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    Faster Than Light Neutrinos And Relativity
    By Paolo Ciafaloni | October 31st 2011 04:37 AM | 32 comments | Print | E-mail | Track Comments
    About Paolo

    I live in Lecce, in the south of Italy. I work as a Particle Physicist for the italian Istituto Nazionale di Fisica Nucleare (INFN)....

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    There is a certain amount of confusion on the relationship between Einstein's Theory of Relativity and the recent experimental results that seem to point towards neutrinos that are faster than light by an amount of about 7 km/s. So let me try to clarify things by answering to the following question:

    If neutrinos travel faster than light by 7 km/s, do we need to modify Relativity?

    The answer is a clear-cut "Yes"; let me explain why.

    Lorentz invariance, which is embodied in the theory of Relativity, has the unescapable consequence that there exists a precise relationship between a free particle's energy E, its momentum p and its mass m:


    c being the speed of light. From this equation one immediately obtains the velocity v as:




    Then, from m2 >0 follows v<c. This holds for all known particles, with the possible exception of neutrinos for which measurements of the involved physical quantities are much harder than, say, for electrons.

    Is it possible to admit v>c in a Lorentz-invariant framework?

    In principle yes, by assuming a negative squared mass (m2<0) and hence an imaginary mass. In this case the particle is called a "tachyon" and its velocity is greater than that of light. Tachyons give me a headache, since they lead to mathematically inconsistent theories and to problems with causality. But even forgetting these (major) problems, from a phenomenological point of view OPERA neutrinos cannot be tachyons. In order to understand this, we need a quantitative argument.

    If we suppose neutrinos are tachyons, then in order to comply with the value published  by OPERA:



    which corresponds to the above mentioned 7 km/s, neutrinos must have an imaginary mass with modulus |m c2 | ~ 0.2 GeV. But such a huge scale is totally incompatible with 80 years of neutrino physics. Just to mention one (of many) constraints, the corresponding value for neutrinos coming from the 1987A Supernova would be of the order of 10-6 GeV: five orders of magnitudes smaller than the value required by OPERA.

    The "tachyonic way" for OPERA neutrinos is dead, and one has to abandon Lorentz invariance, which is at the heart of Einstein's special relativity. So be it.

    If one gives up Lorentz invariance, then the usual relationship between energy and momentum can be broken and one can, in principle, be compatible both with OPERA and with SN1987A. However, then come the real problems as I have signaled in my article:

    http://www.gravita-zero.org/2011/10/neutrini-piu-veloci-della-luce-le.html

    Unfortunately, it is in italian; I plan to write an updated version in English and to post it in Science 2.0.

     

    Comments

    vongehr
    First sentence
    "the recent experimental results that seem to point towards neutrinos that are faster than light by an amount of about 7 km/s."
    already wrong. All the data up to now point consistently towards much faster superluminal velocities than this.
    whitepanther65
    Dear Sascha,
    honestly speaking, the thought that data "point consistently towards ... much faster superluminal velocities" that you say could be "a million times the speed of light" has never occured to me, and I bet it has never occurred to the (many) physicists I now. But then, as you certainly know, science is not democratic; I suggest that you directly contact Cohen and Glashow and/or the members of the OPERA collaboration. I am pretty sure that they will be, like me, astonished when they read your post. For my part, I am not convinced by your arguments and I think there is nothing terribly wrong in what I have written. But, if you think that my first sentence is wrong, then you certainly agree with me that reading (and commenting) the rest of my post would be for you an unexcusable loss of precious time.

       Cheers Paolo
    vongehr
    "reading (and commenting) the rest of my post would be for you an unexcusable loss of precious time"
    Not sure what this is supposed to mean. The first sentence assumes 7km/s and the rest of the post argues on the basis of that.
    If you think it is a good idea to contact Glashow - please try, but do not be disappointed if you discover that physics is not just undemocratic but also not exactly nice sometimes.
    Since you are "not convinced by" my arguments, feel free to comment on them [if you can at all excuse such "loss of precious time" ;-) ].
    Wrong Sascha. All the data up to now point consistently towards a systematic error.

    vongehr
    Dear Anton, feel free to join Science2.0 and write an article of how all the data point consistently toward a systematic error. I have given three different reasons that point towards very high velocities over small initial distances and there has been as of now nobody who pointed out any mistake in my arguments. If you can similarly come up with three independent reasons involving consistently all data that point toward a systematic error, please tell the world. If you have such, it should be quite helpful to actually pinpoint the systematic error. You will be famous soon, Anton. Good luck. The world is waiting impatiently for your article here.
    Although I've a degree in Physics from 1962 I'm certainly not a physicist. I thought that the the Lorentz equations applied only to particles that have a rest mass, that is, particles that are accelerated, unlike photons which are born moving, so to speak and have no rest mass. Could the neutrino be in this category and its mass be only a mass equivalent?

