«Speakable and unspeakable in special relativity. I. Synchronization and clock rhythms. Rodrigo de Abreu and Vasco Guerra Departamento de F´ısica, ...»
Speakable and unspeakable in special relativity. I.
Synchronization and clock rhythms.
Rodrigo de Abreu and Vasco Guerra
Departamento de F´ısica, Instituto Superior T´cnico, Universidade T´cnica de Lisboa,
1049-001 Lisboa, Portugal
Abstract. The traditional presentation of special relativity is made from a rupture
with previous ideas, such as the notion of absolute motion, emphasizing the antagonism
of the Lorentz-Poincar´’s views and Einstein’s ideas. However, a weaker formulation e of the postulates allows to recover all the mathematical results from Einstein’s special relativity and reveals that both viewpoints are merely diﬀerent perspectives of one and the same theory. The apparent contradiction simply stems from diﬀerent procedures for clock “synchronization,” associated with diﬀerent choices of the coordinates used to describe the physical world. Even very fundamental claims, such as the constancy of the speed of light, relativity of simultaneity and relativity of time dilation, are seen to be no more than a consequence of a misleading language adopted in the description of the physical reality, which confuses clock rhythms with clock time readings. Indeed, the latter depend on the “synchronization” adopted, whereas the former do not. As such, these supposedly fundamental claims are not essential aspects of the theory, as reality is not altered by a mere change of coordinates. The relation between the rhythms of clocks in relative motion is derived with generality. This relation, which is not the standard textbook expression, markedly exposes the indeterminacy of special relativity, connected with the lack of knowledge of the value of the one-way speed of light. Moreover, the theory does not collapse and remains valid if some day the one- way speed of light is truly measured and the indeterminacy is removed. It is further shown that the slow transport method of “synchronization” cannot be seen as distinct from Einstein’s procedure.
PACS numbers: 03.30.+p, 01.55.+b Keywords: Special relativity, speed of light, synchronization, simultaneity, clock rhythms, time dilation, slow transport.
Synchronization and clock rhythms 2
1. Introduction In previous works we undertook a reﬂection on the foundations of special relativity [1–5].
An inspiring source in this journey was John Bell’s book “Speakable and unspeakable in quantum mechanics” . More precisely, the book includes a chapter entitled “How to teach special relativity,” in which Bell recommends the use of a Lorentzian pedagogy,
i.e., that special relativity should be taught starting from the idea of a preferred frame:
“I have for long thought that if I had the opportunity to teach this subject, I would emphasize the continuity with earlier ideas. Usually it is the discontinuity which is stressed, the radical break with more primitive notions of space and time. Often the result is to destroy completely the conﬁdence of the student in perfectly sound and useful concepts already acquired.” John Bell’s idea goes much deeper than the questions of “continuity” and “conﬁdence.” Indeed, he continues by adding and stressing an ingredient of special relativity still somewhat unnoticed, namely that all the results from special relativity can be derived either by following the ideas of Lorentz and Poincar´ of the existence of a “preferred e reference frame” or Einstein’s “equivalence of all inertial frames.” He acknowledges a diﬀerence of philosophy – and a diﬀerence of style – on two approaches describing the
The diﬀerence of philosophy is this. Since it is experimentally impossible to say which uniformly moving system is really at rest, Einstein declares the notions ‘really resting’ and ‘really moving’ as meaningless. For him only the relative motion of two or more uniformly moving objects is real. Lorentz, on the other hand, preferred the view that there is indeed a state of real rest, deﬁned by the ‘aether’, even though the laws of physics conspire to prevent us identifying it experimentally. The facts of physics do not oblige us to accept one philosophy than the other.
The last quoted assertion, although well-known by specialists, still startles most physicists. Nevertheless, it is simply the quite obvious aﬃrmation that the study of relative motion can be made without any reference to absolute motion, but is not incompatible with it. A fact well-known by Galileo, Newton, Lorentz and Poincar´ e and deeply connected with the principle of relativity, as carefully debated in , that most textbooks tend to forget.
In his outstanding 1905 relativity paper , Einstein considers the reference to absolute motion as “superﬂuous.” Modern physics lead to a widespread acceptance of a strict operational view of physics, making it easy to identify the word “superﬂuous” with “meaningless.” This constitutes the essence of the “diﬀerence in phylosophy.” However, the bases of Einstein’s special relativity are much less solid than it is generally accepted.
In short, both postulates from special relativity are too strong and can be formulated in weaker forms , while keeping fully compatible with all available observations and experimental results [1–5]. This more general formulation of special relativity, brieﬂy Synchronization and clock rhythms 3 reviewed in sections 2 and 4, strikingly evinces that there is an indeterminacy in the theory , since there are quantities which eventually cannot be measured, such as the one-way speed of light, as noted early by Reichenbach [8, 9] and discussed by many authors [2, 10–19]. As a consequence, a deadlock arises in practical terms – although not in fundamental ones – and some additional assumptions have to be required to cut this Gordian knot. Einstein’s theory solves the problem in an extremely simple and elegant way, with his methodology for “synchronization” of distant clocks, providing a straightforward and eﬀective operational procedure to study physics . Still, other approaches to the problem are possible, fully compatible with Einsteins relativity in practice, but leading to very diﬀerent assertions in fundamental and philosophical terms.
It seems reasonable to conceed that when additional restrictions are included on top of those implied by the physical reality, then it is likely we are describing only part of it. The diﬃculty in transposing this somewhat evident statement into the context of Bell’s two philosophies lays in a misleading interpretation of the “symbols” employed in the mathematical formalism, with t and v on the ﬁrst line. In fact, a negligent use of language, associated with the unclear separation of scientiﬁc results and “philosophical” or “ideological” statements (deﬁned here as statements that are dependent on an arbitrary convention or on an interpretation relying on additional assumptions not imposed by experiment), has led to a terminological confusion and apparent contradictions.
