by Peter Jekel
Time is that quality of nature which keeps events from happening all at once. Lately, it doesn’t seem to be working.
~Anonymous.
Ever do something that you regret? Ever think that if you had only known the implications of your decision you would have looked at the issue differently? What if all of that was changeable? What if you could travel back in time and actually change your fortune, no longer living with regret, no longer wondering what would have happened if you had decided on an issue differently? What about going back in time to visit a deceased relative or friend? This sounds like a pipe dream, but there is a group of scientists who truly believe that time travel is possible, both forward and backward.
The concept of time travel is certainly not new as there is some reference to forward time travel even in some ancient religious and mythological texts. As a genre of literature, time travel tales have undergone an evolution. The earliest tales of time travel were essentially fantasies with no true mechanism of time travel being discussed. Modern tales bring physics along for the ride.
In Washington Irving’s 1819, Rip Van Winkle, the main character lays down for a nap only to awaken twenty years later to a dead wife and grown-up daughter. He had missed a major component of his life to sleep. Ironically, as an aside, we probably sleep away about a third of our lives by the time that we pass away.
Charles Dickens’ 1843 A Christmas Carol shows the main character moving backwards and forwards in time with the assistance of Christmas ghosts. Ebenezer Scrooge, the main character, is able to visualize the past, present and future, but is entirely unable to interact with it. He is merely an observer.
Mark Twain, famous author of Huckleberry Finn and Tom Sawyer, wrote in 1889 the lesser known A Connecticut Yankee in King Arthur’s Court. With his story, time travel took on a new meaning. Like the earlier fantasies there is no real mechanism of transportation into the past that is discussed; the main character is transported through time after a hit by a sledge hammer. However, the major difference between Twain’s tale and earlier time travel stories is that his tale is probably one of the first tales in which the author shows how a time traveler might change the course of history.
Almost concurrently, literary time travel branched into science fiction. Now the authors were exploring the mechanism by which time travel became possible. Many may think that it was Wells who was the first to enlighten us with an actual time machine in his 1897 classic, The Time Machine. That novel was not his first tale of time travel. Wells wrote in 1888 a short story The Chronic Argonauts, which acted as a precursor to his later more famous novel. In spite of an early start to put the science behind time travel, even Wells was not the first to do so. That honour belongs to Edward Page Mitchell, the same man who wrote the famous editorial that is still quoted today, "Yes, Virginia, there is a Santa Claus." He wrote a story that appeared in an edition of an 1881 New York Sun entitled "The Clock That Went Backward," the story of a man who traveled back in time via a time machine.
Not to downgrade Mitchell’s story--he wrote many other science fiction stories that are first attempts at futuristic technologies such as teleportation, faster-than-light travel, invisibility and cyborgs to name a few--it was Wells’ novel that really created a niche for time travel in science fiction. Recently, Stephen Baxter wrote an official sequel to Wells’ classic, called The Time Ships. Joe Haldeman’s The Accidental Time Machine somewhat mirrors Wells’ tale. He writes of a researcher who stumbles across a forward traveling time machine.
With time travel having entered the realm of science fiction, how does science reconcile with the possibility? Some scientists, as has been stated, have seen it as a distinct possibility. In fact, one theoretical physicist and member of the American Physical Society and the National Society of Black Physicists, Ronald Mallett of the University of Connecticut, is actually working on plans for a time machine. The inspiration for his work was his father’s premature death at age 33 of a heart attack. Mallett, inspired by Wells’ time travel classic, vowed to return back in time to save his father. In fact, he has written a best-selling book on his quest, Time Traveler: A Scientist’s Personal Mission to Make Time Travel a Possibility.
Mallett is basing his machine on the ring laser. In Einstein’s theory of general relativity, matter and energy, including a beam of light, can create a gravitational field. As more energy was applied to the ring laser closed timelike curves would result, thus allowing time travel. A closed timelike curve is a worldline that is closed; that is, it returns to its starting point. As with any new theory, especially one that is untested, there are scientists out there that do not agree with Mallett.