    -- I suppose if this were the case dozens of physicists would have mentioned it by now.

    Nate--

    If what you are referring to are the equations of Lorentz transformation, then the energy-momentum 4-vector of all three types of particles, namely bradyons (slower-than-light), luxons (light-like) and tachyons (faster-than-light) particles, should transform according to them. The transformations relate the components of these 4-vectors according to observers in one inertial frame to those as they are seen in another such frame. If the components of the energy momentum 4-vector of a bradyon transform this way, then those for a luxon must also transform this way, if energy-momentum conservation should hold in every frame (which is a very basic assumption), and the same argument would show that tachyonic energy-momenta must also transform in the same way. Inherent in all this is the basic fact that the Lorentz transformations are linear.

    There is something called the Lorentz equation which describes the Force acting on a charged particle (usually a bradyon like an electron) due to a magnetic field, but presumably this is not what you mean, based on the context.

    I'm not sure what you mean by the neutrino's "mass be[ing] only a mass equivalent". Einstein's E=mc^2 really only applies directly to bradyons (1) at rest (in which case the m is the rest mass m_0, and E is the corresponding "potential energy or equivalent energy of the particle), or (2) in motion (in which case m is the relativistic mass \gamma m_0, where \gamma = 1/\sqrt(1-(v/c)^2) and E is the total energy [potential + kinetic] of the particle). It applies to photons only in the limit as v->c and m->0, which you can see in the formula above gives 0/0, an undefined quantity, which means photons can have any real & positive energy. In all cases, however, energy E and momentum p (really a vector, but I'll take it to be in 1 space direction for simplicity) E^2 - (pc)^2 is a constant under Lorentz transformation. The constant is positive for bradyons, zero for luxons, and negative for tachyons. If one takes v>c in these equations, and simultaneously takes m_0 to be imaginary, call it i\mu, then E and p stay real as long as |p|>=\mu c, and the value of the invariant constant is -\mu^2 c^4. The analog to E=mc^2 in the tachyonic case would be the "momentum equivalent" |p'| = \mu c , where |p'| is the minimum possible magnitude of any momentum the tachyon can have (just like the E in E=m_0 c^2 is the minimum possible energy that any bradyon of rest mass m_0 can have).

    Best regards,
    Marek Radzikowski
    Assist. Prof. of Mathematics
    American University of Afghanistan

    whitepanther65
    Lorentz invariance is supposed to hold for all laws of physics; the OPERA measurements put in doubt this validity  for neutrinos. All laws of physics means all laws of physics: free particles, massless or not, but also interacting systems. I don't know what you mean by "mass equivalent".
    All related to Neutrinos is pure speculation and Fantasies because the Neutrino doesn't exisas it is proved by Autodynamics without any doubt:
    Related to "CHINA construction of a New Neutrino Detector" we are sending the following paper to who could be interested.
    Dear Reader:
    It is sad to see that the China’s Government will be deceived because the scientists didn’t learn from History.
    NEVER any Neutrino was detected by any detector around the World, for the simple reason that the Neutrino doesn’t exist as the Experiment by W. W. Bouchner and R. J. Van de Graaff proved, in 1946.

    http://autodinamicaar.org/ad_9_108.htm

    The Neutrino born by a Pauli’s misconception applying the Special Relativity (SR) equations - where the Kinetic Energy (KE) and Mass (m) INCREASE – to a DECAY phenomenon where the KE is CONSTANT and the Mass DECREASE.

    http://autodynamicslborg.blogspot.com/2011/05/project-for-neutrino-non-e...

    http://autodynamicslborg.blogspot.com/2011/05/blog-post_06.html

    This also was proved by another famous deceiving experiment, the SuperKamiokande, by manipulating the DATA (plotting together evens at different angles) - using a Booby Trap – hidden the results behind the deceiving use of a geometrical function (cosine),

    http://autodynamicslborg.blogspot.com/2011/05/super-kamokande-super-proo...
    Lucy Haye Ph. D.
    SAA's representative.
    Long Beach, CA, USA.
    lucyhaye22@gmail.com

