1 - Introduction

2 - What does "special" means?

3 - Sagnac effect

4 - The light’s speed for non-inertial observers

5 - The Principle of (Special) Relativity and the "twins paradox"

6 - Roemer observations

7 - Bradley aberration

8 - Is it true that electromagnetism is relativistic? That

"Classical Physics" is the "limit" of SR for low speeds?
 
 
 
 

1 - Introduction
 
 

During its (rather brief) history, Physics has seen many different theories challenging each other, in the attempt to solve some of the riddles presented by natural phenomena. Between them, a special place is occupied by SR, which, at this century’s beginning, proposed to wipe out all discussions about aether’s (and light’s!) nature, with its bold proposal to change the "usual" space-time structure. We have said that SR must really be considered as "special", since, in the aforesaid situation, advocates of the one or the other confronting parties have always warmly debated, but never with that harshness which characterizes the criticism towards SR. The reason for that is easily understandable, since SR forces to abandon that ordinary intuition which, in a sense or in the other, was always present in both opposing fields: remember for instance the "struggle" between the supporters of Ptolemy and those of Copernicus, or between the proponents the corpuscular nature of light against the defenders of the wave-theoretical approach.

As a matter of fact, SR should more properly be regarded as a kind of revolution with strong conservative aspects, like for instance the proposal to choose as a corner-stone of the "new" theory the "old"¹ Principle of Relativity, joining in such a way the Newton’s point of view of an "empty space" against Descartes’s "plenum" - exactly as it went unmodified since XVIII century, during the development of modern Mechanics.

This "conservative heart" of SR is even noticeable in the metaphysical interpretation of the theory, that matches very well with the deanthropocentrization process which, started with the birth of modern science, found his climax with Darwin’s revolution (1859)². It would have indeed been rather disturbing "modern philosophy", if man’s ordinary categories of space and time, built only under evolution’s pressure on the Earth’s surface, would have shown themselves useful even for a deeper understanding of the largest universe’s structure. Thus, in some sense, the opposition against SR has something to do with the wider fight between Tradition and Modernity, and it shares some characteristic of the reactions to what is considered the Final Age of Moral Dissolution.

According to this conceptual framework, the present author is not objectively indifferent, and he considers himself firmly rooted in one party. For this reason, since the last 20 years, he has looked with great interest at the activity of the opponents of relativity, and has witnessed the resolute obstructionism of the "establishment" against them³. Nevertheless, he must also acknowledge that, sometimes, even referees defending the "orthodox" point of view are not so wrong, since it happens that many anti-relativistic papers are questionable, as they do either ignore the confrontation with the relativistic approach, or do not show a good understanding of it. This circumstance favours the actual holders of the "scientific power", those who dictate the cultural strategies of Western Civilization, allowing them to discard all issues about the experimental validity of SR. Of course, there is no place for questioning the logical validity of the theory, since it presents itself in the guise of a mathematical theory (naturally, a mathematical theory with physical significance, namely, endowed with a set of codification and decodification rules, which allows to transform a physical situation into a mathematical one, and conversely, but a mathematical theory anyhow), and as that one has to confront it.

Thus this paper is born, with the purpose to collect the most common errors of anti-relativistic physicists - in matters which are sometimes misunderstood even by relativity supporters!, as we shall see - and with the hope to contribute in such a way to make criticism against SR grow stronger, and respected, with the purpose to finally restore the dominion of (ordinary) rationality in science, but not only in it...
 
 

Remark 1 - Since the aim of this paper is just to pursue "scientific truth", and not to feed endless (and sometimes useless, but not always) polemics, particularly with travelling companions, I shall not give, generally, references to the opinions I shall try to disprove, even if all of them can be found explicitely written somewhere.
 
 
 
 

2 - What does "special" means?
 
 

Even if it would appear unbelievable, after almost one century of relativity, the first point which needs to be examined, is concerning what SR really is. As a matter of fact, even during the conference it has been said that SR is that part of relativity which takes under consideration only inertial systems and uniform motions, and that one has to introduce General Relativity (GR) is one wishes for instance analyse Sagnac experiment, of course from a relativistic point of view! This opinion is incorrect, as we shall show even in the next section, and we start here our comments by recalling that GR can be defined as the theory of a general space-time, where by this term one simply means a Lorentz 4-dimensional connected time-oriented manifold⁴. Inside GR, SR is just the special case of a flat space-time, and this means, from a phyisical point of view, of a space-time in absence of gravitation, since in Einstein’s theory gravitation is introduced as an effect of space-time curvature. This shows that, as a physical theory, SR can be applied (successfully or not, this is another matter!) when gravitational effects can be ignored (as well as quantistic ones, but this is again another matter), and not just when uniform motion are involved, and that is (almost) all. One should indeed add that, under "mild" mathematical assumptions, there exists only one SR, according to this point of view. As a matter of fact, one can prove that any two simply connected and complete flat space-times are isometric, and then both isometric to the space R⁴ with its canonical Lorentz structure. One calls this unique (up to isometries) space-time the Minkowski space-time, and from now on we shall frame our relativistic considerations in such a space-time, let us call it M. Of course, M is endowed with privileged coordinate mappings, or systems, (called Lorentz coordinate systems), which are the (time-orientation preserving) isometries M ® R⁴. These are physically interpreted as the coordinate mappings introduced by inertial observers, two of which are completely equivalent, in the sense that they differ up just to an isometry of R⁴ into itself (the transformations of the so-called Poincaré group⁵). This can be considered a formulation of the Principle of (Special) Relativity.

Using these Lorentz coordinate systems, the physical phenomenology pertinent to SR can be easily expressed: for instance the light’s speed turns out to be isotropic and equal to the universal constant c (in this mathematical framework, one puts often c=1) everywhere, namely for all inertial observers, but one can also introduce different (and even just local, that is to say, only defined on an open portion of M) coordinate systems (from a physical point of view, accelerated observers), and things can change very much, as we shall see in the next sections: but it will always be special relativity!
 
 

Remark 2 - The fact that one "practically" never has inertial frames, does not mean anything against the applicability of SR, just because one can use, even in SR, general coordinate systems. Of course, if one is "lucky" enough, he can suppose that his natural physical frame is a good approximation of an inertial one, and then use simple mathematics, but this cannot always be the case, and there is nothing "bad" in it.
 
 
 
 

3 - Sagnac effect
 
 

We have said that, when you ask what SR would predict for observers which are not in uniform motion, you have to introduce coordinate systems of M which are different from the Lorentz ones. We shall now study as an example the famous Sagnac experiment, and the wrong claim that it would disprove SR (or that it would necessarily require GR in order to be explained from a relativistic point of view).

