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25:
659:. The Cauchy horizon is generated by closed null geodesics. Associated with each closed null geodesic is a redshift factor describing the rescaling of the rate of change of the affine parameter around a loop. Because of this redshift factor, the affine parameter terminates at a finite value after infinitely many revolutions because the geometric series converges.
411:. For instance, while moving in the vicinity of a star, the star's gravity will "pull" on the object, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the corresponding spacetime diagram. An object in free fall in this circumstance continues to move along its local
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can be used to calculate the complete state of the rest of spacetime. However, in a CTC, causality breaks down, because an event can be "simultaneous" with its cause—in some sense an event may be able to cause itself. It is impossible to determine based only on knowledge of the past whether or not
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A closed timelike curve can be created if a series of such light cones are set up so as to loop back on themselves, so it would be possible for an object to move around this loop and return to the same place and time that it started. An object in such an orbit would repeatedly return to the same
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to move, since its present spatial location would not be in its own future light cone. Additionally, with enough of a tilt, there are event locations that lie in the "past" as seen from the outside. With a suitable movement of what appears to it its own space axis, the object appears to travel
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the light cone is directed forward in time. This corresponds to the common case that an object cannot be in two places at once, or alternately that it cannot move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, timewise. However this
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point in spacetime if it stays in free fall. Returning to the original spacetime location would be only one possibility; the object's future light cone would include spacetime points both forwards and backwards in time, and so it should be possible for the object to engage in
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In extreme examples, in spacetimes with suitably high-curvature metrics, the light cone can be tilted beyond 45 degrees. That means there are potential "future" positions, from the object's frame of reference, that are spacelike separated to observers in an external
252:". A light cone represents any possible future evolution of an object given its current state, or every possible location given its current location. An object's possible future locations are limited by the speed that the object can move, which is at best the
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The lower light cone is characteristic of light cones in flat space—all spacetime coordinates included in the light cone have later times. The upper light cone not only includes other spatial locations at the same time, but also does not include
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There are two classes of CTCs. We have CTCs contractible to a point (if we no longer insist it has to be future-directed timelike everywhere), and we have CTCs which are not contractible. For the latter, we can always go to the
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must, according to such arguments, eventually result in the state that is identical to the original one. This idea has been explored by some scientists as a possible approach towards disproving the existence of CTCs.
366:. On such a diagram, every possible future location of the object lies within the cone. Additionally, every space location has a future time, implying that an object may stay at any location in space indefinitely.
197:, a property which can be called chronological censorship, then that spacetime with event horizons excised would still be causally well behaved and an observer might not be able to detect the causal violation.
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The existence of CTCs would arguably place restrictions on physically allowable states of matter-energy fields in the universe. Propagating a field configuration along the family of closed timelike
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One feature of a CTC is that it opens the possibility of a worldline which is not connected to earlier times, and so the existence of events that cannot be traced to an earlier cause. Ordinarily,
622:, which quantum theory predicts is impossible. If Deutsch's prescription holds, the existence of these CTCs implies also equivalence of quantum and classical computation (both in
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Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka (2011-07-13). "Quantum mechanics of time travel through post-selected teleportation".
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seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of
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599:), as the manifold would not be causally well behaved at that point. The topological feature which prevents the CTC from being deformed to a point is known as a
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Stockum, W. J. van (1937). "The gravitational field of a distribution of particles rotating around an axis of symmetry.". Proc. Roy. Soc. Edinburgh. 57.
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axis, but to an external observer it appears it is accelerating in space as well—a common situation if the object is in orbit, for instance.
389:-axis; if it accelerates, it moves across the x axis as well. The actual path an object takes through spacetime, as opposed to the ones it
639:, and reestablish causality. For the former, such a procedure is not possible. No closed timelike curve is contractible to a point by a
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This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of
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orientation is only true of "locally flat" spacetimes. In curved spacetimes the light cone will be "tilted" along the spacetime's
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for space. Light cones in this representation appear as lines at 45 degrees centered on the object, as light travels at
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439:. From this outside viewpoint, the object can move instantaneously through space. In these situations the object would
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W. Bonnor; B.R. Steadman (2005). "Exact solutions of the
Einstein-Maxwell equations with closed timelike curves".
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Watrous, John; Aaronson, Scott (2009). "Closed timelike curves make quantum and classical computing equivalent".