Mallett sees time travel as being a distinct possibility. He certainly is not the first. When Einstein first hypothesized his theory of general relativity, he may not have realized the new world that he opened up. General relativity is a unified description of gravity as a geometric property in space and time, two things which in the world of physics are now looked upon as inseparable. The observed gravitational attraction between objects results in the warping of spacetime. When others started finding resolutions to the field equations of general relativity, some strange things started to happen. In 1949, Kurt Godel found that one solution to the field equations allowed for the creation of closed timelike curves.
Other physicists also have found similar strange solutions to general relativity that allow for closed timelike curves. On a more abstract level, Misner space, a mathematical interpretation of spacetime first hypothesized by Charles Misner of the University of Maryland, allows for closed timelike curves. Moving the spatial limits of an existing Misner space could lead to potential time travel.
A Kerr ringwarp first described by Roy Kerr, a New Zealand mathematician, in 1963 is essentially a rotating uncharged black hole that allows for closed timelike curves. Instead of collapsing to a point, it flattens out into a ring. If a craft were approach the ring from the side, the curvature of spacetime would be infinite. If, however, the craft were to enter from the top or bottom, the curvature of spacetime would be great but not infinite. Larry Niven wrote Singularities Make Me Nervous about an intrepid explorer orbiting a rotating black hole only to find his younger self. In essence, a Kerr vacuum behaves like a traversable wormhole which also can act like a sort of time machine.
Wormholes are mathematical phenomena that, physically if they exist, would be a tunnel through spacetime. If you enter in, you would come out somewhere and perhaps, somewhen, else. However, wormholes are also very unstable and anything caught crossing the wormhole during its tenuous life would be terminated in the closure. If one were to make a wormhole into a time tunnel, it would have to become traversable by keeping it open. Physicists Kip Thorne and Michael Morris came up with a way to stabilize a wormhole. The only way to prevent the demise of the wormhole is through an application of Einstein’s filed equations; the wormhole is held open by applying an exotic field. An exotic field is a field in which energy density is less than the tensile strength of the filed.
Such exotic fields do exist. In 1948, Dutch physicist, Henrik Casimir, demonstrated the possibility of exotic fields. Take two large uncharged metal plates and place them parallel to one another in a vacuum. One would think that since the plates are neutral that no force exists between them. Casimir, however, demonstrated that the vacuum separating the two plates is teeming with the activity of trillions of particles and anti-particles constantly appearing and disappearing. The net result is an attractive force between the two plates. In 1958, physicist, M. J. Sparnaay actually experimentally demonstrated in the lab the Casimir effect.
To render a wormhole into a time machine, one end of the wormhole would have to be accelerated to a large fraction of the speed of light and then brought back to the point of origin. Another way would be to take one entrance of the wormhole and move it within the influence of a larger gravitational field and then return it to the position near the other entrance. For both methods, time dilation causes the end of the wormhole that has been moved to age less than the stationary end. Robert Forward used one of these wormholes in his novel Timemaster.
Similarly the BTZ rotating black hole--the BTZ is a acronym for the discoverers of this unique solution to a black hole, Maximo Banados, Claudio Teitelboim and Zorge Zarelli---is different but very similar to a Kerr black hole and can create closed time-like curves.
Sergei Krasnikov, a physicist with the University of St. Petersburg, in 1995 proposed the Krasnikov tube which was a speculative method of space travel involving the warping of spacetime into permanent superluminal tunnels. The result is a wormhole structure with endpoints displaced in time and space.
Van Stockum dust is a model of a cylindrical symmetric configuration of dust rediscovered independently by Willem van Stockum, a Dutch mathematician, in 1937, after an initial discovery by Hungarian mathematician, Cornelius Lanczos in 1924. It is seen as one of the simpler solutions to the field equations. However, both van Stockum and Lanczoe did not recognize the formation of closed timelike curves.
