View source for Wormhole

{{short description|Hypothetical topological feature of spacetime}}
{{other uses}}
[[File:Einstein-rosen-bridge-model.png|thumb|upright=1.3|A wormhole visualized as a two-dimensional surface. Route (a) is the shortest path through normal space between points 1 and 2; route (b) is a shorter path through a wormhole.]]
{{General relativity sidebar |phenomena}}
A '''wormhole '''is a hypothetical structure which connects disparate points in [[spacetime]]. It may 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 [[Solutions of the Einstein field equations|solution of the Einstein field equations]].<ref name="NYT-20221010">{{cite news |last=Overbye |first=Dennis |author-link=Dennis Overbye |title=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. |url=https://www.nytimes.com/2022/10/10/science/black-holes-cosmology-hologram.html |date=10 October 2022 |work=[[The New York Times]] |accessdate=10 October 2022 }}</ref> Specifically, they are a transcendental [[bijection]] of the spacetime continuum, an [[asymptote|asymptotic]] projection of the [[Calabi–Yau manifold]] manifesting itself in [[anti-de Sitter space]].<ref>{{Cite journal |last1=Shinkai |first1=Hisa-aki |last2=Torii |first2=Takashi |date=2015-01-21 |title=Wormhole Dynamics |journal=Journal of Physics: Conference Series |volume=574 |pages=012056 |doi=10.1088/1742-6596/574/1/012056 |issn=1742-6588|doi-access=free }}</ref>

Wormholes are consistent with the [[General relativity|general theory of relativity]], but whether they actually exist is unknown. Many scientists postulate that wormholes are merely projections of a [[Four-dimensional space|fourth spatial dimension]], analogous to how a two-dimensional (2D) being could experience only part of a three-dimensional (3D) object.<ref>{{Cite web |url=http://www.nbcnews.com/science/science-news/spooky-physics-phenomenon-may-link-universes-wormholes-f2D11690659 |title=Spooky physics phenomenon may link universe's wormholes |last=Choi |first=Charles Q. |date=2013-12-03 |publisher=NBC News |language=en|access-date=2019-07-30}}</ref> A well-known analogy of such constructs is provided by the [[Klein bottle]], displaying a hole when rendered in three dimensions but not in four or higher dimensions. 

In 1995, [[Matt Visser]] suggested there may be many wormholes in the universe if [[cosmic string]]s with [[negative mass]] were generated in the [[Chronology of the universe|early universe]].<ref>{{cite journal |arxiv=astro-ph/9409051 |bibcode=1995PhRvD..51.3117C |doi=10.1103/PhysRevD.51.3117 |pmid=10018782 |title=Natural wormholes as gravitational lenses |year=1995 |last1=Cramer |first1=John |last2=Forward |first2=Robert |last3=Morris |first3=Michael |last4=Visser |first4=Matt |last5=Benford |first5=Gregory |last6=Landis |first6=Geoffrey |journal=Physical Review D |volume=51 |issue=6 |pages=3117–3120|s2cid=42837620 }}</ref><ref>{{cite press release |url=http://www.geoffreylandis.com/wormholes.htp |title=Searching for a 'Subway to the Stars' |url-status=dead |archive-url=https://web.archive.org/web/20120415100921/http://www.geoffreylandis.com/wormholes.htp |archive-date=2012-04-15 }}</ref> Some physicists, such as [[Kip Thorne]], have suggested how to make wormholes artificially.<ref>{{cite book |last1=Thorne |first1=Kip S. |title=Black holes and time warps : Einstein's outrageous legacy |date=1994 |location=New York |isbn=978-0393312768 |page=493}}</ref>

== Visualization technique ==
For a simplified notion of a wormhole, [[space]] can be visualized as a two-dimensional surface. In this case, a wormhole would appear as a hole in that surface, lead into a [[Three-dimensional space|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|2-Dimensional plane]], the entry and exit points could be visualized as spherical holes in [[3D space]] leading into a four-dimensional "tube" similar to a [[spherinder]].{{cn|date=August 2023}}

Another way to imagine wormholes is to take a sheet of paper and draw two somewhat distant points on one side of the paper. The sheet of paper represents a plane in the [[Spacetime|spacetime continuum]], and the two points represent a distance to be traveled, but theoretically, a wormhole could connect these two points by folding that plane (⁠''i.e.'' the paper) so the points are touching. In this way, it would be much easier to traverse the distance since the two points are now touching. {{cn|date=February 2025}}

== 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;<ref>{{cite journal |last1 = Weyl |first1 = H. |title = Feld und Materie |doi = 10.1002/andp.19213701405 |journal = Annalen der Physik |volume = 65 |issue = 14 |year = 1921 |pages=541–563 |bibcode = 1921AnP...370..541W |url = https://zenodo.org/record/1424373 }}</ref><ref>{{cite book|url=https://books.google.com/books?id=oZLiqDQGnjgC|editor-last=Scholz |editor-first=Erhard|title=Hermann Weyl's Raum – Zeit – Materie and a General Introduction to His Scientific Work|date=2001|page=199|publisher=Springer|series=Oberwolfach Seminars|volume=30|isbn=9783764364762}}</ref> however, he did not use the term "wormhole" (he spoke of "one-dimensional tubes" instead).<ref name=SEP>[https://plato.stanford.edu/entries/weyl/ "Hermann Weyl"]: entry in the ''[[Stanford Encyclopedia of Philosophy]]''.</ref>

