# Wormhole

(Redirected from Wormhole metrics)

A wormhole is a solution of the Einstein field equations having a non-trivial structure linking separate points in spacetime, much like a tunnel with two ends, each at separate points in spacetime. Such connections are consistent with the general theory of relativity, yet their existence remains hypothetical.

A wormhole may connect extremely long distances such as a billion light years or more; short distances such as a few meters; different universes; and/or different points in time.[further explanation needed]

## Terminology

In 1921, Hermann Weyl proposed a wormhole theory of matter in connection with mass analysis of electromagnetic field energy;[1][2] however, he did not use the term "wormhole" (he spoke of "one-dimensional tubes" instead).[3]

The American theoretical physicists John Archibald Wheeler (inspired by Weyl's work)[3] coined the term "wormhole" in a 1957 paper co-authored by Charles Misner:[4]

This analysis forces one to consider situations... where there is a net flux of lines of force, through what topologists would call "a handle" of the multiply-connected space, and what physicists might perhaps be excused for more vividly terming a "wormhole".

— Charles Misner and John Wheeler in Annals of Physics

### Modern definitions

Wormholes have been defined both geometrically and topologically,[further explanation needed]

#### Topological

An intra-universe wormhole (a wormhole between two points in the same universe) is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes (1996).

If a Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S2, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intrauniverse wormhole.

#### Geometric

Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces. For example, in Enrico Rodrigo's The Physics of Stargates, a wormhole is defined informally as:

a region of spacetime containing a "world tube" (the time evolution of a closed surface) that cannot be continuously deformed (shrunk) to a world line (the time evolution of a point).

## Development

The equations of the theory of general relativity have valid solutions that contain wormholes. The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them.

"Embedding diagram" of a Schwarzschild wormhole

### Schwarzschild wormholes

An artist's impression of a wormhole from an observer's perspective, crossing the event horizon of a Schwarzschild wormhole that bridges two different universes. The observer originates from the right, and another universe becomes visible in the center of the wormhole's shadow once the horizon is crossed, the observer seeing light that has fallen into the black hole interior region from the other universe; however, this other universe is unreachable in the case of a Schwarzschild wormhole, as the bridge always collapses before the observer has time to cross it, and everything that has fallen through the event horizon of either universe is inevitably crushed in the singularity.

Lorentzian wormholes (named after Hendrik Lorentz), also known as Schwarzschild wormholes or EinsteinRosen bridges, are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal[clarification needed] black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the space-time should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of a free-falling particle (following a geodesic in the spacetime), unless the trajectory hits a gravitational singularity like the one at the center of the black hole's interior.

In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates.

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') and draw an "embedding diagram" depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". Note that the Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's history, it removes the part of the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe.[5]

The Einstein–Rosen bridge was discovered by Ludwig Flamm in 1916,[6] a few months after Schwarzschild published his solution, and was rediscovered (although it is hard to imagine that Einstein had not seen Flamm's paper when it came out) by Albert Einstein and his colleague Nathan Rosen, who published their result in 1935.[7][8] However, in 1962, John A. Wheeler and Robert W. Fuller published a paper[9] showing that this type of wormhole is unstable if it connects two parts of the same universe, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to the other exterior region.

According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan–Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum (spin) of matter. The minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction that is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity.[clarification needed] Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.[10]

Although Schwarzschild wormholes are not traversable in both directions, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the "throat" of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy).

### Traversable wormholes

The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at a given point.[11] Many physicists, such as Stephen Hawking,[12] Kip Thorne[13] and others,[14][15][16] therefore argue that such effects might make it possible to stabilize a traversable wormhole. Physicists have not found any natural process that would be predicted to form a wormhole naturally in the context of general relativity, although the quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,[17][18] and stable versions of such wormholes have been suggested as dark matter candidates.[19][20] It has also been proposed that, if a tiny wormhole held open by a negative mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.[21]

Image of a simulated traversable wormhole that connects the square in front of the physical institutes of University of Tübingen with the sand dunes near Boulogne sur Mer in the north of France. The image is calculated with 4D raytracing in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.[22]

