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The '''chronology protection conjecture''' is a [[conjecture]] by the [[physicist]] Professor [[Stephen Hawking]] that the [[laws of physics]] are such as to prevent [[time travel]] on all but sub-microscopic scales. Mathematically, the permissibility of time travel is represented by the existence of [[closed timelike curve]]s. The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an [[event horizon]], which might prevent an observer from detecting the causal violation.<ref>http://dx.doi.org/10.1007/s10701-008-9254-9</ref>
The '''chronology protection conjecture''' is a [[conjecture]] by the [[physicist]] Professor [[Stephen Hawking]] that the [[laws of physics]] are such as to prevent [[time travel]] on all but sub-microscopic scales. Mathematically, the permissibility of time travel is represented by the existence of [[closed timelike curve]]s. The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an [[event horizon]], which might prevent an observer from detecting the causal violation.<ref>{{cite web|url=http://dx.doi.org/10.1007/s10701-008-9254-9 |title=Are Causality Violations Undesirable? - Springer |publisher=Dx.doi.org |date=2008-10-29 |accessdate=2014-08-25}}</ref>


==Origin of the term==
==Origin of the term==
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{{quote|It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.}}
{{quote|It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.}}


The idea of the Chronology Protection Agency appears to be drawn playfully from the Time Patrol or Time Police concept, which has been used in many works of [[science fiction]]<ref>http://www.sf-encyclopedia.com/entry/time_police</ref> such as [[Poul Anderson]]'s series of [[Poul Anderson#Time Patrol|Time Patrol]] stories or [[Isaac Asimov]]'s novel ''[[The End of Eternity]]'' (though in Asimov's novel the time travelers were constantly trying to make small changes to history to improve it, while preserving some broad features like a lack of atomic wars).
The idea of the Chronology Protection Agency appears to be drawn playfully from the Time Patrol or Time Police concept, which has been used in many works of [[science fiction]]<ref>{{cite web|url=http://www.sf-encyclopedia.com/entry/time_police |title=Time Police : SFE : Science Fiction Encyclopedia |publisher=Sf-encyclopedia.com |date=December 21, 2011 |accessdate=2014-08-25}}</ref> such as [[Poul Anderson]]'s series of [[Poul Anderson#Time Patrol|Time Patrol]] stories or [[Isaac Asimov]]'s novel ''[[The End of Eternity]]'' (though in Asimov's novel the time travelers were constantly trying to make small changes to history to improve it, while preserving some broad features like a lack of atomic wars).


==General relativity and quantum corrections==
==General relativity and quantum corrections==
Many attempts to generate scenarios for closed timelike curves have been suggested, and the theory of [[general relativity]] ([[Albert Einstein]]'s theory of gravity) does allow them in certain circumstances. Some theoretical solutions in general relativity that contain closed timelike curves would require an infinite universe with certain features which our universe does not appear to have, such as the universal rotation of the [[Gödel metric]] or the rotating cylinder of infinite length known as a [[Tipler cylinder]]. However, some solutions allow for the creation of closed timelike curves in a bounded region of spacetime, with the [[Cauchy horizon]] being the boundary between the region of spacetime where closed timelike curves can exist and the rest of spacetime where they can't.<ref>{{cite book | last = Gott | first = J. Richard | authorlink = J. Richard Gott | title = Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time | publisher = Houghton Mifflin | year= 2001 | page = 117 | isbn = 0-395-95563-7}}</ref> One of the first such bounded time travel solutions found was constructed from a [[Wormhole#Traversable wormholes|traversable wormhole]], based on the idea of taking one of the two "mouths" of the wormhole on a round-trip journey at relativistic speed to create a time difference between it and the other mouth (see the discussion at [[Wormhole#Time travel]]).
Many attempts to generate scenarios for closed timelike curves have been suggested, and the theory of [[general relativity]] ([[Albert Einstein]]'s theory of gravity) does allow them in certain circumstances. Some theoretical solutions in general relativity that contain closed timelike curves would require an infinite universe with certain features which our universe does not appear to have, such as the universal rotation of the [[Gödel metric]] or the rotating cylinder of infinite length known as a [[Tipler cylinder]]. However, some solutions allow for the creation of closed timelike curves in a bounded region of spacetime, with the [[Cauchy horizon]] being the boundary between the region of spacetime where closed timelike curves can exist and the rest of spacetime where they can't.<ref>{{cite book | last=Gott | first=J. Richard | authorlink=J. Richard Gott | title=Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time | publisher=Houghton Mifflin | year= 2001 | page=117 | isbn=0-395-95563-7}}</ref> One of the first such bounded time travel solutions found was constructed from a [[Wormhole#Traversable wormholes|traversable wormhole]], based on the idea of taking one of the two "mouths" of the wormhole on a round-trip journey at relativistic speed to create a time difference between it and the other mouth (see the discussion at [[Wormhole#Time travel]]).


