Wormhole

A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are based on a special solution of the Einstein field equations.[1] Wormholes are consistent with the general theory of relativity, but whether they actually exist is unknown. Many physicists postulate that wormholes are merely projections of a fourth spatial dimension, analogous to how a two-dimensional (2D) being could experience only part of a three-dimensional (3D) object.[2]

In 1995, Matt Visser suggested there may be many wormholes in the universe if cosmic strings with negative mass were generated in the early universe.[3][4] Some physicists, such as Kip Thorne, have suggested how to create wormholes artificially.[5]

## Contents

* 1 Terminology
* 1.1 Modern definitions
* 2 Development
* 2.1 Schwarzschild wormholes
* 2.1.1 Einstein–Rosen bridges
* 2.2 Traversable wormholes
* 3 Raychaudhuri's theorem and exotic matter
* 4 Modified general relativity
* 5 Faster-than-light travel
* 6 Time travel
* 7 Interuniversal travel
* 8 Metrics
* 9 In fiction
* 9.1 Warp portals and higher-dimensional shortcuts
* 10 See also
* 11 Notes
* 12 References
* 12.1 Citations
* 12.2 Sources
* 13 External links

## Terminology

In 1928, German mathematician, philosopher and theoretical physicist Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy;[6][7] however, he did not use the term "wormhole" (he spoke of "one-dimensional tubes" instead).[8]

The term "worm-holes" appears in British astronomer Arthur Eddington's 1928 popular exposition The Nature of the Physical World, in which he uses it metaphorically for "worm-holes" (material particles) crossing the "grain" of spacetime, rather than for a spacetime shortcut.[9]

American theoretical physicist John Archibald Wheeler (inspired by Weyl's work)[8] used the term "wormhole".[10][11][12] In a 1957 paper that he wrote with Charles W. Misner, they write:[13]

> 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. From a topological point of view, 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).[14]

> If a Minkowski spacetime contains a compact region

> Ω

> , and if the topology of

> Ω

> is of the form

> Ω∼S×Σ

> , where

> Σ

> is a three-manifold of the nontrivial topology, whose boundary has the topology of the form

> δΣ∼S2

> , and if, furthermore, the hypersurfaces

> Σ

> are all spacelike, then the region

> Ω

> contains a quasi-permanent intrauniverse wormhole.

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 or observer).

## Development

"Embedding diagram" of a Schwarzschild wormhole

### Schwarzschild 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, were thought to be possible only if exotic matter with negative energy density could be used to stabilize them.[15] Later, physicists reported that microscopic traversable wormholes may be possible and not require any exotic matter, instead requiring only electrically charged fermionic matter with small enough mass that it cannot collapse into a charged black hole.[16][17][18] While such wormholes, if possible, may be limited to transfers of information, humanly traversable wormholes may exist if reality can broadly be described by the Randall–Sundrum model 2, a brane-based theory consistent with string theory.[19][20]

#### Einstein–Rosen bridges

Einstein–Rosen bridges (or ER bridges),[21] named after Albert Einstein and Nathan Rosen,[22] 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 black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime 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).

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.[23] 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 the 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 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') is picked and an "embedding diagram" drawn 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". 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 geography, 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.[24]

The Einstein–Rosen bridge was discovered by Ludwig Flamm in 1916,[25] a few months after Schwarzschild published his solution, and was rediscovered by Albert Einstein and his colleague Nathan Rosen, who published their result in 1935.[22][26] In 1962, John Archibald Wheeler and Robert W. Fuller published a paper[27] 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 dynamic 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 (e.g. a black hole). Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.[28]

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).[29]

Other non-traversable wormholes include Lorentzian wormholes (first proposed by John Archibald Wheeler in 1957), wormholes creating a spacetime foam in a general relativistic spacetime manifold depicted by a Lorentzian manifold,[30] and Euclidean wormholes (named after Euclidean manifold, a structure of Riemannian manifold).[31]

