Faster-than-light

Faster-than-light (also superluminal or FTL) communications and travel refer to the propagation of information or matter faster than the speed of light. Under the special theory of relativity, a slower-than-light particle with nonzero rest mass needs infinite energy to accelerate to the speed of light, although special relativity does not forbid the existence of particles that travel faster than light at all times (see tachyons).

On the other hand, what some physicists refer to as "apparent" or "effective" FTL[1][2][3][4] is the hypothesis that unusually distorted regions of spacetime might permit matter to reach distant locations faster than what it would take light in the normal or undistorted spacetime. However, according to current theories, matter is still required to travel subluminally with respect to the locally distorted spacetime region.

Apparent FTL is not excluded by general relativity. Examples of apparent FTL proposals are the Alcubierre drive and the traversable wormhole, although the physical plausibility of these solutions is uncertain.

Contents

Travel

In the context of this article, FTL is transmitting information or matter faster than c, a constant equal to the speed of light in a vacuum, 299,792,458 meters per second, or about 186,282.4 miles per second. This is not quite the same as traveling faster than light, since:

Neither of these phenomena violates special relativity or creates problems with causality, and thus neither qualifies as FTL as described here.

Possibility

Faster-than-light communication is, by Einstein's theory of relativity, equivalent to time travel. According to Einstein's theory of special relativity, what we measure as the speed of light in a vacuum is actually the fundamental physical constant c. This means that all inertial observers, regardless of their relative velocity, will always measure zero-mass particles such as photons traveling at c in a vacuum. This result means that measurements of time and velocity in different frames are no longer related simply by constant shifts, but are instead related by Poincaré transformations. These transformations have important implications:

Justifications

Despite the established conclusion that relativity precludes FTL travel, some have proposed ways to justify FTL behaviour:

Faster light (Casimir vacuum and quantum tunnelling)

Einstein's equations of special relativity postulate that the speed of light in a vacuum is invariant in inertial frames. That is, it will be the same from any frame of reference moving at a constant speed. The equations do not specify any particular value for the speed of the light, which is an experimentally determined quantity for a fixed unit of length. Since 1983, the unit of length (the meter) has been defined using the speed of light.

The experimental determination has been made in vacuum. However, the vacuum we know is not the only possible vacuum which can exist. The vacuum has energy associated with it, called the vacuum energy. This vacuum energy can perhaps be changed in certain cases.[7] When vacuum energy is lowered, light itself has been predicted to go faster than the standard value 'c'. This is known as the Scharnhorst effect. Such a vacuum can be produced by bringing two perfectly smooth metal plates together at near atomic diameter spacing. It is called a Casimir vacuum. Calculations imply that light will go faster in such a vacuum by a minuscule amount: a photon traveling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 1036.[8] Accordingly there has as yet been no experimental verification of the prediction. A recent analysis[9] argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signalling. However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze. Other authors argue that Scharnhorst's original analysis which seemed to show the possibility of faster-than-c signals involved approximations which may be incorrect, so that it is not clear whether this effect could actually increase signal speed at all.[10]

The physicists Günter Nimtz and Alfons Stahlhofen, of the University of Koblenz, claim to have violated relativity experimentally by transmitting photons faster than the speed of light.[11] They say they have conducted an experiment in which microwave photons—relatively low energy packets of light—travelled "instantaneously" between a pair of prisms that had been moved up to 3 ft (1 m) apart. Their experiment involved an optical phenomenon known as "evanescent modes", and they claim that since evanescent modes have an imaginary wave number, they represent a "mathematical analogy" to quantum tunnelling.[11] Nimtz has also claimed that "evanescent modes are not fully describable by the Maxwell equations and quantum mechanics have to be taken into consideration."[12] Other scientists such as Herbert Winful and Robert Helling have argued that in fact there is nothing quantum-mechanical about Nimtz's experiments, and that the results can be fully predicted by the equations of classical electromagnetism (Maxwell's equations).[13][14]

