Wormhole
General relativity  

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Fundamental concepts




A wormhole, officially known as an Einstein–Rosen bridge, is a hypothetical topological feature of spacetime that would fundamentally be a shortcut through spacetime. A wormhole is much like a tunnel with two ends, each in separate points in spacetime.
For a simplified notion of a wormhole, visualize space as a twodimensional (2D) surface. In this case, a wormhole can be pictured as a hole in that surface that leads into a 3D tube (the inside surface of a cylinder). This tube then reemerges at another location on the 2D surface with a similar hole as 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, a real wormhole's mouths could be spheres in 3D space.
Researchers have no observational evidence for wormholes, but the equations of the theory of general relativity have valid solutions that contain wormholes. Because of its robust theoretical strength, a wormhole is one of the great physics metaphors for teaching general relativity. The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too quickly for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them.
The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at a given point.^{[1]} Many physicists, such as Stephen Hawking,^{[2]} Kip Thorne^{[3]} and others,^{[4]}^{[5]}^{[6]} therefore argue that such effects might make it possible to stabilize a traversable wormhole. Physicists have not found any natural process that would be predicted to form a wormhole naturally in the context of general relativity, although the quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,^{[7]}^{[8]} and stable versions of such wormholes have been suggested as dark matter candidates.^{[9]}^{[10]} It has also been proposed that, if a tiny wormhole held open by a negativemass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.^{[11]}
The American theoretical physicist John Archibald Wheeler coined the term wormhole in 1957; the German mathematician Hermann Weyl, however, had proposed the wormhole theory in 1921, in connection with mass analysis of electromagnetic field energy.^{[12]}
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 multiplyconnected space, and what physicists might perhaps be excused for more vividly terming a "wormhole".—John Wheeler in Annals of Physics
Contents
 Definition 1
 Schwarzschild wormholes 2

Traversable wormholes 3
 Raychaudhuri's theorem and exotic matter 3.1
 Fasterthanlight travel 3.2
 Time travel 3.3
 Interuniverse travel 3.4
 Metrics 4
 In fiction 5
 In Movies 6
 See also 7
 Notes 8
 References 9
 External links 10
Definition
The basic notion of an intrauniverse wormhole is that it 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.
If a Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R x Σ, where Σ is a threemanifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S^{2}, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intrauniverse wormhole.Characterizing interuniverse wormholes is more difficult, with little consideration being given to available technology. For example, one can imagine a baby universe connected to its parent by a narrow umbilicus. One might like to regard the umbilicus as the throat of a wormhole, but the spacetime is simply connected. For this reason, wormholes have been defined geometrically, as opposed to topologically, 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).
Schwarzschild wormholes
Lorentzian wormholes known as Schwarzschild wormholes or Einstein–Rosen bridges are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and which 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": for any possible trajectory of a freefalling particle (following a geodesic) in the spacetime, it should be possible to continue this path arbitrarily far into the particle's future or past, unless the trajectory hits a gravitational singularity like the one at the center of the black hole's interior. In order to satisfy this requirement, it turns out that in addition to the black hole interior region which particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region which allows us to extrapolate the trajectories of particles which an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram which uses Kruskal–Szekeres coordinates.
In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a spacelike separation, giving what is called a 'spacelike surface') and draw an "embedding diagram" depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". Note that the Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's history, it removes the part of the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe.^{[13]}
The Einstein–Rosen bridge was discovered by Albert Einstein and his colleague Nathan Rosen, who first published the result in 1935. However, in 1962 John A. Wheeler and Robert W. Fuller published a paper 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.
