Ardeshir Mehta's Critique of the Web Article Entitled:

"Most common misunderstandings
about Special Relativity (SR)"


Prof. Umberto Bartocci

(N.B.: This Critique is dated January 2002.)

The following is my critique of Prof. Umberto Bartocci's Web article "Most common misunderstandings about Special Relativity", published on his Web site.

I would like to mention right at the outset that Prof. Bartocci is my respected friend. Although we have not (yet) met in person, we have corresponded extensively over the summer and part of the autumn of the year 2001 regarding the Theory of Relativity. Besides, I am immensely indebted to him for having organised the conference "Galileo Back in Italy II" in May 1999, during which many very bright and insightful people from all around the world gathered together in Bologna, Italy, to put forward their ideas about the Theory of Relativity, and why it must be wrong. (I did not attend the conference myself, because at the time I and Prof. Bartocci did not know one another, and so I did not know about the conference, while he did not invite me to it; but its proceedings helped me immensely in refining my own ideas about the Theory of Relativity; and subsequently I corresponded via e-mail and even snail-mail with many of the attendees.)

Also, both Prof. Bartocci and I share a common dislike for "dogmatic science" -- i.e., the acceptance of a scientific theory merely because the person or persons who propound it have great prestige. (If you, dear reader, do not share this sentiment, read no further!) So you will not find any respect for the big names revered by the "scientific establishment" hereunder -- not for Einstein, nor for Newton, nor for Aristotle, nor for Plato.

But you will find a respect for the truth. For as Prof. Bartocci himself is fond of saying, quoting Aristotle -- and with which maxim I wholeheartedly agree -- Amicus Plato, sed magis amica veritas ("Plato is a friend, but truth is an even greater friend".)

I am however convinced that Prof. Bartocci's own writings on the Theory of Relativity are not consonant with the truth. But I am sorry to say that despite my best efforts, I have not been able to convince him of this. I am not sure what the reason is. (It cannot be old age -- and the "hardening of the attitudes" which often accompanies it -- for he is only a few years older than I am, although his hair is much whiter: but then again, mine is more scanty! -- We are both around 60 years of age, he a little older, I a little younger.) He, for his part, thinks that I do not understand Relativity. I, for my part, think that he does not understand logic, mathematics or physics. (Of course his being a professor of mathematics is no barrier to that.)

However, since he has made his arguments public on his own Web site, I feel I have a right to publish my Critique of his arguments on my Web site, so that any rational and interested third party may judge for himself or herself who is right: he or I. So I think that even though I may not be able to change his mind -- even though I have totally deparied of being one day able to do so! -- maybe I will be able to convince some others.

In the following Critique I have quoted the entire text of Prof. Bartocci's Web article word for word, so as to provide the reader with the complete context. I have changed the typeface of those of his words I am specifically critiquing to red (from their original black), so that the reader may more easily spot them. And to distinguish his words from mine, I have left his in their original Times New Roman typeface, while mine are in Arial, and coloured dark blue. Other than that, I have made no changes to his article save for creating some white space around the entire article, for better legibility.

So without further ado, if you are interested in the Theory of Relativity, read on!

                                                                                                                           Ardeshir Mehta.



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"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.

Right here we see the beginnings of what I consider to be Prof. Bartocci's misunderstandings of logic. The fact is that there is indeed a "place" -- as Prof. Bartocci expresses it -- for questioning the logical validity of SR (as there is of GR), because they both are built on the postulate of the constancy of the speed of light regardless of the relative speed of the source of the light relative to the observer. This postulate is itself contrary to logic -- and therefore to mathematics as well (because whatever else mathematics may be, it cannot be illogical.)

It is to be noted -- to begin with -- that there is not a single experiment which has ever been conducted which conclusively proves that the speed of light is a constant for all inertial observers, regardless of the relative speed between the observer and the source of the light. All the experiments and arguments which purport to prove this -- such as De Sitter's argument about binary stars, to which Einstein refers in his book Relativity: The Special and General Theory -- have alternative explanations as well, and there is no reason for accepting one explanation over the others. (To see some alternative explanations, click here and here.)

But even logically speaking the postulate of the constancy of the speed of light is incompatible with the Principle of Relativity -- as Einstein himself realised quite clearly (again see his book Relativity: The Special and General Theory, Chapter VII.)

It is to get around this incompatibility -- which Einstein considered to be merely an "apparent" one -- that he devised his famous "Train" Gedankenexperimente, using which he purports to prove that simultaneity is relative.

But this Gedankenexperimente is logically flawed, for it implicitly assumes that the speed of light is not a constant. One can hardly prove that the speed of light is a constant, using the assumption that the speed of light is not a constant -- at least not without using a reductio ad absurdum type of proof: and Einstein's "Train" Gedankenexperimente is definitely not such a type of proof.

To see how Einstein's "Train" Gedankenexperimente is logically flawed, just consider his thought-experiment in every single detail.  Suppose there is a train, says Einstein, moving at a velocity v in a straight line past a man standing on a railway platform next to the tracks. Suppose a passenger is standing in the train exactly at its mid-point. Suppose two lightning flashes hit the exact front and the exact rear of the train. Suppose the man on the platform sees both the flashes simultaneously. Suppose he then measures the distance from where he was standing to the spots on the track where the two lightning bolts struck, where they left burn marks on the track as evidence of their location. (Einstein does not speak of the burn marks, but we shall introduce them so as to be sure of the locations where the bolts of lightning struck.) Suppose the man finds that he was standing exactly mid-way between the locations on the track where the two lightning bolts struck and left burn marks -- the distance from his position to each of the lightning bolts being exactly d. Using the postulate of the constancy of the speed of light, the man must conclude -- says Einstein -- that the two bolts struck the tracks simultaneously. (Einstein himself does not work it out, but we can easily see that the man must have seen the two flashes at a time instant t = d/c after they struck.)

However -- or so argues Einstein -- the passenger on the train cannot have seen the flashes simultaneously, even though she too was standing exactly mid-way between the locations on the train where the two lightning bolts struck; for she -- along with the train -- was moving towards the flash of light coming from the front end of the train, and moving away from the flash of light coming from the rear of the train. Thus the former flash must have taken less time to reach her than the latter.

But it is easy to see that here, Einstein has implicitly assumed that the speed of light is not a constant for the passenger on the train. We have only to realise that using Einstein's own argument, we could conclude that as observed by the passenger, the light should take longer to go all the way from the rear end of the train to its front end, than it takes to go all the way from the front end of the train to its rear end! One can hardly argue that for the passenger on the train, the distance from the front end of the train to its rear end is not the same as the distance from the rear end of the train to its front end. If for any given observer, light travels the same distance in different amounts of time, then obviously its speed cannot be a constant for that observer!

So what Einstein forgets is that if the postulate of the constancy of the speed of light is in fact correct, it is impossible for the passenger to have seen the flashes non-simultaneously. In her frame, she was always standing exactly half way between the locations where the lightning bolts struck. Remember that her frame of reference is not that of the track but that of the train. (The frame of the track was moving relative to her, and thus cannot be hers, for by its very definition the frame of reference of an observer must be stationary relative to the observer!)

Now in her frame, namely that of the train, the locations where the two lightning bolts struck were not moving away from her, nor were they moving towards her. These locations -- the exact front and exact rear of the rain -- were always equidistant from her: and let us call that distance d'. In that case, the exact length, in her frame, of the train from its front to its rear would be 2d'. Or in other words, if in her frame of reference she was at x, y, and z space co-ordinates given as (0,0,0), then in her frame the lightning bolts struck at co-ordinates (d',0,0) and (-d',0,0). (We need not get into an argument at this stage in our argument about whether d = d' or not: all we need to agree to is the statement that in the passenger's frame, the distance from the passenger to each of the lightning flashes was always the same.)

Thus in her frame, if the postulate of the constancy of the speed of light does indeed hold, the amount of time -- let's call it t' -- which the light took to reach her eyes from each of the lightning flashes must be the same, viz., t' = |d'|/c.

Thus Einstein's conclusion -- namely that the passenger did not see the flashes simultaneously -- is incompatible with the postulate of the constancy of the speed of light! If that postulate does hold, then if the man on the platform saw the two flashes of light simultaneously, the passenger must have seen the two flashes simultaneously as well.

Besides, this thought-experiment is flawed in other ways as well. If the passenger does not see the flashes simultaneously -- as asserted by Einstein -- despite standing, in her frame, exactly at the mid-point of the two locations where the two lightning bolts struck, while the man on the platform does see them simultaneously (and it is to be borne in mind that he too was standing, in his frame, at the mid-point of the two locations where they struck), then we can tell which of the two is moving: the train or the track! But this would contradict the Principle of Relativity, according to which there ought to be no way to tell which object is moving and which is stationary.

