Our notion of causality is not constructed on any axiom that would exclude superluminal speeds a priori. Indeed, I am not even aware of such an axiom (you certainly don't need that axiom to construct special relativity. Tachyons are welcome by the theory, whether we like them or not. The only axiom you need is that the speed of light is invariant, you do not need to postulate that it cannot be exceeded.)

Causality is simply the non-existence of closed timelike loops. The idea that no observer (or bit of information) can intersect the same spacelike hypersurface more than once. And this would indeed by violated by the existence of tachyons, which is why they could, in principle, be used to send messages to the past or otherwise violate our notion of causality.

Whether or not a tachyon can be observed depends on how it interacts with other fields. If a particle does not interact with the electromagnetic field, there'll be no Cherenkov radiation, for instance, regardless of the particle's velocity relative to the local speed of light.

The experiment is easy in principle. Measuring a velocity amounts to measuring a distance with meter sticks and time with clocks. So long as the apparatus is inertial, the clocks can be unambiguously synchronized. You can then measure the velocity of an electromagnetic wave, and compare it to the velocity of the desired experiment. If the latter exceeds the former, you have a superluminal observation.

The one thing that this requires is experimental apparatus that can interact with the presumed tachyon. But that's kind of a given: if you can't interact with something, you won't be able to observe it!

One such observation was _almost_ made about a couple of years ago, by the way, when a very respectable European experiment (OPERA) claimed to have observed neutrinos that traveled faster-than-light. Given the reputation of the OPERA team, their result was taken very seriously. Ultimately though, it turned out to have been the result of two subtle errors in the apparatus: a faulty optical cable connection and an incorrectly calibrated clock.

Robert,

Thanks for the article. However, in my question I mentioned a MATERIAL object. True,

I didn't emphasize enough that I mean an object with the real mass, not an imaginary one. Anyway, I am open to accept mass as possibly being a complex number. Why not? Although Lorentz transformations presents a light barrier, I agree this should not be an obstacle. Bats can have their STR, but that does not exclude supersonic velocities. However, in this analogy a mass does not change its mathematical property as being a real positive number.

Vladimir: I presume that by "real mass" you mean real rest mass. No Lorentz transformation exists that can move a worldline from inside a lightcone to outside and vice versa. Furthermore, not even a discontinuous transformation (e.g., some kind of a reflection with respect to the lightcone) can exchange the simply connected region that is within a lightcone and the not simply connected spacelike region outside as their topologies are different. In other words, within the rules of special relativity it is not possible to accelerate a slower-than-light object to faster-than-light speeds or vice versa. For this reason, it is also not possible to ascribe a rest mass to a faster-than-light particle. However, as faster-than-light particles lose kinetic energy when they accelerate, their lowest energy state is when they move with an infinite velocity; the mass associated with this lowest energy state can serve as a rest mass of sorts, and this quantity can be real. So at least in this sense, a hypothetical tachyon can have real mass.

To go beyond this, it would be necessary to break the rules of special relativity in some manner, e.g., by allowing imaginary velocities.

Superluminal velocities--are they real?

Quantum electrodynamics claims that exist;) ... path integrals include particles trajectories with superluminal velocities as well... and how about quantum teleportation?

Yurij,

Quantum mechanics, and therefore QED, claims the trajectories are not observable. Thus, non-observable quantity may have any velocity whatsoever--it doesn't matter anyway.

In quantum teleportation you cannot assign any trajectory to the teleported particle. Correct me if I'm wrong.

Vladimir, since such trajectories have statistical weight, they in some sence are observable. The second is corect but teleportation somtimes considered as an example of exceeding of light-speed limit.

Yurij, are you aiming at Bohm's interpretation of QM? There the trajectories are observable but the "quantum potential" is non-local i.e. depends on the very same trajectory calculated by potential.

Vladimir, of course I agree that answer on question must be follows: the superluminal velocities are real but nonobservable. The one of examples is expanding of the Universe at large scales. Due to Habble law, v=HR, if R>c/H, then v>c. But this part of the Universe is not observable because located outside of our light cone.

Yurij, no the answer need not necessarily be "the superluminal velocities are real but nonobservable". Let's draw an analogy with bats' echolocation: a bat CAN get echo from a supersonic military jet if the jet is approaching to the bat; a bat can't get echo from a jet moving away from the bat. In one situation a bat sees a jet (in its own way of "seeing" material things), and in the other situation he doesn't see a supersonic material thing. Of course, in the first case, when he "sees" a supersonic object, he is deceived regarding the exact location of the object.

