Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
Constancy of the speed of light and the unit matching problem
Introduction
Albert Einstein based the special theory of relativity (SR) on two principles, or postulates (Einstein, 1905):
- 1.
[Principle of relativity] The laws governing the changes of the state of any physical system do not depend on which one of two coordinate systems in uniform translational motion relative to each other these changes are referred to.
- 2.
[Light principle] Each ray of light moves in the coordinate system “at rest” with the definite velocity V independent of whether this ray of light is emitted by a body at rest or in motion (Einstein, 1905, p. 895, English translation from The Collected Papers, Vol. 2, 1989, p.143).
Homogeneity of space and time and isotropy of space, implicitly assumed, were later explicitly added to the list of postulates.
From as early as 1910, with Ignatowsky (Ignatowsky, 1910, 1911), and Frank and Rothe (Frank and Rothe, 1911), many researchers questioned the necessity of the second postulate (e.g., Weinstock, 1965; Mitvalsky, 1966; Lévy-Leblond, 1976; Srivastava, 1981; Mermin, 1984; Schwartz, 1984; Sen, 1994; Pal, 2003, among others). Lévy-Leblond (1976) expressed their common aim thus:
… I intend to criticize the overemphasized role of the speed of light in the foundations of the theory of special relativity, and to propose an approach to these foundations that dispenses with the hypothesis of the invariance of c.(Lévy-Leblond, 1976, p. 271)
David Mermin's rationale is the most explicit:
Relativity …. is not a branch of electromagnetism and the subject can be developed without any reference whatever to light.(Mermin, 1984, p. 119)
The origin of these authors’ unease seems to be the fact that while the principle of relativity, spacetime homogeneity and spatial isotropy are all fundamental symmetries of nature, the second postulate appears to be too particular and contingent to deserve such elevated company. They then explore the consequences of assuming only the following postulates:
IG1. Homogeneity of space and time: The laws of physics are invariant under a translation of the origin of coordinates of space and time.
IG2. Isotropy of space: The laws of physics are invariant under rotations of the axes in which they are described.
IG3. Principle of relativity: The laws of physics are invariant under a transformation from one reference frame to another, if they are in uniform relative motion.1
It turns out that postulates IG1-IG3 severely restrict the form of the coordinate transformations between two uniformly moving frames of reference, S and . Assuming that the two frames move parallel to the positive X axis with velocity v, one can derive from IG1-IG3 alone the so-called Ignatowsky transformations. Their form is (see the derivation in e.g., Lévy-Leblond, 1976; Drory, 2014a)
Here, are the coordinates of some event as recorded by observer , and are the coordinates of the same event in the frame S. The undetermined parameter k is a universal constant but nothing is known of its value save that it is non-negative (Drory, 2014a; Lévy-Leblond, 1976). In the Lorentz transformations, corresponding to standard SR, , where c is the speed of light in empty space.
The common attitude towards k is well represented by A.M. Srivastava, when he notes:
As we know, the experiments show that [] has a finite value which is equal to the value of the speed of light. (Srivastava, 1981, -p. 505)
Similarly, Mermin (1984) explains:
From this point of view, experiments establishing the constancy of the speed of light are only significant because they determine the numerical value of the parameter k. (Mermin, 1984, -p. 119)
In this view, the value of k is purely contingent. Light holds no fundamental position in the theory, nor does any other signal.
That space-time symmetries and the principle of relativity restrict the form of the coordinate transformations to being Ignatowskian is non trivial, as justly stressed by Pal in his own derivation (Pal, 2003). But this is still a far cry from saying that we have obtained special relativity. Pauli, for one, claimed that the transformations thus obtained lack physical content, but unfortunately he did not detail his arguments (Pauli, 1958, p. 11).
My claim is that the Ignatowsky transformations do indeed lack physical meaning without additional postulates or information. These need not be specifically about the speed of light, but they are no more “general principles of nature” than the second postulate itself. Like the latter, they must refer to specific phenomena and possibly contingent aspects of our world, certainly to detailed properties of matter or signals.
