Novikov self-consistency principle
The Novikov self-consistency principle, also known as the Novikov self-consistency conjecture, is a principle developed by Russian physicist Igor Dmitriyevich Novikov in the mid-1980s to solve the problem of paradoxes in time travel, which is theoretically permitted in certain solutions of general relativity (solutions containing what are known as closed timelike curves). The principle asserts that if an event exists that would give rise to a paradox, or to any "change" to the past whatsoever, then the probability of that event is zero. It would thus be impossible to create time paradoxes.
History of the principle
Physicists have long been aware that there are solutions to the theory of general relativity which contain closed timelike curves, or CTCs—see for example the Gödel metric. Novikov discussed the possibility of CTCs in books written in 1975 and 1983, offering the opinion that only self-consistent trips back in time would be permitted. In a 1990 paper by Novikov and several others, Cauchy problem in spacetimes with closed timelike curves, the authors state:
- The only type of causality violation that the authors would find unacceptable is that embodied in the science-fiction concept of going backward in time and killing one's younger self ("changing the past"). Some years ago one of us (Novikov10) briefly considered the possibility that CTCs might exist and argued that they cannot entail this type of causality violation: Events on a CTC are already guaranteed to be self-consistent, Novikov argued; they influence each other around a closed curve in a self-adjusted, cyclical, self-consistent way. The other authors recently have arrived at the same viewpoint.
- We shall embody this viewpoint in a 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. This principle allows one to build a local solution to the equations of physics only if that local solution can be extended to a part of a (not necessarily unique) global solution, which is well defined throughout the nonsingular regions of the spacetime.
Among the coauthors of this 1990 paper were Kip Thorne, Mike Morris, and Ulvi Yurtsever, who in 1988 had stirred up renewed interest in the subject of time travel in general relativity with their paper Wormholes, Time Machines, and the Weak Energy Condition, which showed that a new general relativity solution known as a traversable wormhole could lead to closed timelike curves, and unlike previous CTC-containing solutions it did not require unrealistic conditions for the universe as a whole. After discussions with another coauthor of the 1990 paper, John Friedman, they convinced themselves that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent through the wormhole.
In response, another physicist named Joseph Polchinski sent them a letter in which he argued that one could avoid questions of free will by considering a potentially paradoxical situation involving a billiard ball sent through a wormhole which sends it back in time. In this scenario, the ball is fired into a wormhole at an angle such that, if it continues along that path, it will exit the wormhole in the past at just the right angle to collide with its earlier self, thereby knocking it off course and preventing it from entering the wormhole in the first place. Thorne deemed this problem "Polchinski's paradox".
After considering the problem, two students at Caltech (where Thorne taught), Fernando Echeverria and Gunnar Klinkhammer, were able to find a solution beginning with the original billiard ball trajectory proposed by Polchinski which managed to avoid any inconsistencies. In this situation, the billiard ball emerges from the future at a different angle than the one used to generate the paradox, and delivers its younger self a glancing blow instead of knocking it completely away from the wormhole, a blow which changes its trajectory in just the right way so that it will travel back in time with the angle required to deliver its younger self this glancing blow. Echeverria and Klinkhammer actually found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case. Later analysis by Thorne and Robert Forward showed that for certain initial trajectories of the billiard ball, there could actually be an infinite number of self-consistent solutions.
Echeverria, Klinkhammer and Thorne published a paper discussing these results in 1991; in addition, they reported that they had tried to see if they could find any initial conditions for the billiard ball for which there were no self-consistent extensions, but were unable to do so. Thus it is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven. This only applies to initial conditions which are outside of the chronology-violating region of spacetime, which is bounded by a Cauchy horizon. This could mean that the Novikov self-consistency principle does not actually place any constraints on systems outside of the region of spacetime where time travel is possible, only inside it.