American [[theoretical physicist]] [[John Archibald Wheeler]] (inspired by Weyl's work)<ref name=SEP/> coined the term "wormhole".<ref>{{Cite journal |last=Blum |first=Alexander S. |date=2022 |title=From Wood Chuck Holes to Worm Holes—A Look into the Notebooks of John A. Wheeler |journal=Annalen der Physik |language=en |volume=534 |issue=8 |pages=2200244 |doi=10.1002/andp.202200244 |issn=1521-3889|doi-access=free }}</ref><ref>{{Cite journal |last1=Misner |first1=Charles W. |last2=Thorne |first2=Kip S. |last3=Zurek |first3=Wojciech H. |date=2009-04-01 |title=John Wheeler, relativity, and quantum information |url=https://pubs.aip.org/physicstoday/article-abstract/62/4/40/390982/John-Wheeler-relativity-and-quantum?redirectedFrom=fulltext |journal=Physics Today |volume=62 |issue=4 |pages=40–46 |doi=10.1063/1.3120895 |issn=0031-9228}}</ref><ref>{{Citation |last1=Novikov |first1=Igor D. |title=Wheeler Wormholes and the Modern Astrophysics |date=2010 |work=General Relativity and John Archibald Wheeler |pages=39–56 |editor-last=Ciufolini |editor-first=Ignazio |url=https://link.springer.com/chapter/10.1007/978-90-481-3735-0_4 |access-date=2025-02-20 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-90-481-3735-0_4 |isbn=978-90-481-3735-0 |last2=Kardashev |first2=N. S. |last3=Shatskiy |first3=A. A. |editor2-last=Matzner |editor2-first=Richard A.}}</ref> In a 1957 paper that he wrote with [[Charles W. Misner]], they write:<ref>{{cite journal |last1 = Misner |first1 = C. W. |last2 = Wheeler |first2 = J. A. |title = Classical physics as geometry |doi = 10.1016/0003-4916(57)90049-0 |journal = Ann. Phys. |volume = 2 |issue = 6 |page = 525 |year = 1957 |bibcode = 1957AnPhy...2..525M }}</ref>

{{blockquote|This analysis forces one to consider situations&nbsp;... where there is a net flux of lines of force, through what [[topologist]]s would call "a [[Handle decomposition|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[[Geometry and topology|'' geometrically'' and ''topologically'']]. {{explain|are these definitions equivalent?|date=March 2017}} From a topological point of view, an intra-universe wormhole (a wormhole between two points in the same universe) is a [[Compact space|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).<ref name="Visser1996" />{{page needed|date=January 2018}}

{{blockquote|If a [[Minkowski space]]time 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 ∂Σ ~ S<sup>2</sup>, and if, furthermore, the [[hypersurface]]s 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: {{blockquote|a region of spacetime containing a "[[world tube]]" (the time evolution of a closed surface) that cannot be [[homotopy|continuously deformed]] (shrunk) to a [[world line]] (the time evolution of a point or observer).}}

== Development ==
[[File:Lorentzian Wormhole.svg|thumb|right|"Embedding diagram" of a Schwarzschild wormhole]]

=== Schwarzschild wormholes ===
<!--'Lorentzian wormhole', 'Lorentzian wormholes', 'Schwarzschild wormhole', 'Lorentzian wormholes', 'Euclidean wormhole', 'Euclidean wormholes', 'Einstein–Rosen bridge', 'Einstein–Rosen bridges', 'Einstein–Rosen bridge', 'Einstein–Rosen bridges', and 'Eternal black hole' redirect here--[[]]''
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''--><!--boldface per WP: R#PLA-->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|traversable wormholes]], were thought to be possible only if [[exotic matter]] with [[negative energy]] [[energy density|density]] could be used to stabilize them.<ref name="Rodrigo2"/> However, physicists later reported that microscopic traversable wormholes may be possible and not require any exotic matter, instead requiring only [[electrically charged]] [[fermion]]ic matter with small enough mass that it cannot collapse into a [[charged black hole]].<ref>{{cite news |title=Microscopic wormholes possible in theory |url=https://phys.org/news/2021-03-microscopic-wormholes-theory.html |access-date=22 April 2021 |work=phys.org |language=en}}</ref><ref>{{cite journal |last1=Blázquez-Salcedo |first1=Jose Luis |last2=Knoll |first2=Christian |last3=Radu |first3=Eugen |title=Traversable Wormholes in Einstein-Dirac-Maxwell Theory |journal=Physical Review Letters |date=9 March 2021 |volume=126 |issue=10 |pages=101102 |doi=10.1103/PhysRevLett.126.101102 |pmid=33784127 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.101102 |access-date=22 April 2021|arxiv=2010.07317 |bibcode=2021PhRvL.126j1102B |hdl=10773/32560 |s2cid=222378921 }}</ref><ref>{{cite journal |last1=Konoplya |first1=R. A. |last2=Zhidenko |first2=A. |title=Traversable Wormholes in General Relativity |journal=Physical Review Letters |date=4 March 2022 |volume=128 |issue=9 |pages=091104 |doi=10.1103/PhysRevLett.128.091104 |pmid=35302821 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.091104 |arxiv=2106.05034 |bibcode=2022PhRvL.128i1104K |s2cid=247245028 }}</ref> 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|Randall–Sundrum model 2]], a [[brane]]-based theory consistent with [[string theory]].<ref>{{cite news |last1=Schirber |first1=Michael |title=Wormholes Open for Transport |url=https://physics.aps.org/articles/v14/s28 |access-date=22 April 2021 |work=Physics |date=9 March 2021 |language=en}}</ref><ref>{{cite journal |last1=Maldacena |first1=Juan |last2=Milekhin |first2=Alexey |title=Humanly traversable wormholes |journal=Physical Review D |date=9 March 2021 |volume=103 |issue=6 |pages=066007 |doi=10.1103/PhysRevD.103.066007 |arxiv=2008.06618 |bibcode=2021PhRvD.103f6007M |doi-access=free }} [[File:CC-BY icon.svg|50px]] Available under [https://creativecommons.org/licenses/by/4.0/ CC BY 4.0].</ref>

