Editing Wormhole (section)

== 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"/> Later, physicists 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> 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. 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.<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> 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.<ref name="Thorne1994" />{{Rp|504}}