Even if self-consistent extensions can be found for arbitrary initial conditions outside the Cauchy Horizon, the finding that there can be multiple distinct self-consistent extensions for the same initial condition—indeed, Echeverria et al. found an infinite number of consistent extensions for every initial trajectory they analyzed—can be seen as problematic, since classically there seems to be no way to decide which extension the laws of physics will choose. To get around this difficulty, Thorne and Klinkhammer analyzed the billiard ball scenario using quantum mechanics, performing a quantum-mechanical sum over histories (path integral) using only the consistent extensions, and found that this resulted in a well-defined probability for each consistent extension. The authors of Cauchy problem in spacetimes with closed timelike curves write:
- The simplest way to impose the principle of self-consistency in quantum mechanics (in a classical space-time) is by a sum-over-histories formulation in which one includes all those, and only those, histories that are self-consistent. It turns out that,14 at least formally (modulo such issues as the convergence of the sum), for every choice of the billiard ball's initial, nonrelativistic wave function before the Cauchy horizon, such a sum over histories produces unique, self-consistent probabilities for the outcomes of all sets of subsequent measurements. ... We suspect, more generally, that for any quantum system in a classical wormhole spacetime with a stable Cauchy horizon, the sum over all self-consistent histories will give unique, self-consistent probabilities for the outcomes of all sets of measurements that one might choose to make.
Potential implications for paradoxes
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The Novikov Principle is able to circumvent most commonly cited temporal paradoxes which are often alleged to exist should time travel be possible (and are often claimed to make it impossible). A common example of the principle in action is the idea of preventing disasters from happening in the past and the potential paradoxes this may cause (notably the idea that preventing the disaster would remove the motive for the traveller to go back and prevent it and so on). The Novikov self-consistency principle states that a time traveller would not be able to do so. An example is the Titanic sinking; even if there were time travelers on the Titanic, they obviously failed to stop the ship from sinking. The Novikov Principle does not allow a time traveler to change the past in any way at all, but it does allow them to affect past events in a way that produces no inconsistencies—for example, a time traveler could rescue people from a disaster, and replace them with realistic corpses if history recorded that bodies of victims had been found. Provided that the rescuees were not known to have survived prior to the date that the time traveler stepped into the time machine (perhaps because they were taken forward in time to a later date, or because their identities were hidden), the time traveler's motivation to travel back in time and save them will be preserved. In this example, it must always have been true that the people were rescued by a time traveler and replaced with realistic corpses, and there would be no "original" history where they were actually killed, since the notion of changing the past is deemed impossible by the self-consistency principle.
Assumptions of the Novikov self-consistency principle
The Novikov consistency principle assumes certain conditions about what sort of time travel is possible. Specifically, it assumes either that there is only one timeline, or that any alternative timelines (such as those postulated by the many-worlds interpretation of quantum mechanics) are not accessible.
Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox. However, the Novikov self-consistency principle is intended to go beyond just the statement that history must be consistent, making the additional nontrivial assumption that the universe obeys the same local laws of physics in situations involving time travel that it does in regions of spacetime that lack closed timelike curves. This is made clear in the above-mentioned Cauchy problem in spacetimes with closed timelike curves, where the authors write:
- That the principle of self-consistency is not totally tautological becomes clear when one considers the following alternative: The laws of physics might permit CTC's; and when CTC's occur, they might trigger new kinds of local physics which we have not previously met. ... The principle of self-consistency is intended to rule out such behavior. It insists that local physics is governed by the same types of physical laws as we deal with in the absence of CTC's: the laws that entail self-consistent single valuedness for the fields. In essence, the principle of self-consistency is a principle of no new physics. If one is inclined from the outset to ignore or discount the possibility of new physics, then one will regard self-consistency as a trivial principle.
Time loop logic
Time loop logic, coined by the roboticist and futurist Hans Moravec, is the name of a hypothetical system of computation that exploits the Novikov self-consistency principle to compute answers much faster than possible with the standard model of computational complexity using Turing machines. In this system, a computer sends a result of a computation backwards through time and relies upon the self-consistency principle to force the sent result to be correct.
A program exploiting time loop logic can be quite simple in outline. For example, to compute one prime factor of the natural number N in polynomial time (no polynomial time factorization algorithm is known in traditional complexity theory; see integer factorization):
- If N is 0 or 1, abort.
- Allocate a communication channel c.
- Receive one prime factor, F, of N from the future on channel c.
- Test that F ≠ N, that F divides N (time complexity O(log N)), and that F is prime (polynomial time; see AKS primality test).
- If so, send F backwards in time on channel c.
- If not, send F + 1 backwards in time on channel c. Note that this results in a paradox, as the number received in step 3 above is not the same as that sent in this step.
The self-consistency principle guarantees that the sequence of events generating the paradox in the nested conditional has zero probability. Note that if N is itself prime, i.e., there is no such prime F ≠ N, then some event will prevent the execution of step 3 that receives the value F from the future. Assuming the machine executing the program itself continues to function, it can detect this failure and abort.