It was Frank Tipler who found, in 1974, within Van Stockum’s solution that closed timelike curves could be created. His approach to creating closed timelike curves was by the hypothetical construction of a dense cylinder of infinite length. If it spins fast enough on its long axis, a spaceship flying around the cylinder could travel back and forth in time. However, because of the density and speed required to do this it could not be created out of ordinary matter. It would have to be made of exotic matter. Tipler further elaborates that if a cylinder of fixed length were spun fast enough, it may produce closed timelike curves. This theory offers us a potential for the actual creation of a time machine. Interestingly enough, in 1982, a pulsar was discovered that rotated once every millisecond. Around it, if we were ever able to get there, we would find some noticeable distortions of time and space.
Tipler cylinders have been found in some science fiction stories. Rotating Cylinders and the Possibility of Global Causality Violation by Larry Niven follows a human explorer who discovers a half finished tippler cylinder. Vernor Vinge in Marooned in Realtime also makes mention of Tipler Cylinders.
Stephen Hawking, as well as other physicists, maintain that it is impossible to build a time machine of any sort, that there is no way that closed timelike curves can exist in the real universe. The physical laws of the universe protect such curves from happening in the first place.
However, there may be a way around the Hawking and company objections. Richard Gott, an astrophysicist with Princeton University, proposed building a mechanism to create closed timelike curves using cosmic strings. Cosmic strings are tiny one-dimensional hypothetical defects of the formation of the early universe and would not violate the restrictions set by Hawking and company. Gott’s machine is based on the fact that the antigravitational tension of the strings would deform space without attracting nearby objects. If a path could be found around the strings the traveler could move back through time.
Others have looked for a way around Hawking’s objections as well. Mark Hadley, a physicist with the University of Warwick, developed the idea that particles that make up the universe may actually be time warps. He treats particles as warps in spacetime, calling them geons. Inside a geon strange things can happen such as the particle’s timeline looping back on itself, thereby allowing it to be influenced by its future as well as its past.
Even more bizarre is the idea put forward by Cumrun Vafa of Harvard. He has come up with a new theory that there is an extra time dimension (as if one were not complex enough). In fact, the theory allows for the solution to more mathematical problems in the physical world than it creates. Where is this extra time? We do not know. Like the string theories that postulate extra spatial dimensions that are bound so tight that they are invisible to us. To unravel the extra time would require enormous energy, but if we could, we could potentially not only move forward, but backwards, up, down and sideways in time. Even Vafa admits it is probably a mathematical anomaly but it opens up new doors to future speculation.
We do know from experimental data that general relatively does allow for time travel through time dilation. We have to account for general relativity when calibrating clocks on the Global Positioning Satellites; the further away an object is from a gravity well, the faster that time will move.
In addition to general relativity, Einstein came up with a special theory of relativity in 1905. It is a physical theory of measurement in an inertial frame of reference. This means that all uniform motion is relative, that there is no well-defined state of rest. It also states that the speed of light is the same for all observers and is the ultimate speed limit in our universe. As an object or person reaches the speed of light, special relativity predicts that a lot of strange things happen including length contraction, mass increases and more interestingly time dilation. In fact, if one were to go to the nearest star at near the speed of light, you would find upon return to earth that you had only aged several years yet your relatives and friends back on earth would be long dead. That is, because in the special theory of relativity, time slows down more the faster one goes. This has been demonstrated experimentally, but with current technologies the velocities we are capable of achieving are nowhere what are needed to create a truly perceptible time dilation effect.
Many writers have explored the nature of time dilation. Forever War and its sequels, by Joe Haldeman, are some of the most poignant tales of time dilation, following the exploits of humans in a war with aliens light years away. Tau Zero, a classic by Poul Anderson, follows a starship crew on an accelerating craft edging toward the end of the universe. Even the rock band, Queen, with its astrophysicist lead guitarist, Brian May, wrote and performed 39, which is about a band of space travelers who have returned from a voyage in space. From the perspective of the travelers in 39 a year has past, but those that they are returning to are dead.