This is the more easily realised if we transpose the thought-experiment to outer space, with two straight and long spaceships being substituted for the train and the track: a very, VERY long spaceship for the track and a (merely) long spaceship for the train. And the lightning bolts could be substituted by mild explosions, which cause no significant damage to the spaceships or alter their trajectory in any measurable way, but do leave burn marks on them, so that we can sure of where they occurred in each frame.

(Note that it cannot be validly argued that the passenger on the train did not see the flashes simultaneously because in her frame they did not occur simultaneously. That would be assuming in advance the very conclusion the thought-experiment is trying to prove.)

Moreover, even if we assume as a hypothesis that the Theory of Relativity is correct, we could still tell which was moving -- the train or the track -- by simply bringing the train back to the railway platform, causing it to come to rest alongside the tracks where the lightning struck, and comparing the distance between the two burn marks on the train to the distance between the two burn marks on the track. If the burn marks on the train are farther apart or closer together than the burn marks on the track, we can tell that it was the train that was moving and not the track, or vice versa, and the discrepancy when both the train and the track are at rest relative to one another is due to Relativistic length contraction. While if on the contrary the burn marks on the train are found to be the same distance apart as those on the track when they are both at rest relative to one another, we can tell that Relativistic length contraction could not have occurred. Either way the Theory of Relativity would be disproved.

But in light of all the above arguments, simultaneity cannot be relative, dependent on the observer, but must be absolute, dependent on no person whatsoever!

But then the earlier incompatibility between the postulate of the constancy of the speed of light and the principle of relativity -- of which Einstein was keenly aware and of which he made his readers acutely aware as well -- is not removed! And thus the logical validity of the postulate of the constancy of the speed of light falls down utterly and completely.

And if the postulate of the speed of light itself is illogical, then all of the so-called "mathematics" of Relativity, which is entirely built upon this postulate, must be illogical. But an illogical mathematics is a contradiction in terms; and thus Relativity cannot be a truly mathematical theory -- as is asserted by Prof. Bartocci.

Prof. Bartocci continues:

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.

Now here again, Prof. Bartocci errs -- this time in his over-simplification of the matter. SR and GR are not just mathematical theories -- or even theories which, as we have shown above, purport to be mathematical, without really being so -- but physical interpretations of theories which purport to be mathematical. Now the physical interpretations of SR flatly contradict those of GR.

Take for example the most basic and famous GR Gedankenexperimente of them all, Einstein's famous "Elevator" thought-experiment. Imagine, he says, that a man stands in a windowless elevator which happens to be in empty space, far away from any measurable gravitational field; and suppose that with the help of some suitable mechanism -- such as a rocket (Einstein doesn't mention rockets, but we know that a rocket would indeed do the needful) -- the elevator is being continuously accelerated at exactly 1g in the direction of an imaginary perpendicular line stretched from its floor to its ceiling. The man would never know -- or so argues Einstein -- from any experiment carried out purely inside the elevator (i.e., without looking outside it), whether he was in such an elevator in empty space, or stationary in the Earth's gravitational field. Whereupon, Einstein concludes, inertial acceleration and acceleration due to gravity must be equivalent.

But Einstein forgets that if this is correct, then SR must be false! If SR is correct, then as time passes, the elevator's speed -- relative to what it was at the beginning of the process of acceleration -- will gradually increase, up to the point that after about ten months or so of accelerating at 1g, its speed will be almost that of light (relative, of course, to what its speed was about ten months earlier, and also relative to everything else with which it was at rest at that earlier date.) But SR also predicts that at such a stage, and for ever after that, the mass of the elevator -- and of everything in it, including the man -- will be enormously greater than it was at the beginning of the acceleration: and this will be readily sensed by the man, because his muscles will not get any stronger. As the months go by he will find it increasingly difficult to stand up, to lift his hands up to his face, and after some more months or years, even to blink his eyelids or to breathe: his own mass will be several tons. Eventually -- say, a few decades later -- his mass will be so great that he will turn into a black hole, and be swallowed up by himself! But none of these weird things would happen if he were stationary in the Earth's gravitational field.

The above conclusion can also be reached if we use the equation E=Mc2. After all, in empty space far away from any measurable gravitational field, the elevator would not accelerate unless some energy were input therein. And this input of energy would show up, according to the equation E=Mc2, as mass. Initially the increase in mass would be hard to detect, of course, but as time went by the increase in the mass of the elevator -- and of everything in it -- would be easily detected by the man in it: he won't even be able to lift up a cup of coffee to drink it!

Note also that the moment the rocket ran out of fuel, the elevator would no longer be accelerating. Of course we can assume a very large rocket with a very large amount of fuel, and if the fuel were a combination of matter/anti-matter generating plasma or light, or if it were an ion rocket, the rocket could run for many years, maybe even centuries; but eventually there is no question that it would run out of fuel. And at that time the man -- or his descendants, if he had any -- would suddenly find himself / themselves weightless: which would certainly not happen if he / they were in a windowless room on Earth.

Indeed in the real world, constant inertial acceleration always has a beginning and an end: and the man inside it will be able to sense one or the other of them, or both. On the other hand, an elevator with a man in it can stand stationary on Earth indefinitely!

So the physical interpretation of GR is obviously incompatible with that of SR. Indeed it is also incompatible with the rest of physics, for as I said earlier, in the absence of any gravitational field, inertial acceleration of a body is impossible to achieve without some input of energy; whereas in the absence of inertial acceleration, gravitational acceleration (such as that which is being imparted to all of us right now on the surface of the Earth by its gravitational field) requires no input of energy whatsoever.

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).

According to Prof. Bartocci, the speed of light is a constant only for inertial observers: those who -- strictly speaking -- move uniformly and rectilinearly in empty space far away from any detectable gravitational fields. For all other observers, Prof. Bartocci asserts (see later on in his paper) that the speed of light is not a constant, and can even be greater than c.

But then let us ask Prof. Bartocci this very simple question: Is the speed of light a constant for us humans on Earth, or is it not? Since we are all of us rotating, along with "Spaceship Earth", every moment of every day, if Prof. Bartocci is right, then the speed of light should not be a constant for us. When we measure the speed of light, it should differ, depending on whether it is coming from the west or the east! (Or indeed from the north or the south too, when compared with its speed when it comes from the east or the west.)

Admittedly we on Earth are subject to a constant gravitational force of 1g. But consider also the system of geo-stationary GPS satellites orbiting the Earth, all of which are in zero-g. Those who design GPS systems acknowledge that these satellites are definitely subject to the Sagnac effect. Yet they are equally definitely not in rectilinear motion, as I am sure even Prof. Bartocci will agree. But will Prof. Bartocci claim that the speed of light is not c with respect to any of these satellites?

If the speed of light coming from Earth to the satellite is c as measured on board each and every one of these satellites, then how is it that the light which ensues from one of the satellites and returns to the same satellite via all the others takes different amounts of time to go one way around the circle of satellites than it takes to go the exact opposite way?

The answer is, that Prof. Bartocci implicitly assumes that the distance all the way around the periphery of a rotating disk is different from the distance around the periphery of the very same disk when it is not rotating.

Now this is like claiming that the distance around the Earth's equator would be different from what it actually is measured to be, if the Earth were not rotating around its own axis: or that on our rotating Earth, the distance from San Francisco to New York is greater than the distance from New York to San Francisco. Indeed taken to the extreme, it is like claiming that the distance from the east wall of my house to its west wall is not the same as the distance from its west wall to its east wall!

Such claims are obviously absurd -- and any five-year-old will laugh uproariously at them.

Yet if a system of mirrors and vacuum tubes were arranged strictly and rigorously along the equator all around the Earth, and light passed through them, then due to the Sagnac effect, light would take a different amount of time to go all around the Earth in the westward direction compared to the time it takes to go around the Earth in the eastward direction. If light travels the same distance in different amounts of time, then obviously its speed cannot be a constant!

It is idle to claim that because the Earth is rotating while the experiment is being conducted, the "actual" distance the light travels in such an experiment is not the measured distance around its equator. If we are to take the rotation of the Earth around its own axis into account, why not take all its other rotational movements into account also? During such an experiment it is not merely rotating around its own axis, but also revolving around the Sun, and even moving around the entire Galaxy, if not around our local group of galaxies. Indeed if we wish to take all the motions of the Earth into account, it is impossible to say just what the "actual" distance is, which is travelled by the light during the experiment: because we don't even know exactly what the "actual path" of the motion of the Earth is!

And that, in turn, is because the question immediately arises: relative to what should we say the Earth is "actually" moving? The entire universe? The distant quasars? The nearest Galaxy? What?

The problem is that Prof. Bartocci is not altogether clear as to what he means by "inertial observer". Are we humans on Earth "inertial observers", in the sense of the Theory of Relativity? Or are at least astronauts in weightless orbit inside the International Space Station "inertial observers"? If not, then until we go very, very far away from our beloved "Spaceship Earth" into empty space, to a region of space where there is no measurable gravitational field, we will never be able to experimentally test the Special Theory of Relativity!

