Wormhole

From Wikipedia, the free encyclopedia

A wormhole is a concept that represents a solution of the Einstein field equations: a non-trivial resolution of the Ehrenfest paradox structure linking separate points in spacetime. A wormhole can be visualized as a tunnel with two ends, each at separate points in spacetime (i.e., different locations and/or different points of time), or by a transcendental bijection of the spacetime continuum. More precisely, it is an asymptotic projection of the Calabi–Yau manifold manifesting itself in Anti-de Sitter space.

Wormholes are consistent with the general theory of relativity, but 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 meters, different universes, or different points in time.^[1]

Visualization

Wormhole visualized

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 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, the 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 continuum, and 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.

Terminology

In 1928, Hermann Weyl proposed a wormhole theory of matter in connection with mass analysis of electromagnetic field energy;^[2][3] however, he did not use the term "wormhole" (he spoke of "one-dimensional tubes" instead).^[4]

American theoretical physicist John Archibald Wheeler (inspired by Weyl's work)^[4] coined the term "wormhole" in a 1957 paper co-authored by Charles Misner:^[5]

This analysis forces one to consider situations... where there is a net flux of lines of force, through what topologists would call "a handle" of the multiply-connected space, and what physicists might perhaps be excused for more vividly terming a "wormhole".

— Charles Misner and John Wheeler in Annals of Physics

Modern definitions

Wormholes have been defined both geometrically and topologically.^{[further explanation needed]} From a topological point of view, an intra-universe wormhole (a wormhole between two points in the same universe) is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes (1996).^{[6][page needed]}

If a Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S², and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intrauniverse wormhole.

Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces. For example, in Enrico Rodrigo's The Physics of Stargates, a wormhole is defined informally as:

a region of spacetime containing a "world tube" (the time evolution of a closed surface) that cannot be continuously deformed (shrunk) to a world line (the time evolution of a point).

Development

"Embedding diagram" of a Schwarzschild wormhole

Schwarzschild wormholes

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,^[7] 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.

An artist's impression of a wormhole from an observer's perspective, crossing the event horizon of a Schwarzschild wormhole that bridges two different universes. The observer originates from the right, and another universe becomes visible in the center of the wormhole's shadow once the horizon is crossed, the observer seeing light that has fallen into the black hole interior region from the other universe; however, this other universe is unreachable in the case of a Schwarzschild wormhole, as the bridge always collapses before the observer has time to cross it, and everything that has fallen through the event horizon of either universe is inevitably crushed in the singularity.

Schwarzschild wormholes, also known as Einstein–Rosen bridges^[7] (named after Albert Einstein and Nathan Rosen),^[8] are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of a free-falling particle (following a geodesic in the spacetime).

In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. 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 that uses Kruskal–Szekeres coordinates.

In this spacetime, it is possible to come up with coordinate systems such that if a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') is picked and an "embedding diagram" drawn depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". 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.^[9]

The Einstein–Rosen bridge was discovered by Ludwig Flamm in 1916,^[10] 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.^[8][11] However, in 1962, John Archibald Wheeler and Robert W. Fuller published a paper^[12] showing that this type of wormhole is unstable if it connects two parts of the same universe, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to the other exterior region.

According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan–Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a 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.^{[clarification needed]} Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.^[13]

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).

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,^[14] and Euclidean wormholes (named after Euclidean manifold, a structure of Riemannian manifold).^[15]

Traversable wormholes

This 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 at a given point.^[16] Many physicists, such as Stephen Hawking,^[17] Kip Thorne,^[18] and others,^[19][20][21] therefore argue that such effects might make it possible to stabilize a traversable wormhole.^[22][23] 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,^{[24]:494–496[25]} and stable versions of such wormholes have been suggested as dark matter candidates.^[26][27] It has also been proposed that, if a tiny wormhole held open by a negative mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.^[28]

Image of a simulated traversable wormhole that connects the square in front of the physical institutes of University of Tübingen with the sand dunes near Boulogne sur Mer in the north of France. The image is calculated with 4D raytracing in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.^[29]