The diﬃculty in accepting Bell’s point is more on the speech or discourse surrounding special relativity, not so much on the calculations actually performed. As a simple example, “relativity of simultaneity,” one of the trademarks of special relativity, presented almost always in the very beginning of any text or media content about relativity, is one debatable “philosophical” statement. As a matter of fact, it depends on the choice of coordinates (cf. section 3) and, therefore, by no means is an intrinsic feature of the theory . The same is true regarding “relativity of time dilation” (cf.
section 5). Despite the correctness of the underlying calculations, these aﬃrmations are repeatedly given an abusive semantics they do not possess, as detailed in the body of this paper, as they mix the notions of clock rhythms (or clock tick rates) and time readings (or time coordinates) displayed by clocks. Indeed, the former are independent of any “synchronization” procedure, whereas the latter are not.
Following , we note that emphasis should be given to the properties that do not depend on the choice of coordinates, or, equivalently, on the “synchronization” procedure adopted. To avoid the problem of coordinate-dependent quantities, Oziewicz [20,21] and Ivezic  have developed “coordinate-free” approaches to special relativity.
The existence of alternative formalisms and broader views of special relativity following the general lines presented above is rather consensual. As illustrations, we can name the works of Edwards , Mansouri and Sexl , Leubner et al  or Selleri . However, the speech surrounding special relativity has gained a strong ideological charge, nearly dogmatic. Thus, if these unconventional theories and the corresponding calculations are widely accepted, their implications remain “unspeakable.” As a Synchronization and clock rhythms 4 consequence, the establishment of a broader and more general view of special relativity was hindered up to now and a minimal interpretation prevails.
The aim of this work is to accentuate the need for a general formulation of special relativity, by reconciling two apparently contradictory discourses. Hence, one should not speak about two philosophies, as they are diﬀerent aspects of one and the same theory. In particular, one should not say that the results from special relativity can be derived either by following the ideas of Lorentz and Poincar´ of the existence of a e “preferred reference frame” or Einstein’s “equivalence of all inertial frames,” but rather use the word both. For instance, Lorentz’s view is usually associated with the sentence “the speed of light in vacuum is c only in one reference frame,” whereas Einstein’s view with the seemingly contradictory sentence “the speed of light in vacuum is c in all inertial frames.” These statements induce to think of a severe incongruity, that could be depicted schematically as in ﬁgure 1a). The conﬂict can be easily elucidated with the simultaneous use of diﬀerent procedures for clock “synchronization,” to which are associated diﬀerent choices of the time coordinates used to describe physical events [2,4].
A key concept is the notion of “Einstein-speed” previously introduced in  and reviewed in section 6. Within the proposed formulation of special relativity, the former sentences have to be rephrased to “the one-way speed of light in vacuum is c in one reference frame; the two-way speed of light in vacuum is c in all inertial frames” and “the one-way Einstein-speed of light in vacuum is c in all inertial frames,” which could be represented as in ﬁgure 1b). One explicit case to exemplify this assertion can be found in section 5 from . It shows that special relativity was developed under the shadow of a false dichotomy and that with a precise language all conﬂicts disappear at the onset.
The structure of this paper is the following. In the next section we present the weak formulation of the postulates. Synchronization procedures and the deﬁnition of simultaneity are discussed in section 3. The mathematical formalism is introduced in section 4, with the presentation of the IST transformation and its relation with the Lorentz transformation. Section 5 contains a debate on the diﬀerence between clock rhythms and clock time readings, in a short and simple subsection, which nevertheless is a cornerstone of this article. The concept of Einstein speed is reviewed in section 6, where the full compatibility of Bell’s two philosophies and the picture of ﬁgure 1b) is deﬁnitely established. The bridge between the two philosophies is completed in section 7, where the role of the Lorentz transformation is further discussed. In section 8 we obtain the relation of rhythms between two clocks in relative motion and its correlation with the usual time dilation expressions, in another central section of this work. The slow transport method of “synchronization” is presented in section 9 and shown to be equivalent to Einstein’s procedure. Finally, section 10 summarizes our main ﬁndings.
2. The weak statement of special relativity postulates
imposed. In this view, we start, exactly as it has been done by Einstein in his 1905 paper , with the deﬁnition of the “rest system.” Einstein deﬁned it as “a system of co-ordinates in which the equations of Newtonian mechanics hold good (i.e. to ﬁrst approximation).” We deﬁne it as a system in which the one-way speed of light in empty space is c in any direction, independently of the velocity of the source emitting the light.
As the name indicates, the one-way speed of light is the speed of light in a path in just one direction.
One may argue that it may be impossible to know which is the rest system. The answer to this remark is somewhat disconcerting, as the issue is of no relevance for the point we are trying to make. If we accept that when a photon travels between two points in space, it does so with a certain speed, regardless of our knowledge of its value, then there is no conceptual diﬃculty at this stage, although there may exist a practical one.
How to deal with this impossibility has been already discussed [2, 4] and the question is readdressed in section 7.
The postulate of the constancy of the speed of light is then stated as follows :
• the two-way speed of light in empty space is c in any inertial frame, independently of the velocity of the source emitting the light.
Here, an inertial frame is any frame moving at constant velocity in relation to the rest system, and the two-way speed of light is its average speed on a round-trip. It is worth to emphasize that what one learns from the Michelson-Morley experiment is the constancy of the two-way speed of light in vacuum and no information can be obtained regarding its one-way value. The deﬁnition of the rest system is important to give a starting reference point to the theory. In addition, it is interesting to note that the existence of the rest system can be deduced from the constancy of the two-way speed of light and the assumption of homogeneity of space .