The experimental situation is well known. Suppose to think, in an inertial frame in M (or from the point of view of an inertial observer w , or, better, of a field of them, as we shall see in the next section), of an "observer" a placed on the border of a circular platform P (let us call it C, and R its radius - of course with respect to - from now on: "wrt" - w ), and suppose that a sends two light’s rays along C, in the two opposite directions. When P is still, the two rays cover all the length of C, and come back simultaneously to a after a time interval 2p R/c. Let us suppose now that P, and then a , is rotating (and again, wrt w ) with some angular speed v . It is obvious then that, from w ’s point of view, one light’s ray, the one which travels in the same sense of the rotation, will arrive to a delayed of a time interval 2p R/c times b /(1-b ) (where we have put, as usual, b =v R/c), while the second one will arrive anticipated of an analogous time interval 2p R/c times b /(1+b ). To make it short, we can introduce the ratio k between the two time intervals forwards and backwards, D T_I = 2p R/c(1-b ) ³ D T_B = 2p R/c(1+b ), and see that SR, as besides any other "classical" theory (we can suppose for instance w to be an "aether-frame", or some other absolute frame), would predict an effect, the so-called Sagnac effect, due to the rotation of P. Qualitatively, this means that the two light’s rays do not arrive simultaneously to a ; quantitatively, that the effect is "measured" by the number k = (1+b )/(1-b ) (k ³ 1, and k = 1 if, and only if, v = 0). This k coincides even with the analogous value computed by a , using a ’s proper time (see next section), since one would have then only to modify both numerator and denominator of that fraction by the same factor.

So far, so good, but then somebody adds that SR is now in "contradiction". If SR is true, he says, then even for the observer a , who is now supposed to move with the speed v = v R wrt w , the light’s speed should always be a constant equal to c, and then, from a ’s point of view, no effect should be predicted at all. In other words, SR would predict a k > 1 effect for w , but a k = 1 effect for a , which would indeed be a patent contradiction.

Of course, the previous argument is wrong, since a is not an inertial observer in M, and what would be the "light’s speed" for a is matter to be wholly decided with carefully rigorous definitions and computations. But then, in order to avoid such complications, one gets further with naïve arguments, saying that, even if a is not an inertial observer, he would become such when R is "very large", and then the contradiction in SR would still hold. In order to put this argument in a more precise, and attractive, set-up, we go back to that value k, which is indeed a function of R and v . We can suppose to let R increase up on infinity, and to let v vary in such a way that the product v R is a constant v. At the limit, we would have the physical situation of a platform rotating "very slowly", and of an observer a which could not be considered other than an "inertial" one. This would, apparently!, imply that SR is forced to predict a limit for k equal to 1 (no effect), while k is actually defined, for each value of R!, as a constant, definitively different from 1!

The simple solution to this objection is that SR predicts indeed even at the limit an effect which is given always by the same constant k¹ 1, without any contradiction, and that the misunderstanding simply arises from a non complete mastery of how in SR one has to introduce general coordinate systems, and concepts like the light’s speed in these ones. We shall give a sketch of the situation in the next section, but we end the present one saying that if a could indeed be locally seen as an inertial observer, the same thing cannot be said globally; that is to say, the whole of C would definitively remain outside any inertial (and then global) coordinate system, even approximately. Perhaps, it would be useful to remark that, if we think of the "observer" a as he was a "single man" lying on the platform, with his own clock, we should distinctly realize that when this man is in one point p of the platform, endowed with some vectorial velocity v wrt w , then in this moment he belongs to an inertial system which is very different from the inertial system to which the same man belongs when he is in the antipodal point q of p, since in q he is endowed with the vectorial velocity -v (always wrt w ). Claiming that v and -v are "almost the same", is poor physics and even worst mathematics, since it would simply mean that v is "almost zero", which is indeed the only case of an "almost one" Sagnac effect!
 
 

Remark 3 - There seems to be only one correct "limit argument" in this framework, which goes as follows. Suppose beforehand that all P travels with an uniform motion wrt w , without any rotation, and call for instance w * the inertial system in which P is still (it is obvious that P would not be any longer a "circular platform" wrt w , if it is such wrt w *, just because of lenght’s contraction). Then in w * there is no Sagnac effect, and in force of the Principle of Special Relativity, there would be no effect even in w , at least according to SR. Suppose now to think of P placed in some "big" platform Q, say near the border of Q, the centre of P far from the centre of Q, and at first suppose that both P and Q are still wrt w . In this case, you have no Sagnac effect at all on P. Then make just Q rotate, dragging P "rigidly" with itself: there would be any Sagnac effect on P? Yes, there would be one, and now it is true that, for a large value of the radius of Q (and not of P!), in such a way that the speed v of the border (namely, of P) is maintained constant, the limit of k is equal to 1; that is to say, the Sagnac effect will progressively reduce, until it will vanish ("at the infinity")!
 
 
 
 

4 - The light’s speed for non-inertial observers
 
 

Now we come, as announced, to the sketch of the question (which is often misunderstood even by "orthodox physicists") of what is in SR the light’s speed (in "empty space"!) wrt to a non inertial observer - and let us point out that we shall often use the convention to put c = 1, namely to use geometrical unities. First of all, let us recall that by "observer", in a general space-time S, we must actually mean a future-pointing (smooth) curve a (t ) : I® S (I an open interval of the real line R), such that ds²(a ’) < 0 (one says that a is time-like). If ds²(a ’) = -1 for all t , then the parameter t is called a proper time of a (and a a normalized observer). Then, it must be clear that we cannot introduce any conception of "light’s speed" with respect just to a single observer. First of all, we need an observer field, namely a future-pointing unit (which really means -1) vector field X, whose integral curves would become "observers" (coordinatized by a proper time), according to the previous definition⁶. Then we must introduce, if it is possible, a coordinate system of S adapted to X, by which we mean, if X is defined on the open set U of S, a coordinate mapping of U such that:

1) the coordinate lines xⁱ = constant, i = 1,2,3, "coincide" with the integral lines of X (namely, in each point-event p, the velocity of these trajectories, wrt to the parameter x⁴, is parallel, and equi-oriented, with X(p));

2) the hypersurfaces x⁴ = constant are orthogonal⁷ to X (and then, in particular, are space-like).

It is not always possible to find a coordinate system adapted to an arbitrarily chosen observer field X, and we refer to O’Neill’s textbook (Chap. 12) for details. For instance instead, given any inertial (from a mathematical point of view, this simply means geodesic) observer in Minkowski space-time M, it is always possible to uniquely "extend" it to an inertial global (and complete) observer field X (all X-observers are inertial), and to find, between the many adapted coordinate systems to X, a Lorentz one.

But let us suppose to take from now on such a "nice" field X, and then ask what the light’s speed could possibly be wrt X, namely wrt to any coordinate system adapted to X. It is clear that the "usual definition" speed = space/time cannot work any longer without some specifications, since there would be problems in giving correct definitions both for numerator than for denominator of that fraction⁸. For instance, the difference between the final and the initial coordinate time x⁴ of a light’s travel would not have a physical meaning; not even would have a physical meaning the difference between the final and the initial proper times of the travel, since the X-observers would in general not be synchronized. What one could think of, is to see whether is it possible to choose adapted coordinates which are properly synchronized, that is to say, such that the coordinate time x⁴ acts as proper time for all X- observers, but this is impossible, unless the field is geodesic and irrotational! This implies for instance in SR, that only inertial observers are "good" in this sense, and that there is no hope to introduce such good coordinate systems in Minkowski space-time for accelerated observers.