==== Einstein–Rosen bridges ====
'''Einstein–Rosen bridges''' (or '''ER bridges'''),<ref name="Dobrev">Vladimir Dobrev (ed.), ''Lie Theory and Its Applications in Physics: Varna, Bulgaria, June 2015'', Springer, 2016, p. 246.</ref> named after [[Albert Einstein]] and [[Nathan Rosen]],<ref name=ER/> are connections between areas of space that can be modeled as [[vacuum solution]]s to the [[Einstein field equations]], and that are now understood to be intrinsic parts of the [[Kruskal–Szekeres coordinates#The maximally extended Schwarzschild solution|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 [[Geodesics_in_general_relativity|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.<ref>{{Cite web|title=Black Holes Explained – From Birth to Death|url=https://www.youtube.com/watch?v=9P6rdqiybaw |archive-url=https://ghostarchive.org/varchive/youtube/20211211/9P6rdqiybaw |archive-date=2021-12-11 |url-status=live|website=[[YouTube]]| date=12 August 2018 }}{{cbignore}}</ref> 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 system]]s 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.<ref>{{cite web|url=http://casa.colorado.edu/~ajsh/collapse.html#kruskal |title=Collapse to a Black Hole |publisher=Casa.colorado.edu |date=2010-10-03 |access-date=2010-11-11}}{{tertiary source|date=June 2023}}</ref>

The Einstein–Rosen bridge was discovered by [[Ludwig Flamm]] in 1916,<ref>{{Cite journal|last=Flamm|title=Beiträge zur Einsteinschen Gravitationstheorie|date=1916|journal=[[Physikalische Zeitschrift]]|volume=XVII|page=448}} ("Comments on Einstein's Theory of Gravity")</ref> 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.<ref name=ER>A. Einstein and N. Rosen, "The Particle Problem in the General Theory of Relativity," ''Phys. Rev.'' '''48'''(73) (1935).</ref><ref name="focus032505">{{cite journal|last1=Lindley|first1=David|title=Focus: The Birth of Wormholes|url=http://physics.aps.org/story/v15/st11|journal=Physics|publisher=American Physical Society|access-date=20 February 2016|date=Mar 25, 2005|volume=15}}</ref> However, in 1962, [[John Archibald Wheeler]] and [[Robert W. Fuller]] published a paper<ref>{{cite journal |last1=Fuller |first1=Robert W. |last2=Wheeler |first2=John A. |title=Causality and Multiply Connected Space-Time |journal=Physical Review |publisher=American Physical Society (APS) |volume=128 |issue=2 |date=1962-10-15 |issn=0031-899X |doi=10.1103/physrev.128.919 |pages=919–929 |bibcode=1962PhRv..128..919F }}</ref> 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 theory|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 (physics)|spin]]) of matter. The minimal coupling between torsion and [[Dirac spinor]]s 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.<ref>{{cite journal |author=Poplawski, Nikodem J. |author-link=Nikodem Popławski |year=2010 |title=Cosmology with torsion: An alternative to cosmic inflation |journal=Phys. Lett. B |volume=694 |issue=3 |pages=181–185 |doi=10.1016/j.physletb.2010.09.056|arxiv = 1007.0587 |bibcode = 2010PhLB..694..181P }}</ref>

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).<ref>{{cite book |last1=Thorne |first1=Kip S. |title=Black holes and time warps : Einstein's outrageous legacy |date=1994 |location=New York |isbn=978-0393312768 |page=488}}</ref>

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]],<ref>{{cite journal |author=J. Wheeler |title=On the nature of quantum geometrodynamics |journal=Ann. Phys. |date=1957 |volume=2 |issue=6 |pages=604–614 |doi=10.1016/0003-4916(57)90050-7|bibcode = 1957AnPhy...2..604W }} (A follow-up paper to Misner and Wheeler (December 1957).)</ref> and ''Euclidean wormholes'' (named after [[Euclidean manifold]], a structure of [[Riemannian manifold]]).<ref>Eduard Prugovecki, ''Quantum Geometry: A Framework for Quantum General Relativity'', Springer, 2013, p. 412.</ref>