On the other hand, if we -- or at least our astronauts -- are inertial observers, then the Sagnac experiment confirms that the speed of light is not constant for us!

The fact is that experimentally speaking, the speed of light is not a constant for us. If we were to measure the time light takes to travel from the west wall of my house to its east wall, using very sensitive instruments, and provided we make sure the light travels in a total vacuum, the times would not be the same. We do not have such sensitive instruments at present, but the above assertion is proven by the consideration that if we were to construct houses exactly like mine adjacent to each other all the way around the Earth, and pass the light successively through each of them, drilling holes in the walls and using mirrors where and when necessary to cause the light to follow the curvature of the Earth, and making sure the light always travels in a vacuum (say by constructing a tube for the purpose and removing all air from inside it), then the light going one way around the Earth would take a little more time to get all the way back to my house than the light going all the way around the Earth the other way. This is the Sagnac effect, which does not need extremely sensitive clocks to measure, because the distance travelled all the way around the Earth is much greater than the distance between my west wall and my east wall. Besides, we have the returning light itself -- which enables interference fringes to be seen -- to accurately measure the delay.

But if the speed of light is not a constant for us, because as Prof. Bartocci claims, we are not truly inertial observers, then would it be constant for all truly inertial observers -- that is, observers far away from any detectable gravitational field -- regardless of their relative uniform rectilinear motion? A simple thought-experiment will show that it absolutely cannot be. And we don't need the Sagnac effect to prove it, either.

For an alternative way to express the postulate of the constancy of the speed of light is to say that the time light takes -- as measured by any single clock located at either the source of light, or at the spot where it arrives eventually, which we may call the "target" -- the time light takes to travel between any given inertial light source and any given inertial target must be the same, regardless of whether, how, and by how much, the target changes its location with respect to the source during the interval between the moment when the light leaves its source and the moment it hits the target!

But this is obviously impossible.

To illustrate the utter impossibility of this, consider the following thought-experiment:

Suppose there is a source of light which gives off very brief flashes, with an observer holding a clock located right next to it and stationary relative to it; and suppose there are two targets close to one another but at a considerable distance from the source, both targets being made up of mirrors which reflect light back to the source whence it came. (To accomplish this, each target consists of two mirrors fitted exactly at right angles to one another, so that light coming from any direction will be reflected back to where it came from. Such a group of two mirrors was left by the Apollo astronauts on the Moon, so as to help us measure its distance from Earth with extreme accuracy. We also often see this phenomenon when walking in a shopping mall, where such mirrors are often found, especially in clothing stores: we find that we cannot get away from our own reflection.)

Initially let both the targets be stationary relative to the source of light. Let there be a second experimenter -- who is not the observer -- situated equidistant from the source and the targets, linked to the source and the targets via identical electrical wires of equal length, which are also stationary relative to the source and the targets. When this experimenter throws a (single) switch, let an electrical current travel along each of the wires, both currents being of identical strength, one of them going to the source of light and the other to the targets. When the currents reach the source and the targets, let the current going to the source cause the source to emit a very brief flash of light, and let the current going to the targets cause one of the targets to move rapidly towards the source: some suitable mechanism -- such as a magnetic rail-gun, or an air-piston -- being triggered for this purpose by the current. Obviously, due to the fact that both the wires are identical and of equal length, and both the currents in them of equal strength, the occurrence of the flash and the beginning of the movement of the target will be truly simultaneous. (If there is any doubt about that, the experiment may be performed twice, with the wires and currents alternated the second time, to see whether there is any difference in the results.)

Now it will obviously take a finite amount of time for the flash of light to travel from its source to the two targets, and during this brief time period one of the targets will have moved a little closer to the source than the other. So the observer, stationary right next to the source of light, should see one of the two reflections -- the one that was reflected by the target which moved a little closer to him -- a brief period of time before he sees the other flash.

But according to the postulate of the constancy of the speed of light, the time it takes for light to travel from the source to both the targets, regardless of how either of the targets moved relative to the source during the brief time period the light took to travel from the source to the targets, should be the same! Thus according to the Relativistic postulate of the constancy of the speed of light, the observer should see both the reflected flashes simultaneously.

To consider how utterly impossible this must be, just consider an extreme example: suppose that the targets are initially quite far away from the source -- say, a light-second or two from the source -- and suppose that the mechanism which moves the target, moves it at such a high velocity that the target almost reaches the source before the light hits it. (In practice of course this is most likely next to impossible without utterly destroying the target, but what the heck -- this is a thought-experiment! So anything at least theoretically possible should be allowed.)

Under such conditions, the observer will obviously see the reflection from that target almost at the same time as he sees the flash from the source, while he will see the reflection from the other target at least two seconds later; and this delay will be easily noticed! To even pretend that he will see both reflections at the same time, under such conditions, is ludicrous.

Note that it is not necessary for the moving target to have been accelerated. If a third target were also moving uniformly and rectilinearly toward the source of light, in such a manner that it and the second target moved together, right next to each other, toward the source, so that they both meet the light wave at the exact same distance from the source, the observer would measure the same delay after the flash for the reflection from the third target to reach his eyes as from the second, while the first target -- the one which remained stationary relative to the source -- would indicate a longer delay for the reflection to be seen by the observer. This proves that acceleration is not a factor in this result, and it can all be explained in terms of uniform rectilinear motion.

So the Sagnac experiment merely proves experimentally what is abundantly clear from a thought-experimental viewpoint: that the speed of light cannot be a constant -- certainly not regardless of how the source of light moves relative to the observer.

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 TI = 2p R/c(1-b ) 3 D TB = 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 it is precisely the light speed for a which can be computed, and even measured with great precision, even with the assumption that a is not, strictly speaking, an initial observer in M!

Indeed if a is one of the GPS satellites orbiting the Earth in geo-synchronous orbit, we don't need to compute anything after "carefully rigorous definitions" -- we can already measure the light speed for any of them. In fact it is easily measured, by taking the distance from the satellite to any given location on Earth, and dividing it by the time a GPS radio signal takes to go from that ground location to the satellite, or vice versa! One could even measure this time by using a mirror -- or two mirrors at right angles to each other, like the aforesaid mirrors left by the Apollo astronauts on the Moon -- placing the mirrors on one of the satellites, and bouncing a light signal off them, measuring the time it took for the signal to go to the satellite and back, and dividing that time by two.

And as mentioned earlier, all of us humans on Earth -- including Prof. Bartocci -- are in the position of a. If we cannot measure the speed of light for our own selves, who else will do it? God?

The fact is that we all know the speed of light for a, when a refers to us humans on Earth: it is 299,792,458 m/s ! It has been measured -- and that too, repeatedly -- right here on planet Earth, which is rotating just like any other Sagnac disc. Prof. Bartocci's argument highlighted in red above, with this fact in mind, seems to warrant a decidedly surprised "Huh!?!"

The only problem is that this speed has been measured when the source and the observer are stationary relative to one another. There is no argument that the speed of light in a vacuum is 299,792,458 m/s when the source of the light is stationary relative to the observer! But when the speed is measured by sending the light all the way around the Earth, as in the Sagnac experiment, it is not 299,792,458 m/s at all. That's because we human beings -- whether on Earth or in orbit around it -- are all in the position of one or another a, and none of us are in the position of w.

Indeed Prof. Bartocci is at least partially right, and the speed of light is different for a than for w, if the light is coming from or going away from the direction of rotational motion of a. But that is not because a is not an inertial observer in M while wis one -- as Prof. Bartocci asserts -- but rather because a is moving relative to w! For note that the speed of light is also different for w than it is for a hypothetical inertial observer a' who is moving in the same direction and at the same speed as a, but situated on and moving along the tangent subtended on the disc C at the position of a -- wherever a happens to be at any given instant. (Note that at that instant, both a and a' would be at rest relative to one another, and thus the speed of light coming from any given direction should be same for each of them, even if the Relativistic postulate of the constancy of the speed of light for all inertial observers were rejected!) Since both a' and w are inertial observers, this proves that the speed of light is not a constant even for strictly inertial observers, if one of them is moving uniformly and rectilinearly relative to the other.

That's because if the speed of light is different for a than for w, and yet the same for a' as for a -- as I am sure any rational person will agree -- then the speed of light must be different for different inertial observers: there is absolutely no other possible conclusion.

By the way, it cannot be validly argued that the speed of light being the same for a as for a' is only an instantaneous phenomenon, and is not valid for any finite period of time, no matter how brief. That's because for every possible position of a, there corresponds an unlimited number of (hypothetical) inertial observers a'1, a'2, a'3, ... (etc.), each of which can be arbitrarily close to the next one and to the previous one. Thus if the postulate of the constancy of the speed of light for all inertial observers were in fact valid, light speed ought to be the same for all of these inertial observers a'1, a'2, a'3, ...  (etc.), no matter by how much the first is separated from the last around the periphery of C during the Sagnac experiment. As a consequence, the phenomenon ceases to be merely an instantaneous one, and becomes valid over a period of time!