Anyway, one can say something even in this case. For instance, one can introduce at least an instantaneous light’s speed, taking into account the "splitting" of the metric form ds², according to the natural decomposition induced by X(p) on any tangent space T_p(M) (for each point-event pÎ U). Indeed, under our actual hypotheses, we have, all along the light’s path, ds² = ds ²+g₄₄dT² = 0 (we put T = x⁴, and call ds ² the spatial component of ds², ds ² = g_ijdxⁱdx^j, i,j = 1,2,3), which implies that the instantaneous light’s speed, ds /dT, would be - in this system, and wrt to the coordinate time T - equal to

Ö -g₄₄, and this value, as a matter of fact, could be almost everything (of course, even greater than 1 - see for instance O’Neill’s textbook, pp. 181-183). But one can make use of the proper time of the X-observer defined by X(p), and we can get in this case ds /dt = 1. This shows that, under suitable definitions of space and time, it will always possible to define the instantaneous light’s speed (in the "empty space"), in presence or in absence of gravitation, always equal to the universal constant c - but let us repeat once again that this time "t " could possibly not coincide with any coordinate time. "Doing in such a way, one extends to any possible physical reference frame what in SR was confined only to inertial systems"⁹.

We have thus seen that, in some sense, one could say that the "light’s speed" is always equal to c, in any coordinate system (both in SR and in GR), and this would seem to be a point in favour of people asserting that Sagnac experiment would disprove SR. But this is not true, since the previous definition does not imply that the ratio = space/difference of proper time wrt to a single observer, in the case of a closed light’s path, is necessarily equal to c! In other words, coming back to Sagnac set-up, even if the light’s speed can be instantaneously measured as c wrt to any observer on the platform’s border, this will not imply that the average light’s speed, as measured by one single observer, when light comes back to him along a closed path, is a constant - and it is not difficult to understand it, when one realizes that all proper times of the observers of the "observer field" are not synchronized between them.

One could give many examples of this situation, even in SR, showing that this average light’s speed could be even greater than c, and be time depending (in the sense that an accelerated observer in SR could measure this speed along a closed path in some moment, and then do it again in another moment, using the "same" closed path - a path of the same spatial length - in such a way to obtain in some cases two different results - of course, this would not be the case for an uniform circular motion), but we hope that just what has been said until now would make the reader at least understand that one should be more careful before making physical assertions about relativity. Anyway, we shall now briefly discuss a paradigmatic example (from O’Neill’s quoted textbook, pp. 181-183).
 
 

Example - Let us introduce a simplified 2-dimensional Minkowski space-time M, and a coordinate system (x,t) in it, inherent to an inertial observer w (thus, we shall have ds²=dx²-dt²). It is well known that, in relativistic kinematics, one cannot introduce uniformly accelerated motions (parabolic) without restrictions, because any speed must always maintain itself less than 1 (c). The analogous of such a motion is an "observer" a defined by the (hyperbolic) equations: x = g^-1cosh(gt ), t = g^-1sinh(gt ) (for each g> 0). One can immediately verify that t is indeed a proper time for a (ds²(a ’) = -1), and that ds²(a ’’) = g². a can be extended to an observer field with the following adapted coordinates (x*,t*) : x* = (Ö (x²-t²))-g^-1, t* = g^-1tgh^-1(t/x) [the inverse transformation is x = (x*+g^-1)cosh(gt*), t = (x*+g^-1)sinh(gt*)]. As a matter of fact, for each value of the parameter t , the Lorentz-orthogonal line to the velocity a ’(t ), in the point-event a (t ), defines the point-events which are simultaneous to a (t ), and the (positive definite) quadratic form ds² restricted to such a line defines spatial "distances" in our accelerated system. The observer a is defined in these new coordinates simply as x* = 0, and if we introduce an analogous observer a _L : x* = L (say for instance L> 0; - a _L is defined as x* = L in the new coordinates, and as x = (L+g^-1)cosh(gt*), t = (L+g^-1)sinh(gt*) in the old ones), then the "distance" L between a and a _L remains constant (in this new coordinate system), as the coordinate time t* varies. Let us now try to compute the speed’s light in this system. Suppose to take a photon’s path, in the coordinates (x,t), x = t+g^-1 (the photon starts at x = g^-1, t = 0, which is exactly a (0)), and to transform this equation in (x*,t*). We shall have x* = (e^gt*-1)/g, and then dx*/dt* = e^gt*, which shows that the actual light’s speed would approach infinity as t* does. As a matter of fact, the ds² expression in the new coordinates is simply ds² = (dx*)²-(gx*+1)²dt*², and then, with reference to the new notations, and to the aforesaid formula ds ²+g₄₄dT² = 0, we will have at last ds /dT = dx*/dt* = Ö -g₄₄ = gx*+1, which gives exactly e^gt* when one makes the substitution x* = (e^gt*-1)/g.

But, as we have said, this "speed" has not a great physical meaning, since the coordinate time t* coincides with a proper time only for the observer a : for instance, as far as a _L is concerning, the coordinate time t* is no longer a proper time (but for L=0), since: ds²((gL+1)sinh(gt*), (gL+1)cosh(gt*)) =

-(gL+1)². It is now obvious that, if we define the instantaneous photon’s speed, when the photon reaches a _L, not as the previous value dx*/dt*, but as the value dx*/dt _L, where t _L is a proper time for a _L, then we shall get dx*/dt _L = 1, as asserted. On the other hand, we have already remarked that it is impossible to find a coordinate time which could act as a proper time for all observers in this non inertial system!

If we want to learn more from this example, we can send a photon from a to a _L, and then back, and try to compute the value 2L/D t , where D t is the elapsed proper time interval wrt a , in such a way to obtain a value for this average light’s speed (which is also called radar speed). An easy computation shows that 2L/D t is independent of t (but dependent of L, and this is rather paradoxical indeed), as it is equal to gL/log(1+gL). Remark that the "limit" of this expression, as g ® 0, is exactly 1, and that this value can be as well greater than 1 (c).

This shows that the average light’s speed, in SR, wrt to an accelerated observer, needs not to be a constant, and can even be greater than c.

We conclude this long digression saying that, if one makes this same computation exchanging rôles between a and a _L, one would obtain the value gL/(1+gL)log(1+gL), which is different from the previous one (one will find "symmetry" only using the coordinate time t*, instead of the proper time t _L). Moreover, if one sends a photon from x*=0 to x*=L, and then back, and makes the same operation wrt x*=0 and x*=-L (0< L< g^-1), then he would get two different values for the average light’s speed, namely gL/log(1+gL) and gL/log(1-gL). All this shows the existence of obvious optical anisotropies in the accelerated system, which would consent to these observers to realize that they are not inertial, without any violation of the Principle of Special Relativity.

