Predestination paradox
A causal loop is a paradox of time travel that occurs when a future event is the cause of a past event, which in turn is the cause of the future event. Both events then exist in spacetime, but their origin cannot be determined.^{[1]}^{[2]} A causal loop is also known as a bootstrap paradox, predestination paradox or ontological paradox in fiction.^{[3]}
Contents
 Bootstrap paradox 1
 Selffulfilling prophecy 2
 Novikov selfconsistency principle 3
 See also 4
 References 5
Bootstrap paradox
The term "bootstrap paradox" refers to the expression "pulling yourself up by your bootstraps"; the use of the term for the time travel paradox was popularized by Robert A. Heinlein's story By His Bootstraps.^{[4]}^{[5]} It is a paradox in the sense that an independent origin of the events that caused each other cannot be determined, they simply exist by themselves.^{[1]} Some works of fiction, for example Somewhere in Time, have a version of the paradox where an object from the future is brought to the past, where it ages until it is brought back to the past again, apparently unchanged from its previous journey. An object making such a circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency.^{[6]}
Selffulfilling prophecy
A selffulfilling prophecy may be a form of causality loop, only when the prophecy can be said to be truly known to occur, since only then events in the future will be causing effects in the past. Otherwise, it would be a simple case of events in the past causing events in the future. Predestination does not necessarily involve a supernatural power, and could be the result of other "infallible foreknowledge" mechanisms.^{[7]} A notable fictional example of a selffulfilling prophecy occurs in classical play Oedipus Rex, in which Oedipus becomes the king of Thebes, whilst in the process unwittingly fulfills a prophecy that he would kill his father and marry his mother. The prophecy itself serves as the impetus for his actions, and thus it is selffulfilling.^{[8]}^{[9]}
Novikov selfconsistency principle
General relativity permits some exact solutions that allow for time travel.^{[10]} Some of these exact solutions describe universes that contain closed timelike curves, or world lines that lead back to the same point in spacetime.^{[11]}^{[12]}^{[13]} Physicist Igor Dmitriyevich Novikov discussed the possibility of closed timelike curves in his books in 1975 and 1983,^{[14]} offering the opinion that only selfconsistent trips back in time would be permitted.^{[15]} In a 1990 paper by Novikov and several others, "Cauchy problem in spacetimes with closed timelike curves",^{[16]} the authors suggested the principle of selfconsistency, which states that the only solutions to the laws of physics that can occur locally in the real Universe are those which are globally selfconsistent. The authors later concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.^{[17]}^{:509}
Physicist Joseph Polchinski argued that one could avoid questions of free will by considering a potentially paradoxical situation involving a billiard ball sent 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".^{[17]}^{:510–511} Two students at Caltech, 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 past 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 selfconsistent 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 selfconsistent solutions.^{[17]}^{:511–513}
Echeverria, Klinkhammer and Thorne published a paper discussing these results in 1991;^{[18]} 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 selfconsistent extensions, but were unable to do so. Thus it is plausible that there exist selfconsistent extensions for every possible initial trajectory, although this has not been proven.^{[19]}^{:184} This only applies to initial conditions which are outside of the chronologyviolating region of spacetime,^{[19]}^{:187} which is bounded by a Cauchy horizon.^{[20]} The authors of Cauchy problem in spacetimes with closed timelike curves write:
 The simplest way to impose the principle of selfconsistency in quantum mechanics (in a classical spacetime) is by a sumoverhistories formulation in which one includes all those, and only those, histories that are selfconsistent. It turns out that, 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, selfconsistent 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 selfconsistent histories will give unique, selfconsistent probabilities for the outcomes of all sets of measurements that one might choose to make.
Thus selfconsist causal loops are suspected to be physically possible.
See also
References
 ^ ^{a} ^{b} Nicholas J.J. Smith (2013). "Time Travel". Stanford Encyclopedia of Philosophy. Retrieved June 13, 2015.
 ^ Francisco Lobo (2002). "Time, Closed Timelike Curves and Causality" (PDF). p. 3.
 ^ Leora Morgenstern (2010), Foundations of a Formal Theory of Time Travel (PDF), p. 6
 ^ D'Ammassa, Don (2004). Encyclopedia of Science Fiction. N.Y.: Facts On File. p. 67.
 ^ Klosterman, Chuck (2009). Eating the Dinosaur (1st Scribner hardcover ed.). New York: Scribner. p. 60.
 ^ Everett, Allen; Roman, Thomas (2012). Time Travel And Warp Drives: A Scientific Guide To Shortcuts Through Time And Space. Chicago: The University of Chicago Press. p. 138.
 ^ Dr. William Lane Craig (1987). "Divine Foreknowledge and Newcomb's Paradox". Philosophia 17 (3): 331–350.
 ^ E.R. Dodds, Greece & Rome, 2nd Ser., Vol. 13, No. 1 (Apr., 1966), pp. 37–49
 ^ Popper, Karl (1985). Unended Quest: An Intellectual Autobiography (Rev. ed.). La Salle, Ill.: Open Court.
 ^ S. Krasnikov (2002) "No time machines in classical general relativity" Class. and Quantum Grav. 19 4109, arXiv:grqc/0111054
 ^ S. Carroll (2004). Spacetime and Geometry. Addison Wesley.
 ^ Kurt Gödel (1949). "An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation". Rev. Mod. Phys. 21 (3): 447.
 ^ W. Bonnor; B.R. Steadman (2005). "Exact solutions of the EinsteinMaxwell equations with closed timelike curves". Gen. Rel. Grav. 37 (11): 1833.
 ^ Friedman et al., "Cauchy problem in spacetimes with closed timelike curves", p. 42, note 10
 ^ Novikov, Evolution of the Universe (1983), p. 169. "The close of time curves does not necessarily imply a violation of causality, since the events along such a closed line may be all 'selfadjusted'—they all affect one another through the closed cycle and follow one another in a selfconsistent way."
 ^ Friedman, John; Michael Morris; Igor Novikov; Fernando Echeverria; Gunnar Klinkhammer; Kip Thorne; Ulvi Yurtsever (1990). "Cauchy problem in spacetimes with closed timelike curves". Physical Review D 42 (6): 1915.
 ^ ^{a} ^{b} ^{c}
 ^ Echeverria, Fernando; Gunnar Klinkhammer; Kip Thorne (1991). "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory". Physical Review D 44 (4): 1077.
 ^ ^{a} ^{b} Earman, John (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press.
 ^ Nahin, Paul J. (1999). Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction. American Institute of Physics. p. 508.
