete! The geodesies branch at each point of uncertainty, but the flow of realization must always take one branch and obliterate the rest. All the geodesies tend to absorb energy; all possible worlds strive for reality. But the energy of probability must always be withdrawn again from all those other worlds that might have been, to create the single one that can be. All the rest must vanish, as their probability fades to zero."
     "And Jonbar is—vanishing?"
     She nodded. "It—and I. We were given creation by the atomic power of the Chronion, bringing you down the geodesies. We are only an illusion of possibility, the reflection of what may be—a reflection that is doomed."

     (Our Heroes escape from Jonbar just as it vanishes from probability, and travel back to the present in order to save the day. Wilmont explains Gyronchi latest attack on Jonbar) "They went back into the past," said the voiceless-man. "Back to the turning point of probability. They found something there—it must have been a small material object, although we got no glimpse of it—which was the very foundation of Jonbar. Using gyrane power, they wrenched the thing, whatever it was, out of its place in time. The broken geodesies cut off the possibility of Jonbar."
     "What became of this object?"
     "They kept it concealed. And they carried it back to Gyronchi. It is guarded, there, in Sorainya's fortress."
     "Guarded?" Lanning echoed. His fingers twisted together in a sudden agony of hope, and his eyes rose to search McLan's wealed face. "Then if we took it—carried it back—would that help Jonbar?"
     "Yes." The bent white head moved to a tiny nod. "If we could recover the object, if we could discover where they found it, in space and time, if we could put it back there, if we could prevent Sorainya from disturbing it again until the turning point has passed in the fifth dimension—then Jonbar would again be possible."

     (Wilmont discovers the key change in time that will decide which future will occur, Jonbar or Gyronchi) "The time is an afternoon in August of the year 1921," whispered Wil McLan. "The broken geodesies of Jonbar had already given us a clue to that. Now I have found the place, with the search beam."
     Lanning gripped his arm. "Where?"
     "It's a little valley in the Ozarks of Arkansas. I'll show you the decisive scene."
     McLan limped to the metal cabinet of the geodesic analyzer. His broken fingers set its dials. A greenish luminescence filled the crystal block, and cleared. Lanning bent forward eagerly, looking into that strange window of probability. An eroded farm, folded in the low and ancient hills. A sagging paintless shack, a broken window gaping and the roof inadequately patched with rusty tin. A rocky cow pasture, its steep slopes scantily covered with useless brush. A small freckled boy in faded overalls and a big ragged straw hat, trudging slowly barefoot down the slope, accompanied by a gaunt yellow dog, driving two lean red-spotted cows home to the milking pen.
     "Watch him," whispered Wil McLan urgently.
     As Lanning watched, the boy stopped to encourage his dog digging furiously after a rabbit. He squatted to observe a colony of ants. He ran to catch a gaudy butterfly, and carefully dissected it with a broken pocket knife. He rose unwillingly to answer the calls of a slatternly woman from the house below, and ambled after the cows again. Wil McLan's gnarled fingers closed on Lanning's arm, urgently.
     The boy paused over something beside a sumac bush, and stopped to pick it up. The object blurred oddly in the crystal screen, so that Lanning could not distinguish it. The scene was erased, as Wil McLan snapped off the mechanism.
     "Well?" Lanning turned to him, in bewilderment. "What has that to do with Jonbar?"
     "That is John Barr," said the voiceless man. "For that metropolis of future possibility will be—or may be—named for him. He is twelve years old in 1921, barefoot son of a tenant farmer. You saw him at the turning point of his life—and the life of the world."
     "But I don't understand!"
     "The geodesies diverge from the thing he stoops to pick up," whispered Wil McLan. "It is either the magnet that we recovered from Sorainya's citadel—or else only an oddly colored pebble that lies beside it. That small choice—which Sorainya sought to decide by removing the magnet—determines which one of two possible John Barrs is to be ultimately established in reality."
     "Just a scrap of iron," Lanning said.
     "The seed of Jonbar," answered McLan. "If he picks up the discarded magnet, he'll discover the mysterious attraction it has for the blade of his knife, and the strange north-seeking power of its poles. He'll wonder, experiment, theorize. His curiosity will deepen. The scientist will be born in him.

     "But at last, in 1980, a tired but triumphant little man of seventy-one, he'll publish his great discovery. The dynatomic tensors—soon shortened to dynat. A totally new law of nature, linking life and mind to atomic probability. I had stumbled on one phase of it, with the hydrogen converter. But his tensors will open up a tremendous new technology for the direct release of atomic energy, under full control of the human will.

     "If we're unable to replace the magnet," McLan whispered again, "the boy John Barr will pick up the pebble instead, and the tide of probability will be turned—as, indeed, it is turned—toward Gyronchi. The boy will toss the pebble in his hand, and throw it in his sling to kill a singing bird. All his life thereafter will want a precious spark. It will remain curiously similar, yet significantly different.
     "John Barr, in this outcome also, will run away from his father's home, but now to become a shiftless migratory worker. He will marry the same woman, raise the same two children, and leave them at last. The same ingenuity, turned to the same basic problems of probability, will lead him to invent a new gambling device, on which he will make and lose a fortune. He will die, equally penniless, in the same year, and lie at last in the same graveyard.
     "The secret of mentally released atomic power will now be discovered nine years later, but with a control far less complete than John Barr would have attained. The discoverer will be one Ivor Gyros, an exiled engineer from Soviet Eurasia, working with a renegade Buddhist priest. Calling their half-mastered secret the gyrane, the two will guard it selfishly, use it to destroy their enemies and impress the superstitious. They'll establish a fanatical new religion, and a new despotic empire. That's the beginning of the cult of the gyrane, and Sorainya's dark dynasty. You have seen the end of them."