Furthermore, I shall argue that the light postulate serves several functions in SR, which likely need to be fulfilled by several distinct postulates in the Ignatowskian view.
Practically all modern derivations of the Lorentz transformations start by assuming a slightly different set of postulates, namely IG1-IG3 and the postulate that the speed of light is frame-invariant (e.g., Rindler, 2006), rather than Einstein's original formulation (quoted above) which refers only to the behavior of light in a system “at rest”. Brown and Maia have analyzed in detail Einstein's motivations for this formulation as well as its implications for the derivation of the Lorentz transformations (Brown and Maia, 1993). However, Einstein's motivation is mainly of historical interest, having to do with connecting as smoothly as possible to ether theories, which are no longer relevant today. Nor is it my aim here to engage in a complete analysis of the logical structure of SR. I only intend to compare the consequences of postulates IG1-IG3 alone, as opposed to IG1-IG3 plus the light postulate. As Brown and Maia note, the numerical invariance of the speed of light in all inertial frames is a direct consequence of IG1-IG3 plus Einstein's second postulate.2 In the following, the numerical invariance of the speed of light will turn out to be easier to use in the analysis, and I therefore intend to follow modern convention in adopting it as equivalent to the second postulate. In order to respect historical usage, however, I shall distinguish between the “light postulate” (Einstein's original formulation) and the “invariance theorem” (the claim that the speed of light is numerically frame-invariant, which is a mathematical consequence from IG1-IG3 together with the light postulate). Although it would be interesting to investigate the unit matching problem directly on the basis of Einstein's original postulate, this would take us too far off the focus of the present work and I will not pursue this direction here.
I already argued earlier that the postulates IG1-IG3 are insufficient because they leave the value of k completely unconstrained (Drory, 2014a). I concluded that the Ignatowsky transformations do not represent a physical theory, but rather form a general framework that accommodates two rival alternatives without deciding between them. Newtonian kinematics is the theory singled out by postulates IG1-IG3 and the affirmation that , which functions as an additional axiom. Special relativity is the theory singled out by the postulates IG1-IG3 and the affirmation that . These two contradictory options disagree on such fundamental questions as whether lengths are frame-independent, whether simultaneity is absolute and many more. The Ignatowsky transformations offer no clear-cut position on such issues, since the answers they provide depend on whether or not.
I further argued that the claim that k should be empirically determined is irrelevant, because postulates IG1-IG3 are also generalizations from experiments. Whether they must be listed as fundamental postulates is a matter of the logical structure of the theory, not of the basis of their confirmation. The second postulate may be empirically derived, but it is nevertheless necessary to distinguish special relativity from Newtonian physics. As such, it deserves its place as one of the theory's fundamental postulates. It is true, of course, that to determine the world-view of the theory, the specific value of is not important. In this sense, and in this sense only, it does not matter that the invariant speed is that of light, rather than, e.g., gravitational waves. The very fact that k does not vanish, however, is essential to the theory's world-view, and this must be raised to the level of a postulate in order to complete the logical structure of the theory.
The present work criticizes the Ignatowsky claim from a different point of view. The coordinate transformations are not merely mathematical relations. They profess to relate empirical measurements of space and time. Such quantities depend on the units used to measure them and after setting up some basic rules in section 2, section 3 presents the problem faced by Ignatowskian and Lorentzian observers alike when it comes to matching the units of different observers. Sections 4 Matching time units, 5 Matching space units explore the way the second postulate solves the problem in special relativity and the difficulty faced by Ignatowskian observers when contending with the same issue. Section 6 analyzes the assumption of boostability, which is usually invoked to solve this problem, while sections 7 Idealizations, 8 Boostability in special relativity discuss the limits of allowed idealizations and how to bypass them in special relativity, with the use of the second postulate. Section 9 shows that even a single Ignatowskian observer faces some trouble when matching the units he uses to measure lengths along different axes. The conclusion, summarized in Section 10, is that Ignatowskian observers require additional postulates in order to make physical sense of their transformations (as opposed to mere mathematical definiteness).