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demands that each event in spacetime is preceded by its cause in every rest frame. This principle is critical in
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part of the Kerr solution is thought to be in some sense generic, so it is rather unnerving to learn that its
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something exists in the CTC that can interfere with other objects in spacetime. A CTC therefore results in a
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1098:"An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation"
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145:, that is "closed", returning to its starting point. This possibility was first discovered by
169:. If CTCs exist, their existence would seem to imply at least the theoretical possibility of
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No CTC can be continuously deformed as a CTC to a point (that is, a CTC and a point are not
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have been proposed, a strong challenge to them is their ability to freely create
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Some of these examples are, like the Tipler cylinder, rather artificial, but the
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contains CTCs. Most physicists feel that CTCs in such solutions are artifacts.
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is the set of points through which CTCs pass. The boundary of this set is the
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Deutsch, David (1991-11-15). "Quantum mechanics near closed timelike lines".
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among timelike curves, as that point would not be causally well behaved.
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World line of a particle in spacetime which returns to its starting point
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Moulick, Subhayan Roy; Panigrahi, Prasanta K. (2016-11-29).
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H. Monroe (2008). "Are
Causality Violations Undesirable?".
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if they differ along the space axis. If the object were in
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states complete knowledge of the universe on a spacelike
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in 1949, who discovered a solution to the equations of
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927:"Timelike curves can increase entanglement with LOCC"
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797:Roy Kerr (Crafoord Prize Symposium in Astronomy):
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369:Any single point on such a diagram is known as an
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256:. For instance, an object located at position
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240:When discussing the evolution of a system in
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377:if they differ along the time axis, or
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185:which would replace GR, an idea which
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47:adding citations to reliable sources
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630:Contractible versus noncontractible
500:(which models a rotating uncharged
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1264:Novikov self-consistency principle
1059:General Relativity and Gravitation
267:can only move to locations within
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446:through time as seen externally.
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699:Poincaré recurrence theorem
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385:, it would travel up the
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653:chronology violating set
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454:under these conditions.
402:In "simple" examples of
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1199:A Primer on Time Travel
472:Einstein field equation
244:, or more specifically
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1433:closed timelike curves
1284:Time travel in fiction
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493:by a discrete boost)
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1329:Grandfather paradox
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1096:Kurt Gödel (1949).
1077:S. Carroll (2004).
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175:grandfather paradox
139:Lorentzian manifold
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1081:. Addison Wesley.
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620:entanglement
613:
605:
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570:
567:Consequences
560:
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484:Misner space
463:
461:
448:
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189:labeled the
159:Gödel metric
130:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1502:Time travel
1456:Kerr metric
1393:Determinism
1334:Causal loop
1240:Time travel
1176:. Penguin.
577:determinism
498:Kerr vacuum
452:time travel
201:Light cones
171:time travel
1496:Categories
1423:Spacetimes
1398:Eternalism
1365:Multiverse
1070:References
944:1511.00538
684:Roman ring
608:worldlines
502:black hole
491:orbifolded
486:(which is
437:rest frame
289:) by time
250:light cone
151:Kurt Gödel
135:world line
69:newspapers
1408:Free will
1304:Time loop
1161:121204248
1015:0808.2669
969:2045-2322
904:1550-7998
879:1007.2615
843:0556-2821
785:119707350
573:causality
396:worldline
383:free fall
143:spacetime
1403:Fatalism
987:27897219
912:15972766
851:10013776
694:Timelike
663:See also
561:interior
557:exterior
409:geodesic
260:at time
1309:in film
1141:Bibcode
1110:Bibcode
1020:Bibcode
978:5126586
949:Bibcode
884:Bibcode
823:Bibcode
765:Bibcode
470:to the
464:locally
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83:scholar
1180:
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1040:745646
1038:
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624:PSPACE
614:While
85:
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71:
64:
56:
1157:S2CID
1036:S2CID
1010:arXiv
939:arXiv
908:S2CID
874:arXiv
781:S2CID
755:arXiv
705:Notes
391:could
371:event
137:in a
90:JSTOR
76:books
1429:that
1178:ISBN
1083:ISBN
983:PMID
965:ISSN
900:ISSN
847:PMID
839:ISSN
651:The
531:the
524:the
519:dust
513:the
496:the
482:the
442:have
346:per
165:and
125:, a
62:news
1425:in
1149:doi
1118:doi
1028:doi
1006:465
973:PMC
957:doi
892:doi
831:doi
773:doi
474:of
131:CTC
121:In
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