My recent article  on the relationship between Einstein's Theory of Relativity and superluminal neutrinos has triggered a series of comments. Some of them were reasonable, some others not. Among the reasonable doubts on this topic, there is a possible concern about the meaning of "limiting velocity", i.e. velocity that cannot be exceeded in the context of the theory. Could it be that we have found a new limiting velocity-  the one of neutrinos- and the theory still stands up with a new value for a fundamental constant c? Could it be that the speed of light is not well measured? Could it be that there are tiny matter effects that alter the speed of light in our measurements? The answer to all these questions is "NO" and the purpose of this article  is to clarify these issues. Let me state once again that:

"If we interpret OPERA results in term of neutrino velocity, then one has to modify Einstein's Relativity"

In order to understand why any discussion about the "real" value of the speed of light is pointless, let me rediscuss the issue of a limiting velocity without any reference to "speed of light" at all. Once again, the relevant formulae are the textbook ones mentioned in my previous article, and constitute a clear prediction of standard Lorentz invariance. For a free object:

$\left(\frac{E}{c^2}\right)^2-\left(\frac{p}{c}\right)^2=m^2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\; (1)$

$v=\frac{\partial E}{\partial p}=c\sqrt{1-\frac{m^2c^4}{E^2}}\;\;\; \;\;\;\;\; \;\;\;\;\;\;(2)$

Here "c" is a universal constant (i.e. not depending on the particular object you use), while m is the mass of the object. E, p,v are well-defined and measurable quantities named respectively  "energy","momentum" and "velocity" of the object.  A free object is a non-interacting object that propagates freely. As an "object" you can take whatever you like: an electron, a proton, a neutrino, but also a Ferrari car if you wish. Needless to say, you shouldn't use Ferrari cars with the purpose of testing the theory of Relativity; more on this later.

The physical meaning of "c" is of a "limiting velocity": the velocity if all objects must be less-than-or-equal-to "c", and "c" is the value to which the velocity tends as the energy E tends to infinity. This statement must hold for all objects, and the measured value of "c" must be the same for all objects. Measuring "c" for a given object implies measuring the values for (E,p) and extracting "c" from eq. (1) or measuring the values for (E,v) and extracting"c" from eq. (2). Usually the measure is repeated many times for different values of E in order to improve statistical accuracy.

All objects must travel slower than "c"...hey but, wait a minute! What if the measured value of "c" is wrong? And what if "c" is the speed of neutrinos, rather than of light? These are unjustified doubts since, as I will now show, the measured values for "c" are significantly different when comparing results from electrons and from OPERA neutrinos. On the other hand, Einstein's Relativity predicts that "c" has a universal value, that cannot depend on whether I use neutrinos or electrons to measure it.  This is a clear contradiction; therefore Einstein's Relativity has to be modified if we interpret OPERA results in terms of neutrinos velocity.

We need now to be more quantitative. In order to compare neutrinos with electrons, let me introduce a reference value, which I call "zuppaspeed":

z=299792458 m/s       (3)

The introduction of this reference value, which has no physical meaning at this stage, is by no means a necessary step, but this what is usually done in order to compare different experimental measures of a single unique quantity: the "universal limiting speed c". Since the "zuppaspeed" is not a measured quantity, but an arbitrarily introduced one, it has the exact value given by eqn (3). Now, experimental data for the quantity
$\delta=(c-z)/z$ as measured for electrons (see for instance arXiv:0905.4346) and for OPERA neutrinos (arXiv:1109.4897) give

$\inline |\delta_e|\le 5\;\; 10^{-15}\;\;\;\;\;\;\ \delta_\nu=(2.48 \pm 0.28 (stat.) \pm 0.30 (sys.)) \times 10^{-5}$

The two values for delta should be compatible, since they refer to the same limiting velocity c; they are not compatible, game over. Relativity must be modified accordingly. And I have never mentioned "speed of light".

You should now not be much surprised to learn that the "zuppaspeed" is also known as the "speed of light in vacuum" as extracted from the Particle Data Group. But this is another story. Einstein has taught us that the speed of light is an invariant, not depending on the status of motion of the observer. It is very well measured, so we'd better use it a standard "ruler" instead of using a poorly defined, observer dependent quantity as the "meter".

Why shouldn't you use a Ferrari to test Relativity? There are probably many possible answers to this questions, but my time is over and I shall give my answer in another article. As for the million dollar bet, it goes as follows. I will give 100 dollars to anyone who, after reading this article, gives me strong arguments that convince me that I am wrong (I would like to bet more, but I am a poor man economically speaking). If
this doesn't happen, then Hank will give me 1 million dollars from the rich revenues he gets from Science 2.0. Since there are 1 million people reading Science 2.0, the probability that I loose is very high and I think this is a fair bet. Hank, are you reading this? Do you accept the bet?