For this same reason, it appears even hopeless try to persuade relativistic physicists to give up the theory, just pointing out at small "anti-relativistic" effects which can be found (or have already been found) in experiments performed in a terrestrial laboratory: these could always be ascribed to the Earth’s diurnal rotation, or to some other non-inertial feature of the aforesaid system¹⁰!
 
 

Remark 4 - The "optical anisotropies" (and other peculiarities) displayed in the previous example are of the same kind which can be found in presence a gravitational field, in force of the so-called Principle of Equivalence. One could moreover observe that the two previous observers a and a _L, far from remaining at the "same distance", are instead approaching each other from the point of view of w (and, as for that matter, from the point of view of any other inertial observer w *), which shows that one should expect also the existence of a Doppler effect in the previous situation. In force of the aforesaid principle, this effect should then be predicted even in the gravitational case.
 
 

Remark 5 - The example above could be used even to point out another common misunderstanding concerning "terminology", which consists of the belief that the "essence" of SR relies in the assertion that any "velocity" cannot be other than "relative" (which is true), and then going from this statement to the belief that an "apparent relative velocity" wrt to some observer (just introduced as an ordinary "difference" between spatial velocities), has an "absolute" meaning in SR¹¹. Once again, one has to be very careful with definitions, since the two previous observers a and b have indeed a "relative velocity" which is equal to zero in the accelerated coordinate system, but is different from zero wrt w (it is even possible to think of two uniformly accelerated observers which have zero relative velocity wrt to w , but different from zero relative velocity wrt to another inertial observer w *)! This is due once again to the fact that, in order to measure this apparent relative velocity, it is necessary in advance to measure a relative distance, and this cannot be done independently of some definition of simultaneity.
 
 

Remark 6 - One should say that, in some sense, the misunderstandings about the use of accelerated systems in SR (and the Sagnac case is one of these) are rooted in a misinterpretation of the meaning of the "special" Principle of Relativity, versus a "general" Principle of Relativity, which as such will hold, as a particular case, even in SR! Both principles never assert that all observers are in all sense "equivalent": there will always be, either in SR or in GR, "privileged" observers and coordinate systems, in which "physical laws" have a "simpler" mathematical expression, and physical phenomenology can be much more easily describable (as we have seen, in a general system one would have anisotropy versus isotropy, presence of Doppler effects in the case of a zero relative velocity, an so on - which would mean, for instance, that some possible assertion about isotropy of light’s speed would not be a "physical law"!). Nevertheless, physical laws will be described for all of them under a Covariance Principle. For instance, in SR, and for general observers, this principle simply states that a photon’s path is a null geodesic, ds² = 0, which admits, but only in Lorentz coordinates, the well known expression dx²+dy²+dz²-dt² = 0 - while, in other coordinates, one would simply have the general equation g_ijdxⁱdx^j = 0, i,j = 1,...,4, with 10 addenda in place of the previous 4. We shall examine this kind of misinterpretations even in the next section.
 
 

Remark 7 - I am fully aware that some of my readers will dislike all these "mathematical details", and even this rather paradoxical Physics - in which for instance a constant instantaneous speed does not give rise to a constant average speed - and as a matter of fact I acknowledge that it is perhaps possible, in principle, to "do" Physics without all this mathematics, or with a different (simpler?! less?!) one; but then these same physicists must also realize that it would be absolutely impossible for them to confront at last their theories with relativity, either special or general.
 
 
 
 

5 - The "Principle of (Special) Relativity" and the "twins

paradox"
 
 

Between the arguments connected to SR which still today enjoy a great popularity, one must indeed include time dilation, which is commonly described as one the most noticeable consequences of the theory. One of the most known formulation of this phenomenon is the so called twins paradox, which was introduced by the French physicist Paul Langevin in 1911¹². The essence of the paradox, and its "explanation", are contained in almost all relativity textbooks, but we take advantage of the set-up given in the previous section in order to describe the situation in a rather unusual fashion.

We consider the same 2-dimensional Minkowski space-time M as before, and the same coordinate system (x,t) inherent to an inertial observer w (we suppose that w is exactly x = 0). For each L>g^-1>0, we introduce the following uniformly accelerated normalized observer a :

x = L-g^-1cosh(gt ), t = g^-1sinh(gt ).

While w remains still in this system, a "shares" with him the point-event e₁ = (0,-Ö (L²-g^-2)) - the intersection between the line x = 0 and the hyperbolic branch (x-L)²-t² = g^-2, x<L, t(e₁)<0; then he goes away from w in the direction of the increasing x, until he reaches the (spatial) point L-g^-1, in the time

Ö (L²-g^-2). Afterwards, he comes back approaching w , and meets him again in the point-event e₂ = (0,Ö (L²-g^-2)), the other intersection between the line x = 0 and the aforesaid hyperbolic branch.

When one compares the elapsed proper times, both for w and a , between these two point-events, one easily discovers that: D t w = t(e₂)-t(e₁) =

2Ö (L²-g^-2); D t a = t a (e₂)-t a (e₁) = 2g^-1sinh^-1(Ö (g²L²-1)), which implies

D t w > D t a . Namely, when a and w meet again, a finds w older than him (putting for instance L=1, and for a large g, D t w is almost equal to 2, while D t a is infinitesimal).

If one introduces the equation dt a = Ö (1-va ²)dt, which connects the infinitesimal proper time dt a with the infinitesimal coordinate time dt, and integrates dt a on the portion of the hyperbolic branch going from e₁ to e₂, one gets, of course, the same result as before, but the formula
 
 

D t a = INT (dt a ) = INT (Ö (1-va ²)dt)
 
 

has now the advantage to emphasize the dependence of the twins effect on the speed va , besides than on the length of the a ’s trajectory.

This can sound "strange" indeed, and undoubtedly far from the ordinary conception of "time"; but this is SR, and one must accept this result as one of the consequences of the relativistic conception of Nature. Thus, why some people says that this argument reveals an inner contradiction of this theory?

Perhaps the most famous form of this objection is the one advanced by Herbert Dingle, under the logical appearance of a syllogism:
 
 

1 - According to the postulate of relativity, if two bodies (for example, two identical clocks) separate and re-unite, there is no observable phenomenon that will show in an absolute sense that one rather than the other has moved.

2 - If on re-union one clock were retarded by a quantity depending on their relative motion, and the other not, that phenomenon would show that the first has moved and not the second.

3 - Hence, if the postulate of relativity is true, the clocks must be retarded equally or not at all: in either case, their readings will agree on re-union if they agreed at separation¹³.
 
 

But, as we have just seen, assertion 3 is plainly false, and then there must be something wrong either in assertion 1 or in assertion 2. The simple solution of this "riddle" is that the "postulate of relativity", either special or general, does not state that presumed complete symmetry between the two clocks¹⁴. The misunderstanding is rooted in the belief that, if a moves away from w and then re-unites, then, from a ’s point of view, it is w instead the one which movea, and then re-unites, exactly in the same symmetric way. This is not true, because of the very different paths which the two observers are travelling in space-time: one is geodesic, the other definitively not¹⁵.