General relativity does not include [[quantum mechanics|quantum]] effects on its own, and a full integration of general relativity and quantum mechanics would require a theory of [[quantum gravity]], but there is an approximate method for modeling quantum fields in the curved spacetime of general relativity, known as [[semiclassical gravity]]. Initial attempts to apply semiclassical gravity to the traversable wormhole time machine indicated that at exactly the moment that wormhole would first allow for closed timelike curves, quantum [[Virtual particle#Virtual particles in vacuums|vacuum fluctuations]] build up and drive the [[energy density]] to infinity in the region of the wormholes. This occurs when the two wormhole mouths, call them A and B, have been moved in such a way that it becomes possible for a particle or wave moving at the speed of light to enter mouth B at some time T<sub>2</sub> and exit through mouth A at an earlier time T<sub>1</sub>, then travel back towards mouth B through ordinary space, and arrive at mouth B at the same time T<sub>2</sub> that it entered B on the previous loop; in this way the same particle or wave can make a potentially infinite number of loops through the same regions of spacetime, piling up on itself.<ref>{{cite book | last = Thorne | first = Kip S. | authorlink = Kip Thorne | title = [[Black Holes and Time Warps]] | publisher = W. W. Norton | year= 1994 | pages = 505-506 | isbn = 0-393-31276-3}}</ref> Calculations showed that this effect would not occur for an ordinary beam of radiation, because it would be "defocused" by the wormhole so that most of a beam emerging from mouth A would spread out and miss mouth B.<ref>Thorne 1994, p. 507</ref> But when the calculation was done for vacuum fluctuations, it was found that they would spontaneously refocus on the trip between the mouths, indicating that the pileup effect might become large enough to destroy the wormhole in this case.<ref>Thorne 1994, p. 517</ref>
General relativity does not include [[quantum mechanics|quantum]] effects on its own, and a full integration of general relativity and quantum mechanics would require a theory of [[quantum gravity]], but there is an approximate method for modeling quantum fields in the curved spacetime of general relativity, known as [[semiclassical gravity]]. Initial attempts to apply semiclassical gravity to the traversable wormhole time machine indicated that at exactly the moment that wormhole would first allow for closed timelike curves, quantum [[Virtual particle#Virtual particles in vacuums|vacuum fluctuations]] build up and drive the [[energy density]] to infinity in the region of the wormholes. This occurs when the two wormhole mouths, call them A and B, have been moved in such a way that it becomes possible for a particle or wave moving at the speed of light to enter mouth B at some time T<sub>2</sub> and exit through mouth A at an earlier time T<sub>1</sub>, then travel back towards mouth B through ordinary space, and arrive at mouth B at the same time T<sub>2</sub> that it entered B on the previous loop; in this way the same particle or wave can make a potentially infinite number of loops through the same regions of spacetime, piling up on itself.<ref>{{cite book | last=Thorne | first=Kip S. | authorlink=Kip Thorne | title=[[Black Holes and Time Warps]] | publisher=W. W. Norton | year= 1994 | pages=505-506 | isbn=0-393-31276-3}}</ref> Calculations showed that this effect would not occur for an ordinary beam of radiation, because it would be "defocused" by the wormhole so that most of a beam emerging from mouth A would spread out and miss mouth B.<ref>Thorne 1994, p. 507</ref> But when the calculation was done for vacuum fluctuations, it was found that they would spontaneously refocus on the trip between the mouths, indicating that the pileup effect might become large enough to destroy the wormhole in this case.<ref>Thorne 1994, p. 517</ref>