Not that it matters, though, because even if the speed of light for a is only equal to that for a' for a mere instant, we are still left without any explanation how even the instantaneous speed of light for a' can be different from that for w, seeing that if w is an inertial observer -- as would be the case if the disc C were in a part of space far away from any measurable gravitational field -- then a' would also, and without any question whatsoever, be an inertial observer also.

Nor can it be argued a is under acceleration while a' is not, and for this reason that even the instantaneous speed of light for a must be different from that for a'. If the radius R of the disc C were increased indefinitely while the tangential velocity of a remained constant, the amount of acceleration under which a would find itself would progressively decrease, with zero being the limit; but the Sagnac effect at a would remain unaltered. So the rate of acceleration of a is obviously not a factor governing the difference in light speed for a.

For example, if we assume that a is moving relative to w at 1/1,296,000,000c -- so that in a period of time in which light completes the full circle plus or minus the little extra to get to a, a itself moves only one-thousandth of a second of arc -- and if we additionally assume that the radius R is large, say 1 light-year -- then light speed for a should not be measurably different from light speed for w. (Here again, one must remind Prof. Bartocci that Relativity is a physical theory, and thus is subject to the limitations of our measuring instruments, none of which are perfect, and indeed none of which ever will be perfect.)

Assuming, therefore, that the disc is in empty space -- say in one of those huge intergalactic voids of which we often hear astronomers speak -- so that it is very far away from any measurable gravitational fields, with the above-mentioned rate of revolution, the g-force generated by the rotation of the disc will be v2/R, which when worked out here comes to 5.66*10-6 m/s2-- or in other words, so small as to be almost non-measurable even with the best of our available instruments; and even if the instruments of a future age are able to measure this, all we need to do to get beyond their measuring capacity is to increase still further the radius R of the disc! As a result, if the Theory of Relativity were correct, light speed for a should not be any different, from a measurable point of view, than for w. And yet the Sagnac experiment proves that in fact it is.

And it is also not correct to suppose -- when taking into consideration the entire Earth as a Sagnac disc -- that a hypothetical observer w at the Earth's centre is in a non-rotational (i.e., inertial) trajectory, while we on its surface are in a rotational trajectory. If we are referring specifically to the Earth, then it is obvious that w is also in a rotational trajectory around the centre of mass of the Solar system. Nor can it be argued that a hypothetical observer w' at the centre of mass of the Solar system would be in non-rotational (i.e., inertial) motion, because even that observer is in rotational motion around the centre of mass of our Galaxy. And we don't even know whether a hypothetical observer w" at the centre of mass of the Galaxy would be in non-rotational (i.e., inertial) motion, because we don't even know whether the Galaxy itself is rotating around a centre which is farther away still!

From a practical point of view, in fact, it would be hard indeed to say of some very large object -- or virtual object -- whether it is rotating or not, even if in actual fact it were doing so, with the tangential velocity at its periphery relative to its centre equal to that of the surface of the Earth relative to its centre. And yet the Sagnac effect for both the Earth and that very large hypothetical disc would be the same.

Thus the Sagnac effect is more than ample experimental evidence disproving Relativity; and if a person is unwilling to accept the clear and logical conclusion in this regard, then most probably nothing will ever convince that person of the falsity of the Theory of Relativity. (But if this is the case, why call it a "Theory" at all? Would it not be more accurate to call it the "Dogma" of Relativity?)

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!

Ah, but what does it matter that v and -v at points p and q are not the same? If the man moves, say, only one-thousandth of a second of arc while light completes a trip around the entire circle -- 360 degrees -- plus or minus one-thousandth of a second of arc, so as to come back to the man, then the only vectorial velocities that need to be compared with one another are the man's vectorial velocity v at his point p and his velocity v' at point p' which is just one thousandth of a second of arc separated from point p -- and there is absolutely NO question that v and v' are indeed almost the same.

Note also that if the disc were so large as to take up a significant portion of the universe -- if it were, say, 10 billion light years in diameter -- then during the life-time of the universe (assuming that the universe did start off with a "Big Bang" about 20 billion years ago, which is of course just speculation, but let's forget that for the moment), and assuming a speed v -- as above -- of 1/1,296,000,000c, the man (or even his descendants) would never have enough time during the life of the universe to come to the antipodal point q of p.

Note also that it is not necessary for the Sagnac experiment to construct a huge rigid disk with a radius R: all that is needed -- as in the case of the GPS satellites around the Earth -- is to have a limited number of bodies (as few as three will do) set in motion around a common geometric centre in such a manner as to constitute a "virtual disc".

The entire argument -- as the keen-eyed reader will have noticed -- is based on a dispute as to whether rectilinear motion is relative or not, given the fact that rotational motion is rather evidently not relative but absolute: or if it is at all relative, then it is relative only to the entire universe. With rectilinear motion we do not readily observe whether a body is moving relative to the entire universe; we can only observe its movement relative to a few bodies in its immediate vicinity -- a purely local observation. But in the case of a small body's rotational movement, we can actually observe its movement relative to the entire universe.

But it is also easily realised from the above considerations that if the rotating disc (or virtual disc) is sufficiently large -- that is to say, if it were almost as large as the entire universe -- it would not be easy to tell whether it is rotating relative to the entire universe, or not. That's because every single one of the objects in the universe is moving relative to almost every other object, and virtually none of these myriads of objects is moving rectilinearly at all (except in special cases, and even then only for brief periods of time). Indeed if the disc were almost as large as the universe itself, it could never be said that it was "rotating" in any absolute sense at all, even if the tangential velocity of any given spot located at its periphery were almost that of light! The rate of rotation, in that case, would not even be measurable.

And similar considerations would enable us to say, that if we were to construct a rectilinearly and uniformly moving "virtual rod" as large as -- or even almost as large as -- the entire universe, then it would be possible to say whether the rod was in motion relative to the entire universe, or not. (I shall let the reader think about this for himself.)

Thus it seems that with rotational motion, the larger the disc -- or virtual disc -- the more difficult it is to tell whether the disc is in absolute rotational motion or not; while with uniform rectilinear motion, the smaller the rod -- or "virtual rod" -- the more difficult it is to tell whether it is in absolute rectilinear motion or not.

Indeed if -- as Prof. Bartocci no doubt believes -- space is "curved" by the presence of mass in it, and if in addition the curvature is positive, then a "virtual rod" the size of the entire universe, moving in either direction along its own length, would never be able to move rectilinearly at all. Such a rod would be "rotating" no matter how it moved along its own length!

Thus we can take Prof. Bartocci's argument of applying the consideration "globally" to rectilinear motion also, and thereby "prove" that rectilinear motion is absolute and not relative. In other words, Prof. Bartocci's argument, if it is intended to be valid for rotational motion, should be valid for rectilinear motion too.

Thus his own "gloablisation" argument about rotational motion, as applied instead to rectilinear motion, would disprove the Theory of Relativity.

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 as for the entire section which follows, Prof. Bartocci is falling into the logical trap known as "begging the question", or assuming that which is to be proved. The so-called "mathematics" he is using to discredit the Sagnac experiment is itself built upon the postulate of the constancy of the speed of light for all inertial observers, which is accepted in it as an axiom. It is absolutely impossible to construct a "geometry" of Minkowski space-time without accepting, as an axiom, the postulate of the constancy of the speed of light for all inertial observers: that is to say, using only the axioms of mathematics -- such as those of Peano, or those of set theory first enunciated by Zermelo and Fraenkel, and later expanded by John von Neumann -- plus of course the definitions, common notions, propositions and postulates of Euclid.

That is to say, even to construct this so-called "geometry" of Minkowski space-time one must a priori assume, as axiomatic, the notion that the speed of light is a constant as measured by any inertial observer.

Now what is the justification for assuming this notion as an axiom? It is certainly not self-evident -- as Euclid's "common notions" are. Take for example Euclid's fifth common notion: "The whole is greater than the part". This notion is so obviously true from our daily experiences that we do not often doubt it, except when discussing the Holy Trinity.

Nor -- as we pointed out earlier too -- is there any experimental evidence conclusively proving the postulate of the constancy of the speed of light for all inertial obervers. And as Einstein himself noted, it is also logically incompatible with the Principle of Relativity.

Since the only argument Einstein has to try to overcome this incompatibility is his so-called "Train" Gedankenexperimente, it seems that Prof. Bartocci is hanging his hat on this Gedankenexperimente alone. The rest of Prof. Bartocci's arguments are mere superstructures built on top of this foundation! So if the foundation falls, the entire superstructure -- the so-called "mathematics" of Relativity -- must fall with it.