Nimtz told New Scientist magazine: "For the time being, this is the only violation of special relativity that I know of." However, other physicists say that this phenomenon does not allow information to be transmitted faster than light. Aephraim Steinberg, a quantum optics expert at the University of Toronto, Canada, uses the analogy of a train traveling from Chicago to New York, but dropping off train cars at each station along the way, so that the center of the ever shrinking main train moves forward at each stop; in this way, the speed of the center of the train exceeds the speed of any of the individual cars.[15] Herbert Winful argues that the train analogy is a variant of the "reshaping argument" for superluminal tunneling velocities, but he goes on to say that this argument is not actually supported by experiment or simulations, which actually show that the transmitted pulse has the same length and shape as the incident pulse.[13] Instead, Winful argues that the group delay in tunneling is not actually the transit time for the pulse (whose spatial length must be greater than the barrier length in order for its spectrum to be narrow enough to allow tunneling), but is instead the lifetime of the energy stored in a standing wave which forms inside the barrier. Since the stored energy in the barrier is less than the energy stored in a barrier-free region of the same length due to destructive interference, the group delay for the energy to escape the barrier region is shorter than it would be in free space, which according to Winful is the explanation for apparently superluminal tunneling.[16][17] A number of authors have published papers disputing Nimtz's claim that Einstein causality is violated by his experiments, and there are many other papers in the literature discussing why quantum tunneling is not thought to violate causality.[18]

It was later claimed by the Keller group in Switzerland that particle tunneling does indeed occur in zero real time. Their tests involved tunneling electrons, where the group argued a relativistic prediction for tunneling time should be 500-600 attoseconds (an attosecond is one quintillionth of a second). All that could be measured was 24 attoseconds, which is the limit of the test accuracy.[19] Again, though, other physicists believe that tunneling experiments in which particles appear to spend anomalously short times inside the barrier are in fact fully compatible with relativity, although there is disagreement about whether the explanation involves reshaping of the wave packet or other effects.[16][17][20]

Give up causality

Another approach is to accept special relativity, but to posit that mechanisms allowed by general relativity (e.g., wormholes) will allow traveling between two points without going through the intervening space. While this gets around the infinite acceleration problem, it still would lead to closed timelike curves (i.e., time travel) and causality violations. Causality is not required by special or general relativity, but is nonetheless generally considered a basic property of the universe that cannot be sensibly dispensed with. Because of this, most physicists expect that quantum gravity effects will preclude this option. An alternative is to conjecture that, while time travel is possible, it never leads to paradoxes; this is the Novikov self-consistency principle.

An important point to note is that in general relativity it is possible for objects to be moving apart faster than light because of the expansion of the universe, in some reasonable choice of cosmological coordinates. This is understood to be due to the expansion of the space between the objects, and general relativity still reduces to special relativity in a "local" sense, meaning that two objects passing each other in a small local region of spacetime cannot have a relative velocity greater than c, and will move more slowly than a light beam passing through the region. (See Option F below)

Give up (absolute) relativity

Because of the strong empirical support for special relativity, any modifications to it must necessarily be quite subtle and difficult to measure. The best-known attempt is doubly-special relativity, which posits that the Planck length is also the same in all reference frames, and is associated with the work of Giovanni Amelino-Camelia and João Magueijo. One consequence of this theory is a variable speed of light, where photon speed would vary with energy, and some zero-mass particles might possibly travel faster than c. However, even if this theory is accurate, it is still very unclear whether it would allow information to be communicated, and appears not in any case to allow massive particles to exceed c.

There are speculative theories that claim inertia is produced by the combined mass of the universe (e.g., Mach's principle), which implies that the rest frame of the universe might be preferred by conventional measurements of natural law. If confirmed, this would imply special relativity is an approximation to a more general theory, but since the relevant comparison would (by definition) be outside the observable universe, it is difficult to imagine (much less construct) experiments to test this hypothesis.

Non-physical realms

A very popular option in space opera is to assume the existence of some other realm (typically called hyperspace, subspace, or slipspace) which is accessible from this universe, in which the laws of relativity are usually distorted, bent, or nonexistent, facilitating rapid transport between distant points in this universe, sometimes with acceleration differences—that is, not requiring as much energy or thrust to go faster. To accomplish rapid transport between points in hyperspace/subspace, special relativity is often assumed not to apply in this other realm, or that the speed of light is higher. Another solution is to posit that distant points in the mundane universe correspond to points that are close together in hyperspace.