Lorentzian traversable wormholes would allow travel in both directions from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another.
The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis^[30] and independently in a 1973 paper by K. A. Bronnikov.^[31] Ellis thoroughly analyzed the topology and the geodesics of the Ellis drainhole, showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole is a solution manifold of Einstein's field equations for a vacuum 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.) The solution depends on two parameters: ${\displaystyle m}$, which fixes the strength of its gravitational field, and ${\displaystyle n}$, which determines the curvature of its spatial cross sections. When ${\displaystyle m}$ 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, unaware of the 1973 papers by Ellis and Bronnikov, manufactured, and in 1988 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 matter, was from 1988 to 2015 exclusively 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 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.^[32] A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,^[28] 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 by accelerating one of its two mouths.^[18] 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'. On the other hand, until this time it could not have been noticed or have been used.^[24]:504

Raychaudhuri's theorem and exotic matter

To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck is of finite size, we would not expect caustics to develop, at least within the vicinity of the neck. According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition. Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature,^[33] but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.^[34] Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,^[35] violations have nevertheless been found,^[36] so it remains an open possibility that quantum effects might be used to support a wormhole.

Modified general relativity

In some theories where general relativity is modified, it is possible to have a wormhole that does not collapse without having to resort to exotic matter. For example, this is possible with R^2 gravity, a form of f(R) gravity.^[37]

Faster-than-light travel

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

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 a path through the space outside the wormhole. However, a light beam traveling through the wormhole would of course beat the traveler.

Time travel

If traversable wormholes exist, they could allow time travel.^[18] A proposed time-travel machine using a transversable wormhole would hypothetically work in the following way: One end of the wormhole is accelerated to some significant fraction of the speed of light, perhaps with some advanced propulsion system, and then brought back to the point of origin. Alternatively, another way is to take one entrance of the wormhole and move it to within the gravitational field of an object that has higher gravity than the other entrance, and then return it to a position near the other entrance. For both 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.^[24]: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;^[24]: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.^[38][39]
According to current theories on the nature of wormholes, construction of a traversable wormhole would require the existence of a substance with negative energy, often referred to as "exotic matter". More technically, the wormhole spacetime requires a distribution of energy that violates various energy conditions, such as the null energy condition along with the weak, strong, and dominant energy conditions. However, it is known that quantum effects can lead to small measurable violations of the null energy condition,^[6]:101 and many physicists believe that the required negative energy may actually be possible due to the Casimir effect in quantum physics.^[40] 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.^[41]

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,^[42] or otherwise prevent information from passing through the wormhole.^[43] Because of this, the two mouths could not be brought close enough for 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.^[44]

Interuniversal travel

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.^[45] 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.^[46][47] 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.^[48]

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^[49] (named after Hugh Everett) in Steven Weinberg's formulation of nonlinear quantum mechanics.^[50]

The possibility of communication between parallel universes has been dubbed interuniversal travel.^[51]

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:^[52]

${\displaystyle ds^{2}=-c^{2}dt^{2}+dl^{2}+(k^{2}+l^{2})(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}),}$

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

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

${\displaystyle ds^{2}=-c^{2}\left(1-{\frac {2GM}{rc^{2}}}\right)dt^{2}+{\frac {dr^{2}}{1-{\frac {2GM}{rc^{2}}}}}+r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}).}$

The original Einstein–Rosen bridge was described in an article published in July 1935.^[53][54]

For the Schwarzschild spherically symmetric static solution

${\displaystyle ds^{2}=-{\frac {1}{1-{\frac {2m}{r}}}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+(1-{\frac {2m}{r}})dt^{2}}$

(ds = proper time, c = 1)

If one replaces r with u according to ${\displaystyle u^{2}=r-2m}$

${\displaystyle ds^{2}=-4(u^{2}+2m)du^{2}-(u^{2}+2m)^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+{\frac {u^{2}}{u^{2}+2m}}dt^{2}}$

The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to u > 0 and u < 0, which are joined by a hyperplane r = 2m or u = 0 in which g vanishes. We call such a connection between the two sheets a "bridge".