=== Traversable wormholes<!--linked from 'Joseph Polchinski'--> ===
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]], and it has been shown theoretically that quantum field theory allows states where energy can be ''arbitrarily'' [[negative energy|negative]] at a given point.<ref>{{cite book |last1=Everett |first1=Allen |last2=Roman |first2=Thomas |title=Time Travel and Warp Drives |publisher=[[University of Chicago Press]] |year=2012 |page=[https://archive.org/details/isbn_9780226224985/page/n182 167] |isbn=978-0-226-22498-5 |url-access=registration |url = https://archive.org/details/isbn_9780226224985 }}</ref> Many physicists, such as [[Stephen Hawking]],<ref>{{cite web |url=http://www.hawking.org.uk/space-and-time-warps.html |title=Space and Time Warps |website=Hawking.org.uk |access-date=2010-11-11 |archive-date=2012-02-10 |archive-url=https://web.archive.org/web/20120210233225/http://www.hawking.org.uk/space-and-time-warps.html |url-status=dead }}</ref> [[Kip Thorne]],<ref name="time travel" /> and others,<ref>{{cite journal |author1=Sopova |author2=Ford |doi=10.1103/PhysRevD.66.045026 |title=The Energy Density in the Casimir Effect |date=2002 |volume=66 |issue=4 |page=045026 |journal=[[Physical Review D]] |arxiv=quant-ph/0204125 |bibcode=2002PhRvD..66d5026S |citeseerx=10.1.1.251.7471 |s2cid=10649139 }}</ref><ref>{{cite journal |author1=Ford |author2=Roman |doi=10.1103/PhysRevD.51.4277 |year=1995 |title=Averaged Energy Conditions and Quantum Inequalities |pages=4277–4286 |issue=8 |volume=51 |journal=[[Physical Review D]] |arxiv=gr-qc/9410043 |bibcode=1995PhRvD..51.4277F |pmid=10018903 |s2cid=7413835 }}</ref><ref>{{cite journal |doi=10.1103/PhysRevLett.81.3567 |author1=Olum |title=Superluminal travel requires negative energies |date=1998 |volume=81 |issue=17 |pages=3567–3570 |journal=[[Physical Review Letters]] |arxiv=gr-qc/9805003 |bibcode=1998PhRvL..81.3567O |s2cid=14513456 }}</ref> argued that such effects might make it possible to stabilize a traversable wormhole.<ref>{{cite web |url = https://www.quantamagazine.org/newfound-wormhole-allows-information-to-escape-black-holes-20171023/|title=Newfound Wormhole Allows Information to Escape Black Holes |website = Quanta Magazine |date=23 October 2017 }}</ref> 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]],<ref name="Thorne1994" />{{rp|494–496}}<ref name="quantumdynamics">{{cite journal |first=Redmount |last=Ian H. |author2=Wai-Mo Suen |title=Quantum Dynamics of Lorentzian Spacetime Foam |journal=[[Physical Review D]] |volume=49 |year=1994 |issue=10 |pages=5199–5210 |doi=10.1103/PhysRevD.49.5199 |pmid=10016836 |arxiv=gr-qc/9309017 |bibcode=1994PhRvD..49.5199R |s2cid=39296197 }}</ref> and stable versions of such wormholes have been suggested as [[dark matter]] candidates.<ref>{{cite journal |last1=Kirillov |first1=A. A. |first2=E. |last2=P. Savelova |title=Dark Matter from a gas of wormholes |journal=[[Physics Letters B]] |volume=660 |issue=3 |pages=93–99 |year=2008 |arxiv=0707.1081 |doi=10.1016/j.physletb.2007.12.034 |bibcode=2008PhLB..660...93K|s2cid=12150385 }}</ref><ref>{{cite journal |last=Rodrigo |first=Enrico |title=Denouement of a Wormhole-Brane Encounter |journal=[[International Journal of Modern Physics D]] |volume=18 |issue=12 |pages=1809–1819 |year=2009 |arxiv=0908.2651 |doi=10.1142/S0218271809015333 |bibcode=2009IJMPD..18.1809R |s2cid=119239038 }}</ref> 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 scale|macroscopic]] size by [[Inflation (cosmology)|cosmic inflation]].<ref name="naturalwormholes">{{cite journal |author=John G. Cramer |author2=Robert L. Forward |author3=Michael S. Morris|author4=Matt Visser |author5=Gregory Benford |author6=Geoffrey A. Landis |name-list-style=amp |date=1995 |title=Natural Wormholes as Gravitational Lenses |journal=[[Physical Review D]] |volume=51 |issue=6 |pages=3117–3120 |arxiv=astro-ph/9409051 |doi=10.1103/PhysRevD.51.3117 |pmid=10018782 |bibcode=1995PhRvD..51.3117C |s2cid=42837620 |url = https://cds.cern.ch/record/268926 }}</ref>

[[File:Wurmloch.jpg|thumb|right|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 [[Ray tracing (graphics)|raytracing]] in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.{{NoteTag|Other computer-rendered images and animations of traversable wormholes can be seen on [http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html this page] by the creator of the image in the article, and [https://web.archive.org/web/20151228145356/https://www.vis.uni-stuttgart.de/~muelleta/MTvis/ this page] has additional renderings.}}]]

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. {{anchor|Ellis–Bronnikov precedence}}<!--'Ellis wormhole' redirects here-->The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis<ref name="ellis1">{{cite journal |author = H. G. Ellis |year = 1973 |title = Ether flow through a drainhole: A particle model in general relativity |journal = Journal of Mathematical Physics |volume = 14 |issue = 1 |pages = 104–118 |bibcode = 1973JMP....14..104E |doi = 10.1063/1.1666161}}</ref> and independently in a 1973 paper by K. A. Bronnikov.<ref name="bron">{{cite journal |author = K. A. Bronnikov |year = 1973 |title = Scalar-tensor theory and scalar charge |journal = Acta Physica Polonica |volume = B4 |pages = 251–266}}</ref> Ellis analyzed the topology and the [[geodesic]]s 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: {{var|m}}, which fixes the strength of its gravitational field, and {{var|n}}, which determines the curvature of its spatial cross sections. When {{var|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 (physicist)|Mike Morris]] independently discovered in 1988 the Ellis wormhole and argued for its use as a tool for teaching general relativity.<ref>{{cite journal |author1=Morris, Michael S. |author2=Thorne, Kip S. |name-list-style=amp |year=1988 |title=Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity |journal=[[American Journal of Physics]] |volume=56 |issue=5 |pages=395–412 |doi=10.1119/1.15620 |bibcode=1988AmJPh..56..395M |doi-access=free }}</ref> 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. 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.<ref>{{cite journal |author1=Elias Gravanis |author2=Steven Willison |doi=10.1103/PhysRevD.75.084025 |year=2007 |title = 'Mass without mass' from thin shells in Gauss-Bonnet gravity |issue=8 |page=084025 |volume=75 |journal=Phys. Rev. D |arxiv=gr-qc/0701152 |bibcode=2007PhRvD..75h4025G |s2cid=53529713 }}</ref> A type held open by negative mass [[cosmic string]]s was put forth by Visser in collaboration with [[John G. Cramer|Cramer]] ''et al.'',<ref name="naturalwormholes" /> 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 [[time travel|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.<ref name="time travel">{{cite journal |title=Wormholes, Time Machines, and the Weak Energy Condition |year=1988 |last1=Morris|first1=Michael|last2=Thorne|first2=Kip|last3=Yurtsever|first3=Ulvi |journal=Physical Review Letters |volume=61|issue=13|pages=1446–1449 |url = http://authors.library.caltech.edu/9262/1/MORprl88.pdf |doi=10.1103/PhysRevLett.61.1446 |pmid=10038800 |bibcode=1988PhRvL..61.1446M }}</ref> 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". Until this time it could not have been noticed or have been used.<ref name="Thorne1994" />{{Rp|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 [[congruence (general relativity)|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,<ref>{{Cite journal |last=Fewster |first=Christopher J. |author2=Ken D. Olum |author3=Michael J. Pfenning |title=Averaged null energy condition in spacetimes with boundaries |journal=Physical Review D |volume=75 |issue=2 |page=025007 |year=2007 |arxiv= gr-qc/0609007 |doi=10.1103/PhysRevD.75.025007 |bibcode=2007PhRvD..75b5007F|s2cid=119726654}}</ref> but calculations in [[semiclassical gravity]] suggest that quantum effects may be able to violate this condition in curved spacetime.<ref>{{Cite journal |last=Visser |first=Matt |title=Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum|journal=Physical Review D |volume=54 |issue=8 |pages=5116–5122 |year=1996 |arxiv=gr-qc/9604008 |doi=10.1103/PhysRevD.54.5116 |pmid=10021199 |bibcode=1996PhRvD..54.5116V |s2cid=31954680}}</ref> Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,<ref>{{Cite journal |last=Graham |first=Noah |author2=Ken D. Olum |title=Achronal averaged null energy condition |journal=Physical Review D |volume=76 |issue=6 |page=064001 |year=2007 |arxiv=0705.3193 |doi=10.1103/PhysRevD.76.064001 |bibcode= 2007PhRvD..76f4001G |s2cid=119285639}}</ref> violations have nevertheless been found,<ref>{{Cite journal |last=Urban |first=Douglas |author2=Ken D. Olum |title=Spacetime averaged null energy condition |journal=Physical Review D |volume=81 |issue=6 |page=124004 |year=2010 |arxiv=1002.4689 |doi=10.1103/PhysRevD.81.124004 |bibcode=2010PhRvD..81l4004U |s2cid=118312373}}</ref> so it remains an open possibility that quantum effects might be used to support a wormhole.