Uncertainty about this conclusion remained, because the semiclassical calculations indicated that the pileup would only drive the energy density to infinity for an infinitesimal moment of time, after which the energy density would die down.<ref>{{cite book | last = Everett | first = Allen |author2=Roman, Thomas | title = Time Travel and Warp Drives | publisher = University of Chicago Press | year= 2012 | page = 190 | isbn = 0-226-22498-8}}</ref> But semiclassical gravity is considered unreliable for large energy densities or short time periods that reach the [[Planck scale]]; at these scales, a complete theory of quantum gravity is needed for accurate predictions. So, it remains uncertain whether quantum-gravitational effects might prevent the energy density from growing large enough to destroy the wormhole.<ref>Everett and Roman 2012, p. 190</ref> Stephen Hawking conjectured that not only would the pileup of vacuum fluctuations still succeed in destroying the wormhole in quantum gravity, but also that the laws of physics would ultimately prevent ''any'' type of time machine from forming; this is the chronology protection conjecture.<ref name="EvRom191">Everett and Roman 2012, p. 191</ref>
Uncertainty about this conclusion remained, because the semiclassical calculations indicated that the pileup would only drive the energy density to infinity for an infinitesimal moment of time, after which the energy density would die down.<ref>{{cite book | last=Everett | first=Allen |author2=Roman, Thomas | title=Time Travel and Warp Drives | publisher=University of Chicago Press | year= 2012 | page=190 | isbn=0-226-22498-8}}</ref> But semiclassical gravity is considered unreliable for large energy densities or short time periods that reach the [[Planck scale]]; at these scales, a complete theory of quantum gravity is needed for accurate predictions. So, it remains uncertain whether quantum-gravitational effects might prevent the energy density from growing large enough to destroy the wormhole.<ref>Everett and Roman 2012, p. 190</ref> Stephen Hawking conjectured that not only would the pileup of vacuum fluctuations still succeed in destroying the wormhole in quantum gravity, but also that the laws of physics would ultimately prevent ''any'' type of time machine from forming; this is the chronology protection conjecture.<ref name="EvRom191">Everett and Roman 2012, p. 191</ref>


Subsequent work in semiclassical gravity provided examples of spacetimes with closed timelike curves where the energy density due to vacuum fluctuations does not approach infinity in the region of spacetime outside the Cauchy horizon.<ref name="EvRom191" /> However, in 1997 a general proof was found demonstrating that according to semiclassical gravity, the energy of the quantum field (more precisely, the expectation value of the quantum stress-energy tensor) must ''always'' be either infinite or undefined on the horizon itself.<ref>{{cite journal | first = Bernard | last = Kay | last2 = Radzikowski | first2 = Marek | last3 = Wald | first3 = Robert | author3-link = Robert Wald | title = Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon | journal = Communications in Mathematical Physics | volume = 183 | year = 1997 | issue = 3 | pages = 533-556 | doi = 10.1007/s002200050042 | arxiv=gr-qc/9603012v2 |bibcode = 1997CMaPh.183..533K }}</ref> Both cases indicate that semiclassical methods become unreliable at the horizon and quantum gravity effects would be important there, consistent with the possibility that such effects would always intervene to prevent time machines from forming.<ref name="EvRom191" />
Subsequent work in semiclassical gravity provided examples of spacetimes with closed timelike curves where the energy density due to vacuum fluctuations does not approach infinity in the region of spacetime outside the Cauchy horizon.<ref name="EvRom191" /> However, in 1997 a general proof was found demonstrating that according to semiclassical gravity, the energy of the quantum field (more precisely, the expectation value of the quantum stress-energy tensor) must ''always'' be either infinite or undefined on the horizon itself.<ref>{{cite journal | first=Bernard | last=Kay | last2=Radzikowski | first2=Marek | last3=Wald | first3=Robert | author3-link=Robert Wald | title=Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon | journal=Communications in Mathematical Physics | volume=183 | year=1997 | issue=3 | pages=533-556 | doi=10.1007/s002200050042 | arxiv=gr-qc/9603012v2 |bibcode=1997CMaPh.183..533K }}</ref> Both cases indicate that semiclassical methods become unreliable at the horizon and quantum gravity effects would be important there, consistent with the possibility that such effects would always intervene to prevent time machines from forming.<ref name="EvRom191" />