The possibility of other universes following distinct physics permitting for this type of travel is the crux of serious scientific conjecture along these lines.

Space-time distortion

Although the theory of special relativity forbids objects to have a relative velocity greater than light speed, and general relativity reduces to special relativity in a local sense (in small regions of spacetime where curvature is negligible), general relativity does allow the space between distant objects to expand in such a way that they have a "recession velocity" which exceeds the speed of light, and it is thought that galaxies which are at a distance of more than about 14 billion light years from us today have a recession velocity which is faster than light.[21] Miguel Alcubierre theorized that it would be possible to create an Alcubierre drive, in which a ship would be enclosed in a "warp bubble" where the space at the front of the bubble is rapidly contracting and the space at the back is rapidly expanding, with the result that the bubble can reach a distant destination much faster than a light beam moving outside the bubble, but without objects inside the bubble locally traveling faster than light. However, several objections raised against the Alcubierre drive appear to rule out the possibility of actually using it in any practical fashion. Another possibility predicted by general relativity is the traversable wormhole, which could create a shortcut between arbitrarily distant points in space. As with the Alcubierre drive, travelers moving through the wormhole would not locally move faster than light which travels through the wormhole alongside them, but they would be able to reach their destination (and return to their starting location) faster than light traveling outside the wormhole.

Dr. Gerald Cleaver, associate professor of physics at Baylor University, and Richard Obousy, a Baylor graduate student, theorize that by manipulating the extra spatial dimensions of string theory around a spaceship with an extremely large amount of energy, it would create a "bubble" that could cause the ship to travel faster than the speed of light. To create this bubble, the physicists believe manipulating the 10th spatial dimension would alter the dark energy in three large spatial dimensions: height, width and length. Cleaver said positive dark energy is currently responsible for speeding up the expansion rate of our universe as time moves on.[22]

Heim theory

In 1977, a controversial paper on Heim theory theorized that it may be possible to travel faster than light by using magnetic fields to enter a higher-dimensional space, and the paper received some media attention in January 2006.[23] However, due to the many unproven assumptions in the paper, there have been few serious attempts to conduct further experiments.

Lorentz symmetry violation

The possibility that Lorentz symmetry may be violated has been seriously considered in the last two decades, and can be partially tested by ultra-high energy cosmic-ray experiments.[24] In some models of broken Lorentz symmetry, it is postulated that the symmetry is still built into the most fundamental laws of physics, but that spontaneous symmetry breaking shortly after the Big Bang could have left a "relic field" throughout the universe which causes particles to behave differently depending on their velocity relative to the field;[25] however there are also some models where Lorentz symmetry is broken in a more fundamental way. If Lorentz symmetry can cease to be a fundamental symmetry at Planck scale or at some other fundamental scale, it is conceivable that particles with a critical speed different from the speed of light be the ultimate constituents of matter. The superbradyon hypothesis considers the possibility that conventional particles be similar to phonons in a condensed medium (the physical vacuum of our Universe), with a critical speed much smaller than the actual fundamental critical speed.

In current models of Lorentz symmetry violation, the phenomenological parameters are expected to be energy-dependent. Therefore, as widely recognized,[26][27] existing low-energy bounds cannot be applied to high-energy phenomena. Lorentz symmetry violation is expected to become stronger as one gets closer to the fundamental scale.