== Modified general relativity ==
In some hypotheses where [[Alternatives to general relativity|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{{sup|2}} gravity, a form of [[f(R) gravity|{{var|f}}({{var|R}}) gravity]].<ref>{{Cite journal|arxiv=1506.00988|title=Exotica ex nihilo: Traversable wormholes & non-singular black holes from the vacuum of quadratic gravity|journal=Physical Review D|volume=92|issue=4|pages=043516|last1=Duplessis|first1=Francis|last2= Easson|first2=Damien A.|year=2015|doi=10.1103/PhysRevD.92.043516|bibcode=2015PhRvD..92d3516D|s2cid=118307327}}</ref>

== Faster-than-light travel ==
{{further|Faster-than-light}}
[[File: Wormhole travel as envisioned by Les Bossinas for NASA.jpg|alt=Wormhole travel as envisioned by Les Bossinas for NASA Digital art by Les Bossinas (Cortez III Service Corp.), 1998|thumb|Wormhole travel as envisioned by Les Bossinas for [[NASA]], {{Circa|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. However, a light beam traveling through the same wormhole would beat the traveler.

== Time travel ==
{{main|Time travel}}

If [[#Traversable wormholes|traversable wormholes]] exist, they might allow [[time travel]].<ref name="time travel"/> 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 [[Vehicle propulsion|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; however, time connects differently ''through'' the wormhole than ''outside'' it, so that [[Synchronization|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.<ref name="Thorne1994">{{cite book |last = Thorne |first = Kip S. |author-link = Kip Thorne |title = Black Holes and Time Warps |publisher = W. W. Norton |year= 1994 |isbn = 978-0-393-31276-8 |title-link = Black Holes and Time Warps }}</ref>{{rp|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;<ref name="Thorne1994" />{{rp|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.<ref>{{cite arXiv |eprint=gr-qc/0503097 |title=Wormholes and Time Travel? Not Likely |last1=Susskind |first1=Leonard |year=2005}}</ref><ref>{{cite book |last = Everett |first = Allen |author2 = Roman, Thomas |title = Time Travel and Warp Drives |publisher = University of Chicago Press |date = 2012 |page = [https://archive.org/details/isbn_9780226224985/page/n150 135] |isbn = 978-0-226-22498-5 |url-access = registration |url = https://archive.org/details/isbn_9780226224985 }}</ref>

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 condition]]s, such as the null energy condition along with the weak, strong, and dominant energy conditions. However, it is known that quantum effects can lead to small measurable violations of the null energy condition,<ref name="Visser1996">{{cite book |last = Visser |first = Matt |author-link = Matt Visser |title = Lorentzian Wormholes |publisher = Springer-Verlag |year= 1996 |isbn = 978-1-56396-653-8}}</ref>{{rp|101}} and many physicists believe that the required negative energy may actually be possible due to the [[Casimir effect]] in quantum physics.<ref name="casimir">{{cite web |url=http://www.npl.washington.edu/av/altvw69.html |title=NASA Goes FTL Part 1: Wormhole Physics |website=Analog Science Fiction & Fact Magazine|year=1994|access-date=December 2, 2006|last1=Cramer|first1=John G. |author-link=John G. Cramer |archive-url = https://web.archive.org/web/20060627211046/http://www.npl.washington.edu/av/altvw69.html <!-- Bot retrieved archive --> |archive-date = June 27, 2006}}</ref> 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.<ref name="negative energy">{{cite journal |first=Matt |last=Visser |author-link = Matt Visser |author2=Sayan Kar |author3=Naresh Dadhich |title=Traversable wormholes with arbitrarily small energy condition violations |journal = [[Physical Review Letters]] |volume = 90 |year=2003 |issue=20 |pages = 201102.1–201102.4 |doi=10.1103/PhysRevLett.90.201102 |arxiv=gr-qc/0301003 |bibcode=2003PhRvL..90t1102V |pmid=12785880 |s2cid=8813962 |url=https://cds.cern.ch/record/598932 }}</ref>