From THE LEGION OF TIME by Jack Williamson (1952)

Multiple Timelines

AKA Multiverse, Alternate Time lines, Parallel Universes, Cross-time, Branching Timelines. In the physics version, each time a time traveler changes history, the time line branches into the changed history line and the unchanged history line. In the literary version, each time any decision is made by any creature in the entire universe, the time line branches into one branch per possible outcome. Immune to temporal paradox

Also called "Parallel Universes" or "Cross-time". In his Discworld novels Terry Pratchett calls it the "trousers of time" theory.

These can be found in Andre Norton's Crosstime novels and "All The Myriad Ways" by Larry Niven.

In many science fiction novels, a new time line is spawned every time a decision is made. In many attempts by physicists to protect causality, a new time line is only spawned when some meddling time traveler alters history (the theory of parallel universes).

Known as the theory of multiple time tracks.

Let there be a myriad of realities, of universes. For every decision made by any form of life, let it be made both ways; or in all possible wars if there are more then two choices. Let universes be created with every choice.

Then conservation of matter and energy holds only for the universe of universes. One can move time machines from one universe to another.

You've got to admit it's flamboyant!

You still can't visit the past. But you can find a universe where things happened more slowly; where Napoleon is about to fight Waterloo, or Nero is about to ascend the throne. Or, instead of changing the past, you need only seek out the universe where the past you want is the one that happened. The universe you want unquestionably exists. (Though you may search a long, weary time before you find it.)


Temporal Paradoxes

A logical paradox is a statement which is both true and false at the same time. Example: "This statement is false". A temporal paradox is when that happens with time travel. Logical paradoxes strike at the heart of logic, while temporal paradoxes strike at the heart of causality and physics.

There are more or less three major classes of events that are commonly called Temporal Paradoxes.

Actually the Grandfather Paradox is the only full-blown causality-destroying paradox. The other two are not paradoxes, they are just terrifically strange.

There are a couple of ways to avoid temporal paradoxes.

In quite a few science fiction stories, dealing with a temporal paradox is pretty much the entire plot.

Grandfather Paradox

The classic Time-travel paradox is the so-called "Grandfather paradox" (though it actually should be called the "Grandmother paradox"). The technical term is "inconsistent causal loop." This is where a future event prevents an event in the past by its backward influence on the past event. Remember that bootstrap paradoxes are when a future event causes an event in the past.

Boris Badenov sneaks into Mr. Peabody's Wayback Machine (actually the WABAC machine, but who cares?) and travels back in time to when Boris' grandfather was a baby. Boris then gives his infant grandfather a lit stick of dynamite then cackles evilly as his grandfather is blown to bits. Bah-hah-hah!

But wait! Boris' grandfather is now smithereens, he'll never grow up, beget Boris' father, who will beget Boris. In other words, Boris will never exist.

But if Boris never exists, then he will never travel back in time to assassinate his grandfather. In which case grand-pop will beget Poppa Boris, who will beget Boris. Who will then proceed to assassinate his grandfather. Start back at the beginning and repeat.

Does Boris' grandfather get blown up? Both Yes and No! A paradox.

Boris says: "Natasha, next time I get fiendish plan, do me big favor? Sharrup my mouth!"

You can tighten this up a bit by having the time traveler going back in time to murder their infant self instead of their grandfather. The technical term is autoinfanticide.

The real key factor in the paradox is not the grandfather, it is the presence of something that eliminates the cause or means of traveling back in time.

Example of eliminating the cause: the Hitler's murder paradox. A common time-travel trope is traveling back in time to murder Hitler before he starts World War 2. The trouble is, if one succeeds, the time traveler will have grown up in a world where Hitler and World War 2 never existed. So the time traveler has no cause or reason to travel back in time to murder some guy he never heard of in order to stop a war that never happened. Which means Hitler was not assassinated, and the paradox is upon us.

Example of eliminating the means: The Self-Destructing Time Machine paradox. Your grandfather invented and build a time machine. You hate the old man, but if you kill him you'll wind up to be the prime suspect. So you steal his time machine and go back in time. If you kill him at a point in time before you were born, obviously the police will never suspect a person who hasn't been conceived yet. Unfortunately, since you are long on anger and short on brains, you neglected to take into account the fact that you will be killing grandpaw before he invents and builds the time machine (i.e, the means for traveling in time). The temporal paradox sanctions Causality with extreme prejudice as the universe tries to figure out where the heck the time machine came from.

The basic Grandfather Paradox is a type of "eliminating the means." It is just that the "means" is not the time machine, the "means" is the time-traveler.

Another thing to keep in mind is that paradoxes do not require somebody to travel back in time. You can make perfectly good causality-destroying paradoxes by just sending information back in time. Send a message to your future self asking it to send a message to your past self telling you not to send any messages in the present.

Bootstrap Paradox

A Causal Loop is where an event in the future causes an event in the past, through time travel. More simply it is where an effect is its own cause.

It is also known as a bootstrap paradox, ontological paradox or a closed time loop. It is very popular in science fiction stories.

This upsets physicists since Causality says it should be cause and effect NOT effect and cause.

The classic science fiction bootstrap stories are Heinlein's By His Bootstraps and All You Zombies.

It really becomes a full blown paradox when you take an object's beginning and ending then merging them so that the object's world-line becomes a world-circle. You wind up with an object that has no origin or ending. Though really this is not a paradox so much as it is just very very weird.

Example. The pocket-watch from the movie Somewhere In Time.

  1. In 1971 the young Richard Collier is approached by an elderly lady. The lady gives him a pocket watch and says "Come Back To Me."
  2. Richard keeps the watch, and is unaware that the lady dies that night.
  3. In 1979 Richard is a playwright, but has writers block. He goes to a charming Grand Hotel.
  4. At the hotel, in its museum, he falls in love with a photo of Elise McKenna.
  5. Doing some research, he is shocked to discover that Elise McKenna was the elderly lady who gave him the watch.
  6. Richard manages to travel back in time to 1912, where he and Miss McKenna fall in love.
  7. Richard gives Miss McKenna the watch

The paradox is: when was the watch constructed? It has no beginning and no end.