A definite theoretical decision on the status of the chronology protection conjecture would require a full theory of [[quantum gravity]]<ref>Thorne 1994, p. 521</ref> as opposed to semiclassical methods (there are also some arguments from [[string theory]] which seem to support chronology protection,<ref>{{cite news | last = Semeniuk | first = Ivan | title = No going back | newspaper = [[New Scientist]] | date = 20 September 2003 | url = http://www.newscientist.com/article/mg17924134.800-no-going-back.html | accessdate = 10 January 2013}}</ref><ref>{{cite journal | first = C.A.R. | last = Herdeiro | title = Special properties of five-dimensional BPS rotating black holes | journal = Nuclear Physics B | volume = 582 | year = 2000 | issue = 1-3 | pages = 363-392 | doi = 10.1016/S0550-3213(00)00335-7 | arxiv=hep-th/0003063 |bibcode = 2000NuPhB.582..363H }}</ref><ref>{{cite journal | first = Marco | last = Caldarelli | last2 = Klemm | first2 = Dietmar | last3 = Silva | first3 = Pedro | title = Chronology protection in anti-de Sitter | journal = Classical and Quantum Gravity | volume = 22 | year = 2005 | issue = 17 | pages = 3461 | doi = 10.1088/0264-9381/22/17/007 | arxiv=hep-th/0411203 |bibcode = 2005CQGra..22.3461C }}</ref><ref>{{cite journal | first = Marco | last = Caldarelli | last2 = Klemm | first2 = Dietmar | last3 = Sabra | first3 = Wafic | title = Causality violation and naked time machines in AdS<sub>5</sub> | journal = Journal of High Energy Physics | year = 2001 | issue = 5 | doi = 10.1088/1126-6708/2001/05/014 | arxiv=hep-th/0103133 |bibcode = 2001JHEP...05..014C }}</ref><ref>{{cite arXiv |last=Raeymaekers |first=Joris |last2=Van den Bleeken |first2=Dieter |last3=Vercnocke |first3=Bert |eprint=0911.3893 |title=Relating chronology protection and unitarity through holography |class=hep-th |year=2009 |accessdate= 10 January 2014}}</ref> but string theory is not yet a complete theory of quantum gravity). Experimental observation of closed timelike curves would of course demonstrate this conjecture to be [[falsifiability|false]], but short of that, if physicists had a theory of quantum gravity whose predictions had been well-confirmed in other areas, this would give them a significant degree of confidence in the theory's predictions about the possibility or impossibility of time travel.
A definite theoretical decision on the status of the chronology protection conjecture would require a full theory of [[quantum gravity]]<ref>Thorne 1994, p. 521</ref> as opposed to semiclassical methods (there are also some arguments from [[string theory]] which seem to support chronology protection,<ref>{{cite news | last=Semeniuk | first=Ivan | title=No going back | newspaper=[[New Scientist]] | date=20 September 2003 | url=http://www.newscientist.com/article/mg17924134.800-no-going-back.html | accessdate=10 January 2013}}</ref><ref>{{cite journal | first=C.A.R. | last=Herdeiro | title=Special properties of five-dimensional BPS rotating black holes | journal=Nuclear Physics B | volume=582 | year=2000 | issue=1-3 | pages=363-392 | doi=10.1016/S0550-3213(00)00335-7 | arxiv=hep-th/0003063 |bibcode=2000NuPhB.582..363H }}</ref><ref>{{cite journal | first=Marco | last=Caldarelli | last2=Klemm | first2=Dietmar | last3=Silva | first3=Pedro | title=Chronology protection in anti-de Sitter | journal=Classical and Quantum Gravity | volume=22 | year=2005 | issue=17 | page=3461 | doi=10.1088/0264-9381/22/17/007 | arxiv=hep-th/0411203 |bibcode=2005CQGra..22.3461C }}</ref><ref>{{cite journal | first=Marco | last=Caldarelli | last2=Klemm | first2=Dietmar | last3=Sabra | first3=Wafic | title=Causality violation and naked time machines in AdS<sub>5</sub> | journal=Journal of High Energy Physics | year=2001 | issue=5 | doi=10.1088/1126-6708/2001/05/014 | arxiv=hep-th/0103133 |bibcode=2001JHEP...05..014C }}</ref><ref>{{cite arXiv |last=Raeymaekers |first=Joris |last2=Van den Bleeken |first2=Dieter |last3=Vercnocke |first3=Bert |eprint=0911.3893 |title=Relating chronology protection and unitarity through holography |class=hep-th |year=2009 |accessdate= 10 January 2014}}</ref> but string theory is not yet a complete theory of quantum gravity). Experimental observation of closed timelike curves would of course demonstrate this conjecture to be [[falsifiability|false]], but short of that, if physicists had a theory of quantum gravity whose predictions had been well-confirmed in other areas, this would give them a significant degree of confidence in the theory's predictions about the possibility or impossibility of time travel.