Section snippets
Definitions, aims and methods
Formalizing physical theories is a useful endeavor when it clarifies their logical structure and underlying assumptions. But formulae are meaningless if they cannot be connected to empirical procedures, and one is likely to forget this in the dangerous glare of mathematical neatness. In special relativity most particularly, one must be cautious regarding measurement procedures, and these must be explicited and analyzed.
In the following I shall compare measurements performed by two observers,
The unit matching problem
We can now formulate the specific problem considered here. Coordinate transformations translate a set of numbers, the space and time coordinates measured by, e.g., Arthur, to another set of numbers, the coordinates of the same event as measured by, e.g., Betty. The comparison is only meaningful if the two observers can agree on the units that they use to measure space and time, however. It obviously makes no sense to compare the number 2 to the number 200 (or 190), unless we know that the first
Matching time units
Since historical units are ill-suited to the task, as we have seen, let us assume that Arthur and Betty's unit of time is some arbitrary “chronon”. The above analysis shows that for either Ignatowskian or Lorentzian observers, matching units to a single external process (e.g., a terrestrial year) fails, at least for the most straightforwards methods.
A much better choice, consistent with the lessons from Einstein's analysis of space and time, is for Arthur and Betty to devise structurally
The second postulate solution
Let us return now to the problem of the unit of length. As shown in section 3, both Lorentzian and Ignatowskian observers cannot simply set their units according to an external standard, such as an official meter rod. The relativity of lengths means that Arthur and Betty ought to attribute different values to the length of such a standard, according to its apparent velocity in their frames. Thus they cannot simply set it as “one”, unlike Galilean observers, in whose universe lengths are
Boostability of rulers
The possibility of transferring a ruler from one frame to another in such a way that it retains its length is a standard assumption in SR. Harvey Brown describes it as:
The boostability of rulers … Any object that can act a a rigid ruler in the frame S when stationary relative to that frame retains that role in its new rest frame S′ when boosted … (Brown, 2005, p.30)
Einstein made this assumption explicit already in 1910:
It should be noted that we will always implicitly assume that the fact of a
Idealizations
Such effects are often considered temporary difficulties, which - literally - straighten themselves out in time. Thus, Brown writes
We assume of course that following the application of the force that has been applied to the rod to boost it, the rod has time to retain an equilibrium configuration. (Brown, 2005, section 2.4, p.28)
The assumption here seems to be that the rod has a single equilibrium configuration to which it will return after whatever influence the accelerating forces have had on
Boostability in special relativity
Too much of a good thing may become as embarrassing as too little and we should worry whether we have not harmed special relativity as well. The above quotation from Brown claims that we need boostabililty in SR in order to relate the transformations to operational measurements. Perhaps in criticizing Ignatowskian observers we are also tainting Lorentzian ones, and our understanding of SR? However, as in the case of the rigidity assumption (in Einstein's words), we should ask ourselves what is
Lateral and longitudinal units
The above analysis raises another, unexpected issue, that to my knowledge has not been discussed before and in which it turns out that Lorentzian observers have an easier time than even Galilean ones. The problem arises if one tries to match length units using the invariance of transverse distances.
As shown in section 4, Arthur and Betty can match the height of their light clocks by simple observation, i.e., the way Galilean observers can match their unit lengths, because lateral distances are
Conclusions
Although mathematically sound, the Ignatowsky transformations must refer to empirically determinable quantities if they are to have physical meaning. Distances and durations are measured by instruments whose structure determines the units used. The mathematical transformations assume as a matter of course that the units used by both observers are identical, but it turns out that making sure this is the case is no trivial task. Simple methods that work for Galilean observers fail for either
Author statement
Alon Drory: Conceptualization, Writing, Reviewing and Editing, Visualization
Acknowledgments
I am very grateful to an anonymous referee whose insightful comments forced me to sharpen and clarify my thinking on several issues.
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