Another example is SkyNet from the Terminator movies. In the future, the evil SkyNet supercomputer creates Terminator robots to exterminate mankind. It sends a T-800 Terminator back in time to kill Sarah Connor. Even though the T-800 is stamped flat in a hydraulic press, enough of it is left for scientists to examine. They discover computer techniques that humans would never had figured out on their own.

And of course they use these techniques to create ... SkyNet. So where did these technique come from? The same place as Elise McKenna's pocket watch.

This works with ideas and theories as well. Example: Floyd studies Einstein's relativity in college, basically learning the theory from Einstein. Floyd then uses a time machine to go visit Einstein in 1890. Floyd then proceeds to teach the young Einstein the theory of relativity. Einstein taught Floyd who then taught Einstein. Who invented the theory in the first place?

TV Tropes calls this Stable Time Loop.

Predestination Paradox

This is not so much a paradox as it is an unfortunate feature of time travel in a universe run by Fixed Timeline or drastically Self-Healing Timelines.

It is vaguely related to the Bootstrap Paradox, and also to a self-fulfilling prophecy. It happens when a time traveler is motivated to travel into the past to stop an awful event from happening. But when they try, it turns out that the time traveler is the one that caused the event in the first place.

If they had not time traveled in the first place, there would not have been an awful event in the first place. Thus then there would not have been any motivation to time travel, because nobody would time travel to stop something that never happened.

Determinism is really depressing sometimes.

TV Tropes calls this You Can't Fight Fate.

Avoiding Paradoxes

The way to keep physicists happy is to postulate a law of science that somehow makes temporal paradoxes impossible.

Jason Hinson shows there are four ways of enforcing a "no-paradox" rule for time travel. Parallel Universes, Consistency Protection, Restricted Space-Time Areas, and Special Frames. In some ways Special Frames is the best, though it directly contradicts part of Relativity (the first postulate of special relativity is that there are no special frames, "no privileged inertial frames of reference"). Oh well. For details, you'd best read the Hinson article.

The latter three are examples of the Novikov self-consistency principle. It states that: "The only solutions to the laws of physics that can occur locally in the real universe are those which are globally self consistent." In other words if you try to use a time machine to make a paradox, the universe will fix things so you automatically fail. This is sometimes summarized as "you can't change recorded history." Note this implies that the Time Protection Corpstm will go out of their way to not record or observe unknown parts of history, in order to give the Corps some wiggle room on their missions.

It seems to me that a time machine ruled by the Novikov self-consistency principle would operate in a very strange and non-intuitive way. It might be that occasionally the chrononaut would set up a trip and the time machine would refuse to operate. Then the chrononaut would know that somehow someway the proposed trip would cause a paradox.

Or even worse, after a time trip, the chrononauts would discover that if they try certain actions the entire universe throws up random events preventing said actions to avoid creating a paradox. By the same token the entire universe might throw up random events forcing a chrononaut to perform some action, since the action is necessary to prevent the creation of a paradox.

Novikov Principle of Self Consistency

Amazingly enough, there are presently a series of papers appearing in the scientific literature that discuss in great mathematical detail the problem of paradoxes created by time machines. The first of these papers are those by Friedman et al., Echeverria et al., and Novikov to be found in Recommended Reading at the end of this chapter. To everyone's amazement (including the amazement of the scientists writing the papers), they find that instead of their mathematics showing that the existence of logical paradoxes proves that time machines cannot exist, their mathematics indicates that the logical paradoxes cannot exist! For every paradox that they can dream up (for example, someone deciding to go back into time to shoot himself), a detailed mathematical analysis of the way that nature will behave according to the known laws of physics shows that nature will automatically adjust itself so that the paradox will not occur!

These results are now embodied in the Novikov Principle of Self Consistency, which states that: "The only solutions to the laws of physics that can occur locally in the real universe are those which are globally self consistent."

In the example of the person deciding to got back into time and shoot himself, either he changes his mind, or the gun doesn't work, or the bullet misses, or he kills someone who looks just like him, or something else happens to prevent the paradox. Admittedly, the event that prevents the paradox may have a low probability of happening, but once a time machine exists, then possible physical events around that time machine are constrained so that the Principle of Self Consistency is observed, and low probability events become high probability events.

Once you have made a decision to jump off a bridge, nature will severely limit your possible future courses of action. In the same way, once you have made a decision to turn on a time machine, nature will severely limit your future courses of action to prevent you from creating any paradoxes.

From Indistinguishable From Magic by Robert Forward (1995)

Recently physicists Daniel Greenberger and Karl Svozil have shown that the laws of quantum mechanics enforces Consistency Protection (paper here). Translated into English, they maintain that time travellers going back into the past cannot alter the past (i.e., the past is deterministic). This is because quantum objects can act sometimes as a wave. When they go back in time, the various probabilities interfere destructively, thus preventing anything from happening differently from that which has already taken place.

In Matt Visser's The quantum physics of chronology protection he notes that physicists see time travel as problematic, if not outright repugnant. They hate paradoxes, and they hate the fact that relativity contains a plethora of paradox-creating closed timelike curves (AKA time machines). Visser says "General Relativity is in fact infested with peculiar geometries that seem to produce time machines."