The motion through a Schwarzschild wormhole connecting two universes is possible in only one direction. The analysis of the radial geodesic motion of a massive particle into an Einstein–Rosen bridge shows that the proper time of the particle extends to infinity. Timelike and null geodesics in the gravitational field of a Schwarzschild wormhole are complete because the expansion scalar in the Raychaudhuri equation has a discontinuity at the event horizon, and because an Einstein–Rosen bridge is represented by the Kruskal diagram in which the two antipodal future event horizons are identified. Schwarzschild wormholes and Schwarzschild black holes are different, mathematical solutions of general relativity and Einstein–Cartan–Sciama–Kibble theory of gravity. Yet for distant observers, both solutions with the same mass are indistinguishable. These results suggest that all observed astrophysical black holes may be Einstein–Rosen bridges, each with a new universe inside that formed simultaneously with the black hole. Accordingly, our own Universe may be the interior of a black hole existing inside another universe.^{[14]}
According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan–Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable. Torsion naturally accounts for the quantummechanical, intrinsic angular momentum (spin) of matter. The minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction which is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity. Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.^{[15]}
Before the stability problems of Schwarzschild wormholes were apparent, it was proposed that quasars were white holes forming the ends of wormholes of this type.
While Schwarzschild wormholes are not traversable in both directions, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the 'throat' of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy).
Traversable wormholes
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 by Kip Thorne and his graduate student Mike Morris in a 1988 paper. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is referred to 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.^{[17]} A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,^{[11]} in which it was proposed that such wormholes could have been naturally created in the early universe.
Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out explicitly how to convert a wormhole traversing space into one traversing time.^{[3]} 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 by accelerating one of its two mouths.^{[18]}
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 reexpands 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,^{[19]} but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.^{[20]} Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,^{[21]} violations have nevertheless been found,^{[22]} so it remains an open possibility that quantum effects might be used to support a wormhole.
Fasterthanlight travel
The impossibility of fasterthanlight relative speed only applies locally. Wormholes might allow superluminal (fasterthanlight) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slowerthanlight) speeds are used. If two points are connected by a wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. However, a light beam traveling through the wormhole would always beat the traveler.
Time travel
The theory of general relativity predicts that if traversable wormholes exist, they could allow time travel.^{[3]} This would be accomplished by accelerating one end of the wormhole to a high velocity relative to the other, and then sometime later bringing it back; relativistic time dilation would result in the accelerated wormhole mouth aging less than the stationary one as seen by an external observer, similar to what is seen in the twin paradox. However, time connects differently through the wormhole than outside it, so that synchronized clocks at each mouth will remain synchronized to someone traveling through the wormhole itself, no matter how the mouths move around.^{[23]} This means that anything which entered the accelerated wormhole mouth would exit the stationary one at a point in time prior to its entry.
For example, consider two clocks at both mouths both showing the date as 2000. After being taken on a trip at relativistic velocities, the accelerated mouth is brought back to the same region as the stationary mouth with the accelerated mouth's clock reading 2004 while the stationary mouth's clock read 2012. A traveler who entered the accelerated mouth at this moment would exit the stationary mouth when its clock also read 2004, in the same region but now eight years in the past. Such a configuration of wormholes would allow for a particle's world line to form a closed loop in spacetime, known as a closed timelike curve. An object traveling through a wormhole could carry energy or charge from one time to another, but this would not violate conservation of energy or charge in each time, because the energy/charge of the wormhole mouth itself would change to compensate for the object that fell into it or emerged from it.^{[24]}^{[25]}
It is thought that it may not be possible to convert a wormhole into a time machine in this manner; the predictions are made in the context of general relativity, but general relativity does not include quantum effects. Analyses using the semiclassical approach to incorporating quantum effects into general relativity have sometimes indicated that a feedback loop of virtual particles would circulate through the wormhole and pile up on themselves, driving the energy density in the region very high and possibly destroying it before any information could be passed through it, in keeping with the chronology protection conjecture. The debate on this matter is described by Kip S. Thorne in the book Black Holes and Time Warps, and a more technical discussion can be found in The quantum physics of chronology protection by Matt Visser.^{[26]} There is also the Roman ring, which is a configuration of more than one wormhole. This ring seems to allow a closed time loop with stable wormholes when analyzed using semiclassical gravity, although without a full theory of quantum gravity it is uncertain whether the semiclassical approach is reliable in this case.
