217:
36:
670:. 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.
422:. 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
460:
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
263:". 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.
377:. 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.
208:, 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,
633:, 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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610:), 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.
400:-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
650:, 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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450:. 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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1109:"An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation"
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156:, that is "closed", returning to its starting point. This possibility was first discovered by
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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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solutions describing laboratory situations such as two spinning balls
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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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938:"Timelike curves can increase entanglement with LOCC"
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380:Any single point on such a diagram is known as an
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251:When discussing the evolution of a system in
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196:which would replace GR, an idea which
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58:adding citations to reliable sources
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641:Contractible versus noncontractible
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664:chronology violating set
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465:under these conditions.
413:In "simple" examples of
158:Willem Jacob van Stockum
1210:A Primer on Time Travel
483:Einstein field equation
255:, or more specifically
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1444:closed timelike curves
1295:Time travel in fiction
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758:Foundations of Physics
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134:mathematical physics
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1340:Grandfather paradox
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1125:1949RvMP...21..447G
1107:Kurt Gödel (1949).
1088:S. Carroll (2004).
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186:grandfather paradox
150:Lorentzian manifold
1438:general relativity
1361:Alternative future
1354:Parallel timelines
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1308:in science fiction
1092:. Addison Wesley.
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804:
761:
757:
743:, chapter 11
738:
733:
724:
700:Time crystal
661:
644:
631:entanglement
624:
616:
605:
581:
578:Consequences
571:
567:
565:
495:Misner space
474:
472:
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405:
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331:
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250:
200:labeled the
170:Gödel metric
141:
137:
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107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
1513:Time travel
1467:Kerr metric
1404:Determinism
1345:Causal loop
1251:Time travel
1187:. Penguin.
588:determinism
509:Kerr vacuum
463:time travel
212:Light cones
182:time travel
1507:Categories
1434:Spacetimes
1409:Eternalism
1376:Multiverse
1081:References
955:1511.00538
695:Roman ring
619:worldlines
513:black hole
502:orbifolded
497:(which is
448:rest frame
300:) by time
261:light cone
162:Kurt Gödel
146:world line
80:newspapers
1419:Free will
1315:Time loop
1172:121204248
1026:0808.2669
980:2045-2322
915:1550-7998
890:1007.2615
854:0556-2821
796:119707350
584:causality
407:worldline
394:free fall
154:spacetime
1414:Fatalism
998:27897219
923:15972766
862:10013776
705:Timelike
674:See also
572:interior
568:exterior
420:geodesic
271:at time
1320:in film
1152:Bibcode
1121:Bibcode
1031:Bibcode
989:5126586
960:Bibcode
895:Bibcode
834:Bibcode
776:Bibcode
481:to the
475:locally
144:) is a
94:scholar
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1051:745646
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635:PSPACE
625:While
96:
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1168:S2CID
1047:S2CID
1021:arXiv
950:arXiv
919:S2CID
885:arXiv
792:S2CID
766:arXiv
716:Notes
402:could
382:event
148:in a
101:JSTOR
87:books
1440:that
1189:ISBN
1094:ISBN
994:PMID
976:ISSN
911:ISSN
858:PMID
850:ISSN
662:The
542:the
535:the
530:dust
524:the
507:the
493:the
453:have
357:per
176:and
136:, a
73:news
1436:in
1160:doi
1129:doi
1039:doi
1017:465
984:PMC
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903:doi
842:doi
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485:of
142:CTC
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280:p
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