Lorentzian traversable wormholes would allow travel in both directions from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis[23] and independently in a 1973 paper by K. A. Bronnikov.[24] Ellis thoroughly analyzed the topology and the geodesics of the Ellis drainhole, showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole is a solution manifold of Einstein's field equations for a vacuum space-time, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity (negative instead of positive). (Ellis specifically rejected referring to the scalar field as 'exotic' because of the antiorthodox coupling, finding arguments for doing so unpersuasive.) The solution depends on two parameters: ${\displaystyle m}$, which fixes the strength of its gravitational field, and ${\displaystyle n}$, which determines the curvature of its spatial cross sections. When ${\displaystyle m}$ is set equal to 0, the drainhole's gravitational field vanishes. What is left is the Ellis wormhole, a nongravitating, purely geometric, traversable wormhole. Kip Thorne and his graduate student Mike Morris, unaware of the 1973 papers by Ellis and Bronnikov, manufactured, and in 1988 published, a duplicate of the Ellis wormhole for use as a tool for teaching general relativity. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, was from 1988 to 2015 exclusively referred to in the literature as a Morris–Thorne wormhole. Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made where the traversing path does not pass through a region of exotic matter. However, in the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology) exotic matter is not needed in order for wormholes to exist—they can exist even with no matter.[25] A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,[21] in which it was proposed that such wormholes could have been naturally created in the early universe.

Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out explicitly how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths.[13] However, according to general relativity, it would not be possible to use a wormhole to travel back to a time earlier than when the wormhole was first converted into a time 'machine'.[26]

## Raychaudhuri's theorem and exotic matter

To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck is of finite size, we would not expect caustics to develop, at least within the vicinity of the neck. According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition. Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature,[27] but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.[28] Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,[29] violations have nevertheless been found,[30] so it remains an open possibility that quantum effects might be used to support a wormhole.

## Modified general relativity

In some theories where general relativity is modified, it is possible to have a wormhole that does not collapse without having to resort to exotic matter. For example, this is possible with R^2 gravity, a form of f(R) gravity.[31]

## Faster-than-light travel

The impossibility of faster-than-light relative speed only applies locally. Wormholes might allow effective superluminal (faster-than-light) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by a wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. However, a light beam traveling through the wormhole would of course beat the traveler.

## Time travel

The theory of general relativity predicts that if traversable wormholes exist, they can also alter the speed of time. They could allow time travel.[13] This would be accomplished by accelerating one end of the wormhole to a high velocity relative to the other, and then sometime later bringing it back; relativistic time dilation would result in the accelerated wormhole mouth aging less than the stationary one as seen by an external observer, similar to what is seen in the twin paradox. However, time connects differently through the wormhole than outside it, so that synchronized clocks at each mouth will remain synchronized to someone traveling through the wormhole itself, no matter how the mouths move around.[32] This means that anything which entered the accelerated wormhole mouth would exit the stationary one at a point in time prior to its entry.

For example, consider two clocks at both mouths both showing the date as 2000. After being taken on a trip at relativistic velocities, the accelerated mouth is brought back to the same region as the stationary mouth with the accelerated mouth's clock reading 2004 while the stationary mouth's clock read 2012. A traveler who entered the accelerated mouth at this moment would exit the stationary mouth when its clock also read 2004, in the same region but now eight years in the past. Such a configuration of wormholes would allow for a particle's world line to form a closed loop in spacetime, known as a closed timelike curve. An object traveling through a wormhole could carry energy or charge from one time to another, but this would not violate conservation of energy or charge in each time, because the energy/charge of the wormhole mouth itself would change to compensate for the object that fell into it or emerged from it.[33][34]

It is thought that it may not be possible to convert a wormhole into a time machine in this manner; the predictions are made in the context of general relativity, but general relativity does not include quantum effects. Analyses using the semiclassical approach to incorporating quantum effects into general relativity have sometimes indicated that a feedback loop of virtual particles would circulate through the wormhole and pile up on themselves, driving the energy density in the region very high and possibly destroying it before any information could be passed through it, in keeping with the chronology protection conjecture. The debate on this matter is described by Kip S. Thorne in the book Black Holes and Time Warps, and a more technical discussion can be found in The quantum physics of chronology protection by Matt Visser.[35] There is also the Roman ring, which is a configuration of more than one wormhole. This ring seems to allow a closed time loop with stable wormholes when analyzed using semiclassical gravity, although without a full theory of quantum gravity it is uncertain whether the semiclassical approach is reliable in this case.