— A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"

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

${\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}={\frac {\epsilon }{4}},}$

${\displaystyle ds^{2}=-{\frac {1}{(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})dt^{2}}$

(${\displaystyle \epsilon }$ = electrical charge)

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

${\displaystyle \phi _{\mu \nu }=\phi _{\mu ,\nu }-\phi _{\nu ,\mu }}$

${\displaystyle g^{2}\phi _{\mu \nu ;\sigma }g^{\nu \sigma }=0}$

${\displaystyle g^{2}(R_{ik}+\phi _{i\alpha }\phi _{k}^{\alpha }-{\frac {1}{4}}g_{ik}\phi _{\alpha \beta }\phi ^{ab})=0}$

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

${\displaystyle u^{2}=r^{2}-{\frac {\epsilon ^{2}}{2}}}$

and with m = 0 one obtains^[55][56]

${\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}=\epsilon /(u^{2}+{\frac {\epsilon ^{2}}{2}})^{\frac {1}{2}}}$

${\displaystyle ds^{2}=-du^{2}-(u^{2}+{\frac {\epsilon ^{2}}{2}})(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+({\frac {2u^{2}}{2u^{2}+\epsilon ^{2}}})dt^{2}}$

The solution is free from singularities for all finite points in the space of the two sheets

— A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity"

In fiction

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

New Stable, Artificial Photosynthesis Produces Clean Hydrogen Fuel

May 14, 2018

By Dan Newman  University of Michigan
Original link:  http://www.newscats.org/?p=14959

A new, stable artificial photosynthesis device doubles the efficiency of harnessing sunlight to break apart both fresh and salt water, generating hydrogen that can then be used in fuel cells. The device could also be reconfigured to turn carbon dioxide back into fuel.

Hydrogen is the cleanest-burning fuel, with water as its only emission. But hydrogen production is not always environmentally friendly. Conventional methods require natural gas or electrical power. The method advanced by the new device, called direct solar water splitting, only uses water and light from the sun.

“If we can directly store solar energy as a chemical fuel, like what nature does with photosynthesis, we could solve a fundamental challenge of renewable energy,” said Zetian Mi, a professor of electrical and computer engineering at the University of Michigan who led the research while at McGill University in Montreal.

Faqrul Alam Chowdhury, a doctoral student in electrical and computer engineering at McGill, said the problem with solar cells is that they cannot store electricity without batteries, which have a high overall cost and limited life.

The colorized electron microscope image shows the gallium nitride towers of the artificial photosynthesis device at 25k magnification. These nanostructures rip water molecules apart into hydrogen and oxygen to produce clean hydrogen fuel.

Image credit: Faqrul A. Chowdhury, McGill University

The device is made from the same widely used materials as solar cells and other electronics, including silicon and gallium nitride (often found in LEDs). With an industry-ready design that operates with just sunlight and seawater, the device paves the way for large-scale production of clean hydrogen fuel.Previous direct solar water splitters have achieved a little more than 1 percent stable solar-to-hydrogen efficiency in fresh or saltwater. Other approaches suffer from the use of costly, inefficient or unstable materials, such as titanium dioxide, that also might involve adding highly acidic solutions to reach higher efficiencies.Mi and his team, however, achieved more than 3 percent solar-to-hydrogen efficiency. To reach this stable efficiency, the team built a nano-sized cityscape of gallium nitride towers that generated an electric field. The gallium nitride turns light, or photons, into mobile electrons and positively charged vacancies called holes. These free charges split water molecules into hydrogen and oxygen.

The colorized electron microscope image shows the gallium nitride towers of the artificial photosynthesis device at 52.5k magnification. These nanostructures rip water molecules apart into hydrogen and oxygen to produce clean hydrogen fuel.

Image credit: Faqrul A. Chowdhury, McGill University“

When this specially engineered wafer is hit by photons, the electric field helps separate photogenerated electrons and holes to drive the production of hydrogen and oxygen molecules efficiently,” Chowdhury said.At present, the silicon backing of the chip does not contribute to its function, but it could be doing more. The next step may be to use the silicon to help capture light and funnel charge carriers to the gallium nitride towers.

“Although the 3 percent efficiency might seem low, when put in the context of the 40 years of research on this process, it’s actually a big breakthrough,” Mi said. “Natural photosynthesis, depending how you calculate it, has an efficiency of about 0.6 percent.”