### 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 matter vacuum energy, but it has been shown theoretically that quantum field theory disallows states where energy can be arbitrarily negative for an arbitrary length of time.[32][33] Some physicists, such as Stephen Hawking,[34] Kip Thorne,[35] and others,[36] argued that such effects might make it possible to stabilize a traversable wormhole.[37] The only known natural process that is theoretically predicted to form a wormhole in the context of general relativity and quantum mechanics was put forth by Juan Maldacena and Leonard Susskind in their ER = EPR conjecture. The quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,[38]: 494–496 [39] and stable versions of such wormholes have been suggested as dark matter candidates.[40][41] 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.[42]

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.[note 1]

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[43] and independently in a 1973 paper by K. A. Bronnikov.[44] Ellis 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 spacetime, 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: m, which fixes the strength of its gravitational field, and n, which determines the curvature of its spatial cross sections. When 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 independently discovered in 1988 the Ellis wormhole and argued for its use as a tool for teaching general relativity.[45] For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is also known 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. In the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions that is sometimes studied in the context of brane cosmology), however, exotic matter is not needed in order for wormholes to exist—they can exist even with no matter.[46] A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,[42] 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 how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths.[35] According to general relativity, however, 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". Until this time it could not have been noticed or have been used.[38]: 504

## 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,[47] but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.[48] Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,[49] violations have nevertheless been found,[50] so it remains an open possibility that quantum effects might be used to support a wormhole.

## Modified general relativity

In some hypotheses 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 R2 gravity, a form of f(R) gravity.[51]

## Faster-than-light travel

Wormhole travel as envisioned by Les Bossinas for NASA, c. 1998

The impossibility of faster-than-light relative speed applies only 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. A light beam traveling through the same wormhole would still beat the traveler.

## Time travel

Main article: Physics:Time travel

If traversable wormholes exist, they might allow time travel.[35] A proposed time-travel machine using a traversable wormhole might hypothetically work in the following way: One end of the wormhole is accelerated to some significant fraction of the speed of light, perhaps with some advanced propulsion system, and then brought back to the point of origin. Alternatively, another way is to take one entrance of the wormhole and move it to within the gravitational field of an object that has higher gravity than the other entrance, and then return it to a position near the other entrance. For both these methods, time dilation causes the end of the wormhole that has been moved to have aged less, or become "younger", than the stationary end as seen by an external observer; time connects differently through the wormhole than outside it, however, so that synchronized clocks at either end of the wormhole will always remain synchronized as seen by an observer passing through the wormhole, no matter how the two ends move around.[38]: 502 This means that an observer entering the "younger" end would exit the "older" end at a time when it was the same age as the "younger" end, effectively going back in time as seen by an observer from the outside. One significant limitation of such a time machine is that it is only possible to go as far back in time as the initial creation of the machine;[38]: 503 it is more of a path through time rather than it is a device that itself moves through time, and it would not allow the technology itself to be moved backward in time.[52][53]

According to current theories on the nature of wormholes, construction of a traversable wormhole would require the existence of a substance with negative energy, often referred to as "exotic matter". More technically, the wormhole spacetime requires a distribution of energy that violates various energy conditions, such as the null energy condition along with the weak, strong, and dominant energy conditions. It is known that quantum effects can lead to small measurable violations of the null energy condition,[14]: 101 and many physicists believe that the required negative energy may actually be possible due to the Casimir effect in quantum physics.[54] Although early calculations suggested a very large amount of negative energy would be required, later calculations showed that the amount of negative energy can be made arbitrarily small.[55]

In 1993, Matt Visser argued that the two mouths of a wormhole with such an induced clock difference could not be brought together without inducing quantum field and gravitational effects that would either make the wormhole collapse or the two mouths repel each other,[56] or otherwise prevent information from passing through the wormhole.[57] Because of this, the two mouths could not be brought close enough for causality violation to take place. In a 1997 paper, however, Visser hypothesized that a complex "Roman ring" (named after Tom Roman) configuration of an N number of wormholes arranged in a symmetric polygon could still act as a time machine, although he concludes that this is more likely a flaw in classical quantum gravity theory rather than proof that causality violation is possible.[58]