One can hope to persuade irreducible critics showing what is the motion of w from a ’s point of view. Acting as before, we can introduce non-Lorentz coordinates (x ,t ), connected with (x,t) by the transformation:

x = Ö ((x-L)²-t²)-g^-1, t = g^-1tgh^-1(t/(L-x)),

whose inverse is:

x = -(x +g^-1)cosh(gt )+L, t = (x +g^-1)sinh(gt ).

x <0 represents the internal portion of the hyperbolic branch under consideration, while t is a ’s proper time (a is actually described by x = 0).

Then w ’s motion, as "seen" by a , is:

x = 0 = -(x +g^-1)cosh(gt )+L ® x cosh(gt ) = L-g^-1cosh(gt ) ®

x = L/cosh(gt )-g^-1.

This parameter t is not w ’s proper time. If we want to know dt w , namely dt, in the new coordinates (x ,t ), we must directly compute:

dt = (¶ t/¶ x )dx +(¶ t/¶ t )dt = (gL/cosh²(gt ))dt ,

which shows that dt is not equal to Ö (1-vw ²)dt, where vw is the speed of w in the new coordinates (this speed is given by dx /dt = -gLsinh(gt )/cosh²(gt ))¹⁶.

It is necessary to explicitely compute the integral dt = (gL/cosh²(gt ))dt , from e₁ to e₂, in order to be persuaded that one gets the same value D t w = t(e₂)-t(e₁) = 2Ö (L²-g^-2), which was obtained before? Namely, as w travels from e₁ to e₂, he takes a time, wrt a , which is bigger than the corresponding D t a , and not smaller, with a complete asymmetry as to the previous case. No "twins effect" from a ’s point of view: w is moving wrt a , but he does not become "younger".

It is not easy to understand the permanence, and the "fortune", of such pseudo-arguments, which every now and then spring up again, even in the following form of a "proof" per absurdum: the Principle of Relativity implies that some phenomenon (time dilation) cannot be true; this phenomenon is experimentally confirmed; ergo, the Principle of Relativity cannot be true¹⁷! I suppose that this conclusion is very likely physically correct (see the final section of this paper), but unfortunately it will not be so easy to prove it!!

In conclusion, if the time dilation is a real natural phenomenon, then "classical physicists" must find a way to explain it (perhaps an effect of an absolute velocity wrt to the aether?!), but they should stop to believe that the twins paradox have an antinomic value inside SR.
 
 

Remark 8 - We have avoided to comment in this paper "critics" of the kind: in relativity c+v must be equal to c, but this is "clearly" impossible if v ¹ 0 (as Einstein would have changed Algebra’s law!), but there is an "analogous" argument which I have seen many times in action, and then I wish to dedicate a few lines to it. With the previous notations, let us suppose that a is an inertial observer too (different from w , and meeting him in the point-event (0,0)). The a ’s speed wrt w is some v ¹ 0, the w ’s speed wrt a is -v, everything is all right. But dt a = (Ö (1-v²))dt, while dt = dt w = (Ö (1-(-v)²))dt a , and from these two relations, which are in some sense both quite correct, it would follow: dt a = (Ö (1-v²))dt = (Ö (1-v²))(Ö (1-v²))dt a , and then the "contradiction": (1-v²) = 1 ® v = 0! It is obvious that here we are just in front of an unfortunate case of incomplete notation, and not of a "logical inconsistency", since one should have written more accurately the two previous equations as dt a (a ’) = (Ö (1-v²))dt(a ’), dt(w ’) = dt w (w ’) =

(Ö (1-v²))dt a (w ’), and these two rigorous formulations would have not allowed such a misunderstanding¹⁸!
 
 
 
 

6 - Roemer observations
 
 

I shall discuss now the wrong claim¹⁹ that SR would not be able to explain Roemer observations (1675).

Let us synthetically recall the situation. In an (inertial) reference frame R centred in the Sun, suppose that the period of a Jupiter’s (J) moon (M) is equal to T. Suppose to observe this period from the Earth (E), when our planet is going away from J, in a position in which the Earth’s orbital velocity is (almost) parallel to the line EJ. Let t₀ be an instant in which M appears from Jupiter’s shadow, and L be the distance between J and E in this very moment. It is obvious that one shall start to observe M from E in the time t₀+x, where x = (L+vx)/c (since E has moved away from J, with speed v, during the time x, and one can suppose J, which is much slower than E, approximately stationary during all these observations). From this equation, one gets cx = L+vx, and then x = L/(c-v).

In the same way, when M appears again, in the instant t₀+T, one shall see it from E after a time interval y such that
 
 

from which one gets y = (L+vT)/(c-v).

The period T’ of M, as seen from E in this configuration, will be then not equal to T, but rather to:

(t₀+T+y)-(t₀+x) = T+(L+vT)/(c-v)-L/(c-v) = T+vT/(c-v) = cT/(c-v),

that is to say:
 
 

Suppose then to do the same observation 6 months later, when one can approximatively say that Jupiter is still in the same position as before, but now the Earth is approaching J, with the same speed v (and velocity v once again almost parallel to the line EJ).

One can argue as before, and get for the new period T", seen from E in this position:
 
 

whence it follows:
 
 

which allowed Roemer to estimate c, knowing v and T (which is the period of M which appears from E in the two positions in which the Earth’s orbital velocity is approximately orthogonal to the line EJ), and after carefully measuring T’ and T".

As one can see, in the previous argument one makes no use at all of any composition of velocities. Moreover, the effect which has been described is exactly the same thing as a Doppler effect, since one is in front of a cyclic phenomenon, which is seen from a moving observer, in the first case moving away from the source, and in the second one approaching the source (of course, this does not mean that the phenomenon which one is actually appreciating is an optical Doppler effect, namely a "shift" in the frequency of the light coming from Jupiter’s moon²⁰!).

What is the relativistic description of this same phenomenon? Almost exactly identical as before, since we have the same inertial reference frame R in which we can do all computations we need. Is there any difference between the relativistic and the classical approach? Yes, there is one, since all the values which we have previously calculated, are referred to R, and this mean for instance that T’ is not exactly the period of M as seen from E, when it is in the first position we have considered, since the observer moving with E belongs to a different (almost) inertial reference frame R’, moving with a scalar velocity v with respect to R. In order to deduce the relativistic values of the (predicted) periods as really measured from E, one has simply to apply a Lorentz transformation, and then get, instead of the previous T’, the value T’Ö (1-b ²), which is the simple effect of the relativistic time dilation. In the same way, one has to take, in the second position, the value T"Ö (1-b ²), instead of T" (E belongs now to a third inertial reference frame, moving with speed -v with respect to R!), and at last one gets the relativistic Roemer effect
 
 

This is equal to the classical Roemer effect D T_c previously calculated, but for the factor Ö (1-b ²), which gives a very small difference (as a matter of fact, a difference only up to b ³):

D T_r = D T_cÖ (1-b ²) = 2b TÖ (1-b ²)/(1-b ²) =

= 2b T/Ö (1-b ²) » 2b T(1+b ²/2) = 2b T+b ³T

D T_c = 2b T/(1-b ²) » 2b T(1+b ²) = 2b T+2b ³T.