## Interuniversal travel

A possible resolution to the paradoxes resulting from wormhole-enabled time travel rests on the many-worlds interpretation of quantum mechanics. In 1991 David Deutsch showed that quantum theory is fully consistent (in the sense that the so-called density matrix can be made free of discontinuities) in spacetimes with closed timelike curves.[36] However, later it was shown that such model of closed timelike curve can have internal inconsistencies as it will lead to strange phenomena like distinguishing non orthogonal quantum states and distinguishing proper and improper mixture.[37][38] Accordingly, the destructive positive feedback loop of virtual particles circulating through a wormhole time machine, a result indicated by semi-classical calculations, is averted. A particle returning from the future does not return to its universe of origination but to a parallel universe. This suggests that a wormhole time machine with an exceedingly short time jump is a theoretical bridge between contemporaneous parallel universes.[39] Because a wormhole time-machine introduces a type of nonlinearity into quantum theory, this sort of communication between parallel universes is consistent with Joseph Polchinski's proposal of an "Everett-Wheeler telephone" in Steven Weinberg's formulation of nonlinear quantum mechanics.[40] Such a possibility is depicted in the science-fiction 2014 movie Interstellar.

## Visualization

For a simplified notion of a wormhole, space can be visualized as a two-dimensional (2D) surface. In this case, a wormhole would appear as a hole in that surface, lead into a 3D tube (the inside surface of a cylinder), then re-emerge at another location on the 2D surface with a hole similar to the entrance. An actual wormhole would be analogous to this, but with the spatial dimensions raised by one. For example, instead of circular holes on a 2D plane, the entry and exit points could be visualized as spheres in 3D space.

## Metrics

Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a (traversable) wormhole metric is the following:[41]

${\displaystyle ds^{2}=-c^{2}dt^{2}+dl^{2}+(k^{2}+l^{2})(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}),}$

first presented by Ellis (see Ellis wormhole) as a special case of the Ellis drainhole.

One type of non-traversable wormhole metric is the Schwarzschild solution (see the first diagram):

${\displaystyle ds^{2}=-c^{2}\left(1-{\frac {2GM}{rc^{2}}}\right)dt^{2}+{\frac {dr^{2}}{1-{\frac {2GM}{rc^{2}}}}}+r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}).}$

The original Einstein-Rosen bridge was described in an article published in July 1935.[42][43]

For the Schwarzschild spherically symmetric static solution

${\displaystyle ds^{2}=-{\frac {1}{1-{\frac {2m}{r}}}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+(1-{\frac {2m}{r}})dt^{2}}$

(ds = proper time, c = 1)

If one replaces r with u according to ${\displaystyle u^{2}=r-2m}$

${\displaystyle ds^{2}=-4(u^{2}+2m)du^{2}-(u^{2}+2m)^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+{\frac {u^{2}}{u^{2}+2m}}dt^{2}}$

The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to u > 0 and u < 0, which are joined by a hyperplane r = 2m or u = 0 in which g vanishes. We call such a connection between the two sheets a "bridge".

— A.Einstein,N.Rosen - The Particle Problem in the General Theory of Relativity, http://adsabs.harvard.edu/abs/1935PhRv...48...73E

For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution

${\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}={\frac {\epsilon }{4}},}$
${\displaystyle ds^{2}=-{\frac {1}{(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})dt^{2}}$

(${\displaystyle \epsilon }$ = electrical charge)

The field equations without denominators in the case when m = 0 can be written

${\displaystyle \phi _{\mu \nu }=\phi _{\mu ,\nu }-\phi _{\nu ,\mu }}$
${\displaystyle g^{2}\phi _{\mu \nu ;\sigma }g^{\nu \sigma }=0}$
${\displaystyle g^{2}(R_{ik}+\phi _{i\alpha }\phi _{k}^{\alpha }-{\frac {1}{4}}g_{ik}\phi _{\alpha \beta }\phi ^{ab})=0}$

In order to eliminate singularities, if one replaces r by u according to the equation:

${\displaystyle u^{2}=r^{2}-{\frac {\epsilon ^{2}}{2}}}$

and with m = 0 one obtains[44][45]

${\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}=\epsilon /(u^{2}+{\frac {\epsilon ^{2}}{2}})^{\frac {1}{2}}}$
${\displaystyle ds^{2}=-du^{2}-(u^{2}+{\frac {\epsilon ^{2}}{2}})(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+({\frac {2u^{2}}{2u^{2}+\epsilon ^{2}}})dt^{2}}$

The solution is free from singularities for all finite points in the space of the two sheets

— A.Einstein,N.Rosen - The Particle Problem in the General Theory of Relativity, http://adsabs.harvard.edu/abs/1935PhRv...48...73E

## In fiction

Wormholes are a common element in science fiction because they allow interstellar, intergalactic, and sometimes even interuniversal travel within human lifetime scales. They have also served as a method for time travel.