Regarding the Hubble law, let us take it as a true law for the discussion' sake. The objects running away from us with v>c still emit light towards us, just like the supersonic military jets emit sound, and we can hear them whether they are approaching us or are moving away from us. Therefore, ALL parts of the Universe are observable as long as the objects emit light.

I forgot to add: In the second case a bat hears (that is "sees") the jet, but is completely ignorant (not even deceived) of object's position.

In the publication below I calculated the trajectory of the G2 cosmic cloud near Sagitarius A*, the orbital part is coincident with actual data, the hypothetical fall trajectory has no data to compare with but it shows how part of the cloud can achieve a velocity slightly higher than c without violating Lorentz transformations.

Article Trajectories in space-time M4: Reinstating the Newtonian Time

A very interesting article on the subject:

Chapter CAUSALITY AND TACHYONS IN RELATIVITY, by P.Caldirola and E.R...

When you measure output beam of a coupled microring optical resonator (of course in very special conditions) is minus

15 nanosecond e.g. 15 nanosecond prior reaching to input , the group signal can be sensed or measured in output. This means we should think about it more seriously and look for more comprehensive answers.

Yes, superluminal velocities are real. The following paper provides evidences on superluminal velocities of solar X-rays and Bharat radiation in 12.87 to 31 nm range from solar spectra reported by Woods et al in 2011.

M. A. Padmanabha Rao,

Discovery of superluminal velocities of X-rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR-JAP), .Volume 4, Issue 4 (Sep. - Oct. 2013), PP 08-14, DOI: 10.9790/4861-0440814

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue4/B0440814.pdf?id=3522

Abstract: The current paper reports discovery of superluminal velocities of X-rays, and Bharat Radiation in 12.87 to 31 nm range from solar spectra. The discovery challenges the 100 year old Albert Einstein’s assertion that nothing can go faster than velocity of light c in vacuum while formulating E = mc^2 in his special theory of relativity reported in 1905. Several solar spectra recorded at various wavelengths by Woods et al in 2011 demonstrated GOES X-rays arriving earlier than 13.5 nm emission, which in turn arriving earlier than 33.5 nm emission. Finally, the investigators faced difficulty in concluding that short wavelengths traveled fast because of lack of information whether all the three emissions originated from the same source and at the same time. Very recently the author has reported GOES X-rays (7.0 nm) cause 13.5 nm (Bharat Radiation), which in turn causes 33.5 nm Extreme ultraviolet (EUV) emission from same excited atoms present in solar flare by Padmanabha Rao Effect. Based on these findings, the author succeeded in explaining how the solar spectral findings provide direct evidences on superluminal velocities of GOES X-ray and 13.5 nm Bharat Radiation emissions, when 33.5 nm EUV emission is considered travelling at velocity of light c. Among X-ray wavelengths, the short wavelength 7.0 nm X-rays traveled faster than 9.4 nm X-rays, while X-rays go at superluminal velocities. Among Bharat radiation wavelengths, short wavelengths showed fast travel, while Bharat Radiation goes at superluminal velocities as compared to 33.5 EUV emission.

Maybe the question is wrong: think in terms of one EM wavelength and its total information content. Is there a need for a transmission velocity for one aspect of the contained information to be connected and aware of any other contained information since all wave-form contained information share a common EM wave field? Think in terms of the near-field of a radiating element, each polarity of its OAM (orbital Angular Momentum) contains differing information content but are still experiencing the overall EM contained field. Is there indeed a need for a transmission with velocity within that near-field waveform since all contained experience the total? Or could its contained experience be seen as instantaneous?

I am envisioning in effect a BEC (Bose Einstein Condensate) occurring within each and every wave-cycle field, and the same within the non-radiating near-field.

Therefore, there is an information content exchange requiring no velocity factor. No superluminal needed.

Actually superluminal velocities are real and following paper gives a method to push electrons above the speed of light using existing accelerators and lab equipment: http://www.researchgate.net/publication/265643274_Extended_principle_of_relativity_beyond_speed_of_light_and_a_method_to_push_particles_beyond_the_speed_of_light

Existence of superluminal velocities and the ability to measure them are two different problems. A frame of reference going above the speed of light will transform into 2 different frames of reference, one going in negative time and other in positive time. For example as I propose in another paper (not yet here), if an electron is pushed above the speed of light it will behave like a positron, which is an electron going into past. Also I propose that all positrons are electrons going above the speed of light, which shows up as going into past. As you can see in this paper, the Doppler Effect of radiation generated from a superluminal frame of reference shows up as negative frequency radiation, which is equivalent to Left hand em wave in vacuum. This radiation finds matter mostly transparent and the only way to detect is the reduction of entropy when it passes through matter.