But the "Train" Gedankenexperimente cannot possibly be a valid logical argument, because as we pointed out earlier, in order to prove that simultaneity is relative, it assumes in advance the opposite of what it purports to prove. Thus either the assumption is false -- in which case the proof cannot be obtained -- or else, if the assumption is correct, the proof (which is the exact negation of the assumption) must be false.


Remark: It is incorrect to say that any proposition whatsoever may to taken as an axiom, arguing that an axiom is by definition a proposition that is unproven, and therefore does not even need to be proven. Although this is indeed true as far as it goes, in order to generate a non-self-contradictory system of mathematics, no axiom can stand by itself, but rather has to be used along with other axioms, with which it must be compatible. Thus no axiom of any system of mathematics may contradict another, nor may it contradict any theorem proven solely with the help of any of the other axioms.

But the postulate of the constancy of the speed of light regardless of the relative motion between the source of the light and the observer contradicts the Galilean theorem of addition of velocities, which can be proven using only the original axioms of mathematics -- including of course the definitions, common notion, propositions and postulates of Euclid, which are essentially the "axioms" of geometry. (To see how this theorem can be so proven, click here.) Thus the postulate of the constancy of the speed of light cannot be taken as an unproven axiom of mathematics. (We use the term "mathematics" here, of course, as including geometry.)


Besides, there are several other ways to see that simultaneity cannot possibly be relative. That is to say, it is easy to see from an entirely practical point of view that if two events are simultaneous for any observer, they must be simultaneous for all observers.

For example, if two events E1 and E2 are simultaneous for an observer O, that is just another way of saying that both E1 and E2 occurred when a clock C held in the hand of the observer O indicated one, and only one, time instant -- let us call that instant t. That is the very meaning of the term "simultaneous".

But in that case, any other clock C' held by any other observer O' would also have to indicate a single time instant t', which would correspond to the single time instant t indicated by the clock C when E1 and E2 occurred!

A single time instant t indicated by the clock C cannot possibly correspond to two time instants -- call them t1' and t2' -- indicated on any other clock C', whether C' be ticking synchronously with C or not. But this is exactly the meaning of the Relativistic assertion that if two events E1 and E2 are simultaneous for an observer O, they are not simultaneous for an observer O' moving uniformly and rectilinearly relative to O.

For instance -- to give hypothetical hard figures -- if according to an observer whom we shall call Adam, two explosions occurred some distance apart in his city when his clock struck 12:00 noon on the dot, then according to another observer, whom we shall call Eve, and who was moving at a uniform rectilinear velocity v relative to Adam, the same two explosions could not possibly have occurred when her clock showed both 11: 59 a.m. and 12:01 p.m.: because when they both occurred, we know that Adam's clock struck 12:00 noon on the dot. And when Adam's clock struck 12:00 noon on the dot, Eve's clock could have indicated only one single time instant, and could not possibly have indicated two instants!

It is clear that if Einstein had given his "simultaneity" argument just a little more thought, he would have realised that simultaneity cuts right across frames of reference. That is to say, two different clocks -- such as Adam's and Eve's above -- in two different frames of reference -- as they could easily be, even in the hypothetical case described above -- can show two different times simultaneously.

For example, it is quite possible for my clock to show 12:00 noon at the same time as when yours shows 11:59 a.m. -- and it doesn't even matter whether my clock runs at the same rate as yours or not, or whether you are moving at a high speed relative to me or not. That's just the nature of clocks -- they don't always agree with each other! Indeed it happens almost all the time, especially in developing countries like India, where the clocks aren't too accurate and the people don't care much about punctuality anyway. But this is something that virtually everyone (except perhaps poor Einstein!) has always understood only too well.

Remember that the oft-used Relativistic phrase:

The time instant t in a frame of reference F corresponds to a time instant t' in a frame of reference F' which is moving uniformly relative to F.

... can also be expressed as follows:

When a clock C in the frame F indicates the time instant t, a clock C' in the frame F' would indicate the time instant t'.

And this in turn can be expressed as follows:

The clock C in the frame F indicates a time instant t and the clock C' in the frame F' a time instant t' simultaneously!

There is absolutely no other meaning to the above oft-used Relativistic phrase.

Simultaneity therefore is not restricted to a single frame of reference, but cuts across all frames of reference, by the simple fact -- which is accepted in Relativity -- that any time instant t indicated by any clock C in any frame F must correspond to a time instant t' indicated by another clock C' in another frame F'. The term "corresponds to" means the same thing, in this context, as "is simultaneous with". There can be absolutely NO other meaning to the term, at least in the context in which it is being used in the Theory of Relativity!

Thus it is not even necessary to chip away bit by bit at the elegant and complicated "mathematical" edifice Prof. Bartocci has built hereunder. All that's necessary is to show that the entire edifice -- the intricacy of which Prof. Bartocci is without doubt justifiably proud -- has unfortunately been built on the quicksand of an axiom which contradicts a well-proven theorem of mathematics (namely the Galilean Theorem of Addition of Velocities) ... and which thus indirectly contradicts the other axioms of mathematics: whereupon Prof. Bartocci's entire edifice comes a-tumbling down.

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 Tp(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, ds2 = gijdxidxj, 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 aL : x* = L (say for instance L> 0; - aL 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* = (egt*-1)/g, and then dx*/dt* = egt*, 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 egt* when one makes the substitution x* = (egt*-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 aL 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 aL, not as the previous value dx*/dt*, but as the value dx*/dtL, where t L is a proper time for aL, then we shall get dx*/dtL = 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 aL, and then back, and try to compute the value 2L/Dt , 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/Dt 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.

Let us also remind ourselves once again that the so-called "Principle of Equivalence" cannot possibly be correct from a purely physical point of view -- as we illustrated earlier when speaking of Einstein's "Elevator" Gedankenexperimente. That is to say, there can be no possible physical equivalence between inertial mass and gravitational mass, for if there were, it would be possible to (inertially) accelerate a body for an indefinite period of time using no energy whatsoever, so as to impart to it a g-force equal to that generated by the gravitational field of a planet such as the Earth on its (rigid) surface. Such a miracle is beyond even Einstein's capability to bring about.

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.

And again we remind ourselves that the so-called "definition" of simultaneity used in Relativity has absolutely NO logical basis. (Indeed in a private correspondence with me Prof. Bartocci has even insinuated that in Relativity, two synchronised clocks will not show a reading of zero simultaneously -- thereby making a mockery of any possible meaning of the terms "synchronised" and "simultaneous"! -- See his e-mail message to me dated Fri, 31 Aug 2001 14:53:31 +0200.)

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 gijdxidxj = 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


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 tw = t(e2)-t(e1) =

2Ö (L2-g-2); Dta = ta (e2)-ta (e1) = 2g-1sinh-1(Ö (g2L2-1)), which implies

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

If one introduces the equation dta = Ö (1-va2)dt, which connects the infinitesimal proper time dta with the infinitesimal coordinate time dt, and integrates dta 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 ta = INT (dt a ) = INT (Ö (1-va2)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.

But there is absolutely no difficulty in ensuring a complete and absolute symmetry between two clocks which are moving relative to one another. Take for instance two clocks in orbit around the Sun, one in the sense of the Earth's orbit and the other in the exactly opposite sense: that is, as seen by an observer situated on the axis of revolution but not in the plane of revolution of the two clocks, one of them will be revolving clockwise and the other anti-clockwise around the Sun. This is a perfectly symmetrical set-up. If in addition the orbits of the clocks are perfectly circular and at the same distance from the Sun as is the average distance of the Earth from the Sun, every six months the clocks will pass very close to one another going in opposite directions at a relative speed twice that of the Earth's average tangential velocity round the Sun. Of course three months after that the two clocks will be stationary relative to one another, having gradually slowed down relative to one another due to their circular motions in opposite senses; but then they will pick up speed once again relative to one another, till they again pass each other on the other side of the Sun at the same relative velocity as they did six months earlier. As the years go by, they will have spent most of their time moving relative to one another -- and moving quite fast, at that: about 1/10,000c on the average, which should give a distinctly measurable time dilation over a hundred years, at least with sensitive modern instruments. So by the equations of Relativity -- and here, whether we are speaking of SR or GR is immaterial -- each of the two clocks should run slower than the other: which of course is impossible.

If such an experiment were conducted with extremely sensitive clocks -- and it could be conducted even today, though it seems rather unnecessary to do it, since whatever the outcome the Theory of Relativity would be disproved -- then if after, say, a hundred years (as measured by an Earthly clock) it was found that one of the orbiting clocks indicated a lesser elapsed time than the other, we would have conclusive proof that the principle of relativity is wrong, and that absolute movement does exist; while if it was found that both clocks indicated the same elapsed time at the end of the experiment, it would prove that there is no such thing as time dilation due to relative motion of clocks. Either way, however, it would prove that the Theory of Relativity is wrong.