## Interuniversal travel

See also: Religion:Multiverse

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.[59] Later, it was shown that such a model of closed timelike curves can have internal inconsistencies as it will lead to strange phenomena like distinguishing non-orthogonal quantum states and distinguishing proper and improper mixture.[60][61] 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.[15]

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 phone[62] (named after Hugh Everett) in Steven Weinberg's formulation of nonlinear quantum mechanics.[63]

The possibility of communication between parallel universes has been dubbed interuniversal travel.[64]

Wormholes can also be depicted in a Penrose diagram of a Schwarzschild black hole. In the Penrose diagram, an object traveling faster than light will cross the black hole and will emerge from another end into a different space, time or universe. This will be an inter-universal wormhole.

## 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:[65]

ds2=−c2dt2+dℓ2+(k2+ℓ2)(dθ2+sin2θdφ2),

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):

ds2=−c2(1−2GMrc2)dt2+dr21−2GMrc2+r2(dθ2+sin2θdφ2).

The original Einstein–Rosen bridge was described in an article published in July 1935.[66][67]

For the Schwarzschild spherically symmetric static solution

ds2=−11−2mrdr2−r2(dθ2+sin2θdφ2)+(1−2mr)dt2,

where ds is the proper time and c=1.

If one replaces r with u according to u2=r−2m

ds2=−4(u2+2m)du2−(u2+2m)2(dθ2+sin2θdφ2)+u2u2+2mdt2

> 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"

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

φ1=φ2=φ3=0,φ4=ε4,

ds2=−1(1−2mr−ε22r2)dr2−r2(dθ2+sin2θdφ2)+(1−2mr−ε22r2)dt2,

where ε is the electric charge.

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

φμν=φμ,ν−φν,μ

g2φμν;σgνσ=0

g2(Rik+φiαφkα−14gikφαβφαβ)=0

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

u2=r2−ε22

and with m=0 one obtains[68][69]

φ1=φ2=φ3=0 and φ4=ε(u2+ε22)1/2

ds2=−du2−(u2+ε22)(dθ2+sin2θdφ2)+(2u22u2+ε2)dt2

> 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"

## In fiction

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

### Warp portals and higher-dimensional shortcuts

In theoretical physics and science fiction, the concept of a warp or warp portal is frequently used to describe shortcuts through space that are only possible by accessing a higher spatial dimension. Just as a two-dimensional space would require a third spatial dimension to bend or fold its plane in order to bring two distant points together, a three-dimensional space must be embedded within a fourth spatial dimension to allow similar manipulation.[70] This analogy underlies many depictions of warp portals, which function by bending or folding three-dimensional space through the fourth spatial axis, allowing distant regions to become adjacent.

The existence of such a mechanism would imply that the universe possesses, or is embedded within, a four-dimensional spatial framework, even if that dimension is not directly observable. The geometry of these constructs is often modeled using solutions to Einstein's field equations, such as wormholes,[71] and the Alcubierre warp bubble,[72] both of which rely on higher-dimensional curvature.

## See also

* Alcubierre drive
* ER = EPR
* Gödel metric
* Krasnikov tube
* Non-orientable wormhole
* Novikov self-consistency principle
* Polchinski's paradox
* Retrocausality
* Ring singularity
* Roman ring

## Notes

1. ↑ 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.