Visser say that physicists tend to have one of four reaction to this unpleasant state of affairs:

  1. Go whole-hog: embrace time travel and try to control the paradoxes. The least radical ones favor the Multiple Timelines solution to paradoxes. More radical ones favor exotic stuff like non-Hausdorff manifolds and multiple coexisting versions of the "present".
  2. Grudgingly allow the existence of time travel but forbid changing history. They want a Fixed Timeline and/or a strong Novikov self-consistency principle. They are not too happy about the implied elimination of free will, but you can't have everything.
  3. Beg quantum physics to come up with reason that time travel is impossible. This is Hawking's Chronology protection conjecture.
  4. Ignore it and hope it goes away. They decide to not think about such repugnant things until the experimental evidence becomes too huge to sweep under the rug. This is sometimes called the "Boring Physics" conjecture. They favor things like global hyperbolicity (no, I do not understand it) and cosmic censorship.

Time Machines

Time travel into the future is relatively easy. After all, we are all moving into the future at the rate of one second per second (local). It is possible to low the rate for yourself (relative to, say, Terra) by moving at relativistic velocities or loitering near a strong source of gravity. For instance, if you and your starship is moving relativistically at 99% the speed of light relative to Terra, then if you do a round trip that takes one day your time (proper time), upon arrival everybody on Terra will say a week has passed.

Time travel into the past is the difficult part.

Current physics admits to two possible methods of time travel into the past:

  • Traveling faster than the speed of light
  • Using one of the weird spacetime predicted in general relativity that can contain closed timelike curves (time machines)

Under Einstein's Relativity theory, all faster-than-light starships are time machines. Actually relativity states that FTL travel and Time Travel are two different terms for the exact same thing. Most physicists categorically reject the possibility of FTL travel out of hand, mainly because they hate time machines so much.

Dr. Marc Rayman tells us that in all time travel theories that seem to be allowed by real science, there is no way a traveler can go back in time to before the time machine was built. This tells me that one should have an army or two around a time machine as it finishes construction, in order to deal with the invasion force(s) from the future that come charging out.

Dr. Matt Visser noted "General Relativity is in fact infested with peculiar geometries that seem to produce time machines." Examples include:

Traversable wormholes
A Traversable wormhole is one where the hole is held open with exotic matter (negative cosmic strings or something like that). It can be converted into a time machine by accelerating one of the wormhole mouths relativistically relative to the other mouth, then bringing them back together. You can find more details here. Stephen Baxter used this in his science fiction novel Timelike Infinity.
Alcubierre metric
The Alcubierre metric is used in the Alcubierre Drive. It is a clever way to use general relativity as a faster than light drive. Objects cannot move faster than light, but Einstein didn't say anything forbidding a piece of spacetime from moving faster than light. It is a time machine because all FTL drives are. The Alcubierre drive has some major difficulties.
BTZ black hole
A BTZ black hole is a black hole solution for (2+1)-dimensional gravity with a negative cosmological constant. Since our universe has 3+1 dimensional gravity with a positive cosmological constant, we cannot use them.
Gödel metric

The Gödel metric was invented by Kurt Gödel specifically to prove that Einstein's equations of spacetime contained a severe flaw. Among other weird things it predicts that every single event inside such a metric will have several closed timelike curves passing through it (a time machine at every corner, so to speak).

Einstein seemed to be of the opinion that given the choice between his equations of spacetime being wrong and the weirdness of the Gödel metric being true, he'd chose the latter. Take that, Gödel!

The Gödel metric only applies to our universe if the value of the cosmological constant precisely matches the density of matter.

van Stockum dust
van Stockum dust is similar to a Gödel metric, but the universe's density has to increase with distance from the rotation point. It allows closed timelike curves, but it does not seem to apply to our universe.
Kerr metric

The Kerr metric describes the warped space around black holes with mass, spin, but no charge (i.e., practically all naturally occurring black holes). There are three other kinds of black holes, but we don't care about them at the moment.

Anyway apparently there is an inner region inside the black hole that contains closed timelike curve. The Cosmic censorship hypothesis suggests that if you actually try to use the closed timelike curves as a time machine, the inner region will collapse. Much like the weakly god-like AI called the Eschaton in Charles Stross's novel Singularity Sky, who commands: "Thou shalt not violate causality within my historic light cone. Or else."

Krasnikov tube
A Krasnikov tube is a speculative mechanism for space travel involving the warping of spacetime into permanent superluminal tunnels. Much the same as using traversable wormholes as time machines, you displace the ends of the tubes in time as well as space.
Misner space
Misner space is an abstract mathematical spacetime that acts like the world in such video games as Asteroids (if you exit off the left edge, you will instantly re-enter from the right). Our universe does not act like that so this is worthless to us.
Tipler cylinder
A Tipler cylinder is a dense massive cylinder of infinite length rotating near the speed of light. Due to dragging of the metric, it creates a region containing close timelike curves. A spacecraft passing through the proper curve can travel back in time. Stephen Hawking pointed out that if the cylinder's length is less than infinity, it will require exotic matter with negative energy (which may not even exist, and in any event nobody knows where to obtain any).

Causal Weapons

If you have a time-travel system where history can be changed (no fixed or self-healing timelines, see Stuart Armstrong's system below), about thirty seconds after someone invents a time machine somebody will have the bright idea of weaponizing it. If you have always been at war with Eastasia, you could pop back in time to a critical juncture and alter things so Eastasia vanishes from the pages of history.

Such time meddling is ill-advised, since there are always law of unintended consequences. And don't even bother trying to assassinate Hitler, that trick never works.

This is why it is so dangerous to use your star fleet to attack the planet Gallifrey, homeworld of the Time Lords. If you do not instantly wipe them out, one will pop into a TARDIS and instantly you and your home planet will have never existed.

A nation with a monopoly on time travel technology is more or less invincible. Examples include L. E. Modesitt, Jr.'s The Fires of Paratime and Isaac Asimov's The End of Eternity.

But if others have the secret of time travel, you will suddenly have a Time War on your hands. Your time warriors travel back to eliminate the nation of Eastasia, only to be confronted by Eastasian time traveling defenders. Meanwhile Eastasian kill teams will be elsewhen trying to erase your nation from the history books.