In conclusion, both SR and "classical theory" qualitatively predict the same effect, with a quantitative difference which is impossible to experimentally detect, and that is all...
 
 

Remark 9 - Of course in SR one can also do computations directly in the two different reference frames R’ and R"! In this case, it is J which, in the first case, moves away from E, and in the second one moves forward E. The light’s speed is always c, according to the II principle of SR (the light’s speed is independent of the source’s motion), and in order to get exactly the same formulas as before one has to make use now of the relativistic Doppler effect (the previous T would actually play the role of what one call the proper period).
 
 

Remark 10 - Talking about mistakes, unfortunately so frequent in Physics literature, one should point out that, both in Sommerfeld’s Optics (Chap. II), and in Bohr’s Einstein’s Theory of Relativity (Chap. IV, Sec. 3), the formulas given for the "Roemer effect" are not precise, since the approximated espressions (up to higher order terms in b ) are considered exact. For instance, Sommerfeld writes, in our terms, T’ = T(1+b ), instead of the correct one:

T’ = T/(1-b ) » T(1+b ).
 
 
 
 

7 - Bradley aberration
 
 

As in the previous section, the annual displacement of a star due to Earth’s motion around the Sun, was quite well explained using classical concepts²¹, which allowed to estimate light’s speed c, once known the orbital Earth’s speed v, or, conversely, to estimate v (or, which is the same, the value of one Astronomical Unity), once known c.

As usual, the possibility to explain aberration using "ordinary" composition of velocities, even in the case in which one of these is the velocity of light, induces some people to believe that SR cannot explain this phenomenon, or must "strain the truth" in order to do it. This is of course not true, even if many modern textbooks do not give indeed an account of aberration which can be considered completely correct. For instance, the well known Italian physicist Piero Caldirola honestly acknowledges: "The study of such phenomenon is given in all textbooks ... But having ascertained that the discussion that usually is given for the [relativistic] computation of the aberration is not quite correct ... we believe useful to briefly exhibit here the relativistic treatment of this phenomenon..."²².

The starting point for understanding relativistic aberration is to carefully distinguish between speed (scalar velocity) and velocity (vectorial velocity), which in some language is not possible - for instance, in Italian we have just one word, "velocità", like in German one has only "geschwindigkeit". With this specification, SR second postulate prescribes the light’s speed to be independent of the motion of the source (in any inertial frame), but not the light’s velocity, which in fact can depend on the velocity of the source.

As a simple example, let us take a photon’s path, travelling backwards along the y-axis (the photon is supposed to start at time t = 0 from some undetermined distance L>0):
 
 

If you imagine the "usual" observer travelling along x-axis with some uniform velocity (v,0,0), endowed with a coordinate system (x’,y’,z’), then you must use in SR a Lorentz transformation in order to connect coordinates (x,y,z) and coordinates (x’,y’,z’):
 
 

and the motion of f becomes, in these new coordinates:

x = 0 ® (x’+vt’) = 0 ® x’ = -vt’,

y = L-ct ® y’ = L - c(t’+vx’/c²)/Ö (1-b ²) ® y’ = L-ct’Ö (1-b ²),

z = 0 ® z’ = 0.

These equations show that, in the new coordinates, the photon’s velocity is

(-v,-cÖ (1-b ²),0) (of course, the photon’s speed is always c, since

v²+c²(1-b ²) = c²!), which clearly does depend on the velocity of the source (in the system (x’,y’,z’), this is obviously equal to (-v,0,0))²³.

This is the reason for relativistic aberration, since the light coming from the source will be received by the "moving observer" shifted under an angle q such that
 
 

and that is (almost) all.
 
 

Remark 11 - When one studies astronomical aberration from the relativistic point of view, one can take the solar system as a first reference frame, and then two more different inertial systems connected with the Earth, for instance at 6 months distance, in such a way that once the speed of the Earth, considered as a "moving observer", is v, and the other -v. This implies that aberration is a function of 2v/c = 2b , and this makes vanish even another common misunderstanding: since aberration in SR must depend only upon the relative velocity between the source and the receiver, how is it possible that, at last, only the Earth’s orbital velocity in the solar system appears responsible for it? Well, if it is true that a possible (transversal²⁵) velocity of a star (with respect to the solar system) would modify the angle between the "real" position of the star and the position in which it is seen from the Earth in a given moment, it is also true that we have no way to know, from our planet, this real position, and that the only one thing which we can possibly appreciate is the difference between two apparent positions of the same star say after a time interval of 6 months. This implies, as we have just said, that the final effect depends only on 2v, and not on the velocity of the star, which can be supposed to be always the same after 6 months. Only in the contrary case we would have a different value for the aberration of some single star.
 
 

Remark 12 - It is paradoxical (for people to whom this section is intended) to underline that it is not so much aberration to be a problem for SR, rather than it is aberration to be a problem for (some) aether theory! As a matter of fact, at first sight it would seem that aberration would prove that Earth is really moving "through the aether", and then it is a widespread opinion that this phenomenon would show that Stokes "aether dragged" theory²⁶ is not maintainable; but we wish to point out that this is not correct, as it has been shown very clearly by G. Cavalleri et al.: "Some special relativity textbooks assert, without giving a detailed history of the question of aberration, that Stokes theory is wrong [...] their argument is grounded on a misunderstanding: precisely, the aether which they consider is not irrotational"²⁷. In conclusion, Bradley aberration does not prove that Earth is really moving through the aether, but just that the Earth (if you prefer, the aether!) is moving around the Sun, perhaps dragged by the "gravitational vortex" imagined by Descartes...
 
 
 
 

8 - Is it true that electromagnetism is relativistic? That

"Classical Physics" is the "limit" of SR for low speeds?
 
 

Until now, this paper could appear a kind of "mémoire en défense" of SR, rather than a critical analysis of it, as his author would have preferred. Thus in this last section we shall discuss some other common convictions concerning SR, which appear no better founded than the others we have hitherto examined, and which will hopefully point out the very heart of relativity, the only one which anti-relativistic physicists (or philosophers) should try to "attack".

As a matter of fact, there is an important epistemological aspect of SR which is often not appraised as instead it would deserve: namely, its (partial) conventionality, which truly removes the theory from the strict realm of an experimental science. This conventional nature of SR is clearly manifested in its second postulate, which one can interpret rather as a methodological suggestion for coordinatizating space-time (that is to say, for synchronizing distant clocks), than as an experimental datum. Since it appears that one has to perform this task before measuring any velocity, here it is that one could propose to conventionally assume the light’s speed to be isotropic, and synchronize clocks in such a way that this characteristic will obviously be experimentally confirmed. Of course, this does not exhaust the content of the II postulate, since for instance it excludes, on a presumed experimental basis, the so-called ballistic hypothesis, but this is certainly "enough", thus inducing most physicists to believe that any alleged experimental confutation of SR is defective under this aspect. As we have already said in footnote 8, anti-relativistic physicists often try to avoid such objections using concepts as "rigid rods" and so on (with whose help introduce surreptitious simultaneity, and as it were "invariant" concepts which are not), but all the same do not succeed in getting that attention which they demand. If this is in some sense true, it is even true that SR consists of two postulates, and that in fact it is this first postulate which appears as a good conceptual foundation even for the second one, granting it trustfulness, even if there is not a strict implication relationship between the two²⁸. This shows that it should be the first one the most questioned on the experimental ground, but that notwithstanding, unfortunately, the anti-relativistic criticism does mostly concern the second²⁹.