    Not directly addressing Sasha's arguments, but generally about "not marginally" superluminal neutrinos: if c were not a limit for speed of (at least) neutrinos, and these could travel at arbitrary speed, this would imply that there is actually no thing like event horizons, wouldn't it? Not in the proximity of "black holes" (which would be "black" for light, but would likely have a neutrino emission spectrum), nor in the far universe, so there would be no "cut" to the influx of neutrinos from the remote galaxies farther than c * (age of the universe). Wouldn't this have implications on the expected density of neutrino radiation on Earth?

    whitepanther65
    Hi Filippo,
    I have never considered seriously phenomenological implication of superluminal neutrinos for astrophysical and cosmological observations (other than SN1987A). I am not an astrophysicist, for one thing, and I believe there are far more restrictive evidences from earth-based experiments. But I am pretty sure that one can also constrain neutrino properties with astrophysical/cosmological evidences similar to the ones you mention.
    vongehr
    Well, the neutrinos may in that case only (depending on energy) go superluminal for about 18 meters. If you tossed CERN into a black hole, you may get a few neutrinos thus escaping for the extremely short amount of time that CERN is only 18 meters behind the event horizon. Astrophysical neutrinos are low energy and created mostly further than a few meters from horizons, so no problem. However, you are correct that this is an interesting thought experiment. Time dilation means that an outside observer never actually observes CERN passing the event horizon, but nevertheless the observer could (all assuming the strongly superluminal mechanism over short distances) observe a very short message from the already fallen in CERN.
    Yes, indeed that is a valid and interesting thought experiment. However Sascha's suggestion about what effect that would have seems a limited conclusion; What I mean is a blackhole would have different interactions on light (which has a constant velocity) and on Neutrinos which you (sascha) have proposed would not have a constant velocity. As also we do need to include the aspect of mass into the considerations. Then again observation of CERN passing the event horizon is based on the information being relayed by light or slower than light objects,etc. So what would be the actual effect of Event Horizon on the Neutrinos? Would it not seem to imply that there will be multiple event horizons for all the different neutrinos that are superluminal based on the various energy and velocities they possess?

    Paolo--

    You said:

    `The "tachyonic way" for OPERA neutrinos is dead, and one has to abandon Lorentz invariance, which is at the heart of Einstein's special relativity. So be it.'

    I disagree with this statement. I would have agreed if you had said that a Lorentz invariant theory of tachyonic neutrinos is dead. With that I would fully agree, and I would guess that it is those theories that might be giving you a headache. But Lorentz symmetry breaking (in the tachyonic sector only, it should be emphasized) goes hand in hand with a consistent picture of tachyonic neutrinos (and tachyons in general), which allows a reasonable concept of causality (no anti-telephones), no instability, and (so far, straightforward, but more mathematically complex) renormalizability when incorporated in interacting theories (based on recent unpublished calculations as of this year). There is also, in the case of Dirac-like tachyons, necessary parity breaking while preserving a notion of lepton number in the free quantum field theory, which is just not seen in the regular massive or massless neutrino QFT models (Dirac or Majorana). Please check out http://arxiv.org/abs/1007.5418, sleep & eat well, and possibly see a doctor if your headaches are still bothering you.