Article Extended principle of relativity beyond speed of light and a...

http://www.scribd.com/doc/143574811/Superluminal-FTL-References

http://www.scribd.com/doc/143572640/Near-Field-Ref

The possibility of measuring of faster than light is real depending on observation. According to Einstein interpretation to the Lorentz transformation in SRT depending on objectivity which leading to continuity,  it is impossible measuring faster than light. But reinterpretation the Lorentz transformation according to Copenhagen school, it is possible without violation Lorentz transformation or causality. Lorentz symmetry  is only for keeping on objectivity and then for continuity, but Lorentz symmetry is not important for Lorentz invariance. Removing objectivity in the Lorentz transformation, will lead to unifying between quantum theory and relativity theory (General and special). Review my paper The Quantization of General Relativity: Photon Mediates Gravitation http://dx.doi.org/10.14299/ijser.2014.10.002 

Yes, superluminal velocities are real. 

Please peruse: 

M. A. Padmanabha Rao,

Discovery of superluminal velocities of X-rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR-JAP), .Volume 4, Issue 4 (Sep. - Oct. 2013), PP 08-14, DOI: 10.9790/4861-0440814 

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue4/B0440814.pdf?id=3522

One more research paper on superluminal velocities of beta gamma and X--rays   in other words low wavelengths go fast and reach more distance. 

 M.A.Padmnabha Rao

All the Sunlight that Earth Receives is not directly from Sun

International Journal of Innovative Research in Science, Engineering and Technology, Vol. 4, Issue 11, November 2015,   DOI 10.15680/IJIRSET.2015.0411050

High Impact factor: 5.442 http://www.ijirset.com/upload/2015/november/50_6_All_the.pdf

So far I understood, superluminal speeds are already accepted in the framework of the "expanding universe". It is about far galaxies which seem to accelerate relatively to us. In a given moment their speed becomes superluminal and we stop seeing them. I am more interested about the tries of opportunistic thinking done in this context in order to remain consistent with the special relativity. One of them, coming from General Relativity, says that differently from Special Relativity, in General Relativity there are only local inertial frames and not global inertial frames. This is a sophism a la how many angels can dance on a needle. The cosmological approach tells that one must make a difference between motion obtained because of the action of forces and/or by own effort (motion relatively to the space) and motion as a result of being embedded in the space as a result of space expansion. The relative motion of far galaxies was apparently  of the second kind and so was not contradicting STR. Well IMHO space cannot expand without moving ist points. Also, if the space expands with us inside, we will not be able to perceive this expansion. The space would remain locally euclidian, isotrop and so on. So I do not really accept this second explanation as well. (I am also kind of skeptic relatively to the Big Bang Theory which does not stay the Occam Rasor Test from many points of view, but this is another story).

Just a short add-on. About the Big Bang, see the motivational text of the following RG question: https://www.researchgate.net/post/Is_Kirchhoffs_law_of_thermal_emission_now_dead/1

@Mihail Prunescu

I agree with your view on the explanation of superluminal expansion of the Universe during inflation--the explanation that the different parts of Universe do not move faster than light but  the space between them is growing. They do not move but the distance between them is growing faster than the speed of light.

Such an explanation is--forgive me my expression--stupid. First, how can we ever define an „influx of space“, if the space is not a physical substance at all? An analogy with physical points on an expanding soap bubble does not work, because the bubble IS a physical substance. And even if we accept this analogy then yes, the points are moving relative to each other, there is no doubt about that. Second, what is the definition of a speed? How can we say that something is not moving if we accept the notion that ANY movement is RELATIVE. The different parts of the Universe are at rest relative to what? Or they are moving relative to what, if not relative to each other? And third, why would there be any restriction on the SPEED, knowing that it is not a principal dynamical quantity after all? Acceleration is the principal dynamical quantity, not speed. The restriction on acceleration would make some physical sense.

It is even worse, Vladimir! They told me that "the co-moving speed" defined for relative speed of galaxies is in fact not a speed, but something different. They said also that the setting of GRT does not allow the definition of an instantaneous or of a mean speed. This is however true: the setting does not allow this, because it is 4-dimensional, because derivative depends of the local mapping, and because is working with events and not with points. In the same time, the Minkowski space is good for nothing in physics. That is because the Minkowski distance between event ""Supernova A explodes"" and event ""Vladimir sees the exposion of supernova A"" is 0. With such a distance you can do at most philosophy of causalities, but not physics. It is also not a distance, because the form is not positive definite.

Last but not least, in this discussion they accepted that there is a last photon that reaches us, and then the accelerating galaxy becomes relatively superluminal and dissapears from our "event horizon". RIP.

On the "expanding universe": I believe that it is more probable that the universe always looked as expanding, exactly like now. That probably in very empty regions matter is created in order to keep something constant. (It suffices if only one electron or one proton is created). Finally by gravity such things condense.