Other proposals which allow for backwards time travel but prevent [[Physical paradox#Causality paradoxes|time paradoxes]], such as the [[Novikov self-consistency principle]] which would ensure the timeline stays consistent, or the idea that a time traveler is taken to a [[Many-worlds interpretation|parallel universe]] while his original timeline remains intact, do not qualify as "chronology protection".
Other proposals which allow for backwards time travel but prevent [[Physical paradox#Causality paradoxes|time paradoxes]], such as the [[Novikov self-consistency principle]] which would ensure the timeline stays consistent, or the idea that a time traveler is taken to a [[Many-worlds interpretation|parallel universe]] while his original timeline remains intact, do not qualify as "chronology protection".

Revision as of 05:33, 25 August 2014

The chronology protection conjecture is a conjecture by the physicist Professor Stephen Hawking that the laws of physics are such as to prevent time travel on all but sub-microscopic scales. Mathematically, the permissibility of time travel is represented by the existence of closed timelike curves. The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an event horizon, which might prevent an observer from detecting the causal violation.[1]

Origin of the term

In a 1992 paper, Hawking uses the metaphorical device of a "Chronology Protection Agency" as a personification of the aspects of physics that make time travel impossible at macroscopic scales, thus apparently preventing time paradoxes. He says:

It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.

The idea of the Chronology Protection Agency appears to be drawn playfully from the Time Patrol or Time Police concept, which has been used in many works of science fiction[2] such as Poul Anderson's series of Time Patrol stories or Isaac Asimov's novel The End of Eternity (though in Asimov's novel the time travelers were constantly trying to make small changes to history to improve it, while preserving some broad features like a lack of atomic wars).

General relativity and quantum corrections

Many attempts to generate scenarios for closed timelike curves have been suggested, and the theory of general relativity (Albert Einstein's theory of gravity) does allow them in certain circumstances. Some theoretical solutions in general relativity that contain closed timelike curves would require an infinite universe with certain features which our universe does not appear to have, such as the universal rotation of the Gödel metric or the rotating cylinder of infinite length known as a Tipler cylinder. However, some solutions allow for the creation of closed timelike curves in a bounded region of spacetime, with the Cauchy horizon being the boundary between the region of spacetime where closed timelike curves can exist and the rest of spacetime where they can't.[3] One of the first such bounded time travel solutions found was constructed from a traversable wormhole, based on the idea of taking one of the two "mouths" of the wormhole on a round-trip journey at relativistic speed to create a time difference between it and the other mouth (see the discussion at Wormhole#Time travel).

General relativity does not include quantum effects on its own, and a full integration of general relativity and quantum mechanics would require a theory of quantum gravity, but there is an approximate method for modeling quantum fields in the curved spacetime of general relativity, known as semiclassical gravity. Initial attempts to apply semiclassical gravity to the traversable wormhole time machine indicated that at exactly the moment that wormhole would first allow for closed timelike curves, quantum vacuum fluctuations build up and drive the energy density to infinity in the region of the wormholes. This occurs when the two wormhole mouths, call them A and B, have been moved in such a way that it becomes possible for a particle or wave moving at the speed of light to enter mouth B at some time T2 and exit through mouth A at an earlier time T1, then travel back towards mouth B through ordinary space, and arrive at mouth B at the same time T2 that it entered B on the previous loop; in this way the same particle or wave can make a potentially infinite number of loops through the same regions of spacetime, piling up on itself.[4] Calculations showed that this effect would not occur for an ordinary beam of radiation, because it would be "defocused" by the wormhole so that most of a beam emerging from mouth A would spread out and miss mouth B.[5] But when the calculation was done for vacuum fluctuations, it was found that they would spontaneously refocus on the trip between the mouths, indicating that the pileup effect might become large enough to destroy the wormhole in this case.[6]