Much as science may argue on the possibility of time travel and the various ways that it may be plausible, few writers have ventured into the physics of time travel itself. That is the reason that some literary critics who state emphatically that time travel stories are not true science fiction (that is, science is not a key plot element) but rather a form of fantasy. Perhaps the noblest attempt to explain the science of time travel in a fictional story is Gregory Benford’s Timescape which looks at the use of theoretical tachyons (theoretical particles that move faster than the speed of light) to communicate with the past.
More often than not, though, writers gloss over the science probably due to it mind-bending complexity in favor of the paradoxes that time travel may create since it is the paradoxes that provide the drama. The main form of paradox due to time travel is known as the grandfather paradox. This conceivably could happen if one were to go back in time and kill your grandfather before ever meeting your grandmother. You go back in time and kill your grandfather. Therefore, you are never born. But then how do you go back in time to kill your grandfather if you were never born? It hurts just thinking about it. There are many hypotheses to deal with the paradox, however
The Norvikov Self Consistency Principle formulated by Igor Novikov, a Russian physicist in the 1980’s states that anything that a time traveler from the future does to the past actually has a causal influence on the past. Therefore it is impossible for anything the time traveler to do to change history. If you go back in time to kill your grandfather before he meets your grandmother, your gun will jam, you will miss, or something will happen to prevent you from doing your dirty deed. No grandfather paradox is allowed through this principle, but the theory allows for the "bootstrap paradox". Many science fiction authors have looked at this paradox with some very bizarre outcomes.
A bootstrap paradox occurs whereby a person, information or a thing can exist without having been created. After a person, information or an object is sent back in time, he, she or it is recovered in the present and becomes the very object or information that was initially brought back in time in the first place.
The paradox gets its unusual name from one of the first tales of the paradox written by Robert Heinlein in 1941 entitled By His Bootstraps. In the story, the protagonist goes through a time portal after being asked by one stranger and a second stranger tries to stop him. All three start to fight and the protagonist is still pushed through. We discover that all three are actually the same person. Ultimately the first stranger is the protagonist’s future self and the second is an even older future self trying to prevent the loop from happening. Robert Heinlein took the paradox to a new level with his 1959 All You Zombies. In this story, the protagonist turns out to be his own mother, father, son, daughter, lover and kidnapper.
Other writers have looked at the human element of the bootstrap paradox. David Gerrold’s 1973 The Man who Folded Himself looks at a time-traveling protagonist being his own mother, father and other relatives as well. Philip K. Dick in 1975 published the short story, A Little Something for Us Tempunauts, where our protagonist finds himself in a closed time loop where he is forced to relive a fatal journey.
Michael Moorcock added a real controversy to the bootstrap paradox in his Behold the Man about a time traveler who goes back in time to meet the historical Jesus. He is somewhat disappointed to see that this is not the person that history portrays him to be. As a result the individual finds himself making the ultimate sacrifice to allow history’s vision of Jesus to be the one that is remembered.
Information transported into the past can also create a bootstrap paradox. Harlan Ellison first wrote of this in Soldier From Tomorrow, about a soldier from the future warning the present to steer clear of the path that will lead to global catastrophe. In The Hundred Light Year Diary, Greg Egan describes an invention that allows us to send messages back in time. Soon the history of the future becomes common knowledge and everybody knows their fate.
Other writers have looked at the concept of things going back in time and exploring the outcomes of the bootstrap paradox. Isaac Asimov in his novel The End of Eternity writes of the transportation of goods across centuries. Frederick Brown’s short story "Experiment" is about the sending of a small brass cube back in time to the previous century resulting in disastrous effects. Robert Charles Wilson in The Chronoliths describes monuments from the future that appear in the early 21st century thus precipitating a global collapse. Wilson also wrote Mysterium which is about a United States government project set in motion by the discovery of an artifact at an archeological dig in Turkey.