There are many other ways in which absolute and complete symmetry can be assured between moving clocks. Even Prof. Bartocci's requirement that both clocks travel along paths that are "geodesic" can easily be met. That's because it is not necessary to have the clocks reunite at a single location in order for us to be able to compare their readouts: all that is needed is a set of photographs of their readouts reunited in a single location. The photographs can even be transmitted electronically to a single location, the way the Hubble Space Telescope transmits its photographs to Earth.

For example, we can have two clocks, each of which rectilinearly and uniformly passes two space buoys in opposite directions, both of the buoys being stationary relative to one another, and situated far away from any detectable gravitational field. (This last requirement of course could not yet be met by NASA, so it is not an experiment which could be performed today; but as I said above, it seems unnecessary to perform it, since whatever the outcome, the Theory of Relativity would be disproved.) The buoys being stationary relative to one another, the distance between them, as observed by any single inertial observer, obviously cannot change over time. The clocks could be made to pass close to the buoys in uniform rectilinear motion at speeds v and -v, respectively: those speeds being of course relative to the buoys. (Thus the clocks' uniform speed relative to one another would be 2v -- or a little less if v were almost as great as c and if the Lorentz transformation equations are indeed correct -- but it would definitely not be zero: indeed it must be greater than v.) As each of the clocks passed each of the buoys in turn, a fine antenna jutting out from the buoy could be caused to delicately touch the vessel which is carrying the clock, and thereby electrically trigger a camera affixed next to the clock in the same vessel -- so obtaining a photograph of that clock's readout at the moment when the clock passed that particular buoy. This could be done with both the clocks and both the buoys, obtaining four photographs as a result. Comparing the four photographs, electronically "beamed" to a single location after the experiment is over, would tell us whether one of the clocks ran slower than the other or not, since they both would have covered the same distance in a straight line and at the same speed relative to the buoys, but at more than that speed relative to one another. If the photographs show that one of the clocks ran slower than the other, it would prove that time dilation does exist, but then so does absolute motion; while if the photographs show that the clocks ran at the same rate, it would prove that time dilation due to uniform rectilinear motion of one clock relative to another does not exist. But in either case, the Theory of Relativity would be disproved.

There is no question, therefore, that Prof. Dingle's argument has not been refuted by Prof. Bartocci. Indeed the latter does not seem to have thought of the extreme ease with which any competent engineer can devise such an experiment satisfying every single requirement of symmetry. (The above are only two of the great many set-ups capable of providing complete symmetry between the clocks, which could be devised by any reasonably competent engineer.)

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 notw ’s proper time. If we want to know dtw , 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 Dtw = 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 Dta , 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!!

Actually, it is not all that difficult, if we combine length contraction with time dilation. This is because according to Relativity, length contraction is supposed to be unidirectional (rods are only supposed to contract in the direction of motion, not perpendicularly to that direction), while time dilation is supposed to work uniformly for all possible orientations of the clocks. Thus all that's needed is to measure, say, the time it takes for an electrical current to flow along a wire of given length in two different cases: (A) when the wire is oriented in the direction of motion and (B) when it is oriented perpendicular to the direction of motion. If length contraction does indeed exist, the length of the wire, as observed by a given inertial observer, should be different in (A) compared with (B), while for the very same observer, the time dilation factor -- if time dilation does indeed exist -- should be the same in both cases. Thus the time intervals measured by that observer for the current to flow right through the wire when it is oriented in the two different ways should be different from one another after the vessel carrying the wire is set in motion, while the time intervals should be the same  before setting the vessel in motion. (By "motion" here we are referring only to uniform rectilinear motion, of course. The vessel's velocity would be measured relative to its own velocity, comparing the velocity it had before being set in motion with that it acquired after having been set in motion.)

The same experiment could alternatively be performed with two identical wires of equal length in the vessel, each oriented at right angles to the other. Note however that there is no need for more than one vessel, since after the first set of measurements has been made, the vessel is set in motion for the second set of measurements, and thus its relative velocity in both cases cannot possibly be zero.

But if after the vessel is set in motion, the measured time intervals for the current to flow through the wire in the two different orientations -- or through two identical wires in the two different orientations -- are different from one another, the result would contradict the Principle of Relativity, since it would be possible in that case to tell if the vessel is in motion or not!

On the other hand, if no time difference exists when the current flows through the same wire in the two different orientations (or through two identical wires in the two different orientations), then it can be concluded definitively that there is no such thing as time dilation and/or length contraction.

But either way, the Theory of Relativity would be disproved, and that too, very easily -- contrary to Prof. Bartocci's assertions.

(As a matter of fact such an experiment can be performed even now: though as I said, it seems rather unnecessary to perform it, since whatever the outcome, the Theory of Relativity would be disproved. Six months from today the entire Earth will be headed in the opposite direction compared to the direction toward which it is headed at the present moment, and the relative speed between the two cases will be about 1/5,000c, which should afford a very easily detectable time dilation in a wire of fairly long length (say, 1 km). If needed, the wire can be doubled back upon itself -- and that too, several times -- so that an even longer wire can be used; and moreover the start of the flow of current and the end thereof would be at the same location, so that the time it takes for the current to flow along the given wire -- or both the wires, if two wires are being used -- can be measured by a single clock. As far as I know, no time dilation such as this has ever been detected or even mentioned in the literature. But perhaps no one has ever thought of conducting the experiment properly, even though it would disprove the Theory of Relativity even more conclusively than the Michelson-Morley experiment disproved the "stationary aether" theory! And note that in such a case the effects of rotational acceleration, if any exist at all, would cancel each other out: the experiment would be perfectly symmetrical. -- I just mention this as a possible experimental test to falsify Relativity: something for which Prof. Bartocci has been asking for a long time.)

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 = dtw = (Ö (1-(-v)2))dta , and from these two relations, which are in some sense both quite correct, it would follow: dta = (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 dta (a ’) = (Ö (1-v2))dt(a ’), dt(w ’) = dtw (w ’) =

(Ö (1-v2))dta (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

y = (L+vT+vy)/c,

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:

T’ = T/(1-b ) (where, as usual, b =v/c).

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:

T" = T/(1+b ),

whence it follows:

D T = T’-T" = 2b T/(1-b 2),

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-b2), 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

D Tr = T’Ö (1-b 2) - T"Ö (1-b 2).

This is equal to the classical Roemer effect D Tc 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 b3):

D Tr = D TcÖ (1-b2) = 2b TÖ (1-b2)/(1-b2) =

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

D Tc = 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):

f : x = 0, y = L-ct, z = 0

(velocity (0,-c,0), speed c).

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’):

x = (x’+vt’)/Ö (1-b2), y = y’, z = z’, t = (t’+vx’/c2)/Ö (1-b 2),

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-b2),

z = 0 ® z’ = 0.

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

(-v,-cÖ (1-b2),0) (of course, the photon’s speed is always c, since

v2+c2(1-b2) = 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

tg(q ) = v/cÖ (1-b 2) = b /Ö (1-b 2) »b (up to higher order terms in b )24,

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 showed 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 balistic 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.

The question of synchronisation of distant clocks is another matter which Prof. Bartocci does not seem to understand properly. And yet it is clearly crucial for Relativity, because without it, the logical contradiction between the Principle of Relativity and the postulate of the constancy of the speed of light for all inertial observers is not removed.

It is not at all necessary -- as implicitly claimed by Einstein -- that the second postulate of Relativity (the postulate of the constancy of the speed of light) is indispensable for synchronising clocks at a distance from each other. There are many other ways to synchronise distant clocks, and any competent engineer should be able to describe several such methods. Most of them don't even require light in any guise, shape or form.

For example, one could measure the speed of an electrical signal along a wire of given length. Since a wire is flexible, we can use a single clock for measuring the time the signal takes to travel along a known length of the wire. These two data will be all that's necessary to find the speed of the signal along the wire. (If one is in doubt about the speed of the signal being independent of the orientation of the wire, or of the direction of the signal along it, one can coil and orient the wire in several different ways when taking the measurements, and send the signal in one direction along the wire in some cases, and in the opposite direction in others: then if all the measurements come out the same -- within the limits of accuracy of the clock, of course -- then one can be assured that the speed of the signal is independent of the orientation of the wire or of the direction of the signal along it.)

Now one can place two identical clocks, which are known to tick at the same rate when placed close to one another, at a distance less than or equal to the length of that very same wire, and with the help of that wire and an identical electrical signal sent through it -- the time the signal takes to travel the length of the wire being known -- synchronise the clocks perfectly (again, of course, within the limits of accuracy of the clocks.) All one would need to do is allow for the time the signal takes to travel the length of the wire -- and that time would be known with precision.