He adds that 5 percent efficiency is the threshold for commercialization, but his team is aiming for 20 or 30 percent efficiency.

Mi conducts similar research to strip carbon dioxide of its oxygen to turn the resulting carbon into hydrocarbons, such as methanol and syngas. This research path could potentially remove carbon dioxide from the atmosphere, like plants do.

“That’s the truly exciting part,” Mi said.

The device is documented in the study, “A photochemical diode artificial photosynthesis system for unassisted high efficiency overall pure water splitting,” published in Nature Communications. Along with Mi and Chowdhury, co-authors include Michel Trudeau of the Center of Excellence in Transportation Electrification and Energy Storage, Hydro-Québec, and Hong Guo of McGill University.

The work was supported by the Fuel Cell Technologies Office of the U.S. Department of Energy and Emissions Reduction Alberta.

Contacts and sources:
Dan Newman

Gravitational singularity

From Wikipedia, the free encyclopedia

 

Animated simulation of gravitational lensing caused by a Schwarzschild black hole passing in a line-of-sight planar to a background galaxy. Around and at the time of exact alignment (syzygy) extreme lensing of the light is observed.

A gravitational singularity or spacetime singularity is a location in spacetime where the gravitational field of a celestial body becomes infinite in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite within the singularity, the laws of normal spacetime cannot exist.^[1][2]

Gravitational singularities are mainly considered within general relativity, where density apparently becomes infinite at the centre of a black hole, and within astrophysics and cosmology as the earliest state of the universe during the Big Bang. Physicists are undecided whether the prediction of singularities means that they actually exist (or existed at the start of the Big Bang), or that current knowledge is insufficient to describe what happens at such extreme densities.

General relativity predicts that any object collapsing beyond a certain point (for stars this is the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon) would be formed.^[3] The Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner.^[4] The termination of such a geodesic is considered to be the singularity.

The initial state of the universe, at the beginning of the Big Bang, is also predicted by modern theories to have been a singularity.^[5] In this case the universe did not collapse into a black hole, because currently-known calculations and density limits for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not necessarily apply in the same way to rapidly expanding space such as the Big Bang. Neither general relativity nor quantum mechanics can currently describe the earliest moments of the Big Bang,^[6] but in general, quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths.^[7]

Interpretation

Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the Ultraviolet Catastrophe, re-normalization, and instability of a hydrogen atom predicted by the Larmor formula.

Some theories, such as the theory of loop quantum gravity suggest that singularities may not exist.^[8] This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. Further in a dynamic Newtonian adaptation of the equations of gravity (DNAg) the singularity is entirely avoided.^[9] The idea can be stated in the form that due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.

Types

There are different types of singularities, each with different physical features which have characteristics relevant to the theories in which they originally emerged from, such as the different shape of the singularities, conical and curved. They have also been hypothesized to occur without  Event Horizons, structures which delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked.

Conical

A conical singularity occurs when there is a point where the limit of every diffeomorphism invariant quantity is finite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere if a suitable coordinate system is used.

An example of such a conical singularity is a cosmic string and a Schwarzschild black hole.^[10]

Curvature

A simple illustration of a non-spinning black hole and its singularity

Solutions to the equations of general relativity or another theory of gravity (such as supergravity) often result in encountering points where the metric blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates.

An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the Kretschmann scalar, being the square of the Riemann tensor i.e. ${\displaystyle R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }}$, which is diffeomorphism invariant, is infinite.

While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a Kerr black hole, the singularity occurs on a ring (a circular line), known as a "ring singularity". Such a singularity may also theoretically become a wormhole.^[11]

More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the Big Bang cosmological model of the universe contains a causal singularity at the start of time (t=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.