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,<ref name="visser_1">{{cite journal |first = Matt |last = Visser |author-link = Matt Visser |title = From wormhole to time machine: Comments on Hawking's Chronology Protection Conjecture |journal = Physical Review D |volume = 47 |year = 1993 |issue = 2 |pages = 554–565 |doi = 10.1103/PhysRevD.47.554 |pmid = 10015609 |arxiv=hep-th/9202090 |bibcode = 1993PhRvD..47..554V |s2cid = 16830951 }}</ref> or otherwise prevent information from passing through the wormhole.<ref>{{cite book |arxiv=gr-qc/0204022 |title=The quantum physics of chronology protection |url=https://archive.org/details/arxiv-gr-qc0204022 |last1=Visser |first1=Matt |year=2002 |bibcode=2003ftpc.book..161V}}</ref> Because of this, the two mouths could not be brought close enough for [[causality (physics)|causality]] violation to take place. However, in a 1997 paper, 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.<ref name="visser_2">{{cite journal |first = Matt |last = Visser |author-link = Matt Visser |title = Traversable wormholes: the Roman ring |journal = Physical Review D |volume = 55 |year = 1997 |issue = 8 |pages = 5212–5214 |doi = 10.1103/PhysRevD.55.5212 |arxiv=gr-qc/9702043|bibcode = 1997PhRvD..55.5212V |s2cid = 2869291 }}</ref>

== Interuniversal travel<!--'Interuniversal travel', 'Inter-universal travel' and 'Everett phone' redirect here--> ==
{{See also|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.<ref>{{Cite journal|last=Deutsch|first=David |title=Quantum Mechanics Near Closed Timelike Lines|date=1991|journal=[[Physical Review D]]|volume=44|issue=10|pages=3197–3217 |doi= 10.1103/PhysRevD.44.3197|pmid=10013776 |bibcode=1991PhRvD..44.3197D}}</ref> However, 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.<ref>{{Cite journal|author=Brun|display-authors=etal |title=Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States|date=2009|journal=[[Physical Review Letters]]|volume=102|issue=21|page=210402|doi= 10.1103/PhysRevLett.102.210402|bibcode=2009PhRvL.102u0402B|pmid=19519086|arxiv=0811.1209|s2cid=35370109 }}</ref><ref>{{Cite journal|author=Pati |author2=Chakrabarty|author3=Agrawal|title=Purification of mixed states with closed timelike curve is not possible|date=2011|journal=[[Physical Review A]]|volume=84|issue=6|page=062325|doi= 10.1103/PhysRevA.84.062325|bibcode = 2011PhRvA..84f2325P |arxiv = 1003.4221 |s2cid=119292717}}</ref> 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.<ref name="Rodrigo2">{{cite book|last=Rodrigo|first=Enrico|title=The Physics of Stargates|publisher=Eridanus Press|date= 2010|page=281|isbn= 978-0-9841500-0-7}}</ref>

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'''<!--boldface per WP:R#PLA--><ref>{{Cite journal|last=Polchinski|first=Joseph |title=Weinberg's Nonlinear quantum Mechanics and the Einstein–Podolsky–Rosen Paradox|journal=[[Physical Review Letters]]|volume=66|issue=4|pages=397–400 |date=1991 |bibcode=1991PhRvL..66..397P|doi=10.1103/PhysRevLett.66.397 |pmid=10043797}}</ref> (named after [[Hugh Everett III|Hugh Everett]]) in [[Steven Weinberg]]'s formulation of nonlinear quantum mechanics.<ref>Enrico Rodrigo, ''The Physics of Stargates: Parallel Universes, Time Travel, and the Enigma of Wormhole Physics'', Eridanus Press, 2010, p. 281.</ref>

The possibility of communication between parallel universes has been dubbed '''interuniversal travel'''.<!--boldface per WP:R#PLA--><ref>Samuel Walker, [https://www.newscientist.com/letter/mg23331111-100-14-interuniversal-travel-i-wouldnt-start-from-here/ "Inter-universal travel: I wouldn't start from here], ''[[New Scientist]]'' (1 February 2017).</ref>

Wormhole can also be depicted in a [[Penrose diagram]] of a [[Schwarzschild Black Hole|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 tensor|metric]] is the following:<ref>{{cite book |last1=Raine |first1=Derek |last2=Thomas |first2=Edwin |date=2009 |title= Black Holes: An Introduction |url=https://archive.org/details/blackholesintrod00rain |url-access=limited |publisher=Imperial College Press |page=[https://archive.org/details/blackholesintrod00rain/page/n156 143] |isbn=978-1-84816-383-6 |edition=2nd|doi=10.1142/p637 }}</ref>
{{block indent|<math>ds^2= - c^2 \, dt^2 + d\ell^2 + (k^2 + \ell^2)(d \theta^2 + \sin^2 \theta \, d\varphi^2),</math>}}

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

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

{{block indent|<math>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\varphi^2).</math>}}

The original Einstein–Rosen bridge was described in an article published in July 1935.<ref>{{cite journal |last1=Einstein |first1=A. |last2=Rosen |first2=N. |title=The Particle Problem in the General Theory of Relativity |journal=Physical Review |date=1 July 1935 |volume=48 |issue=1 |pages=73–77 |doi=10.1103/PhysRev.48.73 |bibcode = 1935PhRv...48...73E |doi-access=free}}</ref><ref>{{Cite web |url=https://www.youtube.com/watch?v=OBPpRqxY8Uw |archive-url=https://ghostarchive.org/varchive/youtube/20211211/OBPpRqxY8Uw |archive-date=2021-12-11 |url-status=live |title=Leonard Susskind &#124; "ER = EPR" or "What's Behind the Horizons of Black Holes?" |date=4 November 2014 |via=www.youtube.com}}{{cbignore}}</ref>

For the Schwarzschild spherically symmetric static solution 
{{block indent|<math>ds^2 = - \frac{1}{1 - \frac{2m}{r}} \, dr^2 - r^2(d\theta^2 + \sin^2 \theta \, d\varphi^2) + \left(1 - \frac{2m}{r} \right) \, dt^2,</math>}}
where <math>ds</math> is the proper time and <math>c = 1</math>.