    I would stress that the notion of Lorentz symmetry breaking in the above tachyonic models is of a benign nature as far as relativity is concerned and should not be thrown out on an ad hoc basis or purely for aesthetic reasons. Full Lorentz symmetry is adhered to for light-like or slower than light particles just as in the case of special relativity as we know it. Lorentz transformations correctly describe the relationships between physical quantities in different frames for all particle types (bradyonic, luxonic, and tachyonic). They must do so for tachyons if they do for bradyons and luxons in order that conservation of energy and momentum be preserved in each inertial frame (as is usual everywhere else in physics). The only way symmetry breaking is implemented is in the choice of *interpretation* of a particle as either a positive energy, forward-in-time moving particle, or as a negative energy, backward-in-time moving anti-particle (with opposite quantum numbers). This may be done in an invariant way for bradyons and luxons (as is done in conventional quantum field theory) but must be done in a Lorentz *covariant* (but not invariant) way for tachyons. I also wish to point out that the cutoff in the one-particle energy-momentum spectrum (in the QFT) which implements this symmetry breaking (down to a conjugate of SO(3)) is at or near 0 energy, so that it would not be noticed as a high energy effect (as would seem to be the case with most Lorentz symmetry breaking investigations at present). Finally, the classical Lagrangian which is the starting point for the quantum tachyonic model is invariant with respect to proper Lorentz transformations, and so the model "flies under the radar", as it were, compared to other models with Lorentz symmetry breaking (LSB), which display that LSB right away in the Lagrangian. (The LSB in the models above is introduced only in the second quantization process to distinguish "positive energy" modes associated with annihilation operators from "negative energy" modes associated with creation operators. It solves at once the dual problems of causality and renormalizability [from what I've seen so far], that would remain intractable in a completely Lorentz symmetric model of tachyons.)

    Please also note that one should distinguish between total abandonment of an underlying Lorentz symmetry of the spacetime, and quantum field models with *spontaneously broken Lorentz symmetry* living on Minkowski spacetime, which inherently retains an underlying Lorentz symmetry.

    Also, one must keep in mind that to prove that something is dead, one must procure a death certificate, and in this case, it would mean a rigorous proof that no reasonable quantum field theory of any kind of tachyon *can possibly* exist, which requires considering *every* model (including those with LSB). Sorry, but I think (especially since I think I having a working model in hand having quite reasonable properties) that producing such a "proof of death" would require a *lot* of work and is presently a very long way off, or may well be impossible. In any case, if you have such a proof, I'd like to see it, and it would be really nice if it would be clear.

    Sincerely and with best regards,
    Marek Radzikowski
    Assist. Prof. of Math.
    American University of Afghanistan

    whitepanther65
    Dear Marek,
    By a "tachyon" I mean exactly what is described in my article: a particle with negative mass squared in a Lorentz invariant framework. This explanation is phenomenologically ruled out as an explanation for OPERA neutrinos.
    This is probably equivalent to your sentence "a Lorentz invariant theory of tachyonic neutrinos is dead", so I think we fully agree on that, and maybe there is a problem of language. Maybe we could agree on calling  "a Lorentz invariant theory of tachyonic neutrinos"  a conventional model for tachyons. Then your model could be an "unconventional model for tachyons".  Notice that I am not concerned here with the mathematical properties of the theories, but with quantitative experimental tests, in particular the OPERA neutrinos and SN1987 data. Once possibile ambiguities with the meaning of words are clarified, I insist that what I have written is corrrect.
    Paolo--

    Your phenomenological assumptions seem somewhat simplistic. You assume that the neutrinos observed in SN1987A are of the same flavour as the ones observed at OPERA. They could be different, and therefore, in principle, have different tachyonic mass parameters.

    The point has been noted elsewhere that a tachyonic neutrino with the same mass parameter as purportedly observed by OPERA would have been seen about 4 years before the SN1987A neutrinos. However, looking at Section 3.4 of M. Koshiba's "Observational Neutrino Astrophysics", Phys. Rep. 220, Nos 5 & 6 (1992), 229--381 reveals that most of the neutrino detectors at the time of SN1987A had come into operation only a few years before Feb. 1987. Kamiokande II itself started taking neutrino data only a month before, in January 1987. Mont Blanc was operational starting in October 1984, only about 2.5 years earlier. The IMB detector did start taking data in time to possibly detect such neutrinos (in August 1982), but went through some upgrades in the mean time, and was ready to go in 1987 (except for a bank of photomultiplier tubes that were out due to a power failure at that time). Perhaps a strong signal may have come in during the upgrade time, or may have been discounted as being completely unrelated to SN1987A which was to come 4 years down the road. I'd have to look more carefully at the dates when the other detectors came online, but the ones mentioned above were the ones which reported the largest numbers of neutrino bursts just before SN1987A was seen.

    Also, you may recall that the Mont Blanc LSD detector recorded its set of bursts 4.7 hrs before the Kamiokande II and IMB signals. Kamiokande II reported only electron events, when they had the capacity to distinguish them from muonic events. If the three flavours of neutrinos have very close to the same mass, then why weren't muon neutrinos also detected from among this bunch? Kamiokande reports having searched for muon events long after the time of their electron ones, and found nothing. The Mont Blanc neutrinos could have been muon events (they couldn't tell the difference) and the different properties of the two detectors may have been why one group observed nothing (or an insignificant number) of signals at the other's time. The muon neutrino's mass parameter in this case would be less than 1 keV. There is a third flavour -- the tau neutrino -- so there is still the possibility that OPERA is detecting this kind of neutrino.