Naked singularity

Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the naked singularity look like a Black Hole.^[12][13][14]

Disappearing event horizons exist in the Kerr metric, which is a spinning black hole in a vacuum. Specifically, if the angular momentum is high enough, the event horizons could disappear. Transforming the Kerr metric to Boyer–Lindquist coordinates, it can be shown ^[15] that the  coordinate (which is not the radius) of the event horizon is, ${\displaystyle r_{\pm }=\mu \pm (\mu ^{2}-a^{2})^{1/2}}$, where ${\displaystyle \mu =GM/c^{2}}$, and ${\displaystyle a=J/Mc}$. In this case, "event horizons disappear" means when the solutions are complex for ${\displaystyle r_{\pm }}$, or ${\displaystyle \mu ^{2}$.

Disappearing event horizons can also be seen with the Reissner–Nordström geometry of a charged black hole. In this metric, it can be shown^[16] that the singularities occur at ${\displaystyle r_{\pm }=\mu \pm (\mu ^{2}-q^{2})^{1/2}}$, where ${\displaystyle \mu =GM/c^{2}}$, and ${\displaystyle q^{2}=GQ^{2}/(4\pi \epsilon _{0}c^{4})}$. Of the three possible cases for the relative values of ${\displaystyle \mu }$ and ${\displaystyle q}$, the case where ${\displaystyle \mu ^{2}$ causes both ${\displaystyle r_{\pm }}$ to be complex. This means the metric is regular for all positive values of ${\displaystyle r}$, or in other words, the singularity has no event horizon.

Entropy

Before Stephen Hawking came up with the concept of Hawking radiation, the question of black holes having entropy was avoided. However, this concept demonstrates that black holes can radiate energy, which conserves entropy and solves the incompatibility problems with the second law of thermodynamics. Entropy, however, implies heat and therefore temperature. The loss of energy also suggests that black holes do not last forever, but rather "evaporate" slowly. Small black holes tend to be hotter whereas larger ones tend to be colder. All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, so they are all gaining energy. They will not begin to lose energy until a cosmological redshift of more than one million is reached, rather than the thousand or so since the background radiation formed.

Tachyon

From Wikipedia, the free encyclopedia

Because a tachyon would always move faster than light, it would not be possible to see it approaching. After a tachyon has passed nearby, an observer would be able to see two images of it, appearing and departing in opposite directions. The black line is the shock wave of Cherenkov radiation, shown only in one moment of time. This double image effect is most prominent for an observer located directly in the path of a superluminal object (in this example a sphere, shown in grey). The right hand bluish shape is the image formed by the blue-doppler shifted light arriving at the observer—who is located at the apex of the black Cherenkov lines—from the sphere as it approaches. The left-hand reddish image is formed from red-shifted light that leaves the sphere after it passes the observer. Because the object arrives before the light, the observer sees nothing until the sphere starts to pass the observer, after which the image-as-seen-by-the-observer splits into two—one of the arriving sphere (to the right) and one of the departing sphere (to the left).

A tachyon /ˈtæki.ɒn/ or tachyonic particle is a hypothetical particle that always moves faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics.^[1][2] If such particles did exist, they could be used to build a tachyonic antitelephone and send signals faster than light, which (according to special relativity) would lead to violations of causality.^[2]

The possibility of particles moving faster than light was first proposed by O. M. P. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan in 1962, although the term they used for it was "meta-particle".^[3] In the 1967 paper that coined the term,^[4] Gerald Feinberg proposed that tachyonic particles could be quanta of a quantum field with imaginary mass. However, it was soon realized that excitations of such imaginary mass fields do not in fact propagate faster than light,^[5] and instead represent an instability known as tachyon condensation.^[1] Nevertheless, in modern physics the term "tachyon" often^[1][6] refers to imaginary mass fields rather than to faster-than-light particles. Such fields have come to play a significant role in modern physics.

The term comes from the Greek: ταχύ, tachy, meaning "rapid". The complementary particle types are called luxons (which always move at the speed of light) and bradyons (which always move slower than light); both of these particle types are known to exist.

Despite theoretical arguments against the existence of faster-than-light particles, experiments have been conducted to search for them, but no compelling evidence of their existence has been found.