If one replaces <math>r</math> with <math>u</math> according to <math>u^2 = r - 2m</math>

{{block indent|<math>ds^2 = -4(u^2 + 2m)\,du^2 - (u^2 + 2m)^2(d\theta^2 + \sin^2 \theta \, d\varphi^2) + \frac{u^2}{u^2 + 2m} \, dt^2 </math>}}

{{Blockquote|text=The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to <math>u>0</math> and <math>u< 0</math>, which are joined by a hyperplane <math>r = 2m</math> or <math>u= 0</math> in which <math>g</math> vanishes. We call such a connection between the two sheets a "bridge".|author=A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"|source=}}

For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution
{{block indent|<math>\varphi_1 = \varphi_2 = \varphi_3 = 0, \varphi_4 = \frac{\varepsilon}{4},</math>}}
{{block indent|<math>ds^2 = - \frac{1}{\left( 1 - \frac{2m}{r} - \frac{\varepsilon^2}{2 r^2}\right)} \, dr^2 - r^2 (d\theta^2 + \sin^2 \theta \, d\varphi^2) + \left(1 - \frac{2m}{r} - \frac{\varepsilon^2}{2 r^2}\right) \, dt^2,</math>}}
where <math>\varepsilon</math> is the electric charge.

The field equations without denominators in the case when <math>m = 0</math> can be written
{{block indent|<math>\varphi_{\mu \nu} = \varphi_{\mu,\nu} - \varphi_{\nu,\mu}</math>}}
{{block indent|<math>g^2 \varphi_{\mu\nu;\sigma}g^{\nu\sigma} = 0</math>}}
{{block indent|<math>g^2 (R_{ik} + \varphi_{i\alpha}\varphi_k^\alpha - \frac{1}{4} g_{ik} \varphi_{\alpha\beta}\varphi^{\alpha\beta}) = 0</math>}}

In order to eliminate singularities, if one replaces <math>r</math> by <math>u</math> according to the equation:
{{block indent|<math>u^2 = r^2 - \frac{\varepsilon^2}{2}</math>}}

and with <math>m = 0</math> one obtains<ref>{{Cite web|url=https://www.sciencedaily.com/releases/2015/09/150903081506.htm|title=Magnetic 'wormhole' connecting two regions of space created for the first time|website=ScienceDaily}}</ref><ref>{{Cite web|url=http://www.uab.cat/web/newsroom/news-detail/magnetic-wormhole-created-for-first-time-1345668003610.html?noticiaid=1345689132054|title=Magnetic wormhole created for first time|website=UAB Barcelona}}</ref>

{{block indent|<math>\varphi_1 = \varphi_2 = \varphi_3 = 0</math> and <math>\varphi_4 = \frac{\varepsilon}{\left( u^2 + \frac{\varepsilon^2}{2} \right)^{1/2}}</math>}}

{{block indent|<math>ds^2 = - du^2 - \left(u^2 + \frac{\varepsilon^2}{2}\right)(d \theta^2 + \sin^2 \theta \, d\varphi^2) + \left(\frac{2 u^2}{2 u^2 + \varepsilon^2}\right) \, dt^2</math>}}

{{Blockquote|text=The solution is free from singularities for all finite points in the space of the two sheets|author=A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"}}

== In fiction ==
{{Main|Wormholes 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]].

== See also ==
{{Portal|Physics|Star}}
{{Div col |colwidth = 25em }}
* [[Alcubierre drive]]
* [[ER = EPR]]
* [[Gödel metric]]
* [[Krasnikov tube]]
* [[Non-orientable wormhole]]
* [[Novikov self-consistency principle]]
* [[Polchinski's paradox]]
* [[Retrocausality]]
* [[Ring singularity]]
* [[Roman ring]]
{{div col end}}