Tachyons

Main article: Tachyon

In special relativity, while it is impossible in an inertial frame to accelerate an object to the speed of light, or for a massive object to move at the speed of light, it is not impossible for an object to exist which always moves faster than light. The hypothetical elementary particles that have this property are called tachyons. Their existence has not been proven, but even so, attempts to quantise them show that they may not be used for faster-than-light communication.[28] Physicists sometimes regard the existence of mathematical structures similar to tachyons arising from theoretical models and theories as signs of an inconsistency or that the theory needs further refining.[29]

General relativity

General relativity was developed after special relativity to include concepts like gravity. It maintains the principle that no object can accelerate to the speed of light in the reference frame of any coincident observer. However, it permits distortions in spacetime that allow an object to move faster than light from the point of view of a distant observer. One such distortion is the Alcubierre drive, which can be thought of as producing a ripple in spacetime that carries an object along with it. Another possible system is the wormhole, which connects two distant locations as though by a shortcut. Both distortions would need to create a very strong curvature in a highly localized region of space-time and their gravity fields would be immense. To counteract the unstable nature, and prevent the distortions from collapsing under their own 'weight', one would need to introduce hypothetical exotic matter or negative energy.

General relativity also agrees that any technique for faster-than-light travel could also be used for time travel. This raises problems with causality. Many physicists believe that the above phenomena are in fact impossible, and that future theories of gravity will prohibit them. One theory states that stable wormholes are possible, but that any attempt to use a network of wormholes to violate causality would result in their decay. In string theory Eric Gimon and Petr Hořava have argued[30] that in a supersymmetric five-dimensional Gödel universe quantum corrections to general relativity effectively cut off regions of spacetime with causality-violating closed timelike curves. In particular, in the quantum theory a smeared supertube is present that cuts the spacetime in such a way that, although in the full spacetime a closed timelike curve passed through every point, no complete curves exist on the interior region bounded by the tube.

FTL phenomena

In these examples, certain influences may appear to travel faster than light, but they do not convey energy or information faster than light, so they do not violate special relativity.

Daily motion of the Heavens

For an earthbound observer objects in the sky complete one revolution around the earth in 1 day. Proxima Centauri, which is the nearest star outside the Solar system, is about 4 light years away.[31] On a geostationary view Alpha Centauri has a speed many times greater than "c" as the rim speed of an object moving in a circle is a product of the radius and angular speed.[32] It is also possible on a geostatic view for objects such as comets to vary their speed from subluminal to superluminal and vice versa simply because the distance from the earth varies. Comets may have orbits which take them out to more than 1000 AU.[33] Circumference of a circle radius 1000 AU is greater than one light day. In other words, a comet at such a distance is superluminal in a geostatic frame.

Light spots and shadows

If a laser is swept across a distant object, the spot of light can easily be made to move at a speed greater than c.[34] Similarly, a shadow projected onto a distant object can be made to move faster than c.[35] In neither case does any information travel faster than light.

Closing speeds

An observer may conclude that two objects are moving faster than the speed of light relative to each other, by adding their velocities according to the principle of Galilean relativity.

For example, two fast-moving particles approaching each other from opposite sides of a particle accelerator will appear to be moving at slightly less than twice the speed of light, relative to each other, from the point of view of an observer standing at rest relative to the accelerator. This correctly reflects the rate at which the distance between the two particles is decreasing, from the observer's point of view and is called the closing speed. However, it is not the same as the velocity of one of the particles as would be measured by a hypothetical fast-moving observer traveling alongside the other particle. To obtain this, the calculation must be done according to the principle of special relativity. If the two particles are moving at velocities v and −v, or expressed in units of c, β and −β, where

\beta \equiv v/c \,\!

then this relative velocity (again in units of the speed of light c) is

\beta_{rel} = { \beta - (-\beta) \over 1 + \beta ^2 } = { 2\beta \over 1 + \beta^2 },

which will always turn out to be less than the speed of light, regardless of the velocities of the two particles.

Proper speeds

If a spaceship travels to a planet one light year (as measured in the Earth's rest frame) away from Earth at high speed, the time taken to reach that planet could be less than one year as measured by the traveller's clock (although it will always be more than one year as measured by a clock on Earth). The value obtained by dividing the distance travelled, as determined in the Earth's frame, by the time taken, measured by the traveller's clock, is known as a proper speed or a proper velocity. There is no limit on the value of a proper speed as a proper speed does not represent a speed measured in a single inertial frame. A light signal that left the Earth at the same time as the traveller would always get to the destination before the traveller.