Examples of science fiction stories featuring time war include Poul Anderson's The Corridors of Time, Fritz Leiber's The Big Time and Andre Norton's Time Trader series.

However, your team might be suddenly confronted by time warriors from 40,000 AD whose existence depends upon Eastasia and take a dim view of your short-sighted actions. This can quickly get out of hand. Your time warriors might suddenly be faced with several thousand other time-traveling groups from all sorts of historical eras. Some might even want to help you.

Even worse, the Time Police might show up. This is an organization(s) existing in your far future who (for whatever reason) frown upon history alterations. They usually have the equipment, training, and backing of an organization huge enough to make your time warrior's lives really difficult. Examples include Keith Laumer's Dinosaur Beach and Poul Anderson's Time Patrol series.

And if your time warrior team is really flat out of luck, they will find themselves being attacked by Clock Roaches.

Time travel system


I’ve been looking to develop a system of time travel in which it’s possible to actually have a proper time war. To make it consistent and interesting, I’ve listed some requirements here. I think I have a fun system that obeys them all.

Time travel/time war requirement:

  • It’s possible to change the past (and the future). These changes don’t just vanish.
  • It’s useful to travel both forwards and backwards in time.
  • You can’t win by just rushing back to the Big Bang, or to the end of the universe.
  • There’s no “orthogonal time” that time travellers follow; I can’t be leaving 2015 to go to 1502 “while” you’re leaving 3015 to arrive at the same place.
  • You can learn about the rules of time travel while doing it; however, time travel must be dangerous to the ignorant (and not just because machines could blow up, or locals could kill you).
  • No restrictions that make no physical sense, or that could be got round by a human or a robot with half a brain. Eg: “you can’t make a second time jump from the arrival point of a first.” However, a robot could build a second copy of a time machine and of itself, and that could then jump back; therefore that initial restriction doesn’t make any particular sense.
  • Similarly, no restrictions that are unphysical or purely narrative.
  • It must be useful to, for instance, leave arrays of computers calculating things for you then jumping to the end to get the answer.
  • Ideally, there would be only one timeline. If there are parallel universes, they must be simply describable, and allow time-travellers to interact with each other in ways they would care about.
  • A variety of different strategies must be possible for fighting the war.

Consistent time travel

Earlier, I listed some requirements for a system of time travel – mainly that it be both scientifically consistent and open to interesting conflicts that aren’t trivially one-sided. Here is my proposal for such a thing, within the general relativity format.

So, suppose you build a time machine, and want to go back in time to kill Hitler, as one does. Your time machine is a 10m diameter sphere, which exchanges place with a similarly-size sphere in 1930. What happens then? The graph here shows the time jump, and the “light-cones” for the departure (blue) and arrival (red) points; under the normal rules of causality, the blue point can only affect things in the grey cone, the red point can only affect things in the black cone.

The basic idea is that when you do a time jump like this, then you “fix” your points of departure and arrival. Hence the blue and red points cannot be changed, and the universe rearranges itself to ensure this. The big bang itself is also a fixed point.

All this “fixed point” idea is connected to entropy. Basically, we feel that time advances in one direction rather than the other. Many have argued that this is because entropy (roughly, disorder) increases in one direction, and that this direction points from the past to the future. Since most laws of physics are symmetric in the past and the future, I prefer to think of this as “every law of physics is time-symmetric, but the big bang is a fixed point of low entropy, hence the entropy increase as we go away from it.”

But here I’m introducing two other fixed points. What will that do?

Well, initially, not much. You go back into time, and kill Hitler, and the second world war doesn’t happen (or maybe there’s a big war of most of Europe against the USSR, see configuration 2 in “A Landscape Theory of Aggregation”). Yay! That’s because, close to the red point, causality works pretty much as you’d expect.

However, close to the blue point, things are different.

Here, the universe starts to rearrange things so that the blue point is unchanged. Causality isn’t exactly going backwards, but it is being funnelled in a particular direction. People who descended from others who “should have died” in WW2 start suddenly dying off. Memories shift; records change. By the time you’re very close to the blue point, the world is essentially identical to what it would have been had there been no time travelling.

Does this mean that you time jump made no difference? Not at all. The blue fixed point only constrains what happens in the light cone behind it (hence the red-to-blue rectangle in the picture). Things outside the rectangle are unconstrained – in particular, the future of that rectangle. Now, close to the blue point, the events are “blue” (ie similar to the standard history), so the future of those events are also pretty blue (similar to what would have been without the time jump) – see the blue arrows. At the edge of the rectangle, however, the events are pretty red (the alternative timeline), so the future is also pretty red (ie changed) – see the red arrows. If the influence of the red areas converges back in to the centre, the future will be radically different.

(some people might wonder why there aren’t “changing arrows” extending form the rectangle into the past as well as the future. There might be, but remember we have a fixed point at the big bang, which reduces the impact of these backward changes – and the red point is also fixed, exerting a strong stabilising influence for events in its own backwards light-cone).

So by time travelling, you can change the past, and you can change part of the future – but you can’t change the present.

But what would happen if you stayed alive from 1930, waiting and witnessing history up to the blue point again? This would be very dangerous; to illustrate, let’s change the scale, and assume we’ve only jumped a few minutes into the past.

Maybe there you meet your past self, have a conversation about how wise you are, try and have sex with yourself, or whatever time travellers do with past copies of themselves. But this is highly dangerous! Within a few minutes, all trace of future you’s presesence will be gone; you past self will have no memory of it, there will be no physical or mental evidence remaining.

Obviously this is very dangerous for you! The easiest way for there to remain no evidence of you, is for there to be no you. You might say “but what if I do this, or try and do that, or…” But all your plans will fail. You are fighting against causality itself. As you get closer to the blue dot, it’s as if time itself was running backwards, erasing your new timeline, to restore the old one. Cleverness can’t protect you against an inversion of causality.