== Notes ==
{{NoteFoot}}
== References ==
=== Citations ===
{{Reflist}}
=== Sources ===
{{refbegin}}
* {{cite journal |author=DeBenedictis, Andrew |author2=Das, A. |name-list-style=amp |title=On a General Class of Wormhole Geometries |doi=10.1088/0264-9381/18/7/304 |year=2001 |volume=18 |issue=7 |pages=1187–1204 |journal= [[Classical and Quantum Gravity]] |arxiv=gr-qc/0009072 |bibcode = 2001CQGra..18.1187D |citeseerx=10.1.1.339.8662 |s2cid=119107035 }}
* {{cite journal |author= Dzhunushaliev, Vladimir |title = Strings in the Einstein's paradigm of matter |doi= 10.1088/0264-9381/19/19/302 |year = 2002 |volume= 19 |issue= 19 |pages = 4817–4824 |journal = [[Classical and Quantum Gravity]] |arxiv=gr-qc/0205055 |bibcode = 2002CQGra..19.4817D |citeseerx=10.1.1.339.1518 |s2cid = 976106 }}
* {{cite journal |author1=Einstein, Albert |author2=Rosen, Nathan |name-list-style=amp |year=1935 |title = The Particle Problem in the General Theory of Relativity |journal=[[Physical Review]] |volume=48 |issue=1 |page=73 |doi=10.1103/PhysRev.48.73 |bibcode=1935PhRv...48...73E |doi-access=free }}
* {{cite journal |author1=Fuller, Robert W. |author2=Wheeler, John A. |name-list-style=amp |year=1962 |title=Causality and Multiply-Connected Space-Time |journal=[[Physical Review]] |volume=128 |issue=2 |page=919 |doi= 10.1103/PhysRev.128.919 |bibcode = 1962PhRv..128..919F }}
* {{cite journal |doi= 10.1142/S0217732304015658 |author= Garattini, Remo |title = How Spacetime Foam modifies the brick wall |year = 2004 |volume= 19 |issue= 36 |pages = 2673–2682 |journal = [[Modern Physics Letters A]] |arxiv=gr-qc/0409015 |bibcode= 2004MPLA...19.2673G |s2cid= 119094239 }}
* {{cite journal |author=González-Díaz, Pedro F. |title = Quantum time machine |doi=10.1103/PhysRevD.58.124011 |year=1998 |volume=58 |issue=12 |page=124011 |journal=[[Physical Review D]] |arxiv=gr-qc/9712033|bibcode = 1998PhRvD..58l4011G |hdl=10261/100644 |s2cid = 28411713 }}
* {{cite journal |author=González-Díaz, Pedro F. |title = Ringholes and closed timelike curves |doi=10.1103/PhysRevD.54.6122 |year=1996 |volume=54 |issue=10 |pages=6122–6131 |journal=[[Physical Review D]] |pmid = 10020617 |arxiv=gr-qc/9608059 |bibcode = 1996PhRvD..54.6122G |s2cid = 7183386 }}
* {{cite journal |author=Khatsymosky, Vladimir M. |title=Towards possibility of self-maintained vacuum traversable wormhole |doi=10.1016/S0370-2693(97)00290-6 |year=1997 |pages=215–222 |issue=3–4 |volume=399 |journal=[[Physics Letters B]] |arxiv=gr-qc/9612013 |bibcode = 1997PhLB..399..215K |s2cid=13917471 }}
* {{cite journal |author=Krasnikov, Serguei |title=Counter example to a quantum inequality |year=2006 |volume=46 |issue=2006 |page=195 |journal=[[Gravity and Cosmology]] |arxiv=gr-qc/0409007|bibcode = 2006GrCo...12..195K }}
* {{cite journal |author=Krasnikov, Serguei |title=The quantum inequalities do not forbid spacetime shortcuts |doi=10.1103/PhysRevD.67.104013 |year=2003 |volume=67 |issue=10 |page=104013 |journal=[[Physical Review D]] |arxiv=gr-qc/0207057 |bibcode = 2003PhRvD..67j4013K |s2cid=17498199 }}
* {{cite journal |author=Li, Li-Xin |title=Two Open Universes Connected by a Wormhole: Exact Solutions |doi=10.1016/S0393-0440(01)00028-6 |year=2001 |volume=40 |issue=2 |pages=154–160 |journal=[[Journal of Geometry and Physics]] |arxiv=hep-th/0102143|bibcode = 2001JGP....40..154L |citeseerx=10.1.1.267.8664 |s2cid=44433480 }}
* {{cite journal |author1=Morris, Michael S. |author2=Thorne, Kip S. |author3=Yurtsever, Ulvi |name-list-style=amp |year=1988 |title= Wormholes, Time Machines, and the Weak Energy Condition |journal=[[Physical Review Letters]] |volume=61 |issue=13 |pages=1446–1449 |doi=10.1103/PhysRevLett.61.1446 |bibcode=1988PhRvL..61.1446M |pmid=10038800 |url = https://authors.library.caltech.edu/9262/1/MORprl88.pdf }}
* {{cite journal |author1=Morris, Michael S. |author2=Thorne, Kip S. |name-list-style=amp |year=1988 |title=Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity |journal=[[American Journal of Physics]] |volume=56 |issue=5 |pages=395–412 |doi=10.1119/1.15620 |bibcode=1988AmJPh..56..395M |doi-access=free }}
* {{cite journal |author1=Nandi, Kamal K. |author2=Zhang, Yuan-Zhong |name-list-style=amp |title = A Quantum Constraint for the Physical Viability of Classical Traversable Lorentzian Wormholes |year = 2006 |volume=9 |issue=2006 |pages=61–67 |journal=[[Journal of Nonlinear Phenomena in Complex Systems]] |arxiv=gr-qc/0409053 |bibcode = 2004gr.qc.....9053N }}
* {{cite journal |author= Ori, Amos |title=A new time-machine model with compact vacuum core |doi= 10.1103/PhysRevLett.95.021101 |pmid=16090670 |year = 2005 |journal= [[Physical Review Letters]] |volume= 95 |issue= 2 |pages=021101 |arxiv=gr-qc/0503077|bibcode = 2005PhRvL..95b1101O }}
* {{Cite journal |author=Roman, Thomas A. |title=Some Thoughts on Energy Conditions and Wormholes |arxiv=gr-qc/0409090 |year=2004 |doi=10.1142/9789812704030_0236 |journal=The Tenth Marcel Grossmann Meeting |pages=1909–1924 |isbn=978-981-256-667-6 |s2cid=18867900 }}
* {{cite journal |author=Teo, Edward |title=Rotating traversable wormholes |doi=10.1103/PhysRevD.58.024014 |year=1998 |issue=2 |pages=024014 |volume=58 |journal=[[Physical Review D]] |arxiv=gr-qc/9803098 |bibcode = 1998PhRvD..58b4014T |citeseerx=10.1.1.339.966 |s2cid=15316540 }}
* {{cite arXiv |author=Visser, Matt |title = The quantum physics of chronology protection by Matt Visser |eprint=gr-qc/0204022 |year=2002 }} An excellent and more concise review.
* {{cite journal |author=Visser, Matt |year=1989 |title=Traversable wormholes: Some simple examples |journal=[[Physical Review D]] |volume=39 |issue=10 |pages=3182–3184 |doi=10.1103/PhysRevD.39.3182 |pmid=9959561 |bibcode=1989PhRvD..39.3182V |arxiv = 0809.0907 |s2cid=17949528 }}
{{refend}}
== External links ==
{{Commons category|Wormholes}}
* [https://www.scientificamerican.com/article/follow-up-what-exactly-is/ "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
* [http://homepages.mcs.vuw.ac.nz/~visser/general.shtml#why-wormholes "Why wormholes?"] by Matt Visser (October 1996)
* {{webarchive |url=https://web.archive.org/web/20120222034225/http://www.bun.kyoto-u.ac.jp/~suchii/wormholes.html |date=February 22, 2012 |title=Wormholes in General Relativity by Soshichi Uchii }}
* [http://www.webfilesuci.org/WormholeFAQ.html Questions and Answers about Wormholes]{{snd}}A comprehensive wormhole FAQ by Enrico Rodrigo
* [http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html animation that simulates traversing a wormhole]
* [https://web.archive.org/web/20151228145356/https://www.vis.uni-stuttgart.de/~muelleta/MTvis/ Renderings and animations of a Morris-Thorne wormhole]

{{Black holes}}
{{time travel}}
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[[Category:Wormhole theory|*]]
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