## Notes

1. ^ Weyl, H. (1921). "Feld und Materie". Annalen der Physik. 65: 541–563. doi:10.1002/andp.19213701405.
2. ^ Scholz, Erhard (ed.), Hermann Weyl's Raum - Zeit - Materie and a General Introduction to His Scientific Work, 2001, p. 199.
3. ^ a b "Hermann Weyl": entry in the Stanford Encyclopedia of Philosophy.
4. ^ Misner, C. W.; Wheeler, J. A. (1957). "Classical physics as geometry". Ann. Phys. 2: 525. doi:10.1016/0003-4916(57)90049-0.
5. ^ "Collapse to a Black Hole". Casa.colorado.edu. 2010-10-03. Retrieved 2010-11-11. This tertiary source reuses information from other sources but does not name them.
6. ^ Flamm (1916). "Beiträge zur Einsteinschen Gravitationstheorie". Physikalische Zeitschrift. XVII: 448. ("Comments on Einstein's Theory of Gravity")
7. ^ A. Einstein and N. Rosen, "The Particle Problem in the General Theory of Relativity," Phys. Rev. 48(73) (1935).
8. ^ Lindley, David (Mar 25, 2005). "Focus: The Birth of Wormholes". Physics. American Physical Society. Retrieved 20 February 2016.
9. ^ R. W. Fuller and J. A. Wheeler, "Causality and Multiply-Connected Space-Time," Phys. Rev. 128(919) (1962).
10. ^ Poplawski, Nikodem J. (2010). "Cosmology with torsion: An alternative to cosmic inflation". Phys. Lett. B. 694 (3): 181–185. arXiv:. Bibcode:2010PhLB..694..181P. doi:10.1016/j.physletb.2010.09.056.
11. ^ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 167. ISBN 0-226-22498-8.
12. ^ "Space and Time Warps". Hawking.org.uk. Retrieved 2010-11-11.
13. ^ a b c Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (1988). "Wormholes, Time Machines, and the Weak Energy Condition" (PDF). Physical Review Letters. 61 (13): 1446–1449. Bibcode:1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800.
14. ^ Sopova; Ford (2002). "The Energy Density in the Casimir Effect". Physical Review D. 66 (4): 045026. arXiv:. Bibcode:2002PhRvD..66d5026S. doi:10.1103/PhysRevD.66.045026.
15. ^ Ford; Roman (1995). "Averaged Energy Conditions and Quantum Inequalities". Physical Review D. 51 (8): 4277–4286. arXiv:. Bibcode:1995PhRvD..51.4277F. doi:10.1103/PhysRevD.51.4277.
16. ^ Olum (1998). "Superluminal travel requires negative energies". Physical Review Letters. 81 (17): 3567–3570. arXiv:. Bibcode:1998PhRvL..81.3567O. doi:10.1103/PhysRevLett.81.3567.
17. ^ Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. pp. 494–496. ISBN 0-393-31276-3.
18. ^ Ian H., Redmount; Wai-Mo Suen (1994). "Quantum Dynamics of Lorentzian Spacetime Foam". Physical Review D. 49 (10): 5199. arXiv:. Bibcode:1994PhRvD..49.5199R. doi:10.1103/PhysRevD.49.5199.
19. ^ Kirillov, A.A.; E.P. Savelova (21 February 2008). "Dark Matter from a gas of wormholes". Physics Letters B. 660 (3): 93. arXiv:. Bibcode:2008PhLB..660...93K. doi:10.1016/j.physletb.2007.12.034.
20. ^ Rodrigo, Enrico (30 November 2009). "Denouement of a Wormhole-Brane Encounter". International Journal of Modern Physics D. 18 (12): 1809. arXiv:. Bibcode:2009IJMPD..18.1809R. doi:10.1142/S0218271809015333.
21. ^ a b John G. Cramer; Robert L. Forward; Michael S. Morris; Matt Visser; Gregory Benford & Geoffrey A. Landis (1995). "Natural Wormholes as Gravitational Lenses". Physical Review D. 51 (6): 3117–3120. arXiv:. Bibcode:1995PhRvD..51.3117C. doi:10.1103/PhysRevD.51.3117.
22. ^ Other computer-rendered images and animations of traversable wormholes can be seen on this page by the creator of the image in the article, and this page has additional renderings.