Interuniverse travel
A possible resolution to the paradoxes resulting from wormholeenabled time travel rests on the manyworlds interpretation of quantum mechanics. In 1991 David Deutsch showed that quantum theory is fully consistent (in the sense that the socalled density matrix can be made free of discontinuities) in spacetimes with closed timelike curves.^{[27]} However, later it was shown that such model of closed timelike curve can have internal inconsistencies as it will lead to strange phenomena like distinguishing non orthogonal quantum states and distinguishing proper and improper mixture.^{[28]}^{[29]} Accordingly, the destructive positive feedback loop of virtual particles circulating through a wormhole time machine, a result indicated by semiclassical 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.^{[30]} Because a wormhole timemachine introduces a type of nonlinearity into quantum theory, this sort of communication between parallel universes is consistent with Joseph Polchinski’s discovery of an “Everett phone” in Steven Weinberg’s formulation of nonlinear quantum mechanics.^{[31]}
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:
 ds^2=  c^2 dt^2 + dl^2 + (k^2 + l^2)(d \theta^2 + \sin^2 \theta \, d\phi^2).
Z^2=(d \theta^2 + \sin^2 \theta \, d\phi^2)
One type of nontraversable wormhole metric is the Schwarzschild solution (see the first diagram):
 ds^2=  c^2 \left(1  \frac{2GM}{rc^2}\right)dt^2 + \frac{dr^2}{1  \frac{2GM}{rc^2}} + r^2(d \theta^2 + \sin^2 \theta \, d\phi^2).
In fiction
Wormholes are a common element in science fiction as they allow interstellar, intergalactic, and sometimes interuniversal travel within human timescales. They have also served as a method for time travel.
In Movies
 Donnie Darko (2001, directed by Richard Kelly)
 Interstellar (2014, directed by Christopher Nolan)
See also
 Black hole
 Closed timelike curve
 Fasterthanlight
 Exotic star
 Gödel metric
 Krasnikov tube
 Nonorientable wormhole
 Selfconsistency principle
 Retrocausality
 Ring singularity
 Roman ring
 White hole
 Universe
Notes
 ^ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 167.
 ^ "Space and Time Warps". Hawking.org.uk. Retrieved 20101111.
 ^ ^{a} ^{b} ^{c} Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review Letters 61 (13): 1446–1449.
 ^ Sopova; Ford (2002). "The Energy Density in the Casimir Effect".
 ^ Ford; Roman (1995). "Averaged Energy Conditions and Quantum Inequalities".
 ^ Olum (1998). "Superluminal travel requires negative energies".
 ^
 ^ Ian H., Redmount; WaiMo Suen (1994). "Quantum Dynamics of Lorentzian Spacetime Foam".
 ^ Kirillov, A.A.; E.P. Savelova (21 February 2008). "Dark Matter from a gas of wormholes".
 ^ Rodrigo, Enrico (30 November 2009). "Denouement of a WormholeBrane Encounter".
 ^ ^{a} ^{b} John G. Cramer, Robert L. Forward, Michael S. Morris, Matt Visser, Gregory Benford, and Geoffrey A. Landis (1995). "Natural Wormholes as Gravitational Lenses".
 ^ Coleman, Korte, Hermann Weyl's Raum  Zeit  Materie and a General Introduction to His Scientific Work, p. 199
 ^ "Collapse to a Black Hole". Casa.colorado.edu. 20101003. Retrieved 20101111. This is a tertiary source that clearly includes information from other sources but does not name them.
 ^
 ^
 ^ Other computerrendered 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.
 ^ Elias Gravanis; Steven Willison (2007). "`Mass without mass' from thin shells in GaussBonnet gravity". Phys. Rev. D 75 (8).
 ^
 ^ Fewster, Christopher J.; Ken D. Olum; Michael J. Pfenning (10 January 2007). "Averaged null energy condition in spacetimes with boundaries". Physical Review D 75 (2): 025007.