John Varley’s Millennium is about a futuristic dying society that travels back in time to collect people to build up civilization. They do this by collecting humans from the present whose disappearance will have no effect on the timeline such as those who are about to die. The story takes a twist when a weapon is lost in the course of extracting people from a doomed airplane thus setting up a paradox.
Another way to avoid the grandfather paradox and the bootstrap paradox is the multiple universes hypothesis. The theory states that there are an infinite number of universes. If a person is about to travel back into time, he will end up in a parallel timeline. If he kills his grandfather, a paradox would not happen because the grandfather that he killed is the grandfather who lives in the universe that he currently is i--not in the one that he ends up in.
In fact, there exists a whole new genre of speculative fiction known as alternate histories. Harry Turtledove is probably the most prolific author of alternate histories, visualizing alternate earths where the Confederates win the Civil War, where Homo erectus populates and never becomes extinct in the New World, where the Byzantine Empire survives and where the Axis powers have won the Second World War, just to name a few of his ideas.
The alternate histories of the Axis powers winning the Second World War are almost a subgenre of the alternate history genre itself. One of the best and the first novel of such an alternate universe is Philip K. Dick’s The Man in the High Castle. It also has a bit of a twist as several of the characters in the story read a popular book banned by the Germans wherein the Axis powers lose the Second World War. Norman Spinrad’s The Iron Dream also has a twist to it in that it is portrayed as a story written by Adolph Hitler fleeing to America in the 1920’s following the Great War.
Other authors have looked at the alternate universe of the Axis powers winning the Second World War. James Hogan’s The Proteus Operation is a fast-paced novel about intervention in World War II, to prevent the Axis powers from winning the war. Fatherland by Richard Harris is about the Nazis having won the war. The story follows a murder investigation in America which is beginning to unravel a massive conspiracy around the Nazi regime.
Some authors have looked at other moments in history to determine a totally new alternate world. Kim Stanley Robinson’s The Years of Rice and Salt is about a world where 99% of Europe was decimated by the plague during medieval times thus allowing Islam to take a foothold on the continent. Roma Eterna by Robert Silverberg is about a world where the Roman Empire never collapsed.
Alternate history literature has also morphed into another branch of speculative fiction. Instead of changing an event in history to alter it, why not insert a piece of modern technology back in a lesser developed society and look at the results? Steampunk is a whole new offshoot genre of alternate histories, involving a modern invention being a part of a Victorian culture.
Anti-Ice, by Stephen Baxter, is about the invention of a new form of ice and how it impacts the world of the 19th century. The Difference Engine, by Bruce Sterling and William Gibson, is about the computer having been invented in Victorian times. We could even make the argument that Wells’ The Time Machine and even Jules Verne’s novels such as 20,000 Leagues under the Sea are actually forms of steampunk writing.
Another way to overcome the grandfather paradox is by not overcoming it at all. There are those who feel that the grandfather paradox will happen and that there will be an impact on a given timeline. It is known as the timeline corruption hypothesis. It is more commonly known as the Butterfly Effect after a famous 1952 short story by Ray Bradbury, The Sound of Thunder. In the story, a big game hunter goes back in time only to inadvertently kill a butterfly (the big game hunting itself is well-controlled and participants must follow all instructions) which sets off a change in the course of history. When he returns to the present, he begins to notice very subtle changes. Robert Heinlein’s 1956 novel Door into Summer is about a man who travels back in time to exact revenge on a friend who took over his business. Poul Anderson wrote a series of novels about the Time Patrol whose mission was to protect the integrity of the past.
There is also a temporal merging hypothesis which is similar the multiverse hypotheis. In this theory each action committed in time travel will overlap one reality with another. Two events would merge into the nearest event so as not to create a paradox. Ward Moore in his 1953 novel Bring the Jubilee wrote of a time traveler from an alternate reality that appears at the Battle of Gettysburg and alters his own future into ours. One could even look at the Neanderthal Parallax by Robert Sawyer as a temporal merging tale. It is a tale of two alternate worlds, one that is familiar to us and the other where the Neanderthal humans survived to become the dominant species of the day, coming together in an underground laboratory in Sudbury, Canada.