23. ^ H. G. Ellis (1973). "Ether flow through a drainhole: A particle model in general relativity". Journal of Mathematical Physics. 14: 104–118. Bibcode:1973JMP....14..104E. doi:10.1063/1.1666161.
24. ^ K. A. Bronnikov (1973). "Scalar-tensor theory and scalar charge". Acta Physica Polonica. B4: 251–266.
25. ^ Elias Gravanis; Steven Willison (2007). "'Mass without mass' from thin shells in Gauss-Bonnet gravity". Phys. Rev. D. 75 (8): 084025. arXiv:. Bibcode:2007PhRvD..75h4025G. doi:10.1103/PhysRevD.75.084025.
26. ^ Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. p. 504. ISBN 0-393-31276-3.
27. ^ Fewster, Christopher J.; Ken D. Olum; Michael J. Pfenning (10 January 2007). "Averaged null energy condition in spacetimes with boundaries". Physical Review D. 75 (2): 025007. arXiv:. Bibcode:2007PhRvD..75b5007F. doi:10.1103/PhysRevD.75.025007.
28. ^ Visser, Matt (15 October 1996). "Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum". Physical Review D. 54 (8): 5116. arXiv:. Bibcode:1996PhRvD..54.5116V. doi:10.1103/PhysRevD.54.5116.
29. ^ Graham, Noah; Ken D. Olum (4 September 2007). "Achronal averaged null energy condition". Physical Review D. 76 (6): 064001. arXiv:. Bibcode:2007PhRvD..76f4001G. doi:10.1103/PhysRevD.76.064001.
30. ^ Urban, Douglas; Ken D. Olum (1 June 2010). "Spacetime averaged null energy condition". Physical Review D. 81 (6): 124004. arXiv:. Bibcode:2010PhRvD..81l4004U. doi:10.1103/PhysRevD.81.124004.
31. ^ Duplessis, Francis; Easson, Damien A. (2015). "Exotica ex nihilo: Traversable wormholes & non-singular black holes from the vacuum of quadratic gravity". Physical Review D. 92 (4). arXiv:. doi:10.1103/PhysRevD.92.043516.
32. ^ Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. p. 502. ISBN 0-393-31276-3.
33. ^ Susskind, Leonard (2005). "Wormholes and Time Travel? Not Likely". arXiv:.
34. ^ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 135. ISBN 0-226-22498-8.
35. ^ Visser, Matt (2002). "The quantum physics of chronology protection". arXiv:.
36. ^ Deutsch, David (1991). "Quantum Mechanics Near Closed Timelike Lines". Physical Review D. 44 (10): 3197. Bibcode:1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197.
37. ^ Brun; et al. (2009). "Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States". Physical Review Letters. 102 (21): 210402. Bibcode:2009PhRvL.102u0402B. doi:10.1103/PhysRevLett.102.210402. PMID 19519086.
38. ^ Pati; Chakrabarty; Agrawal (2011). "Purification of mixed states with closed timelike curve is not possible". Physical Review A. 84 (6): 062325. arXiv:. Bibcode:2011PhRvA..84f2325P. doi:10.1103/PhysRevA.84.062325.
39. ^ Rodrigo, Enrico (2010). The Physics of Stargates. Eridanus Press. p. 281. ISBN 0-9841500-0-5.
40. ^ Polchinski, Joseph (1991). "Weinberg's Nonlinear quantum Mechanics and the Einstein-Podolsky-Rosen Paradox". Physical Review Letters. 66 (4): 397. Bibcode:1991PhRvL..66..397P. doi:10.1103/PhysRevLett.66.397. PMID 10043797.
41. ^ Raine, Derek; Thomas, Edwin (2009). Black Holes: An Introduction (2nd ed.). Imperial College Press. p. 143. doi:10.1142/p637. ISBN 978-1-84816-383-6.
42. ^ Einstein, A.; Rosen, N. (1 July 1935). "The Particle Problem in the General Theory of Relativity". Physical Review. 48 (1): 73–77. Bibcode:1935PhRv...48...73E. doi:10.1103/PhysRev.48.73.
43. ^
44. ^
45. ^