## References

### Citations

1. ↑ Overbye, Dennis (10 October 2022). "Black Holes May Hide a Mind-Bending Secret About Our Universe – Take gravity, add quantum mechanics, stir. What do you get? Just maybe, a holographic cosmos.". The New York Times. https://www.nytimes.com/2022/10/10/science/black-holes-cosmology-hologram.html.
2. ↑ Choi, Charles Q. (2013-12-03). "Spooky physics phenomenon may link universe's wormholes" (in en). NBC News. http://www.nbcnews.com/science/science-news/spooky-physics-phenomenon-may-link-universes-wormholes-f2D11690659.
3. ↑ Cramer, John; Forward, Robert; Morris, Michael; Visser, Matt; Benford, Gregory; Landis, Geoffrey (1995). "Natural wormholes as gravitational lenses". Physical Review D 51 (6): 3117–3120. doi:10.1103/PhysRevD.51.3117. PMID 10018782. Bibcode: 1995PhRvD..51.3117C.
4. ↑ "Searching for a 'Subway to the Stars'" (Press release). Archived from the original on 2012-04-15.
5. ↑ Thorne, Kip S. (1994). Black holes and time warps: Einstein's outrageous legacy. New York. p. 493. ISBN 978-0-393-31276-8.
6. ↑ Weyl, H. (1921). "Feld und Materie". Annalen der Physik 65 (14): 541–563. doi:10.1002/andp.19213701405. Bibcode: 1921AnP...370..541W. https://zenodo.org/record/1424373.
7. ↑ Scholz, Erhard, ed (2001). Hermann Weyl's Raum – Zeit – Materie and a General Introduction to His Scientific Work. Oberwolfach Seminars. 30. Springer. p. 199. ISBN 978-3-7643-6476-2. https://books.google.com/books?id=oZLiqDQGnjgC.
8. ↑ 8.0 8.1 "Hermann Weyl": entry in the Stanford Encyclopedia of Philosophy.
9. ↑ Eddington, Arthur Stanley (1928). "Time". The Nature of the Physical World. https://www.gutenberg.org/cache/epub/72963/pg72963-images.html.
10. ↑ Blum, Alexander S. (2022). "From Wood Chuck Holes to Worm Holes—A Look into the Notebooks of John A. Wheeler" (in en). Annalen der Physik 534 (8). doi:10.1002/andp.202200244. ISSN 1521-3889. Bibcode: 2022AnP...53400244B.
11. ↑ Misner, Charles W.; Thorne, Kip S.; Zurek, Wojciech H. (2009-04-01). "John Wheeler, relativity, and quantum information". Physics Today 62 (4): 40–46. doi:10.1063/1.3120895. ISSN 0031-9228. Bibcode: 2009PhT....62d..40M. https://pubs.aip.org/physicstoday/article-abstract/62/4/40/390982/John-Wheeler-relativity-and-quantum?redirectedFrom=fulltext.
12. ↑ Novikov, Igor D.; Kardashev, N. S.; Shatskiy, A. A. (2010), Ciufolini, Ignazio; Matzner, Richard A., eds., "Wheeler Wormholes and the Modern Astrophysics" (in en), General Relativity and John Archibald Wheeler (Dordrecht: Springer Netherlands) 367: 39–56, doi:10.1007/978-90-481-3735-0_4, ISBN 978-90-481-3735-0, Bibcode: 2010ASSL..367...39N, https://link.springer.com/chapter/10.1007/978-90-481-3735-0_4, retrieved 2025-02-20
13. ↑ Misner, C. W.; Wheeler, J. A. (1957). "Classical physics as geometry". Ann. Phys. 2 (6): 525. doi:10.1016/0003-4916(57)90049-0. Bibcode: 1957AnPhy...2..525M.
14. ↑ 14.0 14.1 Visser, Matt (1996). Lorentzian Wormholes. Springer-Verlag. ISBN 978-1-56396-653-8.
15. ↑ 15.0 15.1 Rodrigo, Enrico (2010). The Physics of Stargates. Eridanus Press. p. 281. ISBN 978-0-9841500-0-7.
16. ↑ "Microscopic wormholes possible in theory" (in en). phys.org. https://phys.org/news/2021-03-microscopic-wormholes-theory.html.