    As a footnote, due to the current explanation of neutrino oscillations being grounded in the assumption of bradyonic neutrinos, alternative explanations in terms of tachyonic neutrinos should be searched for. For example, decoherence of the neutrino from the emission at the source may not occur before the neutrino is detected, in which case, it would be interesting to do a QFT calculation to see on what (besides the energy and mass squared) the matrix element involved might depend. In any case, the phenomenological argument you allude to does not utilize 6 sigma measurements all the way, hence for me it is not as convincing as the OPERA result by itself. I admit it does rule out some scenarios, but not all, and those are the ones I'm interested in.

    Thank you for at least agreeing with me that Lorentz invariant QFTs of tachyons are ruled out (I would say for many, and some quite obvious reasons, viz. lack of causality and non-renormalizability). But how would the LSB ones which I'm dealing with, which have all the desirable properties I mentioned (including parity breaking from the get-go) not be interesting?

    By the way, elsewhere, I've been reading that people are assuming that when a tachyon loses energy, it slows down. You should be aware that this is a mistake. They would speed up. Velocity v = p/E (in natural units), but p^2 = mu^2 + E^2, so you can see how that would work when E is positive. If Cohen and Glashow assume this about tachyons, then they are ruling out a theory of tachyonic neutrinos which is flawed for other, much more basic reasons.

    Best regards,
    Marek

    whitepanther65
    Marek,
    I am not assuming any "simplistic phenomenological assumptions". I am aware of all the points you mention here, but they do not play a significant role in the game. This blog is supposed to be directed to a general public, so I chose not discuss the enormous amount of details which could obscure the discussion of my main point. Which is, I repeat : "If neutrinos travel faster than light by 7 mk/s as OPERA seems to suggest, a modification of Einstein's relativity (and, namely, of standard Lorentz invariance) is necessary". Do you agree with this or not?

    In any case, you raise a series of interesting questions, so let me try to answer in the limited form allowed by this context.

    -neutrino flavor
    It is true that neutrinos come in three flavors - muon, electron and tau  neutrinos -  but a difference between the velocities of the three flavors is excluded by the observations on neutrino oscillations at a relative level of 10-20 (!!).  The OPERA anomaly is of the order of 10-5, so flavor doesn't play any role here.

    -SN1987A
    A difference of a few hours in the arrival times of neutrinos of photons form SN1987A is irrelevant, since in this case neutrinos have travelled for hundreds of thousands of years. Even if we wish to interpret this difference as an indication of superluminal velocity for neutrinos (which means forcing the meaning of data imho), then supernova neutrinos would be faster than light by a relative amount of 10-9; in terms of mass scales this would mean five order of magnitudes less than OPERA neutrinos, as I have signaled in my article. On the other hand, if you believe that the observed neutrinos have nothing to do with SN1987A, then this is certaily a way out. A very implausible one, I must say. I can only suggest that you contact some astrophysicist, which I am not. In any case there is an enormous amount of ground-based experiments that tell us that the mass scale of neutrinos is less than/of the order of 1 eV; the 0.2 GeV required by OPERA are off by several orders of magnitude.

    -v vs. E
    I have never heard that someone assumes that "when a tachyon loses energy, it slows down". What people generically assume is energy-momentum conservation. THIS could be violated, of course.

    -QFT, LSB, decoherence, causality, renormalizability and so on
    You still miss my point, which is a very general one, and completely independent of all of this stuff.

    Cheers Paolo
    Nice post, Paulo. I came to similar conclusions after OPERA's announcement (see http://t.co/PpSokwji ), which is why I have suggested that it's not the neutrinos per se, but rather something strange happening near the point of production (see http://t.co/lOzF0IYF ).