The temporal modification theory allows for time travel but it is impossible to violate the grandfather paradox. Other changes can occur but anything that negates the time travel is not allowed. Therefore, you cannot murder yourself but you could cause the death of someone else without affected the reason for the journey.
Another resolution to the paradoxes produced by time travel is through doomed timeline theory. This states that the universe will destroy any timeline that deviates from a main timeline to prevent the construction of multiple universes. Any time travel deviating from a stable time loop is doomed to be destroyed. Dannie Plachta wrote the 1966 short story "The Man From When" about a man who destroys the world in order to travel back eighteen minutes in time. In James Hogan’s Thrice Upon a Time messages are sent back through time to delete the timeline in which they exist. A really interesting novella by Stephen King, The Langoliers, is about an airplane that accidentally flies through a rip in space/time and appears in the used time of yesterday. The characters of the novella struggle to stay ahead of a universe that is constantly reinventing itself.
There is also a fantastical version of time travel called a timeslip which is closer to the original fantasy time travel stories. It is a paranormal phenomenon where a person or group of people travel through time via a some paranormal, non-technical means. There are several cases in history where people have appeared and claim to be travelers from another time; however, these claims do for the most part appear to be highly suspect. This does not take away from some of the excellent literature around this fictional form of time travel, though.
Time Slip by Lynne Ellison is about a teenage girl who finds an ancient mirror in the sand and looks into it and is transported back into time to Rome. Daphne du Maurier in The House on the Strand describes a drug induced trip back to 14th century Cornish Village. Jack Finney, author of the famous Invasion of the Body Snatchers, wrote a more subdued speculative novel with Time and Again in which hypnosis is used to travel through time. In Replay Ken Grimwood writes of a man who in 1988 suffers a heart attack only to find himself in 1963. In Audrey Niffenegger’s The Time Travellers’ Wifea rare genetic disorder causes a man to unpredictably travel in time, living his life out of sequence. Like Frank Baum’s Wizard of Oz in which a storm thrust Dorothy into the magical land of Oz, in Malorie Blackman’s Thief! a storm traps a girl accused of being a thief and takes her into the future. Stephen King's ambitious 11/22/63 is about a man who travels through time portal to 1958 and prevents the shooting of President Kennedy. Connie Willis’ combination novel Blackout and All Clear has her time travelers from 2048 go to the London Blitz of WWII only to find themselves a part of it.
The limits of a time travel novel are only as far as the imagination of the writer will take them. We may never realize time travel due to a number of physical barriers. However, we can be sure that the possibility of time travel will always allow for a hugely entertaining story about what might be or what might have been. Then again, somewhere or somewhen in the twisting mathematics of quantum physics and relativity the secret to time travel will be found. Never say never again.
Further Reading
Banados, M., Teitelboim, C., and Zanelli, J. 1992. The Black Hole in Three Dimensional Space Time. Physical Review Letters. 69:1849-1851.
Bonner, W. and Steadman, B. 2005. Exact solutions of the Einstein-Maxwell equations with closed timelike curves. General Relativity and Gravity. 37(11):1833-1844.
Carroll, S. 2004. Spacetime and Geometry. Addison-Wesley.
Casimir, H. 1948. On the attraction between two perfectly conducting plates. Koninklijke Nederlandse Akademie van Wettenschappen. B51:793.
Chou, C., Hume, D. et al. 2010. Optical Clocks and Relativity. Science. 329(5999):1630-1633.
Deutsch, David. 1991. Quantum mechanics near closed timelike lines. Physical Review D. 44(10):3197-3217.
Everett, Allen. 2004. Time travel paradoxes, path integrals, and the many worlds interpretation of quantum mechanics. Physical Review D. 69:123023.
Friedman, J., Morris, M. et al. 1980. Cauchy problem in spacetimes with closed timelike curves. Physical Review D. 42(6):1915-1930.