Note that with this method -- which does not make use of light at all -- it is also possible to synchronise two clocks which are not merely separated by a distance, but even moving relative to one another at any arbitrary velocity v. (And it wouldn't even matter whether they are moving rectilinearly and uniformly, or not!) All that's needed is two flexible wires connecting each of the clocks which need to be synchronised with one another, to a single third clock which is moving at velocity +(0.5v) relative to one of the two clocks which need to be synchronised, and at velocity -(0.5v) relative to the other. Electrical signals carrying the ticks of the third clock to the other two would enable the latter to be synchronised perfectly (until and unless, of course, the two clocks got farther apart than the length of the wires -- but if the wires were very long, the period during which the clocks would be perfectly synchronised could last for quite some time.)

This proves that it is not at all necessary -- as Relativity asserts -- to think of the rates of flow of time in two different inertial frames which are moving relative to one another as being different, depending on the motion. It can be quite absolute.

And of course once absolute time is shown to be possible, even for a limited period, the Theory of Relativity (both SR and GR) must utterly and completely collapse.

Indeed theoretically one could synchronise all the clocks on all the planets of the Solar system using this method. (Of course the wires would have to be enormously long, and also properly arranged so that they do not get entangled with each other: but that need not be actually impossible, just a tad difficult.) The planets are all in motion relative to each other and are also separated by varying distances; but it wouldn't matter, because the time each electrical signal takes to get from one clock to the other would be known with precision, and could be accounted for when synchronising the clocks.

Alternatively, in order to synchronise two clocks, one can take two identical clocks which are known to be ticking synchronously when placed close to one another, and transport one of them very slowly to another location at a given speed, accelerating and decelerating it very gently. The speed, and the rate of acceleration/deceleration, can be so calculated that when transporting the clock to the other location at that particular speed using that particular rate of acceleration/deceleration, the so-called Relativistic time dilation effect will be smaller than the limit of accuracy of the clocks. (It is to be remembered that there is no such thing as a perfect clock: every clock has a limit of accuracy!)

For instance, if the clocks are accurate to one part in a billion -- i.e., they gain or lose at most one second in a billion seconds, or one hour in a billion hours -- and if the readouts of the clocks are limited to the sixth decimal place after the full second (i.e., the indicators show at best one microsecond, but no lesser interval of time), then it can be calculated, using the Lorentz transformation equations, that if one of them is transported at a uniform velocity of 10 km/h to a distance of ten light-seconds from its original location -- i.e., 2,997,924.58 km, or about five times the distance from the Earth to the Moon -- the time dilation effect would be too small for it to be indicated on the moved clock. (To see the calculations in this regard, click here.) Thus the two clocks will still be synchronised, up to the limits of their accuracy, when they are ten light-seconds apart. (And it is to be noted that these limits of accuracy are quite high.)

Yet alternatively, if both clocks are moved away from a common location in opposite directions perfectly symmetrically, using identical rates of acceleration and deceleration and speeds attained thereupon (as observed, of course, by an observer stationary relative to their point of departure), then any Relativistic time dilation -- if it exists at all -- would have to be the same for both clocks ... again, of course, as observed by the observer who is stationary relative to their point of departure. Thus  even when the two clocks are far away from one another and from the observer, they would have to be running synchronously, despite being at a great distance from the observer, and even if they are moving at a high speed relative to him and to one another. (In this case the acceleration / deceleration rates, and final speed attained thereupon, can even be large: it wouldn’t matter!)

Alternatively still, one could put two identical objects into perfectly circular orbits in a single plane, each orbit being identical in diameter but in opposite sense to the other, around a large source of gravity like the Sun. As we said earlier, the two objects will pass one another with regularity at two points each diametrically opposite the other. If we situate our two clocks at these two diametrically opposite points, we could synchronise them using the regular events when the two objects pass one another.

None of the above methods of synchronising clocks makes use of light at all. But we can even make use of light for this purpose, provided we do it judiciously.

For example, we could take advantage of the fact that light always spreads out radially from its source. For instance, if two clocks (whether stationary or rectilinearly moving relative to one another) were at quite a large distance from one another -- even a distance as great as the diameter of the orbit of Jupiter -- but if the line between them, at its mid-point, were at right angles to the line thereto from a much more distant pulsar, then each pulse of the pulsar would arrive simultaneously at each of the clocks, and they could be synchronised using the pulses. If the pulsar were tens of thousands, or maybe even a hundred-thousand, light-years away from the clocks, the distance between the clocks, even if it were as large as the diameter of the orbit of Jupiter, would be so small in comparison as to be negligible as far as subtending a measurable angle at the pulsar goes; and it would be utterly unreasonable to suppose that photons (or light waves) from the pulsar with such a small angle between their trajectories would have travelled all the way for tens of thousands of years at different speeds: even a tiny difference in speed at that distance would amount to a huge and easily detectable time difference at our end. And any four such pulsars, not all of them being in a single flat plane, would enable clocks to be distributed anywhere within a sphere of diameter the size of the orbit of Jupiter, or even larger, with perfect synchronisation (within, of course, the limit of accuracy of the clocks.)

I could mention many more such methods, none of which absolutely requires the second postulate of Relativity. Admittedly any one of these methods may have some flaw in it which I might have overlooked, but if all the methods were applied to a distant pair of clocks, and if they all agreed with one another, then one could be virtually certain that the clocks were indeed synchronised. And if one of the methods disagreed with the others, it could be discarded as unreliable. It is most unreasonable to assume that all of them would disagree with all the others.

So you see, it is NOT necessary to have the second postulate of Relativity to synchronise clocks (and thereupon to obtain a universal time, valid for all of space.)

And as I said earlier, once a universal time is obtained, the rest of Relativity -- both the Special and General Theories -- must perforce collapse!

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.

It is also to be mentioned that Prof. Bartocci does not touch upon several other criticisms of Relativity which are also valid. For example, just like the "Train" thought-experiment, Einstein's "Elevator" thought-experiment is also flawed in more ways than one. Consider an elevator with a man in it, accelerating at a constant rate in a straight line past a ray of light which is being propagated at right angles to the line of acceleration; then if that ray were admitted into the elevator through a small hole in one of its walls, it would hit the opposite wall a little lower -- as judged by the man in the elevator -- than the spot directly across the elevator from the small hole. From this consideration, and applying the "Principle of Equivalence" (i.e., of inertial and gravitational acceleration), Einstein concludes that gravity should also bend light in precisely the same way and in precisely the same measure.

But Einstein forgets that if the experiment were repeated, with the sole change that this time the elevator were being decelerated rather than accelerated, but moving along the same trajectory in the same direction past the same ray of light as before, with the light entering the same hole as before, it would hit the opposite wall a little higher -- again, of course, as judged by the man in the elevator -- than the spot directly opposite the hole! The man would, in this case, be pushed toward what was previously the ceiling of the elevator -- that is to say, what was the ceiling when the first experiment was conducted; and if both the drawings of the experiment were drawn on a single piece of paper, with the lines of the light path shown as being horizontal on the paper, and the trajectory of the elevator shown as going vertically upward, when depicting the elevator decelerating instead of accelerating, the man would have to be drawn standing on the inside of the top line representing the elevator, rather than the bottom line -- all other things being exactly the same as before. (I need not draw the two pictures, for any competent reader should be easily able to satisfy himself -- or herself -- that what I say is correct.)

This proves that if the Principle of Equivalence is correct, the path of light ought to bend in a gravitational field in either direction, depending on whether one day the field decides to think of itself as being "equivalent to inertial acceleration" or on another day, "equivalent to inertial deceleration"! And it is to be noted that by the Principle of Relativity, there ought to be absolutely no difference whatsoever between inertial acceleration and deceleration, for if there were, we would be able to know whether a body was speeding up or slowing down.

This proves that either the Principle of Relativity is wrong or the Principle of Equivalence is wrong. But whatever the case, the Theory of Relativity is hereby disproved.

(I am indebted for this argument to Thomas Weber, who very kindly sent me his book in which this argument is outlined in the very initial pages.)

The reason for this is, of course, the fact that the "Elevator" thought-experiment tells us something about the properties of geometry, and nothing at all about the properties of matter. For it is to be noted that the same argument as Einstein's will apply even to an imaginary point-like particle moving rectilinearly and uniformly at right angles to the line of acceleration of an accelerating (or, for that matter, decelerating) grid! Although the imaginary particle is, by our very hypothesis, moving rectilinearly, yet as seen by an observer stationary relative to that accelerating (or decelerating) grid, its path will be curved.

It is ludicrous to suppose therefrom, though, that gravity too will attract even imaginary point-like particles! What GR claims is that gravity "bends" space. But if it does, then so does an elevator under constant rectilinear acceleration -- one which is accelerating in a straight line through that very same "bent" space!

Of course it doesn't do anything of the sort. The above argument is a purely geometric one, and not a physical one at all. It is a trick geometry plays on us, very similar to the trick it plays on us by making rain which is falling vertically downward appear to fall at an acute angle to the road when we observe it through the windshield of a car moving horizontally along the road. Just as we know that the rain is in fact coming down perpendicularly to the roadway, even though when observed from a moving car it appears to be coming down at an angle to it -- and the faster the car's speed the closer to the horizontal does the rain appear to "fall"! -- so too the path of a uniformly moving point, in pure geometry, will appear to be curved or straight depending upon whether the vehicle or Cartesian grid relative to which it is moving is accelerating (or decelerating), or instead is itself moving uniformly and rectilinearly.