    I'd be interested in your comments.

    whitepanther65
    Thank you John (but my name is Paolo). I have read your suggestion, and it doesn't look apriori senseless, differently from some of the comments here. But I have to say that it also looks like an ad hoc explanation and, for this reason, featuring a lack of predictivity. The really difficult thing is to find a new predictive framework that includes the huge amount of information that we have from ground-based experiments and from astophyical observations. Life is hard....
    meopemuk
    Here is another theory about superluminal neutrinos. In fact, it is similar to what Sascha was suggesting: http://vixra.org/pdf/1110.0052v2.pdf

    Eugene.
    whitepanther65
    "vixra"?!?
    Hank
    You've never seen it?  It was started by someone who thinks arXiv is not open enough(imagine that!) so any crank can submit to vixra.   With arXiv, there is at least some chance you can find something good without reading it as a fulltime job - I don't think anyone serious reads vixra. 

    Nice people, though, and they mean well, but when you let in cranks serious people tend to disappear.
    vongehr
    There are many cranks on vixra, but they do point out some valid issues, one being that the arXiv has also plenty of cranks. Joy Christian is there for example, and he just put yet another pseudoscientific paper on the archive. Nobody objects while many serious scientists are not allowed to submit for all kinds of reasons that depend mainly on some administrators mood the given day. When I used it many years ago, it was still open. Now I cannot even put gravity/cosmology stuff into anything but the history section (you know how much I care about history - not at all). Not sure why they do it, but there seems to be different degrees of censure. Other gravity specialists do not care about the history section either. A reclassified paper gets thus onto the archive, but it is still invisible, buried where nobody of your target audience finds it. Suppression has many subtle faces.
    whitepanther65
    Actually, in view of some of the comments in reply to my article, I was about to disappear from Science 2.0 as well. I have decided to ignore arrogant and aggressive comments instead. But the issue of "open vs. filtered" is by no means a trivial one in ane case. For instance, peer reviewing means that your article is tipically  published roughly six months after you submit it. This delay is unacceptable in the Internet era.
    Hank
    You'll find over time that you silently reach a lot of people who get a little smarter but have no meaningful comment to make.  1% is about the comment rate and those will be split between literate and not literate, but the 99% you reach who never talk are the important part in making society more knowledgeable about science.
    Apologies, Paolo, that is what I meant to type. I think my fingers must be pre-programmed (my middle name is Paul).

    It will be interesting to see how this pans out ...

    whitepanther65
    No problem about my name John. We will see how what experiments and OPERA upgrades have to say about that. However, I am afraid we will have to wait at least one year....
       Cheers Paolo
    PS "Costella" looks like an italian name. Is it? 
    Yes, my father came from Turin to Australia when he was 4, in 1927.

    With the change to OPERA with 3 ns proton pulses with a 500 ns gap, I thought there were going to be results before the end of 2011. In principle every single detected neutrino can be used to test the earlier results. You have different information?

    Cheers
    John

    whitepanther65
    St. Margaret River, Australia is one of the places I would like definitely to visit. We shall see...

    John, as far as I know you're perfectly right about the bunches. I was referring to the fact that there will  probably be also another major upgrade on the CERN side. One of the problems is that energy and production time of neutrinos are not measured directly. One tracks protons and simulate the physics of chain decays  along the beamlime. At CERN they plan to track muons instead, which correspond 1:1 to neutrinos. This upgrade, if it will be done, will take longer. If you put together muon tracking and shortened bunches, you significantly reduce the probability of systematical errors. Then, one may still question the precision of time and distance measurements through GPS; they are trying to make independent measurements also on this.
    But if they measure it from the muons at the far side of the hadron stop, then they eliminate the possibility that the 60 ns advance is across the hadron stop itself (a la my suggestion).

    Hopefully they will do all three -- protons, muons after the hadron stop, and neutrinos at Gran Sasso. That will allow the best determination of (a) whether there is an anomaly left to explain at all, and (b) if so, whether it is across the hadron stop or in the 730 km trip to Italy.

    FWIW, the assumptions about the proton decay look fine to me -- at those energies everything is practically going at the speed of light.

    John

    whitepanther65
    You're right that everything is practically going at the speed of light of course, but I have not understood your concern about "a 60 ns advance across the hadron stop". Could you explain this better?
    In any case, it looks like you're very well informed about experimental details. Informations apparently travel faster than neutrinos from Europe to Australia these days :)
    whitepanther65
    There are two follow-up articles to this one. The first one is about  the same subject and shows that it can be discussed without no reference to the "speed of light". The second one I think is more important. You are warmly encouraged to post comments there.

       Cheers Paolo