Your only real chance of survival (unless you do a second time jump to get out of there) is to rush away from the red point at near light-speed, getting yourself to the edge of the rectangle and ejecting yourself from the past of the blue point.

Right, that’s the basic idea!

Multiple time travellers

Ok, the previous section looked at a single time traveller. What happens when there are several? Say two time travellers (blue and green) are both trying to get to the red point (or places close to it). Who gets there “first”?

Here is where I define the second important concept for time-travel, that of “priority”. Quite simply, a point with higher priority is fixed relative to the other. For instance, imagine that the blue and green time travellers appear in close proximity to each other:

This is a picture where the green time traveller has a higher priority than the blue one. The green arrival changes the timeline (the green cone) and the blue time traveller fits themselves into this new timeline.

If instead the blue traveller had higher priority, we get the following scenario:

Here the blue traveller arrives in the original (white) timeline, fixing their arrival point. The green time traveller arrives, and generates their own future – but this has to be put back into the first white timeline for the arrival of the blue time traveller.

Being close to a time traveller with a high priority is thus very dangerous! The green time traveller may get erased if they don’t flee-at-almost-light-speed.

Even arriving after a higher-priority time traveller is very dangerous – suppose that the green one has higher priority, and the blue one arrives after. Then suppose the green one realises they’re not exactly at the right place, and jump forwards a bit; then you get:

(there’s another reason arriving after a higher priority time traveller is dangerous, as we’ll see).

So how do we determine priority? The simplest seems time-space distance. You start with a priority of zero, and this priority goes down proportional to how far your jump goes.

What about doing a lot of short jumps? You don’t want to allow green to get higher priority by doing a series of jumps:

This picture suggests how to proceed. Your first jump brings you a priority of -70. Then the second adds a second penalty of -70, bringing the priority penalty to -140 (the yellow point is another time traveller, who will be relevant soon)

How can we formalise this? Well, a second jump is a time jump that would not happen if the first jump hadn’t. So for each arrival in a time jump, you can trace it back to the original jump-point. Then your priority score is the (negative) of the volume of the time-space cone determined by the arrival and original jump-point. Since this volume is the point where your influence is strongest, this makes sense (note for those who study special relativity: using this volume means that you can’t jump “left along a light-beam”, then “right along a light-beam” and arrive with a priority of 0, which you could do if we used distance travelled rather than volume).

Let’s look at that yellow time traveller again. If there was no other time traveller, they would jump from that place. But because of the arrival of the green traveller (at -70), the ripples cause them to leave from a different point in space time, the purple one (the red arrow shows that the arrival there prevents the green time jump, and cause the purple time jump):

So what happens? Well, the yellow time jump will still happen. It has a priority of 0 (it happened without any influence of any time traveller), so the green arrival at -70 priority can’t change this fixed point. The purple time jump will also happen, but it will happen with a lower priority of -30, since it was caused by time jumps that can ultimately be traced back to the green 0 point. (note: I’m unsure whether there’s any problem with allowing priority to rise as you get back closer to your point of origin; you might prefer to use the smallest cone that includes all jump points that affected you, so the purple point would have priority -70, just like the green point that brought it into existence).

What other differences could there be between the yellow and the purple version? Well, for one, the yellow has no time jumps in their subjective pasts, while the purple has one – the green -70. So as time travellers wiz around, they create (potential) duplicate copies of themselves and other time travellers – but those with the highest priority, and hence the highest power, are those who have no evidence that time jumps work, and do short jumps. As your knowledge of time travel goes up, and as you explore more, your priority sinks, and you become more vulnerable.

So it’s very dangerous even having a conversation with someone of higher priority than yourself! Suppose Mr X talks with Mrs Y, who has higher priority than him. Any decision that Y does subsequently has been affected by that conversation, so her priority sinks to X’s level (call her Y’). But now imagine that, if she wouldn’t have had that conversation, she would have done another time jump anyway. The Y who didn’t have the conversation is not affected by X, so retains her higher priority.

So, imagine that Y would have done another time jump a few minutes after arrival. X arrives and convinces her not to do so (maybe there’s a good reason for that). But the “time jump in an hour” will still happen, because the unaffected Y has higher priority, and X can’t change that. So if the X and Y’ talk or linger too long, they run the risk of getting erased as they get close to the “point where Y would have jumped if X hadn’t been there”. In graphical form, the blue-to-green square is the area in which X and Y’ can operate in, unless they can escape into the white bands:

So the greatest challenge for a low priority time-traveller is to use their knowledge to evade erasure by higher priority ones. They have a much better understanding of what’s going on, they may know where other time jumps likely end up at or start, they might have experience at “rushing at light speed to get out of cone of danger while preserving most of their personality and memories” (or technology that helps them do so), but they are ever vulnerable. They can kill or influence higher priority time-travellers, but this will only work “until” the point where they would have done a time jump otherwise (and the cone before that point).

From Time Travel System by Stuart Armstrong (2015)

Time travel can also be used for short term gain. A good example is Cordwainer Smith's short story The Crime and the Glory of Commander Suzdal. The commander is duped into traveling to the planet Arachosia, which is a trap. The natives capture his ship, and start burning their way into the airlock. Oh Noes! Whatever will the commander do?

As it turns out the commander is a tricky son-of-a-spacer. Agent ships of the Instrumentality have all sort of top-secret equipment. One of them is the chronopathic device, a time distorter that can send a eighty thousand ton starship one second into the past in order to escape destruction. Another was a life-bank containing fertilized ova of a selection animals. Suzdal noticed that the moon of Arachosia was habitable.