Tachyons in relativity theory

In special relativity, a faster-than-light particle would have space-like four-momentum,^[4] in contrast to ordinary particles that have time-like four-momentum. Although in some theories the mass of tachyons is regarded as imaginary, in some modern formulations the mass is considered real,^[7][8][9] the formulas for the momentum and energy being redefined to this end. Moreover, since tachyons are constrained to the spacelike portion of the energy–momentum graph, it could not slow down to subluminal speeds.^[4]

Mass

In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation:

${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}\;}$

(where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle:

${\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}$

This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When v is larger than c, the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real,^{[dubious – discuss]} the numerator must also be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.

In some modern formulations of the theory, the mass of tachyons is regarded as real.^[7][8][9]

Speed

One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases. In particular, ${\displaystyle E}$ approaches zero when ${\displaystyle v}$ approaches infinity. (For ordinary bradyonic matter, E increases with increasing speed, becoming arbitrarily large as v approaches c, the speed of light). Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below.

As noted by Einstein, Tolman, and others, special relativity implies that faster-than-light particles, if they existed, could be used to communicate backwards in time.^[10]

Neutrinos

In 1985, Chodos proposed that neutrinos can have a tachyonic nature.^[11] The possibility of standard model particles moving at superluminal speeds can be modeled using Lorentz invariance violating terms, for example in the Standard-Model Extension.^[12][13][14] In this framework, neutrinos experience Lorentz-violating oscillations and can travel faster than light at high energies. This proposal was strongly criticized.^[15]

Cherenkov radiation

A tachyon with an electric charge would lose energy as Cherenkov radiation^[16]—just as ordinary charged particles do when they exceed the local speed of light in a medium (other than a hard vacuum). A charged tachyon traveling in a vacuum, therefore, undergoes a constant proper time acceleration and, by necessity, its worldline forms a hyperbola in space-time. However reducing a tachyon's energy increases its speed, so that the single hyperbola formed is of two oppositely charged tachyons with opposite momenta (same magnitude, opposite sign) which annihilate each other when they simultaneously reach infinite speed at the same place in space. (At infinite speed, the two tachyons have no energy each and finite momentum of opposite direction, so no conservation laws are violated in their mutual annihilation. The time of annihilation is frame dependent.)

Even an electrically neutral tachyon would be expected to lose energy via gravitational Cherenkov radiation, because it has a gravitational mass, and therefore increase in speed as it travels, as described above. If the tachyon interacts with any other particles, it can also radiate Cherenkov energy into those particles. Neutrinos interact with the other particles of the Standard Model, and Andrew Cohen and Sheldon Glashow recently used this to argue that the faster-than-light neutrino anomaly cannot be explained by making neutrinos propagate faster than light, and must instead be due to an error in the experiment.^[17]

Causality

Causality is a fundamental principle of physics. If tachyons can transmit information faster than light, then according to relativity they violate causality, leading to logical paradoxes of the "kill your own grandfather" type. This is often illustrated with thought experiments such as the "tachyon telephone paradox"^[10] or "logically pernicious self-inhibitor."^[18]

The problem can be understood in terms of the relativity of simultaneity in special relativity, which says that different inertial reference frames will disagree on whether two events at different locations happened "at the same time" or not, and they can also disagree on the order of the two events (technically, these disagreements occur when the spacetime interval between the events is 'space-like', meaning that neither event lies in the future light cone of the other).^[19]

If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event.^[19] However, in the case of a hypothetical signal moving faster than light, there would always be some frames in which the signal was received before it was sent so that the signal could be said to have moved backward in time. Because one of the two fundamental postulates of special relativity says that the laws of physics should work the same way in every inertial frame, if it is possible for signals to move backward in time in any one frame, it must be possible in all frames. This means that if observer A sends a signal to observer B which moves faster than light in A's frame but backwards in time in B's frame, and then B sends a reply which moves faster than light in B's frame but backwards in time in A's frame, it could work out that A receives the reply before sending the original signal, challenging causality in every frame and opening the door to severe logical paradoxes.^[20] Mathematical details can be found in the tachyonic antitelephone article, and an illustration of such a scenario using spacetime diagrams can be found in Baker, R. (2003)^[21]

Reinterpretation principle

The reinterpretation principle^[4][3][20] asserts that a tachyon sent back in time can always be reinterpreted as a tachyon traveling forward in time, because observers cannot distinguish between the emission and absorption of tachyons. The attempt to detect a tachyon from the future (and violate causality) would actually create the same tachyon and send it forward in time (which is causal).