Godel, Kurt. 1949. An Example of a New Type of Cosmological Solution of Einstein’s Field Equations of Gravitation. Reviews of Modern Physics. 21(3):447-450.
Gott, J. Richard 2002. Time Travel in Einstein’s Universe. Mariner Books.
Hadley, Mark. 1997. The Logic of Quantum Mechanics Derived from Classical General Relativity. Foundations of Physics Letters. 10(1):43-60.
Hawking, Stephen, 1992. Chronology protection conjecture. Physical Review D. 46(2):603-611.
Hawking, Stephen, Thorne, Kip et al. 2002. The Future of Spacetime. Norton.
Ives, H. and Stilwell, G. 1941. An experimental study of the rate of a moving clock. Journal of the Optical Society of America. 31:369-374.
Kerr, R. 1963. Gravitational field of a spinning mass as an example of algebraically special metrics. Physical Review Letters. 11(5):237-238..
Krasnikov, N. 1987. On supersymmetry breaking in superstring theories. Physics Letters B. 193(1):37-40.
Lanczos, K. 1924, republished in 1997). On the Stationary Cosmology in the Sense of Einstein’s Theory of Gravitation. General Relativity and Gravitation. 29(3):363-394.
Lewis, David. 1976. Troubles with Time Travel. Philosophy. 74(1):55-70.
Li, Li-Xin. 1996. Must Time Machines Be Unstable against Vacuum Fluctuations. Classical and Quantum Gravity. 13(9):2563.
Mallett, R. 2000. Weak gravitational field of the electromagnetic radiation in a ring laser. Physical Letters A. 269(4):214-217.
Mallett, R. 2003. The gravitation field of a circulating light beam. Foundations of Physics. 33(9):1307-1314.
Mallett, Ronald. 2006. Time Traveler: A Scientist’s Personal Mission to Make Time Travel a Reality. Thunder’s Mouth Press.
Misner, C., Thorne, K. and Wheeler, J. 1973. Gravitation. W.H. Freeman.
Morris, M. and Thorne, Kip. 1988. Wormholes in spacetime and their use in interstellar travel: A tool for teaching general relativity. American Journal of Physics. 56(5):395-412.
Nahin, Paul. 2001. Time Machines: Time Travel in Physics, Metaphysics and Science Fiction. Springer.
Neitzke, A. and Vafa, C. 2005. Topological Strings and Their Physical Applications. arXiv:hep-th/0410178v2
Novikov, I. 1995. Black Holes and the Universe. Cambridge University Press.
Olum, K. and Everett, Al. 2005. Can a circulating light beam produce a time machine? Foundations of Physics Letters. 18(4):379-385.
Ori, Amos. 1999. Oscillatory Null Singularity inside Realistic Spinning Black Holes. Physical Review Letters. 83(26):5423-5426.
Penrose, Roger. 1999, The Question of Cosmic Censorship. Journal of Astrophysics and Astronomy. 20(3-4):233-248.
Thorne, Kip 1994. Black Holes and Time Warps. Norton.
Thorne, Kip. 1995. Black Holes and Time Warps: Einstein’s Outrageous Legacy. Norton.
Thorne, Kip, Morris, M. and Yurtsever, U. 1988. Wormholes, Time Machines, and the Weak Energy Condition. Physical Review Letters. 61(13):1446-1449.
Tipler, Frank. 1974. Rotating Cylinders and the Possibility of Global Causality Violation. Physical Review D. 9(8);2203-2206.
Van Stockum, Willem 1936. The Gravitational Field of a Distribution of Particles Rotating about an Axis of Symmetry. Proceeding of the Royal Society of Edinburgh. 57:135.
Visser, M. 1995. Lorentzian Wormholes: From Einstein to Hawking. AIP Press.
Visser, M. 1997. Traversable wormholes: the Roman ring. Physical Review D. 55(8):5212-5214.
Visser, Matt, Karr, S. and Dadhich, N. 2003. Traversable wormholes with arbitrarily small energy condition violations. Physical Review Letters. 90 (20):201102.1-201102.4.