Suzdal took ferilized ova of eight pairs of cats. He genetically encoded them to [a] serve man, [b] evolve into an intelligence species and [c] invent high technology. He then adjusts the time distorter so instead of sending eighty thousand tons one second into the past, it instead sends four kilograms two million years into the past. He projects the cat ova in artificial wombs to the moon.

Instantly the evil Arachosians are attacked by a huge Cat starfleet that appears out of nowhere. The cats scream their allegiance to to commander Suzdal, and proceed to shoot the living snot out of the Arachosians. Suzdal escapes in the confusion.

Time-Machine Computer

First off, remember that "closed-timelike curves" (CTC) is a code word for "time machine", that is, a way to travel backwards in time.

As it turns out, if you have a way of sending things backwards in time, you can use this to make a really fast quantum computer. "Really fast" is defined as "able to quickly solve problems that would take a conventional computer longer than the lifespan of the universe".

In 2002 Dr. Todd A. Brun wrote a paper suggesting that a computer having access to CTC could actually solve solve NP-complete (NP + NP-hard) and PSPACE-complete problems. In 2003 Dr. Dave Bacon wrote a paper suggesting that a quantum computer using CTC could easily solve NP-hard problems.

But anytime closed-timelike curves are involved, the dread spectre of Temporal Paradox rears its ugly head.

In late-breaking news, Dr. Xiao Yuan et al came up with a method to use closed-timelike curves that avoided the horror of temporal paradox. In a 2015 paper they demonstrated that a computer can use quantum entanglement to prevent a time-traveling particle from interacting with the past (thus avoiding temporal paradox) yet the particle's entanglement with another particle in the present will create a gain in computational power.

You see, if you prevent the particle from interacting with the past, it becomes what is called a Deutschian open timelike curve, free from temporal paradox. But if the particle does not interact, there is no gain in computer power.

Dr. Yuan's technique prevents the particle from interacting (thus preventing paradox) but cleverly uses quantum entanglement to get a gain in computer power.

Closed-Timeline-Curve Computer

(ed note: in this story the CTC computer just rampantly breaks causality all to heck. They already have problems with causality being shattered by their faster-than-light drive. So they figure causality is highly over-rated)

     But Nilis had assigned her to another part of the project, the development of his "CTC computer," as he called it. his closed-timelike-curve time-travel computing machine. Nilis made it clear that he considered the CTC-processor work just as important as experiments with the Xeelee ship, and she had to accept the assignment.

     "I'm one of his advisors, yes." Pila waved a hand at the prototype. "Very impressive. And it's all based on time travel?"
     "Closed-timelike-curves, yes." Torec pointed to the ducts. "Pilot Officer Pirius—Pirius Blue—defeated the Xeelee because his fellow pilot used her FTL drive to bring tactical information back from the future. So we have miniature bots in those tubes. The bots are the components of the processor. They fly back and forth, and actually jump through short FTL loops."
     "Little starships in tubes! And these bots travel back in time and tell you the answer before you even pose the question?"
     "Something like that."
     "How marvelous."
     The dummy problem they were hoping to run today concerned protein folding. Proteins were the structural elements of life, but remained beyond the capability of humans to design optimally. There were more proteins a hundred components long than there were electrons in the universe; to work out how many ways a long protein molecule could fold up was an ancient problem, previously insoluble even in principle. "But we hope to crack it," Torec said. She pointed to a large blank Virtual screen. "The results will be displayed there."

     "This is only the first step. A proof of principle. Eventually we will have to cram this down into a unit small enough to be carried on a greenship."
     "A clever answer," the woman murmured. "And what is your key problem?" Torec shrugged. "Control of those flying bots. obviously. We've a list of issues."

     "Describe your algorithm."
     Torec took a breath. Despite the way she had hammered away at her techs to get them to talk to her comprehensibly, the theory of the CTC software was still her weakest point. "We give the system a problem to solve, in the case of our prototype to find a particular protein geometry. And we give it a brute-force way to solve the problem. In the case of protein folding, we instruct the processor simply to start searching through all possible protein geometries. And we have a time register, a special cache that stores a flag if a signal has been received from the future.
     "The basic CTC program has three steps. When the processor starts, the first step is to check the time register. If a signal has been received—if the solution to the problem is already in memory—then stop. If not, we go to step two, which says to carry out the calculation by brute force, however long it takes. When the answer is finally derived, we go to step three: go back in time, deliver the solution and mark the time register."
     Luru nodded. "So the timeline is redrafted. In the first draft timeline, the problem is solved by brute force. In the final version of the timeline, the answer is sent back through time to the moment when the question is posed. So it isn't necessary to run the computation at all."
     Luru sighed. "The joy of time-travel paradoxes. You can get the answer to a problem without needing to work it out! But there must be a good deal more to your design. Your closed-timelike-curves must be pretty short."
     "Actually just milliseconds."
     "Surely you can solve no problem which would take longer to solve than that length of time."
     Torec smiled, her confidence growing. "No. By breaking a problem down into pieces you can solve anything." She described how the problem was broken up into a hierarchy of nested subcomponents. At the base level were calculations so trivial they could be handled within the processor's short CTC periods. The answers were passed back in time to become the input for the next run-through, and so on. That way an answer was assembled piece by piece and looped back repeatedly to the zero instant, until the overall problem was resolved. "The technical challenge is actually decomposing the problem in the first place, and controlling the information flow back up the line," she said.
     Luru laughed, an odd. hollow sound. "You're computing with multiple time loops, and you think that's the only challenge? Ensign, you're a true pragmatist. ... I think it's nearly time."
     Over the glittering, much-patched array of the prototype processor, the bots hovered, utterly motionless against the greater lunar stillness. Behind the prototype, the blank Virtual screen hovered, waiting to display the solution.
     The last seconds wore away.
     And at zero, the screen filled with a molecular diagram. Just like that, with no time elapsed. It was almost anticlimactic, Torec thought.