Physicist David Deutsch showed in 1991 that this model of computation could solve NP problems in polynomial time, and Scott Aaronson later extended this result to show that the model could also be used to solve PSPACE problems in polynomial time.
Claims, arguments, or philosophical principles logically equivalent to the Novikov self-consistency principle have been published before Novikov's own publication. This makes the principle an example of Stigler's law of eponymy.
- Something resembling the idea can be found in Greek mythology, in the story of Cassandra. Cassandra was given the gift of prophecy by Apollo but also cursed such that no one would believe her predictions. This left her unable to avert any of the disastrous events she foresaw. The metaphor has been adopted in modern times into the notion of a "Cassandra Complex".
- In "Paradox" by Charles Cloukey, from the Summer 1929 issue of Amazing Stories Quarterly, a scientist named Endicott Hawkinson discovers a manuscript on time travel written in the handwriting of his friend Raymond Cannes, who is a non-scientist and does not know where the manuscript came from. The scientist uses the manuscript to build a time machine and Cannes volunteers to try it out, and travels to the year 2930 where he meets a scientist named Bonn, who has discovered a method of travel into the past. Bonn uses a copying machine to write up the plans for the machine in Cannes' handwriting, and Cannes then travels back to 1928 to deposit the manuscript in Hawkinson's desk drawer. Cannes later returns to 2930 and has a series of adventures, during the course of which he discovers records showing that he was killed in an automobile accident. He tells his friends at the club he has cheated fate by returning to his own era a day after the accident was supposed to take place, but in fact he had made a mistake about the date, and when he steps out of the club he is killed by an automobile.
- H. P. Lovecraft discussed this idea of time travel in a 1930 letter to Clark Ashton Smith, where he wrote:
Your idea for a time-voyaging machine is ideal—for in spite of Wells, no really satisfactory thing of this sort has ever been written. The weakness of most tales with this theme is they do not provide for the recording, in history, of those inexplicable events in the past which were caused by the backward time-voyagings of persons of the present and future. It must be remembered that if a man of 1930 travels back to B.C. 400, the strange phenomenon of his appearance actually occurred in B.C. 400, and must have excited notice wherever it took place. Of course, the way to get around this is to have the voyager conceal himself when he reaches the past, conscious of what an abnormality he must seem. Or rather, he ought simply to conceal his identity—hiding the evidences of his "futurity" and mingling with the ancients as best he can on their own plane. It would be excellent to have him know to some extent of his past appearance before making the voyage. Let him, for example, encounter some private document of the past in which a record of the advent of a mysterious stranger—unmistakably himself—is made. This might be the provocation for his voyage—that is, the conscious provocation.
- In "Via the Time Accelerator" by F. J. Bridge (a pseudonym of Francis J. Brueckel), from the January 1931 issue of Amazing Stories, a time traveler in 1930 wonders if he should travel to the future, and then he sees himself returning from the future, which reassures him about the success of his voyage. Later, in a situation where he finds himself in danger, he tells himself "I would escape ... It was so decreed. Had I not, with my own eyes, seen myself appear out of the fourth dimension back there in the Twentieth Century, and glide down to my landing-field? Surely, then, I was destined to return to my own age safe and sound." The time traveler eventually arrives in a ruined city in A.D. 1,001,930, where he is met by an old man who claims to be the Last Man still alive, and who says he knew the time traveler was coming because he read in an ancient history book that he himself (the Last Man) had arrived from the future in A.D. 502,101 in the same time machine the time traveler was using. When the time traveler goes to sleep, the Last Man does indeed take the time machine back to A.D. 502,101, leaving the time traveler stranded. The time traveler then wanders around the ruined city until he finds a museum, where preserved in a glass case is his time machine, which had been put there after it had appeared in A.D. 502,101. The time traveler adds some oil to its engine, and uses it to travel back to 1930, arriving there just as he had seen himself do.
- Robert A. Heinlein's By His Bootstraps (1941) features a plot in which a man interacts with different older versions of himself that travel by way of a "Time Gate", with all the interactions revisited later in the story from the perspective of the now-older man, everything being tied together in a completely self-consistent way. Heinlein later revisited a similar theme in his 1958 story —All You Zombies—, in which the main character's interactions with sex-changed versions of himself/herself at various points in his/her life result in a bizarre version of the ontological paradox in which the character becomes his/her own mother and father.