Phase velocities above c

The phase velocity of an electromagnetic wave, when traveling through a medium, can routinely exceed c, the vacuum velocity of light. For example, this occurs in most glasses at X-ray frequencies.[36] However, the phase velocity of a wave corresponds to the propagation speed of a theoretical single-frequency (purely monochromatic) component of the wave at that frequency. Such a wave component must be infinite in extent and of constant amplitude (otherwise it is not truly monochromatic), and so cannot convey any information.[37] Thus a phase velocity above c does not imply the propagation of signals with a velocity above c.

Group velocities above c

The group velocity of a wave (e.g. a light beam) may also exceed c in some circumstances. In such cases, which typically at the same time involve rapid attenuation of the intensity, the maximum of the envelope of a pulse may travel with a velocity above c. However, even this situation does not imply the propagation of signals with a velocity above c[38], even though one may be tempted to associate pulse maxima with signals. The latter association has been shown to be misleading, basically because the information on the arrival of a pulse can be obtained before the pulse maximum arrives. For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind, the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than c without this effect.

Universal expansion

The expansion of the universe causes distant galaxies to recede from us faster than the speed of light, if comoving distance and cosmological time are used to calculate the speeds of these galaxies. However, in general relativity, velocity is a local notion, so velocity calculated using comoving coordinates does not have any simple relation to velocity calculated locally.[39] Rules that apply to relative velocities in special relativity, such as the rule that relative velocities cannot increase past the speed of light, do not apply to relative velocities in comoving coordinates, which are often described in terms of the "expansion of space" between galaxies. This expansion rate is thought to have been at its peak during the inflationary epoch thought to have occurred in a tiny fraction of the second after the Big Bang (models suggest the period would have been from around 10−36 seconds after the Big Bang to around 10−33 seconds), when the universe may have rapidly expanded by a factor of around 1020 to 1030.[40]

Astronomical observations

Apparent superluminal motion is observed in many radio galaxies, blazars, quasars and recently also in microquasars. The effect was predicted before it was observed by Martin Rees and can be explained as an optical illusion caused by the object partly moving in the direction of the observer,[41] when the speed calculations assume it does not. The phenomenon does not contradict the theory of special relativity. Interestingly, corrected calculations show these objects have velocities close to the speed of light (relative to our reference frame). They are the first examples of large amounts of mass moving at close to the speed of light.[42] Earth-bound laboratories have only been able to accelerate small numbers of elementary particles to such speeds.

Quantum mechanics

Certain phenomena in quantum mechanics, such as quantum entanglement, appear to transmit information faster than light. According to the No-communication theorem these phenomena do not allow true communication; they only let two observers in different locations see the same event simultaneously, without any way of controlling what either sees. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality or allow FTL it follows that neither does the additional effect of wavefunction collapse, whether real or apparent.

The uncertainty principle implies that individual photons may travel for short distances at speeds somewhat faster (or slower) than c, even in a vacuum; this possibility must be taken into account when enumerating Feynman diagrams for a particle interaction. .[43] However, macroscopically these fluctuations average out, so that photons do travel in straight lines over long (i.e., non-quantum) distances, and they do travel at the speed of light on average. Therefore, this does not imply the possibility of superluminal information transmission.

There have been various reports in the popular press of experiments on faster-than-light transmission in optics—most often in the context of a kind of quantum tunneling phenomenon. Usually, such reports deal with a phase velocity or group velocity faster than the vacuum velocity of light. But, recall from above, that a superluminal phase velocity cannot be used for faster-than-light transmission of information. There has sometimes been confusion concerning the latter point.

Quantum teleportation transmits quantum information at whatever speed is used to transmit the same amount of classical information, likely the speed of light. This quantum information may theoretically be used in ways that classical information can not, such as in quantum computations involving quantum information only available to the recipient. In science fiction, quantum teleportation is either used as a basis for teleportation of physical objects at the speed of light, presumably preserving some important aspect of the entanglement between the particles of the object, or else is misrepresented as allowing faster-than-light communication.