 ^ Visser, Matt (15 October 1996). "Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum". Physical Review D 54 (8): 5116.
 ^ Graham, Noah; Ken D. Olum (4 September 2007). "Achronal averaged null energy condition". Physical Review D 76 (6): 064001.
 ^ Urban, Douglas; Ken D. Olum (1 June 2010). "Spacetime averaged null energy condition". Physical Review D 81 (6): 124004.
 ^
 ^ "Wormholes and Time Travel? Not Likely". Retrieved 4 October 2014.
 ^ Everett, Allen; Roman, Thomas (2012). Time Travel and Warp Drives. University of Chicago Press. p. 135.
 ^ "The quantum physics of chronology protection". Retrieved 4 October 2014.
 ^ Deutsch, David (1991). "Quantum Mechanics Near Closed Timelike Lines".
 ^ Brun et.al (2009). "Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States".
 ^ Pati, Chakrabarty, Agrawal (2011). "Purification of mixed states with closed timelike curve is not possible".
 ^ Rodrigo, Enrico (2010). The Physics of Stargates. Eridanus Press. p. 281.
 ^ Polchinski, Joseph (1991). "Weinberg’s Nonlinear quantum Mechanics and the EinsteinPodolskyRosen Paradox".
References
 DeBenedictis, Andrew and Das, A. (2001). "On a General Class of Wormhole Geometries".
 Dzhunushaliev, Vladimir (2002). "Strings in the Einstein's paradigm of matter".
 Einstein, Albert and Rosen, Nathan (1935). "The Particle Problem in the General Theory of Relativity".
 Fuller, Robert W. and Wheeler, John A. (1962). "Causality and MultiplyConnected SpaceTime".
 Garattini, Remo (2004). "How Spacetime Foam modifies the brick wall".
 GonzálezDíaz, Pedro F. (1998). "Quantum time machine".
 GonzálezDíaz, Pedro F. (1996). "Ringholes and closed timelike curves".
 Khatsymosky, Vladimir M. (1997). "Towards possibility of selfmaintained vacuum traversable wormhole".
 Krasnikov, Serguei (2006). "Counter example to a quantum inequality".
 Krasnikov, Serguei (2003). "The quantum inequalities do not forbid spacetime shortcuts".
 Li, LiXin (2001). "Two Open Universes Connected by a Wormhole: Exact Solutions".
 Morris, Michael S., Thorne, Kip S., and Yurtsever, Ulvi (1988). "Wormholes, Time Machines, and the Weak Energy Condition".
 Morris, Michael S. and Thorne, Kip S. (1988). "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity".
 Nandi, Kamal K. and Zhang, YuanZhong (2006). "A Quantum Constraint for the Physical Viability of Classical Traversable Lorentzian Wormholes".
 Ori, Amos (2005). "A new timemachine model with compact vacuum core".
 Roman, Thomas, A. (2004). "Some Thoughts on Energy Conditions and Wormholes". arXiv:grqc/0409090 [grqc].
 Teo, Edward (1998). "Rotating traversable wormholes".
 Visser, Matt (2002). "The quantum physics of chronology protection by Matt Visser". arXiv:grqc/0204022 [grqc]. An excellent and more concise review.
 Visser, Matt (1989). "Traversable wormholes: Some simple examples".
External links
 What exactly is a 'wormhole'? answered by Richard F. Holman, William A. Hiscock and Matt Visser.
 Why wormholes? by Matt Visser.
 Wormholes in General Relativity by Soshichi Uchii.
 White holes and Wormholes provides a very good description of Schwarzschild wormholes with graphics and animations, by Andrew J. S. Hamilton.
 Questions and Answers about Wormholes a comprehensive wormhole FAQ by Enrico Rodrigo.
 Large Hadron Collider – Theory on how the collider could create a small wormhole, possibly allowing time travel into the past.
 animation that simulates traversing a wormhole
 renderings and animations of a MorrisThorne wormhole
 N.A.S.A's current theory on wormhole creation


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