As a matter of fact, the Principle of Relativity only seems commonplace when we forget about the peculiarity of asserting "no matter what is your reference frame" for light phenomena: in principle, it would instead be reasonable to claim that light does have a preferred reference frame (like sound), and only very strong evidence should impose the contrary viewpoint. What evidence was available to Einstein (and, as for that matter, to us nowadays)? Apart from the "unsuccessful attempts do discover any motion of the Earth relatively to the light-medium", so sparingly referred to, he lead the main emphasis on the well known induction phenomenon. But is it really true, as he affirmed in the first lines of his 1905 paper, that electromagnetic phenomenology is not affected by uniform motions? Have we enough experimental evidence to this regard? The example that he pointed out (which was not really founded on an experimental ground, as it was nothing more than an exercise in Maxwell Theory - MT), indeed predicts identity of the two inductions, but this could have been accidental. Apart the fact, ignored by most textboks, that there exist different electromagnetic theories challenging each other, and that it is not so clear that Maxwell’s approach will definitively prevail against all the others, even in MT, classically interpreted (namely, without introducing a priori length contractions and time dilations) symmetry is not the norm, and that of the induction is more an exception, than a rule. This "classically interpreted" MT is not relativistic in essence³⁰, and relativity can spring up in it only when relativistic ingredients are inserted in it in advance. We cannot here but send the interested reader to the already quoted U. Bartocci & M. Mamone Capria’s paper (which has been used in this section without explicit references), in which it is clearly shown also that the validity of the Principle of Relativity in electromagnetism could be checked with low-speed experiments, and this is once again in disagreement with which one generally believes³¹. The reason for this "misunderstanding" is quite evident: in the fundamental relativistic equation F = d/dt(mv), it is true that for low speeds the right-hand side of this equation is "almost equal" to the classical one, but the most important point is concerning the left-hand side, namely the expression of the forces. There are forces which are predictable in some theories, and not in others, for instance there are electromagnetic forces which arise in classical MT from a uniform motion, and which obviously do not arise in SR, and then the final word in this field must just be left to direct experiments³².
 
 

^* Dipartimento di Matematica Università, Via Vanvitelli 1, 06100 Perugia - Italy (bartocci@dipmat.unipg.it). The author wishes to thank most heartily the Cartesian philosopher and good friend Rocco Vittorio Macrì, for very stimulating discussions, and Prof. Giancarlo Cavalleri, for his friendly revision of a preliminary version of this paper.

¹ As it is well known, the conviction that an uniform motion does not affect any "physical phenomenology" was already perfectly exposed by Galileo in his celebrated "ship’s argument", but we can find the same conviction (using moreover the same argument!) in Giordano Bruno, Johannes Rheticus, Nicholas Krebs,... and even in ancient science (Lucretius, Seneca,...). But does this principle really express some essential "natural truth", or is it just a theoretical expedient, which was brushed up during the great debate about the Copernican system, in order to give some reason for the unperceived Earth’s motion? This historical utilization has still today psychological influence ("Eppur si muove"), for instance in the interpretation of the famous Michelson-Morley or Trouton-Noble experiments, which tried to look at the elusive "absolute velocity" of a terrestrial laboratory. Could one say instead: "Eppur non si muove"?! On the contrary, this principle very likely holds just approximately in Mechanics, and its extension to optical and electromagnetic phenomena appears the more arbitrary the less it is supported by suitable direct experimental evidence.

² Of course, when we make use here of the term "conservative", we intend just to emphasize the continuity’s relationship between SR and the "modern science" of the last 4 or 5 centuries, and not to mean that SR is really conservative in a most proper sense.

³ One can find many evidences of this obstructionism in the famous dissident physicist Stefan Marinov’s publications.

⁴ As a good reference textbook one can quote B. O’Neill’s Semi-riemannian geometry (Academic Press, 1983).

⁵ The Lorentz group is a (large) subgroup of Poincaré group, consisting of those transformations which fix the point-event (0,0,0,0) (homogeneous isometries).

⁶ Of course, it is quite clear that the extension from one single observer to a "continuous infinity" of them is far from being unique, and that this extension is very often decided by the "physical problem" one is trying to adapt to the theory. Anyway, there is always a local unique extension, which uses as a "coordinate time" an extension of the proper time of the primitive observer (we shall very soon see an application of this concept).

⁷ This assumption is not strictly necessary, but we are trying to make things the simplest as possible.

⁸ One example is given by the question known as the Ehrenfest paradox: what would be the "length" of the platform’s border C as "measured" by the observer a ? The "radius" of C should not change, but the border of P should experience length contraction. Would that means that the value of p would change on the rotating system? This is not really a "problem" for SR, in which one talks just of transformations between "measured values", and could always assert that some global measure is not defined; but this could be a problem for some "relativistic aether theory", like for instance Lorentz theory, in which true Lorentz-Fitzgerald contractions are predicted. It is this kind of problems that naïve anti-relativistic physicists try often to avoid, by means of introducing in their arguments "rigid" bodies as trains, spaceships, platforms, and so on (on this point, see also the final section).

⁹ From C. Cattaneo’s Introduzione alla teoria einsteiniana della gravitazione, Roma, 1961, p. 158.

¹⁰ As for that matter, according to SR, even the fact that the Earth is not inertial in its motion around the Sun, should indeed be experimentally detected! (and in fact it is so, see the next section 7, concerning aberration). Going further in this direction, we would enter into the realm of a Physics of precision, in which one is compelled to compare very small effects, and this does not seem the best strategy to fight the theory - even because physicists know very well what the real value of experiments is, at a very near to zero quantitative level. As Einstein once said, or at least one says so: nobody believes in a theory, but its promoter, all believe in experiments, but their performers (we could remark that: all declare to believe in experiments)...

¹¹ This could be perhaps the reason for another common misinterpretation of the "relativity principle", in its different degrees of generalizations, which confuses this principle with the relative motion principle. We shall not make a comparison of these two principles in this paper, sending the interested reader to: U. Bartocci, M. Mamone Capria, "Symmetries and Asymmetries in Classical and Relativistic Electrodynamics", Foundations of Physics, 21, 7, 1991, pp. 787-801.

¹² "L’évolution de l’espace et du temps", Scientia, Vol. 10, 1911, pp. 31-54. Nevertheless, Einstein himself had already paid attention to this "phenomenon" in his first fundamental relativistic essay (1905), and in a subsequent paper (1911).