Hartman effect

The Hartman effect is the tunnelling effect through a barrier where the tunnelling time tends to a constant for large barriers.[44] This was first described by Thomas Hartman in 1962.[45] This could, for instance, be the gap between two prisms. When the prisms are in contact, the light passes straight through, but when there is a gap, the light is refracted. There is a finite probability that the photon will tunnel across the gap rather than follow the refracted path. For large gaps between the prisms the tunnelling time approaches a constant and thus the photons appear to have crossed with a superluminal speed.[46]

However, an analysis by Herbert Winful from the University of Michigan suggests that the Hartman effect cannot actually be used to violate relativity by transmitting signals faster than c, because the tunnelling time "should not be linked to a velocity since evanescent waves do not propagate".[47]

Casimir effect

In physics, the Casimir effect or Casimir-Polder force is a physical force exerted between separate objects due to resonance of vacuum energy in the intervening space between the objects. This is sometimes described in terms of virtual particles interacting with the objects, due to the mathematical form of one possible way of calculating the strength of the effect. Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small.

EPR Paradox

We can also quote the spectacular case of the thought experiment of Einstein, Podolski and Rosen (EPR paradox) which could be realized in experiments for the first time by Alain Aspect in 1981 and 1982 in the Aspect experiment. In this case, the measurement of the state on one of the quantum systems of an entangled pair forces the other system to be measured in the complementary state. Thus functions quantum teleportation.

An experiment performed in 1997 by Nicolas Gisin at the University of Geneva has demonstrated nonlocal quantum correlations between particles separated by over 10 kilometers.[48] But as noted earlier, the nonlocal correlations seen in entanglement cannot actually be used to transmit classical information faster than light, so that relativistic causality is preserved; see no-communication theorem for further information. A 2008 quantum physics experiment also performed by Nicolas Gisin and his colleagues in Geneva, Switzerland has determined that the "speed" of the quantum non-local connection (what Einstein called spooky action at a distance) is at least 10,000 times the speed of light.[49]

Delayed choice quantum eraser

Delayed choice quantum eraser (an experiment of Marlan Scully) is a version of the EPR paradox in which the observation or not of interference after the passage of a photon through a double slit experiment depends on the conditions of observation of a second photon entangled with the first. The characteristic of this experiment is that the observation of the second photon can take place at a later time than the observation of the first photon, [50] which may give the impression that the measurement of the later photons "retroactively" determines whether the earlier photons show interference or not, although the interference pattern can only be seen by correlating the measurements of both members of every pair and so it can't be observed until both photons have been measured, ensuring that an experimenter watching only the photons going through the slit does not obtain information about the other photons in an FTL or backwards-in-time manner.

Variable speed of light

In conventional physics, the speed of light in a vacuum is assumed to be a constant. There exist theories which postulate that the speed of light is not a constant. The interpretation of this statement is as follows.

The speed of light is a dimensional quantity and so, as has been emphasized in this context by João Magueijo, it cannot be measured.[51] Measurable quantities in physics are, without exception, dimensionless, although they are often constructed as ratios of dimensional quantities. For example, when you measure the height of a mountain you really measure the ratio of its height to the length of a meterstick. The conventional SI system of units is based on seven basic dimensional quantities, namely distance, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.[52] These units are defined to be independent and so cannot be described in terms of each other. As an alternative to using a particular system of units, one can reduce all measurements to dimensionless quantities expressed in terms of ratios between the quantities being measured and various fundamental constants such as Newton's constant, the speed of light and Planck's constant; physicists can define at least 26 dimensionless constants which can be expressed in terms of these sorts of ratios and which are currently thought to be independent of one another.[53] By manipulating the basic dimensional constants one can also construct the Planck time, Planck length and Planck energy which make a good system of units for expressing dimensional measurements, known as Planck units.

Magueijo's proposal used a different set of units, a choice which he justifies with the claim that some equations will be simpler in these new units. In the new units he fixes the fine structure constant, a quantity which some people, using units in which the speed of light is fixed, have claimed is time dependent. Thus in the system of units in which the fine structure constant is fixed, the observational claim is that the speed of light is time-dependent.