However, this principle is not widely accepted as resolving the paradoxes.^[10][20][22] Instead, what would be required to avoid paradoxes is that unlike any known particle, tachyons do not interact in any way and can never be detected or observed, because otherwise a tachyon beam could be modulated and used to create an anti-telephone^[10] or a "logically pernicious self-inhibitor".^[18] All forms of energy are believed to interact at least gravitationally, and many authors state that superluminal propagation in Lorentz invariant theories always leads to causal paradoxes.^[23][24]

Fundamental models

In modern physics, all fundamental particles are regarded as excitations of quantum fields. There are several distinct ways in which tachyonic particles could be embedded into a field theory.

Fields with imaginary mass

In the paper that coined the term "tachyon", Gerald Feinberg studied Lorentz invariant quantum fields with imaginary mass.^[4] Because the group velocity for such a field is superluminal, naively it appears that its excitations propagate faster than light. However, it was quickly understood that the superluminal group velocity does not correspond to the speed of propagation of any localized excitation (like a particle). Instead, the negative mass represents an instability to tachyon condensation, and all excitations of the field propagate subluminally and are consistent with causality.^[5] Despite having no faster-than-light propagation, such fields are referred to simply as "tachyons" in many sources.^{[1][6][25][26][27][28]}

Tachyonic fields play an important role in modern physics. Perhaps the most famous is the Higgs boson of the Standard Model of particle physics, which has an imaginary mass in its uncondensed phase. In general, the phenomenon of spontaneous symmetry breaking, which is closely related to tachyon condensation, plays an important role in many aspects of theoretical physics, including the Ginzburg–Landau and BCS theories of superconductivity. Another example of a tachyonic field is the tachyon of bosonic string theory.^[25][27][29]

Tachyons are predicted by bosonic string theory and also the Neveu-Schwarz (NS) and NS-NS sectors, which are respectively the open bosonic sector and closed bosonic sector, of RNS Superstring theory prior to the GSO projection. However such tachyons are not possible due to the Sen conjecture, also known as tachyon condensation. This resulted in the necessity for the GSO projection.

Lorentz-violating theories

In theories that do not respect Lorentz invariance, the speed of light is not (necessarily) a barrier, and particles can travel faster than the speed of light without infinite energy or causal paradoxes.^[23] A class of field theories of that type is the so-called Standard Model extensions. However, the experimental evidence for Lorentz invariance is extremely good, so such theories are very tightly constrained.^[30][31]

Fields with non-canonical kinetic term

By modifying the kinetic energy of the field, it is possible to produce Lorentz invariant field theories with excitations that propagate superluminally.^[5][24] However, such theories, in general, do not have a well-defined Cauchy problem (for reasons related to the issues of causality discussed above), and are probably inconsistent quantum mechanically.

History

The term "tachyon" was coined by Gerald Feinberg in a 1967 paper titled "Possibility of Faster-Than-Light Particles".^[4] He had been inspired by the science-fiction story "Beep" by James Blish.^[32] Feinberg studied the kinematics of such particles according to special relativity. In his paper he also introduced fields with imaginary mass (now also referred to as "tachyons") in an attempt to understand the microphysical origin such particles might have.

The first hypothesis regarding faster-than-light particles is sometimes attributed to German physicist Arnold Sommerfeld in 1904,^[33] and more recent discussions happened in 1962^[3] and 1969.^[34]

In September 2011, it was reported that a tau neutrino had traveled faster than the speed of light in a major release by CERN; however, later updates from CERN on the OPERA project indicate that the faster-than-light readings were resultant from "a faulty element of the experiment's fibre optic timing system".^[35]

In fiction

Tachyons have appeared in many works of fiction. They have been used as a standby mechanism upon which many science fiction authors rely to establish faster-than-light communication, with or without reference to causality issues. The word tachyon has become widely recognized to such an extent that it can impart a science-fictional connotation even if the subject in question has no particular relation to superluminal travel (a form of technobabble, akin to positronic brain).