17. ↑ Blázquez-Salcedo, Jose Luis; Knoll, Christian; Radu, Eugen (9 March 2021). "Traversable Wormholes in Einstein-Dirac-Maxwell Theory". Physical Review Letters 126 (10). doi:10.1103/PhysRevLett.126.101102. PMID 33784127. Bibcode: 2021PhRvL.126j1102B. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.101102. Retrieved 22 April 2021.
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20. ↑ Maldacena, Juan; Milekhin, Alexey (9 March 2021). "Humanly traversable wormholes". Physical Review D 103 (6). doi:10.1103/PhysRevD.103.066007. Bibcode: 2021PhRvD.103f6007M. 50px Available under CC BY 4.0.
21. ↑ Vladimir Dobrev (ed.), Lie Theory and Its Applications in Physics: Varna, Bulgaria, June 2015, Springer, 2016, p. 246.
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24. ↑ "Collapse to a Black Hole". Casa.colorado.edu. 2010-10-03. http://casa.colorado.edu/~ajsh/collapse.html#kruskal. Template:Tertiary source
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27. ↑ Fuller, Robert W.; Wheeler, John A. (1962-10-15). "Causality and Multiply Connected Space-Time". Physical Review (American Physical Society (APS)) 128 (2): 919–929. doi:10.1103/physrev.128.919. ISSN 0031-899X. Bibcode: 1962PhRv..128..919F.
28. ↑ Poplawski, Nikodem J. (2010). "Cosmology with torsion: An alternative to cosmic inflation". Phys. Lett. B 694 (3): 181–185. doi:10.1016/j.physletb.2010.09.056. Bibcode: 2010PhLB..694..181P.
29. ↑ Thorne, Kip S. (1994). Black holes and time warps: Einstein's outrageous legacy. New York. p. 488. ISBN 978-0-393-31276-8.
30. ↑ J. Wheeler (1957). "On the nature of quantum geometrodynamics". Ann. Phys. 2 (6): 604–614. doi:10.1016/0003-4916(57)90050-7. Bibcode: 1957AnPhy...2..604W. (A follow-up paper to Misner and Wheeler (December 1957).)
31. ↑ Eduard Prugovecki, Quantum Geometry: A Framework for Quantum General Relativity, Springer, 2013, p. 412.
32. ↑ Ford; Roman (1995). "Averaged Energy Conditions and Quantum Inequalities". Physical Review D 51 (8): 4277–4286. doi:10.1103/PhysRevD.51.4277. PMID 10018903. Bibcode: 1995PhRvD..51.4277F.
33. ↑ Sopova; Ford (2002). "The Energy Density in the Casimir Effect". Physical Review D 66 (4). doi:10.1103/PhysRevD.66.045026. Bibcode: 2002PhRvD..66d5026S.
34. ↑ "Space and Time Warps". http://www.hawking.org.uk/space-and-time-warps.html.
35. ↑ 35.0 35.1 35.2 Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review Letters 61 (13): 1446–1449. doi:10.1103/PhysRevLett.61.1446. PMID 10038800. Bibcode: 1988PhRvL..61.1446M. http://authors.library.caltech.edu/9262/1/MORprl88.pdf.
36. ↑ Olum (1998). "Superluminal travel requires negative energies". Physical Review Letters 81 (17): 3567–3570. doi:10.1103/PhysRevLett.81.3567. Bibcode: 1998PhRvL..81.3567O.
37. ↑ "Newfound Wormhole Allows Information to Escape Black Holes". 23 October 2017. https://www.quantamagazine.org/newfound-wormhole-allows-information-to-escape-black-holes-20171023/.
38. ↑ 38.0 38.1 38.2 38.3 Thorne, Kip S. (1994). Black Holes and Time Warps. W. W. Norton. ISBN 978-0-393-31276-8.
39. ↑ Ian H., Redmount; Wai-Mo Suen (1994). "Quantum Dynamics of Lorentzian Spacetime Foam". Physical Review D 49 (10): 5199–5210. doi:10.1103/PhysRevD.49.5199. PMID 10016836. Bibcode: 1994PhRvD..49.5199R.