Uncertainty about this conclusion remained, because the semiclassical calculations indicated that the pileup would only drive the energy density to infinity for an infinitesimal moment of time, after which the energy density would die down.[7] But semiclassical gravity is considered unreliable for large energy densities or short time periods that reach the Planck scale; at these scales, a complete theory of quantum gravity is needed for accurate predictions. So, it remains uncertain whether quantum-gravitational effects might prevent the energy density from growing large enough to destroy the wormhole.[8] Stephen Hawking conjectured that not only would the pileup of vacuum fluctuations still succeed in destroying the wormhole in quantum gravity, but also that the laws of physics would ultimately prevent any type of time machine from forming; this is the chronology protection conjecture.[9]

Subsequent work in semiclassical gravity provided examples of spacetimes with closed timelike curves where the energy density due to vacuum fluctuations does not approach infinity in the region of spacetime outside the Cauchy horizon.[9] However, in 1997 a general proof was found demonstrating that according to semiclassical gravity, the energy of the quantum field (more precisely, the expectation value of the quantum stress-energy tensor) must always be either infinite or undefined on the horizon itself.[10] Both cases indicate that semiclassical methods become unreliable at the horizon and quantum gravity effects would be important there, consistent with the possibility that such effects would always intervene to prevent time machines from forming.[9]

A definite theoretical decision on the status of the chronology protection conjecture would require a full theory of quantum gravity[11] as opposed to semiclassical methods (there are also some arguments from string theory which seem to support chronology protection,[12][13][14][15][16] but string theory is not yet a complete theory of quantum gravity). Experimental observation of closed timelike curves would of course demonstrate this conjecture to be false, but short of that, if physicists had a theory of quantum gravity whose predictions had been well-confirmed in other areas, this would give them a significant degree of confidence in the theory's predictions about the possibility or impossibility of time travel.

Other proposals which allow for backwards time travel but prevent time paradoxes, such as the Novikov self-consistency principle which would ensure the timeline stays consistent, or the idea that a time traveler is taken to a parallel universe while his original timeline remains intact, do not qualify as "chronology protection".

See also

Notes

  1. ^ "Are Causality Violations Undesirable? - Springer". Dx.doi.org. 2008-10-29. Retrieved 2014-08-25.
  2. ^ "Time Police : SFE : Science Fiction Encyclopedia". Sf-encyclopedia.com. December 21, 2011. Retrieved 2014-08-25.
  3. ^ Gott, J. Richard (2001). Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time. Houghton Mifflin. p. 117. ISBN 0-395-95563-7.
  4. ^ Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. pp. 505–506. ISBN 0-393-31276-3.
  5. ^ Thorne 1994, p. 507
  6. ^ Thorne 1994, p. 517
  7. ^ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 190. ISBN 0-226-22498-8.
  8. ^ Everett and Roman 2012, p. 190
  9. ^ a b c Everett and Roman 2012, p. 191
  10. ^ Kay, Bernard; Radzikowski, Marek; Wald, Robert (1997). "Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon". Communications in Mathematical Physics. 183 (3): 533–556. arXiv:gr-qc/9603012v2. Bibcode:1997CMaPh.183..533K. doi:10.1007/s002200050042.
  11. ^ Thorne 1994, p. 521
  12. ^ Semeniuk, Ivan (20 September 2003). "No going back". New Scientist. Retrieved 10 January 2013.
  13. ^ Herdeiro, C.A.R. (2000). "Special properties of five-dimensional BPS rotating black holes". Nuclear Physics B. 582 (1–3): 363–392. arXiv:hep-th/0003063. Bibcode:2000NuPhB.582..363H. doi:10.1016/S0550-3213(00)00335-7.
  14. ^ Caldarelli, Marco; Klemm, Dietmar; Silva, Pedro (2005). "Chronology protection in anti-de Sitter". Classical and Quantum Gravity. 22 (17): 3461. arXiv:hep-th/0411203. Bibcode:2005CQGra..22.3461C. doi:10.1088/0264-9381/22/17/007.
  15. ^ Caldarelli, Marco; Klemm, Dietmar; Sabra, Wafic (2001). "Causality violation and naked time machines in AdS5". Journal of High Energy Physics (5). arXiv:hep-th/0103133. Bibcode:2001JHEP...05..014C. doi:10.1088/1126-6708/2001/05/014.
  16. ^ Raeymaekers, Joris; Van den Bleeken, Dieter; Vercnocke, Bert (2009). "Relating chronology protection and unitarity through holography". arXiv:0911.3893 [hep-th]. {{cite arXiv}}: Unknown parameter |accessdate= ignored (help)

References

External links

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