While it may be mathematically possible to construct such a system, it is not clear what additional explanatory power or physical insight such a system would provide, assuming that it does indeed accord with existing empirical data.

Notes

  1. ↑ Gonzalez-Diaz, Pedro F. (2000). "Warp drive space-time" (PDF). Physical Review D 62: 044005–1–044005–7. doi:10.1103/PhysRevD.62.044005. http://omnis.if.ufrj.br/~mbr/warp/etc/PRD62_44005.pdf.  arXiv:gr-qc/9907026
  2. ↑ http://arxiv.org/pdf/gr-qc/0107097
  3. ↑ PDF for gr-qc/9810026
  4. ↑ PDF for gr-qc/9908023
  5. ↑ Einstein, A, Relativity:the special and the general theory, pp25-27, Methuen & Co, 1927.
  6. ↑ Gott, J. Richard (2002). Time Travel in Einstein's Universe.  pp82-83
  7. ↑ "What is the 'zero-point energy' (or 'vacuum energy') in quantum physics? Is it really possible that we could harness this energy?". Scientific American. 1997-08-18. http://www.scientificamerican.com/article.cfm?id=follow-up-what-is-the-zer. Retrieved 2009-05-27. 
  8. ↑ Klaus Scharnhorst (1990-05-12). "Secret of the vacuum: Speedier light". http://www.nat.vu.nl/~scharnh/m16scine.htm. Retrieved 2009-05-27. 
  9. ↑ Matt Visser; Stefano Liberati, Sebastiano Sonego (2001-07-27). "Faster-than-c signals, special relativity, and causality". Annals Phys. http://arxiv.org/abs/gr-qc/0107091. Retrieved 2009-05-27. 
  10. ↑ [0706.0553] Can Light Signals Travel Faster than c in Nontrivial Vacuua in Flat space-time? Relativistic Causality II
  11. ↑ 11.0 11.1 Macroscopic experiments with virtual photons
  12. ↑ Superluminal Tunneling Devices by Günter Nimtz (2001)
  13. ↑ 13.0 13.1 Herbert Winful (2007-09-18). "Comment on "Macroscopic violation of special relativity" by Nimtz and Stahlhofen". http://arxiv.org/abs/0709.2736. 
  14. ↑ "Faster than light or not", from the blog of Robert C. Helling
  15. ↑ Anderson, Mark (August 18–24, 2007). "Light seems to defy its own speed limit". New Scientist 195 (2617): pp. 10. http://www.eurekalert.org/pub_releases/2007-08/ns-lst081607.php 
  16. ↑ 16.0 16.1 Winful, Herbert G. (December 2006). "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox". Physics Reports 436 (1-2): 1–69. doi:10.1016/j.physrep.2006.09.002. http://sitemaker.umich.edu/herbert.winful/files/physics_reports_review_article__2006_.pdf. 
  17. ↑ 17.0 17.1 For a summary of Herbert Winful's explanation for apparently superluminal tunneling time which does not involve reshaping, see http://spie.org/x18001.xml?ArticleID=x18001
  18. ↑ A number of papers are listed at Literature on Faster-than-light tunneling experiments
  19. ↑ P. Eckle, et al, "Attosecond Ionization and Tunneling Delay Time Measurements in Helium", Science, 322 (2008) 1525
  20. ↑ Sokolovski, D. (8 February 2004). "Why does relativity allow quantum tunneling to 'take no time'?". Proceedings of the Royal Society A 460 (2042): 499–506. doi:10.1098/rspa.2003.1222. http://rspa.royalsocietypublishing.org/content/460/2042/499.full.pdf. 
  21. ↑ Charles H. Lineweaver and Tamara M. Davis (March 2005). "Misconceptions about the Big Bang". Scientific American. http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03. 
  22. ↑ Traveling Faster Than the Speed of Light: A New Idea That Could Make It Happen Newswise, Retrieved on August 24, 2008.
  23. ↑ http://www.newscientist.com/article/mg18925331.200-take-a-leap-into-hyperspace.html