40. ↑ Kirillov, A. A.; P. Savelova, E. (2008). "Dark Matter from a gas of wormholes". Physics Letters B 660 (3): 93–99. doi:10.1016/j.physletb.2007.12.034. Bibcode: 2008PhLB..660...93K.
41. ↑ Rodrigo, Enrico (2009). "Denouement of a Wormhole-Brane Encounter". International Journal of Modern Physics D 18 (12): 1809–1819. doi:10.1142/S0218271809015333. Bibcode: 2009IJMPD..18.1809R.
42. ↑ 42.0 42.1 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. doi:10.1103/PhysRevD.51.3117. PMID 10018782. Bibcode: 1995PhRvD..51.3117C. https://cds.cern.ch/record/268926.
43. ↑ H. G. Ellis (1973). "Ether flow through a drainhole: A particle model in general relativity". Journal of Mathematical Physics 14 (1): 104–118. doi:10.1063/1.1666161. Bibcode: 1973JMP....14..104E.
44. ↑ K. A. Bronnikov (1973). "Scalar-tensor theory and scalar charge". Acta Physica Polonica B4: 251–266.
45. ↑ Morris, Michael S.; Thorne, Kip S. (1988). "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity". American Journal of Physics 56 (5): 395–412. doi:10.1119/1.15620. Bibcode: 1988AmJPh..56..395M.
46. ↑ Elias Gravanis; Steven Willison (2007). "'Mass without mass' from thin shells in Gauss-Bonnet gravity". Phys. Rev. D 75 (8). doi:10.1103/PhysRevD.75.084025. Bibcode: 2007PhRvD..75h4025G.
47. ↑ Fewster, Christopher J.; Ken D. Olum; Michael J. Pfenning (2007). "Averaged null energy condition in spacetimes with boundaries". Physical Review D 75 (2). doi:10.1103/PhysRevD.75.025007. Bibcode: 2007PhRvD..75b5007F.
48. ↑ Visser, Matt (1996). "Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum". Physical Review D 54 (8): 5116–5122. doi:10.1103/PhysRevD.54.5116. PMID 10021199. Bibcode: 1996PhRvD..54.5116V.
49. ↑ Graham, Noah; Ken D. Olum (2007). "Achronal averaged null energy condition". Physical Review D 76 (6). doi:10.1103/PhysRevD.76.064001. Bibcode: 2007PhRvD..76f4001G.
50. ↑ Urban, Douglas; Ken D. Olum (2010). "Spacetime averaged null energy condition". Physical Review D 81 (6). doi:10.1103/PhysRevD.81.124004. Bibcode: 2010PhRvD..81l4004U.
51. ↑ 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). doi:10.1103/PhysRevD.92.043516. Bibcode: 2015PhRvD..92d3516D.
52. ↑ Susskind, Leonard (2005). "Wormholes and Time Travel? Not Likely". arXiv:gr-qc/0503097.
53. ↑ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 135. ISBN 978-0-226-22498-5. https://archive.org/details/isbn_9780226224985.
54. ↑ Cramer, John G. (1994). "NASA Goes FTL Part 1: Wormhole Physics". http://www.npl.washington.edu/av/altvw69.html.
55. ↑ Visser, Matt; Sayan Kar; Naresh Dadhich (2003). "Traversable wormholes with arbitrarily small energy condition violations". Physical Review Letters 90 (20): 201102.1–201102.4. doi:10.1103/PhysRevLett.90.201102. PMID 12785880. Bibcode: 2003PhRvL..90t1102V. https://cds.cern.ch/record/598932.
56. ↑ Visser, Matt (1993). "From wormhole to time machine: Comments on Hawking's Chronology Protection Conjecture". Physical Review D 47 (2): 554–565. doi:10.1103/PhysRevD.47.554. PMID 10015609. Bibcode: 1993PhRvD..47..554V.
57. ↑ Visser, Matt (2002). The quantum physics of chronology protection. Bibcode: 2003ftpc.book..161V. https://archive.org/details/arxiv-gr-qc0204022.