- The 1961 story, The Dandelion Girl by Robert F. Young states the theory in the words of the heroine, Julie, as she describes her father's concept of time travel.
- In Chris Marker's film La jetée (1962) the protagonist remembers, when he was a child, seeing a man killed at an airport jetty. He later travels back in time and is killed at the same airport jetty, and realizes that, as a child, he had seen his future self being killed. This motif is being reused in the 1995 film 12 Monkeys: As a young boy, the protagonist witnesses a man being killed at the airport; it turns out during the events of the film that it was in fact he himself who was killed, after travelling back in time.
- In Harry Harrison's The Technicolor Time Machine (1967), main characters go back in time to shoot a movie about founding a Viking colony in North America, only to discover to their surprise that the colony they founded turned out to be written into the history as the original Viking colony in North America—and some of them are even featured in Norse sagas.
- In Michael Moorcock's Behold the Man (1969), a time traveler goes back to 28 A.D. in hopes of meeting Jesus, only to end up playing the role of Jesus himself, just as described in the Bible.
- Science fiction author Larry Niven called this idea the "law of conservation of history" in an essay titled "The Theory and Practice of Time Travel," which was published in his book All the Myriad Ways in 1971.
Later fictional usage
- The "Bill and Ted" movies play this completely straight. Since the protagonists know that they will succeed in saving the world and they have a time machine (they were told this by a time-traveller from the future, who also gave them their time machine, which was only invented because they'd already saved the world), their standard response to problems becomes, once they think of it, "after we win, we'll go back in time to yesterday and set things up so that we can win now". Their reliance on the universe behaving self-consistently becomes more and more blatant as the movies go on. For example, early in the film Ted's father suspects that he has stolen a spare set of keys to the police station, which he has not. Later in the film, when they need to break into the police station, Ted realizes that his future self must have gone back in time and stolen the keys, and then left them somewhere his current self could easily find - at which he finds the keys right where he suggests they later hide them. Ted stresses that after they win, they have to remember to use the time machine to go back to the beginning of the movie and steal his father's keys, or else none of this will happen.
- The sequel takes the principle even further, as the villain also has access to a time machine. This turns into a running duel as both sides postulate how they will set up their victory by using the time machine after they have won. Bill and Ted say they will hang a cage from the ceiling to drop on the villain, which then falls from the ceiling as soon as they think up this plan. The villain, however, then says that two can play at that game, and after he has vanquished them, he can simply use his time machine to leave a key to the cage at that exact spot on the floor, as well as another gun - both of which he then pulls out. The gun does not fire when he pulls the trigger, however, leading Bill and Ted to declare that because they're the heroes of this story, they will always beat him, and they left a sabotaged gun on the floor just to trick him (though given that the specific reason the villain traveled back in time was to kill Bill and Ted, it would have been an impossible paradox for him to actually succeed, as this would remove his motivation for travelling back in time in the first place).
- The movie 12 Monkeys appears to obey Novikov's principle. All attempts by the main character James Cole (played by Bruce Willis) to change the past prove unsuccessful, and in the end his death is witnessed by his own childhood self in exactly the way he had remembered earlier in the movie. Additionally, the scientists in charge of the time travel mission have no interest in attempting to avert the release of the deadly virus which killed most of the human population, and are instead only trying to obtain a pure strain of the original virus, in hopes that it will help them to cure the disease in their own time.
- The movie Millennium (based on the 1977 short story "Air Raid" by John Varley) uses the aspect of the Novikov Principle that does not allow a time traveler to change the past in any way at all, but it does allow them to affect past events in a way that produces no inconsistencies—for example, a time traveler could rescue people from a disaster, and replace them with realistic corpses if history recorded that bodies of victims had been found. Needing to repopulate the world due to an infertile future population, time travelers save passengers on doomed flights by abducting people who will not be able to affect the future any further and replacing them with copies of those who would have died.
- The first-season 2002 TV series Twilight Zone episode "Cradle of Darkness" featured a woman who traveled from the modern day back in time to kill the infant Adolf Hitler to prevent his future atrocities. The woman, posing as the family nurse, disposes of the Hitlers' baby and replaces him with a beggar woman's child so as not to arouse suspicion. This replacement infant is the child who then grows up to become the infamous Adolf Hitler.
- The second-season opener of the 1985 revival of Twilight Zone featured a story about an Elvis impersonator thrown back in time and becoming Elvis himself after an accident kills the real one.
