Most common misunderstandings
about Special Relativity (SR)
(Umberto Bartocci*)
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"1Principle 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) isometriesM 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 , or, better, of a field of them, as we shall see in the next section), of an "observer" 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" - ), and suppose that 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 after a time interval 2 R/c. Let us suppose now that P, and then , is rotating (and again, wrt ) with some angular speed . It is obvious then that, from ’s point of view, one light’s ray, the one which travels in the same sense of the rotation, will arrive to delayed of a time interval 2 R/c times /(1- ) (where we have put, as usual, = R/c), while the second one will arrive anticipated of an analogous time interval 2 R/c times /(1+ ). To make it short, we can introduce the ratio k between the two time intervals forwards and backwards, TI = 2 R/c(1- ) TB = 2 R/c(1+ ), and see that SR, as besides any other "classical" theory (we can suppose for instance 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 ; quantitatively, that the effect is "measured" by the number k = (1+ )/(1- ) (k 1, and k = 1 if, and only if, = 0). This k coincides even with the analogous value computed by , using ’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 , who is now supposed to move with the speed v = R wrt , the light’s speed should always be a constant equal to c, and then, from ’s point of view, no effect should be predicted at all. In other words, SR would predict a k 1 effect for , but a k = 1 effect for , which would indeed be a patent contradiction.
Of course, the previous argument is wrong, since is not an inertial observer in M, and what would be the "light’s speed" for 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 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 . We can suppose to let R increase up on infinity, and to let vary in such a way that the product R is a constant v. At the limit, we would have the physical situation of a platform rotating "very slowly", and of an observer 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 k 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 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" 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 , 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 ). Claiming that vand -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 , without any rotation, and call for instance * the inertial system in which P is still (it is obvious that P would not be any longer a "circular platform" wrt , if it is such wrt *, just because of lenght’s contraction). Then in * there is no Sagnac effect, and in force of the Principle of Special Relativity, there would be no effect even in , 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 . 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 ( ) : IS (I an open interval of the real line R), such that ds2( ’) 0 (one says that is time-like). If ds2( ’) = -1 for all , then the parameter is called a proper time of (and 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 Sadapted 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.