     The bots descended on the prototype complex, checking its physical integrity. But ironically. Torec knew, there should be little for them to find, for as the processor's paradoxical operation had worked, there had been no need for the problem to be brute-force solved, and no need even for the little toy ships to chatter back and forth on their FTL hops—in this draft of the timeline anyhow. The curves in time had served their purpose—and had rendered their own existence unnecessary. It was another peculiar advantage of a time-travel computer. If it worked correctly, it never actually ran at ail—and so it should never wear out. Some of the techs had even debated whether they could get away with the economy of making the processors shoddily, almost at the point of failure—for that failure would never be tested.
From Exultant by Stephen Baxter (2004)

Literary Crutches

There are some standard limits on time travel used by science fiction authors. These do not exist because physics demand them, they exist because the SF writers and readers would become hopelessly confused otherwise.

A common one is the Linearity Principle or San Dimas Time. This makes a scientifically ridiculous (but reader comforting) one-to-one correspondence between past time and present time. If the time travelers go back in time to meet Alexander the Great and spend a week in the past, they will return to the present exactly one week after they left. If Time Patrol agents are dispatched from Project Tic-Toc to 1914 (monitoring the start of World War 1), the agents used their Time RadioTM to call Tic-Toc Control in the present to request some vital historical research, and the research takes two days, when the info is sent back to the agents in 1914 the agents will get the message exactly two days after they made the request.

In reality travelers should be able to return to any time they want (even an hour or so before they left), and Tick-Tock should be able to sent the info to the agents before the agents ask for it. But this gets very confusing, and quickly sets up some nasty temporal paradoxes.

A related literary crutch is the Delayed Ripple Effect. So the moron in A Sound of Thunder goes back in time to the late Cretaceous Era for some big-game T-Rex hunting, and like a total idiot steps on a butterfly and thus alters history. In reality all of history should be changed "instantly" (if that means anything) as soon as the butterfly is crushed.

However in movies such as Millennium there is a ridiculous delay. The Time Control Central in the far future sends time agents back to 1989, where they screw up and create a paradox that will result in Time Control Central never existing in the first place. But from the viewpoint on Time Control Central, the "time quake" created by the paradox progresses up the timeline at the rate of a few second of delay per timeline year. The end result is that Time Control Central has about ten minutes to do something before they cease to exist. Scientifically bogus, but dramatically exciting.

Left unanswered is the question of exactly when the ten minute count-down starts at Time Control Central. Especially considering that the time agents screwed up in 1989, several thousand years prior to the current "Now" at Time Control Central.

In the computer game The Journeyman Project after the invention of the worlds first time machine the government locks it down under tight security, and establishes the Temporal Security Annex to make sure nobody uses time travel technology to alter history. Their problem is, if anybody alters history, how the heck will they know? Everything will instantly change, and everybody knows there was never a person named Hitler nor was their ever something called World War 2.

The answer was a high density CD-ROM (the game came out in 1993, OK?) containing all of recorded history. It was placed back in Dinosaur times, presumably before the time of any evil time-travel history-changing. This means even if history is changed, the CD-ROM will stay unchanged.

Every day the Temporal Security Annex dispatches an agent with a new CD-ROM (appended with last day's history) back in time to the Dinosaur cache. There the new CD-ROM will be added to the cache. But first, the agent runs a comparison between the new disc and the old disc. If there is a discrepancy, it is a red alert that history has been changed. The place where the two discs diverge will pin point the historical moment that the villains did their dirty work.

Sometimes in literary time-travel, the universe has a Dynamic timeline but it acts like a Self-Healing timeline or even a Fixed timeline. This means the timeline contains Clock Roaches or even Time Police. This creates an Enforced Fixed timeline.

Remember the weakly god-like AI called the Eschaton in Charles Stross's novel Singularity Sky, who commands: "Thou shalt not violate causality within my historic light cone. Or else."

Yet another common trope is a time traveler bent on changing history learning the hard way the law of unintended consequences. A sub trope is somebody else having to clean up the mess you made.

A literary trope that I have not managed to figure out is the "You Can't Use Time Travel To Meet Your Older/Younger Self". In reality there is nothing to prevent this. In fact, that trope is routinely violated in Poul Anderson's novel There Will Be Time and violently violated in Robert Heinlein's All You Zombies.

A good summary of the various ways time-travel seems to work in various SF novel can be found in the Guide To SF CHRONOPHYSICS and of course the infinite time-sink TV Tropes.

Time Traders

(ed note: In Andre Norton's Time Trader series the time agents sent back in time thousands of years are doing everything possible not to alter history. They need a way to cover up so that the natives do not discover that they are impostors. One good technique is to pretend that you are a trader from a far away land.)

     “Trade, eh?” Renfry nodded. “Heard how you boys on the time runs play that angle.”
     “Its’ a good cover, one of the best there is. A trader moves around without question in a primitive world. Any little strangeness in his speech, his customs, his dress, can be legitimately accounted for by his profession. He is supposed to come from a distance, his contacts don’t expect him to be like their fellow tribesmen. And a trader picks up news quickly. Yes, trade was a cover the project used from the first.”
     “You were a trader, back in time?” Travis asked.
     Ashe appeared willing enough to talk of his previous ventures. “D’you ever hear of the Beaker Folk? There were traders for you — had their stations from Greece to Scotland during the early Bronze Age. That was my cover, in early Britain, and again in the Baltic. You can really be fascinated by such a business. My first partner might have retired a millionaire — or that period’s equivalent to one.”

From Galactic Derelict by Andre Norton ()

Atomic Rockets notices

This week's featured addition is Pulsed Solid Core Nuclear Thermal Rocket

This week's featured addition is Convair Electria spacecraft

This week's featured addition is the Lewis Research Center Mars Mission

Atomic Rockets

Support Atomic Rockets

Support Atomic Rockets on Patreon