- Each episode of the Irwin Allen series The Time Tunnel (less the few set in the future) depicted the time-traveling duo arriving at the scene of an historic tragedy, days or hours before the event, and invariably failing in their attempts to prevent it. This was also true in the 1976 revival pilot, and the similar 1982–83 series, Voyagers!.
- In the Star Trek: The Next Generation episode "Time's Arrow", a 500-year-old copy of the android Data's head is unearthed in an excavation near San Francisco on Earth. In the sequence of events that ensues upon investigation, Data is sent back in time to 19th century San Francisco. Later in this time frame, the shock of another temporal event causes his head to be split from his body and remain in the 19th century, while the rest of his body travels forward to the original point in time, when it is re-attached to the 500-year-old head. Thus Data remains, as does the causality of time. However, the series does not consistently obey the Novikov principle in episodes featuring time travel, as episodes like Time Squared and Yesterday's Enterprise show history being changed as a result of time travel.
- Likewise in Star Trek: Voyager two-part episode "Future's End", Captain Braxton, a member of the Time Federation, attempts to destroy Voyager in order to prevent a future where Earth's entire solar system becomes annihilated by a massive temporal shockwave allegedly caused by the actions of Voyager's crew in the late 20th century. His attempt to destroy Voyager, however, is what causes the sequences of events where the temporal shockwave would happen in the first place. This paradox ultimately leads Captain Braxton to experience severe emotional trauma, feeling as though his actions are pointless and meaningless in the face of the paradox - that he is ultimately to blame for the temporal shockwave which destroys 20th century Earth. However, like in previous time travel episodes in Star Trek containing this paradox, last-second actions by the crew of Voyager ultimately lead them to break the causality loop and reset the timeline to its original state.
- While predating Novikov's thesis, the film Timerider: The Adventure of Lyle Swann presents a similar predestination paradox, wherein the title character becomes his own great-great-grandfather thus causing his own existence and time-travel. The same film, however, contains an ontological paradox in that Swann's heirloom necklace which his great-great-grandmother took from his great-great-grandfather (i.e., himself) is never created and is perpetually in the time-loop.
- The events of the first Terminator film appear to respect the self-consistency principle. In the film, a sentient computer called Skynet attempts to exterminate the human race, but faces difficulty in dealing with a human resistance effort led by a man called John Connor. In a last-ditch attempt to win the war, Skynet sends a cybernetic assassin called a Terminator back through time to murder Connor's mother Sarah Connor before he is born, thereby preventing Connor's existence and the success of his future rebellion. Connor sends a soldier named Kyle Reese back to the same time to protect Sarah. Rather than altering history, these time travellers end up creating the timeline as it was meant to be. While on the run from the Terminator, Kyle and Sarah engage in sexual intercourse and conceive the child who will become John Connor. Likewise, the Terminator is destroyed in a factory and its remains are claimed by Cyberdyne Systems, the factory's owner. This company uses the Terminator's remains as the basis for a research project that will ultimately result in the creation of the malevolent computer Skynet. (It should be noted, however, that only the first film respects this principle. In the sequels, the main characters are able to significantly alter the timeline. Also, even in the first film Skynet's plan would not really make sense unless it believed that history could be changed, and when Sarah Connor asks Kyle Reese if he's saying the Terminator is from the future, he responds 'One possible future. From your point of view. I don't know tech stuff').
- In Roswell That Ends Well, episode 19 of season 3 of the series Futurama, the protagonist Fry returns to the past and accidentally kills his young grandfather in an attempt to protect him from harm. Fry visits his young grandmother whom he'd been introduced to earlier as his grandfather's fiancee, and convincing himself that she can't actually be his grandmother as he is still alive, allows himself to be seduced by her. The explanation at the end of the show is that Fry has become his own grandfather. By the laws of genetic relatedness, one might find a paradox here, in that Fry being his own grandfather means his father is both 50% related to him (since he is Fry's father) and 62.5% related to him (since Fry's father is also his son and the son of Fry's grandmother, who by virtue of being Fry's grandmother, is 25% related to Fry). However, these laws are probabilistic and Fry being his own grandfather is not necessarily a violation of Novikov's principle. The reason is that while Fry may have obtained 50% of his SNPs from his father, his grandmother might have been similar enough to him (his grandfather) that his father is 62.5% related to him. Regardless, it's clear that Fry's just very inbred.