Remember that in the case of the rain example -- which is the same thing as aberration -- even if we were not speaking of rain, but of totally imaginary points moving vertically downwards at the same speed as the raindrops, we would "observe" the same thing: the points would appear to be moving, not at right angles to the roadway, but at an acute angle to it. But the car's horizontal motion does not change the fact that in reality, the raindrops -- as well as the imaginary points -- are in fact falling at right angles to the roadway. It is just a geometrical illusion that the points appear to be moving at an angle to the roadway rather than perpendicularly to it. (If the reality and the illusion were equivalent, then Pythagoras's Theorem would be disproved, for a moving point which traces out a right-angled triangle would trace out a triangle which is not right-angled when observed by an inertial observer moving uniformly and rectilinearly relative to the Cartesian grid in which that point is tracing out the right-angled triangle! And indeed if the observer were not inertial but were under either acceleration or deceleration -- or both alternatively -- for such an observer the very same point would not appear to trace out a triangle at all, but a figure made up of three curved lines.)

We should not let ourselves be taken in by such treicks geometry plays on us -- for if we did, we would lay ourselves open to the same sort of criticism we heap upon medieval artists who, having no knowledge of perspective, painted things that were farther away the same size as those which were closer.

Pure geometry is entirely in the mind, and considered by itself has no counterpart in the physical world: in the material realm there is no such thing as a strictly dimensionless entity called a point, or a strictly one-dimensional entity called a line -- or even, perhaps arguably, a strictly two-dimensional entity called a plane. One has to be very careful to apply a theorem of pure geometry to the material world: it may not be applicable, as illustrated by the absurdity of the above conclusion that if Einstein is right, gravity should bend the paths of strictly imaginary entitles like geometrical points.

(I am indebted for this argument to Dr Christoph von Mettenheim, who in his book Popper versus Einstein -- which is written in English, and published at Tübingen by the German publisher Mohr -- illustrates that the "Elevator" thought experiment is not only a strong argument, but, as Dr von Mettenheim wrote to me in a private e-mail, is "too strong", applicable to imaginary entitles like geometrical points -- and I would add, for that matter, even to elves, goblins and unicorns; and thus cannot possibly prove that in the physical world inertial acceleration is equivalent to gravitational.)

One very serious problem with thought-experiments -- and it is to be noted that both SR and GR are founded upon nothing else -- is the fact that one is never sure whether one has missed something crucial in one's line of thought! (Mind you, this doesn't mean that ordinary physical experiments are without flaws, for even in them, one can never be sure that one has not missed something crucial in interpreting the data which are the outcome of the experiment -- as we clearly see when we read Prof. Bartocci's interpretation of the Sagnac experiment.)


In conclusion, the following is clear from a careful perusal of Prof. Bartocci's lines of thought expressed in his paper:

  1. The so-called "mathematics" of Relativity (both SR and GR) is nothing but a pseudo-mathematics, being based on the postulate of the constancy of the speed of light for all inertial observers, which is taken as an unproven axiom. Since in mathematics no axiom may contradict any other, nor may it contradict any theorem proven with the help of the other axioms alone, and since the postulate of the constancy of the speed of light contradicts the (Galilean) theorem of addition of velocities, which can be proven with the help of the other axioms of mathematics, the postulate of the constancy of the speed of light cannot, logically speaking, be a part of genuine mathematics. (In the term "mathematics" here we include geometry, and consequently in the meaning of the term "axiom" we include the definitions, common notions, propositions and postulates of Euclid.)

  2.  .
  3. Both SR and GR are based on two thought-experiments (Gedankenexperimenten) of Einstein, both of which are logically flawed. Since these two Gedankenexperimenten are absolutely crucial to the development of SR and GR, both these theories must also be logically flawed as a result.

  4.  .
  5. The physical interpretation of SR contradicts that of GR: in particular, the so-called phenomenon of Relativistic mass increase contradicts the so-called Principle of Equivalence of inertial and gravitational acceleration.

  6.  .
  7. There is absolutely no reason to think that simultaneity is relative. The only -- and it cannot be emphasised enough, ONLY -- argument offered to show that simultaneity is relative is Einstein's famous "Train" thought-experiment, which as demonstrated above is logically flawed, and that too, in more ways than one. And if simultaneity is not relative, time also cannot be relative, because it would be possible to synchronise clocks all over the universe, using a single master clock.

  8.  .
  9. The Sagnac experiment conclusively proves that the speed of light for all of us humans on Earth is not a constant -- and Prof. Bartocci admits as much, though inadvertently, when he claims that for an observer a situated on the rim of a rotating disc, the speed of light must be different from that for an observer w stationary relative to the centre of that rotating disc.

  10.  .
  11. Nor can the speed of light be a constant for two inertial observers moving uniformly and rectilinearly relative to one another, because of the fact that if at the moment the light leaves its source, the two observers are equidistant from the source, then when the light reaches the observers one of them will have moved relative to the other, and thus the light will have travelled different distances to reach each of them. The same light wave cannot, obviously, take the same amount of time -- as measured by any single clock -- to travel two or more different distances ... those distances being measured, of course, by the very observer in whose hands the aforesaid clock is being held!

  12.  .
  13. There is no problem at all providing a completely symmetrical set-up for a very genuine "twin paradox", showing that if two clocks are moving uniformly and rectilinearly relative to one another, then either one will be running slower than the other, by indicating a shorter elapsed time to cover a fixed distance, a distance which moreover is identical in the frame of reference of each of the clocks -- in which case it will be possible to know which clock is moving, thereby contradicting the Principle of Relativity -- or both the clocks will run at the same rate, by indicating the same elapsed time to cover the same fixed distance, thereby showing that there is no such thing as time dilation.

  14.  .
  15. It is easy to show -- even experimentally -- that there is no such thing as time dilation and/or length contraction, by passing an electrical current of known strength along a wire of fixed length, and measuring the time it takes for the current to go all the way through the wire, orienting the wire in one case in the direction of motion of the Earth around the Sun, and in the other case orienting it perpendicular to that direction. Since the motion of the Earth around the Sun is about 1/10,000c, and since the experiment could be repeated six months later when the Earth is moving in the exact opposite direction, this should easily indicate a difference in times for the two orientations of the wire, if time dilation and length contraction were real physical phenomena. But if such a difference were observed, that would not suffice to prove the Theory of Relativity, for then it would be possible to know that the Earth was moving in an absolute sense, which would contradict the Principle of Relativity. So whatever the outcome of the experiment, the Theory of Relativity would be disproved.

  16.  .
  17. It is also easy to see that Einstein's "Elevator" thought-experiment is logically flawed, and that too, in more ways than one. In addition to the flaws pointed out earlier, it also implies that acceleration is different from deceleration, whereas according to the Principle of Relativity there should be absolutely no difference between acceleration and deceleration. Thus either the Principle of Relativity must be wrong or the "Elevator" thought-experiment -- or both -- must be wrong.

  18.  .
  19. And finally, it is easy also to see that neither Einstein nor -- I am sorry to say -- my friend Prof. Bartocci, has actually thought through their arguments thoroughly. The Theory of Relativity has more holes in it than a sieve, and that is probably the reason why it is intuitively so unappealing. It is just that those who support it, do so out of a sense of awe in which the name of Einstein is held, and not at all out of a logically reasoned conviction as genuine scientists. Whereas Prof. Bartocci, for his part, who to his credit is not overawed by the name of Einstein, appears however to be overawed instead by the names of Lorentz, Minkowski, Gauss and Riemann, and seems to put all his faith in the so-called "mathematics" of Relativity developed by them, without even bothering to ask himself: "Is the development of this so-called 'mathematics' consistent with the axioms upon which the rest of mathematics is founded?" Perhaps he would do well to repeat to himself his own maxim Amicus Plato, sed magis amica veritas, substituting in turn the names Fitzgerald, Lorentz, Minkowski, Bolyai, Gauss and Riemann -- and perhaps Lobachevsky as well -- for the name Plato.

  20.  .
And as far as I am concerned, my own maxim is well expressed by the following quote from an author whose name I do not know:
Fear not a clash between truth and falsehood,
because if what you believe is true,
the clash will strengthen your conviction;
while if it is false, then you are afforded the inestimable value
of exchanging a wrong idea for a correct one!

If the reader has any comments or questions, I can be contacted at .

Ardeshir Mehta.



* Dipartimento di Matematica Università, Via Vanvitelli 1, 06100 Perugia - Italy ( 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 = dtw 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 = dtw = (Ö (1-(-v)2))dta = (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.