Another terrible possibility would be that the red-shift is missinterpreted, and just far-away light becomes "tired". We just don't know enough to understand this universe. Sorry.

@Dananic: "Second, the possible answers aiming at the breakdown of causality, and thus impossibility of superluminal velocities, are not logically valid, simply because the current notion of causality is constructed on the axiom that there are no real superluminal speeds."

That is not quite correct. If you accept the relativity principle and the universality of the speed of light -- two axioms that do not invoke causality in any way, then it is possible to demonstrate that superluminal messages can be used to send information to the past. The proof is well-known, so I shall not repeat it here.

So there is no circularity in arguing that superluminal message passing or superluminal energy transport violate causality within the framework of special relativity. This does not mean that shadows cannot be superluminally fast. Also superluminal tunneling experiments do not violate causality, because it can be shown that the speed of information transport does not exceed the speed of light there. And Einstein-Podolsky-Rosen type experiments do not help in superluminal message sending either.

So superluminal velocities are possible in cases, where neither information nor energy travels at superluminal speed. Otherwise, if you can assign a direction with the information transport, superluminal velocities would violate causality -- or relativity would be wrong....

@Kassner is there a citation to the proof? It might be well known, but not to me.

I saw nice discussions on the issue of superluminal velocity. By now a few thousands of papers might have been published on this issue. TO UNDERSTAND IT, NUCLEAR PHYSICS BACKGROUND IS NEEDED. Secondly, thorough knowledge on nuclear fission is needed. Otherwise, impossible to understand solar physics and superluminal velocities. Since knowledgeable scientists on fission are rare, my papers quoted in this discussion are not discussed. Particularly my paper published in 2015 provides experimental evidence X-rays going faster than EUV from the measurements of solar spectrum by Woods et al from University of Colorado.

M.A.Padmanabha Rao, Discovery of superluminal velocities of X-¬rays and Bharat Radiation challenging the validity of Einstein’s formula E= mc^2, IOSR Journal of Applied Physics (IOSR¬JAP), .Volume 4, Issue 4 (Sep. ¬ Oct. 2013), PP 08¬14, DOI:10.9790/4861-0440814

http://www.iosrjournals.org/iosr-jap/papers/Vol4-issue4/B0440814.pdf?id=3522

My General CED theory predicts a 'superluminal' velocity for longitudinal vacuum waves. It predicts a different energy/momentum ratio for wave "quanta", and therefore a different interaction with matter, in comparison with luminal TEM waves. Mainly the intrinsic potential energy of particles is changed, while kinetic energy changes of massive particles are less substantial, in comparison with luminal radiation interaction with matter. I assume superluminal (longitudinal) radiation exists only above a certain frequency. I agree with Vladimir that there is no "violation of causality" considering superluminal velocities, there is only a violation of special/general relativity THEORY. Eugene Podkletnov produced/detected a signal with measured signal velocity of v = 64c (it is not a "shadow" nor a 'phase velocity' nor a 'group velocity', Podkletnov measured a SIGNAL velocity, which is supposed to be not greater than 'c', therefore it falsifies SR and GR).

I suppose that superluminal velocities may be achivable for objects made of superdense material (like hypothetical neutronium) or having special protection against "friction" of the physical vacuum (aether). Mechanical analogy- movement of different supersonic objects. Superluminal propagation speeds of the electric and magnetic fields (electroynamic and magnetodynamic waves) are also possible.

The difficulty of achieving superluminal velocity of elementary particles in accelerators can be explained by an effect similar to sound barrier in aerodynamics. Experimental verification of electric and magnetic field speeds is actual up to this day also.

Initial but not so simple measurements of the propagation velocity of the Coulomb field show that this velocity is higher than 'c'; it is superluminal. Some physicists concluded it is 'instantaneous' (and therefore the Coulomb field is a form of "quantum correlation of virtual photons on a macroscopic scale"), and others concluded it is 'higher than c', just that. It is certainly within the financial and technical limits of any physics institute to test the propagation velocity of the Coulomb field (the near electric field), and yet there are little or no attempts made to repeat this experiment. One rather publishes 'theoretical physics' papers that extrapolate "well known data" into the realm of fantasy and SF, than to perform a low budget (but not that simple) experiment. It is obvious that these important experimental results falsify both non-relativistic and relativistic Maxwellian Classical Electrodynamics theory: whatever the selected "gauge condition", the Coulomb electric 'near' field is predicted to have a propagation velocity equal to 'c', in disagreement with experimental results. Therefore definitely YES, superluminal velocities are real, the Coulomb field is a fine example.