- In The Final Countdown, the aircraft carrier USS Nimitz, based at Pearl Harbor, is sent back in time from 1980 to December 6, 1941. Aboard is a civilian consultant from one of the companies which contributed to the design of the ship. One of the ship's officers, a historian, is lost and presumed dead while transporting a rescued senator and his secretary to a small island for safety in advance of the coming battle; according to history, the pair and his chief of staff were believed to have been killed by the Japanese. The senator dies trying to escape the island. Just as the ship's complement prepare to go into battle the next morning to stop the Japanese attack, the ship returns to 1980 – their minor involvement having had no net effect on history. Back in port, the consultant meets his reclusive boss who helped design the ship – he and his wife are the pilot and the senator's secretary who assumed false names during the war.
- Throughout the plot of Harry Potter and the Prisoner of Azkaban, two unexplained events that greatly affect the plot are later seen to be caused by the protagonists using a time-turner, a small hourglass allowing its user to travel back in time. Hermione Granger uses the time-turner to go back in time and do a different lesson than the current Hermione is doing and thus appears to be in multiple places at once. In the same book, Harry Potter saved himself from a Dementor attack, realizing at the last moment that his savior was not his father as he had initially thought, but was in fact, himself. The film adaptation adds further details to emphasise the consistent role played by the time travellers: Hermione throws a stone through Hagrid's window, and almost accidentally catches the eye of her earlier self while trying to see what her own hair looks like from the back, and later imitates the cry of a wolf in the distance, all events that the characters and audience had already observed from different perspectives.
- Novikov is alluded to by name in the comic Nth Man: The Ultimate Ninja. One of the characters is Colonel Vavara Novikova (Russian surnames having masculine and feminine forms), who accompanies the protagonist and antagonist back in time to their birth so that they can be delivered to the orphanage where their story begins.
- In the book (and later the film) The Time Traveler's Wife, the main character's mother is killed in a car crash while he is young. He escapes injury by time travelling out of the car. Although he frequently revisits the scene and time of the accident throughout his life, he can never alter the outcome as to do so would remove his desire to be there.
- In Season 4 episode 2 of the TV series Eureka The Novikov Self Consistency Principle is explicitly mentioned to justify the many similarities between the "alternative timeline" experienced by the protagonists and their "original timeline". In the brief discussion the character Dr. Henry Deacon (Joe Morton), using the analogy of ripples in a pond, explains that the further one travels from the point where the timeline is changed the less noticeable are the effects of that change. This is not however an accurate depiction of the principle, since the principle actually forbids any changes to the timeline.
- Principle of non-contradiction
- Causality in physics
- Probability distribution
- Predestination paradox
- Many-worlds interpretation
- Self-fulfilling prophecy
- Blinovitch Limitation Effect (fictional)
- The chicken or the egg
- Cassandra (metaphor)
- Time viewer
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- Earman, John (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press. p. 184. ISBN 0-19-509591-X.
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- Hans Moravec (1991). "Time Travel and Computing". Retrieved 2008-07-28.
- Deutsch, David (1991). "Quantum mechanics near closed timelike lines". Physical Review D 44 (10): 3197–3217. Bibcode:1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197.
- "The Limits of Quantum Computers". Scientific American: 68–69. March 2008.
- Aaronson, Scott; John Watrous (2009). "Closed Timelike Curves Make Quantum and Classical Computing Equivalent". Proceedings of the Royal Society A 465 (2102): 631–647. arXiv:0808.2669. Bibcode:2009RSPSA.465..631A. doi:10.1098/rspa.2008.0350.
- Science-Fiction: The Gernsback Years by Everett F. Bleiler and Richard Bleiler (1998), p. 65
- Lord of a Visible World: An Autobiography in Letters, edited by S. T. Joshi (2000), p. 328.
- Science-Fiction: The Gernsback Years by Everett F. Bleiler and Richard Bleiler (1998), p. 36
- The plot of this story is outlined on p. 206 of Paul J. Nahin's Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction (1993), which also discusses many other early time travel stories, along with discussions of time travel paradoxes.
- Notion of the Past & Can We Change It? – speech by Novikov
- From wormhole to time machine: Comments on Hawking's Chronology Protection Conjecture, which also addresses the Novikov self-consistency principle
- Einstein Physics prevent paradoxical time travel
- Time Travel and Modern Physics