58. ↑ Visser, Matt (1997). "Traversable wormholes: the Roman ring". Physical Review D 55 (8): 5212–5214. doi:10.1103/PhysRevD.55.5212. Bibcode: 1997PhRvD..55.5212V.
59. ↑ Deutsch, David (1991). "Quantum Mechanics Near Closed Timelike Lines". Physical Review D 44 (10): 3197–3217. doi:10.1103/PhysRevD.44.3197. PMID 10013776. Bibcode: 1991PhRvD..44.3197D.
60. ↑ Brun (2009). "Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States". Physical Review Letters 102 (21). doi:10.1103/PhysRevLett.102.210402. PMID 19519086. Bibcode: 2009PhRvL.102u0402B.
61. ↑ Pati; Chakrabarty; Agrawal (2011). "Purification of mixed states with closed timelike curve is not possible". Physical Review A 84 (6). doi:10.1103/PhysRevA.84.062325. Bibcode: 2011PhRvA..84f2325P.
62. ↑ Polchinski, Joseph (1991). "Weinberg's Nonlinear quantum Mechanics and the Einstein–Podolsky–Rosen Paradox". Physical Review Letters 66 (4): 397–400. doi:10.1103/PhysRevLett.66.397. PMID 10043797. Bibcode: 1991PhRvL..66..397P.
63. ↑ Enrico Rodrigo, The Physics of Stargates: Parallel Universes, Time Travel, and the Enigma of Wormhole Physics, Eridanus Press, 2010, p. 281.
64. ↑ Samuel Walker, "Inter-universal travel: I wouldn't start from here, New Scientist (1 February 2017).
65. ↑ 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. https://archive.org/details/blackholesintrod00rain.
66. ↑ Einstein, A.; Rosen, N. (1 July 1935). "The Particle Problem in the General Theory of Relativity". Physical Review 48 (1): 73–77. doi:10.1103/PhysRev.48.73. Bibcode: 1935PhRv...48...73E.
67. ↑ "Leonard Susskind | "ER = EPR" or "What's Behind the Horizons of Black Holes?"". 4 November 2014. https://www.youtube.com/watch?v=OBPpRqxY8Uw.
68. ↑ "Magnetic 'wormhole' connecting two regions of space created for the first time". https://www.sciencedaily.com/releases/2015/09/150903081506.htm.
69. ↑ "Magnetic wormhole created for first time". http://www.uab.cat/web/newsroom/news-detail/magnetic-wormhole-created-for-first-time-1345668003610.html?noticiaid=1345689132054.
70. ↑ Greene, Brian (2004). The Fabric of the Cosmos. Knopf. ISBN 978-0375727207.
71. ↑ Morris, Michael S.; Thorne, Kip S. (1988). "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity". American Journal of Physics. 56 (5): 395–412. doi:10.1119/1.15620.
72. ↑ Alcubierre, Miguel (1994). "The warp drive: hyper-fast travel within general relativity". Classical and Quantum Gravity. 11 (5): L73–L77. doi:10.1088/0264-9381/11/5/001.

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## External links

* "What exactly is a 'wormhole'? Have wormholes been proven to exist or are they still theoretical??" answered by Richard F. Holman, William A. Hiscock and Matt Visser
* "Why wormholes?" by Matt Visser (October 1996)
*   * Questions and Answers about Wormholes - A comprehensive wormhole FAQ by Enrico Rodrigo
* animation that simulates traversing a wormhole
* Renderings and animations of a Morris-Thorne wormhole
* Wormholes on In Our Time at the BBC (audio)

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Spacetimes in general relativity that
can contain closed timelike curves|

* Alcubierre metric
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