  24. ↑ See, for instance, Luis González-Mestres (February 2009), AUGER-HiRes results and models of Lorentz symmetry violation, http://arxiv.org/abs/0902.0994 , Proceedings of CRIS (Cosmic Ray International Seminar), La Malfa, September 15–19, 2008, Nuclear Physics B - Proc. Suppl., Volume 190, May 2009, Pages 191-197 , and references therein
  25. ↑ http://web.archive.org/web/20040405031103/http://physicsweb.org/article/world/17/3/7
  26. ↑ Nick E. Mavromatos (August 2002), Testing models for quantum gravity, CERN Courier, http://cerncourier.com/cws/article/cern/28696
  27. ↑ Dennis Overbye (December 2002), Interpreting the Cosmic Rays, The New York Times, December 31, 2002, http://www.nytimes.com/2002/12/31/science/interpreting-the-cosmic-rays.html?n=Top/News/Science/Topics/Space
  28. ↑ Feinberg, Gerald (1967). "Possibility of Faster-Than-Light Particles". Physical Review 159: 1089–1105. doi:10.1103/PhysRev.159.1089. 
  29. ↑ Gates, S. James. Superstring Theory: The DNA of Reality. 
  30. ↑ Gimon, Eric G.; Petr Horava (May 2004). "Over-rotating black holes, Gödel holography and the hypertube". http://xxx.lanl.gov/abs/hep-th/0405019. Retrieved 2006-06-05. 
  31. ↑ See Salters Horners Advanced Physics A2 student Book, Oxford etc (Heinemann) 2001, pp 302 and 303
  32. ↑ See Salters Horners Advanced Physics A2 student Book, Oxford etc (Heinemann) 2001, pp 175, 302 and 303
  33. ↑ see http://www.oarval.org/furthest.htm
  34. ↑ Gibbs, Philip (1997). Is Faster-Than-Light Travel or Communication Possible?. University of California, Riverside. http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html. Retrieved 20 August 2008 
  35. ↑ Wertheim, M. (June 20, 2007). "The Shadow Goes". New York Times. http://www.nytimes.com/2007/06/20/opinion/20wertheim.html?_r=1&scp=1&sq=%27the%20shadow%20goes%27&st=cse&oref=slogin 
  36. ↑ Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. pp. 62. ISBN 0-201-11609-X. 
  37. ↑ "MathPages - Phase, Group, and Signal Velocity". http://www.mathpages.com/home/kmath210/kmath210.htm. Retrieved 2007-04-30. 
  38. ↑ Brillouin, Léon. Wave Propagation and Group Velocity (Academic Press, 1960)
  39. ↑ Cosmology Tutorial - Part 2
  40. ↑ Inflationary Period from HyperPhysics
  41. ↑ Rees, M. J., "Appearance of relativistically expanding radio sources", Nature 211, 468, 1966.
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  43. ↑ Feynman. "Chapter 3". QED. pp. 89. 
  44. ↑ J.C. Martinez, and E. Polatdemir, "Origin of the Hartman effect", Physics Letters A, Vol 351, Iss 1-2, 20 February 2006, pp31-36.
  45. ↑ T. E. Hartman, "Tunneling of a wave packet", J. Appl. Phys. 33, 3427 (1962).
  46. ↑ G. Nimtz, A. A. Stahlhofen, "Macroscopic violation of special relativity", 5 August 2007 available as eprint [1]
  47. ↑ Winful, H, "Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox", Physics Reports, Vol 436, Iss 1-2, December 2006, pp1-69.
  48. ↑ History
  49. ↑ Testing Spooky Action at a Distance:
  50. ↑ Delayed Choice Quantum Eraser
  51. ↑ Magueijo, João (1999). "A time varying speed of light as a solution to cosmological puzzles". http://arxiv.org/abs/astro-ph/9811018. Retrieved 2006-06-05. 
  52. ↑ "SI base units". http://physics.nist.gov/cuu/Units/units.html. 
  53. ↑ "constants". http://math.ucr.edu/home/baez/constants.html. 

See also

Fiction

References

External links

Scientific links

Proposed FTL Methods links