¹³ From H. Dingle’s Science at the Crossroads, Martin Brian & O’Keeffe, London, 1972, p. 190.

¹⁴ This obvious comment was immediately put forward to Dingle by W.H. McCrea (ibidem, pp. 190 and 240-245): "In Professor Dingle’s letter, his statement (1) is demonstrably false".

¹⁵ Sometimes, "critics" try to avoid this difficulty considering a motion the "most possible inertial", introducing a polygonal path (a broken geodesic): but this is still non geodesic, even is made up with geodesic components!

¹⁶ The infinitesimal proper time interval is defined (only for trajectories with ds² < 0) as dt = Ö (-ds²), and, from this expression, the well known relation follows immediately, but only in a Lorentz coordinate system: dt = Ö (-ds²) = Ö (-dx²+dt²) = (Ö (-v²+1))dt.

¹⁷ With the real paradoxical consequence, from a logical point of view, that the principle of relativity would be able to predict a true natural phenomenon, but this fact, instead of being considered a point in favour of the theory, would on the contrary prove that the theory is wrong!

¹⁸ If we call, as usual, (x’,t’) the Lorentz coordinates inherent to a , we would have indeed dt = dt w and dt’ = dt a . From the Lorentz transformation x = (x’+vt’)/Ö (1-v²), t = (t’+vx’/c²)/Ö (1-v²) (with the actual notations b = v/c = v), we would have for instance dt = (¶ t/¶ x’)dx’+(¶ t/¶ t’)dt’ = v/Ö (1-v²)dx’ + (1/Ö (1-v²))dt’, which implies the previous equation dt = dt w = (Ö (1-(-v)²))dt a = (Ö (1-v²))dt’ only if x’ = -vt’, dx’ = -vdt’ (which is indeed the motion of w wrt a ), and that is all.

¹⁹ It would have been perhaps just enough to say that, if this was true, then relativity would have not been born at all!

²⁰ Of course, even this kind of Doppler effect should have to manifest itself, but it would be very small indeed...

²¹ But one often forgets that the original "simple" Bradley’s explanation (1728) required the corpuscular nature of light (as Bradley belonged to the "Newtonian" party!), and that one had to wait till 1804, before Young succeeded in giving a wave-theoretical explanation! (see for instance E.T. Whittaker’s History of the Theories of Aether and Electricity, Dublin University Press Series, 1910, Chap. IV, p. 115).

²² "Applications and experimental verifications of Special Relativity", at p. 395 of the volume dedicated to the 50 years of relativity, Sansoni, Firenze, 1955.

²³ This phenomenon could perhaps be the conceptual ground for some experiment aimed to compare SR predictions with analogous aether-theoretic expectations, since one could suppose that it would be natural, in an aether-frame, to have total independence of light’s velocity of the velocity of the source. One could think for instance to use the circular platform of section 3 for sending a light’s ray from a directional laser source, placed in the border of the platform, towards the centre, and then to check whether this ray arrives exactly in this point, or it is instead "dragged" from the velocity, as SR would predict!

²⁴ And since b is of course "very small", one can directly approximate q » b , as it is usual.

²⁵ The radial velocity would be responsible instead for the relevant part of the Doppler effect, which is known, in the general case, as the red-shift.

²⁶ With this term one could designate any aether theory which claims that the relative velocity aether-Earth is equal to zero - at least at Earth’s surface, and possibly not taking into account the Earth’s diurnal rotation velocity - even if in the Descartes-Leibniz vortex theory it would be better to speak instead of a "dragging aether" theory!

²⁷ "Esperimenti di ottica classica ed etere - Experiments of classical optics and aether", Scientia, Vol. 111, 1976, pp. 667-673. By kind courtesy of Prof. G. Cavalleri, we hereafter quote this remark: "However, Stokes aether is unable of explaining the transversal property of electromagnetic waves. On the contrary, Stokes-Planck compressible aether would imply the existence of longitudinal electromagnetic waves. That is why we can exclude the theory of an irrotational, compressible aether, as said at the end of the previous reference".

²⁸ By the way, if one accepts the validity of Maxwell electromagnetism (in inertial frames), then the invariance of light’s speed (in these same frames!) directly follows from Maxwell equations plus principle of relativity.

²⁹ In general, it appears that people is willing to readily accept the first SR postulate (in force of the historical suggestions pointed out in the previous footnote 1?!), while being critical on the second, which is admittedly counter-intuitive, and difficult to "understand".

³⁰ It should be made indeed very clear for instance that the finite speed of propagation is not an exclusive character of the relativistic point of view, but simply of Maxwell equations, which, as we have said, can as well been used in a "classical" context. As for that matter, anyway, the prediction of the retardation in the interactions is quite typical of an "aether theory", rather than of a theory which does not introduce such a concept, since then the effects are transmitted through the underlying medium, and the characteristic speed c can be for instance interpreted as a function of some of its physical properties. In other words, instantaneous actions at a distance are much more coherent with the Newtonian point of view of an "empty space", rather than with the Cartesian one, and this shows once more that SR is really a sort of an unpleasant hybrid between these two.

³¹ With some important exception: this circumstance was emphasized for instance in A.P. French’s known textbook, p. 259 (Special Relativity, MIT Press, 1968). The point is that the actual "history of Physics" is prejudiced by some evolutionistic postulate, according to which there are no confutations of "old" theories (for instance, Newtonian versus relativistic Mechanics), but just simple improvements in the degree of approximation ("Principle of Correspondence"), or extensions of the phenomenology. Of course, this point of view is rather questionable, and we could refer the interested reader to the important M. Mamone Capria’s studies: "The Theory of Relativity and the Principle of Correspondence", Physics Essays, 8, 1994, pp. 78-81; "La crisi delle concezioni ordinarie di spazio e di tempo: la teoria della relatività", in La costruzione dell’immagine scientifica del mondo, La Città del Sole, Napoli, 1999, pp. 265-416; "Newtonian Physics and General Relativity: Reflections on Scientific Change", in La scienza e i vortici del dubbio, Proceedings of the International Conference "Cartesio e la scienza", Università di Perugia, 1996.

³² In the quoted Bartocci-Mamone Capria’s paper, an experiment is proposed, which has been recently performed by the Italian physicist Fabio Cardone, in L’Aquila’s laboratories. The experiment has put in evidence some "anisotropy" of difficult interpretation, but was undoubtedly coherent with all the until now "unsuccessful attempts to discover any motion of the Earth relatively to the light-medium" (and let us quote another kind remark of Prof. G. Cavalleri: "All the extremely accurate electronics on space ships, satellites, shuttles, etc., show that the aether drag is not present"). This fact points once again at the possibility that the relative velocity Earth-aether, at Earth’s surface, is zero, or very near to zero (on this argument see also the previous footnote 26). In order to find decisive elements in favour, or contrary, to SR, one should stop to emphasize the certainly successful but indirect consequences of the theory, and start, at long last, to perform experiments in two different reference frames, one in real uniform motion with respect to the other.