Lorentzian wormholes: From Einstein to Hawking. Matt Visser (Washington U., St. Louis). ISBN: , Lorentzian wormholes: from Einstein to Hawking. Front Cover. Matt Visser. AIP Press, American Institute of Physics, – Science – pages. @ 23 January ; LATEX-ed Keywords: traversable wormholes, Lorentzian wormholes. Physical Review D
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A wormhole or Einstein—Rosen bridge is a speculative structure linking disparate points in spacetimeand is based on a special solution of the Einstein field equations solved using a Jacobian matrix and determinant. A wormhole can be visualized as a tunnel with two ends, each at separate points in spacetime i. More precisely it is a transcendental bijection of the spacetime continuum, an asymptotic projection of the Calabi—Yau manifold manifesting itself in Anti-de Sitter space.
Wormholes are consistent with the general theory of relativitybut whether wormholes actually exist remains to be seen. A wormhole could connect extremely long distances such as a billion light years or more, short distances such as a few metersdifferent universesor different points in time.
For a simplified notion of a wormhole, space can be visualized as a two-dimensional 2D surface. In this case, a wormhole would appear as a hole in that surface, lead into a 3D tube the inside surface of a cylinderthen 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 planethe entry and exit points could be visualized as spheres in 3D space. 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 continuumand the two points represent a distance to be traveled, however theoretically a wormhole could connect these two points by folding that plane so the points are touching. In this way it would be much easier to traverse the distance since the two points are now touching.
InHermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy;   however, he did not use the term “wormhole” he spoke of “one-dimensional tubes” instead. American theoretical physicist John Archibald Wheeler inspired by Weyl’s work  coined the term “wormhole” in a paper co-authored by Charles Misner: This analysis forces one to consider situations Wormholes have been defined both geometrically and topologically.
Formalizing this idea leads to definitions such as the following, taken from Matt Visser ‘s Lorentzian Wormholes Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces. The equations of the theory of general relativity have valid solutions that contain wormholes. The first type of wormhole solution discovered was the Schwarzschild wormhole which would be present in the Schwarzschild metric describing an eternal black holebut 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 wormholeswould only be possible if exotic matter with negative energy density could be used to stabilize them. Schwarzschild wormholes, also known as Einstein—Rosen bridges  named after Albert Einstein and Nathan Rosen are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equationsand that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation.
Here, “maximally extended” refers to the idea that the spacetime should not have any “edges”: In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon.
And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different “universes”, with the second universe wormhoels 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 womrholes 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 oneand likewise particles from the interior white hole region can escape into either universe.
All four regions can be seen in a spacetime diagram that uses Kruskal—Szekeres coordinates. In this spacetime, it is possible to come up with coordinate systems such that if a hypersurface of constant time a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a ‘space-like surface’ is picked and an “embedding diagram” drawn depicting the curvature of space at that time, the embedding diagram will look like lorentzia 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 lorentzjan the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe.
The Einstein—Rosen bridge was discovered by Ludwig Flamm in lorentzia,  a few months after Schwarzschild published his solution, and was rediscovered by Albert Einstein and wormhols colleague Nathan Rosen, who published their result in Fuller published a paper  showing that this type of wormhole is lorentzzian 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. This theory extends general relativity by removing a constraint of the lorentaian of the affine connection and regarding its antisymmetric part, the torsion tensor loeentzian, as a dynamical variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum spin of matter.
The minimal coupling between torsion and Dirac spinors generates a repulsive spin—spin interaction that is significant in fermionic matter at extremely high densities.
Such an interaction prevents the formation of a gravitational singularity. Other non-traversable wormholes include Lorentzian wormholes first proposed by John Archibald Wheeler inwormholes creating a spacetime foam in a general relativistic spacetime manifold depicted by a Lorentzian manifold and Euclidean wormholes named after Euclidean manifolda structure of Riemannian manifold. This Casimir wormhkles shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary matter vacuum energyand it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at a given point.
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 visserr traversable wormholes in general relativity was first demonstrated in a paper by Homer Ellis  and independently in a paper by K. The drainhole is a solution manifold of Einstein’s field equations for a vacuum space-time, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity negative instead of positive.
Ellis specifically rejected referring to the scalar field as ‘exotic’ because of the antiorthodox coupling, finding arguments for doing so unpersuasive.
Matt Visser: Book
The solution depends on two parameters: What is left is the Ellis wormholea nongravitating, purely geometric, traversable wormhole. Kip Thorne and his graduate student Mike Morrisunaware of the papers by Ellis and Bronnikov, manufactured, and in published, a duplicate of the Ellis wormhole for use as a tool for teaching general relativity. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matterwas from to referred to in the literature as a Morris—Thorne wormhole.
Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a paper by Matt Visserin 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.
Wormholes connect two points in spacetime, which means that they would in principle allow travel in timeas well as in space. InMorris, Thorne and Yurtsever worked out how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths. Until this time it could not have been noticed or have been used.
To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck is of finite size, we would not expect caustics to vjsser, at least within the vicinity of the neck. According to the optical Raychaudhuri’s theoremthis 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 lorenntzian of space with zero curvature,  but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime. In some hypotheses where general relativity is modifiedit is possible to wormholws a wormhole that does not collapse without having to resort to exotic matter. The impossibility of faster-than-light relative speed only applies locally.
Wormholes might allow effective superluminal faster-than-light travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal slower-than-light speeds are used.
If two points are connected by a wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took lorentziah path through the space outside the wormhole. However, a light beam traveling through the same wormhole would of course beat the traveler. If traversable dormholes exist, they could allow time travel.
Lorentzian wormholes: from Einstein to Hawking – Matt Visser – Google Books
One end of the wormhole is accelerated to some significant fraction of the speed of light, perhaps with some advanced propulsion systemand 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 of 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 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.
One significant limitation of such a time machine is that lorentzixn is only possible to go as far lorentzjan in time as the initial creation of the machine; : 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 conditionssuch 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 woormholes null energy condition, : InMatt 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,  or otherwise prevent information from passing through the wormhole.
However, in a 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.
A possible resolution to the paradoxes resulting from wormhole-enabled time travel rests on the many-worlds interpretation of quantum mechanics. In 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.
A particle returning from the future does not return lorentxian 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.
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  named after Hugh Everett in Steven Weinberg ‘s formulation of nonlinear quantum mechanics.
The possibility of communication between parallel universes has been dubbed interuniversal travel. Theories of wormhole metrics describe viser spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a traversable wormhole metric is the following: One type of non-traversable wormhole metric is the Schwarzschild solution see the first diagram:. The original Einstein—Rosen bridge was described in lorentziian article published in July We call such a connection between the two sheets a “bridge”.
For the combined field, gravity and electricity, Einstein and Rosen derived the following Schwarzschild static spherically symmetric solution.
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. From Wikipedia, the free encyclopedia.
For other uses, see Wormhole disambiguation. Principle of relativity Theory of relativity Frame of reference Inertial frame of reference Rest frame Center-of-momentum frame Equivalence principle Mass—energy equivalence Special relativity Doubly special relativity de Sitter invariant special relativity World line Riemannian geometry.
Varna, Bulgaria, JuneSpringer,p. This tertiary source reuses information from other sources but does not name them. The Birth of Wormholes”. Retrieved 20 February An alternative to cosmic inflation”. A follow-up paper to Misner and Wheeler December Time Travel wormhlles Warp Drives. University of Chicago Press. Black Holes and Wprmholes Warps.