Length Contraction, Passive and Active

Relativistic length contraction is often described as a relationship between the spatial extents of a given configuration of entities in terms of two relatively moving systems of inertial coordinates. In other words, it is described as an attribute of a passive transformation. For example, consider a solid measuring rod, consisting of an equilibrium configuration of bound particles at rest in an inertial coordinate system K. (When we say “at rest” we neglect the agitations of the constituent particles on the atomic scale.) We may imagine such a measuring rod floating freely in space, far from any external influence, and suppose it is aligned with the x axis of K, and that the magnitude of the spatial interval (i.e., the simultaneous spatial distance) between its end points is L in terms of system K. Now, without in any way disturbing the rod or subjecting it to any external influences whatsoever, we can also determine the simultaneous spatial distance L′ between its end points in terms of another system of inertial coordinates K′ moving with speed v in the positive x direction relative to K. Given that inertial coordinate systems are related by Lorentz transformations, it is trivial to show that L′ = L(1−v²)^1/2.  

Of course, it’s also possible to describe relativistic length contraction in a (at least superficially) different way, in terms of an active rather than passive transformation. By this we mean the following: Consider again the solid measuring rod initially at rest in the system K. We will call this State 1 of the measuring rod. Then suppose we apply a sufficiently gentle external force to the rod in the positive x direction, until it is at rest in the system K′ and then allowed to reach equilibrium. We will call this State 2 of the rod. The phrase “sufficiently gentle” signifies that the limit of plastic deformation of the rod is not exceeded, so that after coming to rest and reaching equilibrium in K′, the description of the rod in State 2 in terms K′ is intrinsically identical to the description of the rod in State 1 in terms of K, which must be the case according to the principle of relativity. Letting L₁ and L₂ denote the spatial extents of the rod in States 1 and 2 respectively, both in terms of the original coordinate system K, it follows again from the Lorentz transformation that L₂ = L₁(1−v²)^1/2.

Thus the passive transformation consists of the relation L₁′ = L₁(1−v²)^1/2 between the spatial extents of the rod in a single state but in terms of two different coordinate systems, whereas the active transformation consists of the relation L₂ = L₁(1−v²)^1/2 between the spatial extents of the rod in two different states of motion but in terms of a single coordinate system.

Needless to say, the fact that these two contractions are formally identical is not merely a coincidence. They actually represent the same physical symmetry, because inertial coordinate systems are defined by the behavior of measuring rods and clocks. It follows that any comparison between two different inertial coordinate systems is actually tantamount to a comparison between intrinsically identical equilibrium configurations of physical entities. This is entailed by the principle of relativity, which asserts that the equations of physics take the same simple symmetrical form when expressed in terms of any system of inertial coordinates. Hence, by definition, the equilibrium configuration of any identifiable stable structure, such as a measuring rod, will have the same description in terms of any system of inertial coordinates in which it is at rest. Indeed this was the tacit justification for Einstein’s use of “measuring rods and clocks” in his derivation of the Lorentz transformation. The relations between two relatively moving systems of rods and clocks is obviously under-specified without stipulating that those (ideal) rods and clocks are intrinsically “the same”, a proposition which has physical meaning only if the attributes of those entities are persistent under changes in their states of motion. In other words, the symmetry between different states of (inertial) motion has physical meaning only if it is possible for identifiable and stable configurations to change from one state of motion to another.

Surprisingly, it is sometimes claimed that the standard derivations of the Lorentz transformation and length contraction do not involve the actual behavior of equilibrium configurations of physical objects when their states of motion are changed. This is clearly untrue, as can easily be seen by examining the derivations of the Lorentz transformation in Einstein’s own early papers. He explicitly bases that derivation on the behavior of actual measuring rods and clocks when their states of motion are changed. For example, in §2 of his 1905 paper (the section entitled “On the Relativity of Lengths and Times”) he writes

Let there be given a stationary rigid rod; and let its length be L as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the x axis of the stationary system of coordinates, and that a uniform motion of parallel translation with velocity v along the x axis in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod…

Thus he is explicitly considering the lengths of a single rod in two different states of motion relative to a single coordinate system, because he first determines the spatial length of the rod while it is at rest in the “stationary system”, and then he considers the length of the same rod in terms of the same coordinate system after a velocity v has been imparted to the rod. Likewise in §3 of the same paper he writes

Let there be given two coordinate systems in space “at rest”… Each system shall be supplied with a rigid measuring rod and a number of clocks, and the two measuring rods and all the clocks of the two systems should be exactly alike. The origin of one of the two systems shall now be imparted a (constant) velocity v in the direction of increasing x of the other system, which is at rest, and this velocity shall also be imparted to the coordinate axes, the corresponding measuring rod, and the clocks.

Hence the two relatively moving systems of coordinates, which Einstein then proceeds to show are related by Lorentz transformations, are defined by the behavior of a solid measuring rod when its state of motion is changed. Consequently, any description of length contraction as a passive comparison between relatively moving coordinate systems ultimately represents an active comparison between the spatial extents of a solid rod in two different states of motion. The point is made just as clearly by Einstein in §3 of his 1907 Jahrbuck article, where he writes

Let S and S′ be equivalent reference systems, i.e., these systems shall have unit measuring rods of the same length and clocks running at the same rate when these objects are compared with each other in a state of relative rest. It is then obvious that all physical laws that hold with respect to S will hold in exactly the same form for S′ too, if S and S′ are at rest relative to each other. The principle of relativity requires such total equivalence also if S′ is in uniform translational motion with respect to S.

Thus the correspondence between the two coordinate systems is (again) explicitly based on those systems being constructed using physical measuring rods that have the same length when compared to each other at mutual rest, and then the same measuring rods are used to define their respective frames when they are in relative motion. To make this even more unmistakable, in a footnote later in the same derivation Einstein says

This conclusion is based on the physical assumption that the length of a measuring rod or the rate of a clock do not undergo any permanent changes if these objects are set in motion and then brought to rest again.

Clearly his derivation is based on the behavior of the equilibrium configurations of identifiable solid objects when their states of motion are changed, with the stipulation that the limit of plastic deformation is not exceeded. It should come as no surprise that length contraction applies to individual solid objects placed into various states of motion, because (for example) the arms of Michelson’s interferometer must undergo the appropriate contractions and expansions as the apparatus is actively rotated, in order to account for the null result. Indeed this was the very phenomenon that originally led to the recognition of relativistic length contraction.

An example from Euclidean geometry is useful to illustrate the importance of the operational basis of our standard (relativistic) coordinate systems. Consider a solid rod of length L aligned with the x axis of a standard Cartesian coordinate system K, and let X denote the projection of this rod onto the x axis. Now, without disturbing the rod in any way, consider another standard Cartesian coordinate system K′ oriented in such a way that the rod makes an angle θ with the x′ axis. The extent X′ of the rod in the x′ direction is X′ = Xcos(θ). This describes a passive transformation. But we can also consider applying a (sufficiently gentle) force to the rod, to change its orientation, so that it makes an angle θ with the x axis of the original coordinate system. Letting X₁ and X₂ denote the x projections of the rod in it original and final orientations, we have X₂ = X₁cos(θ). This describes an active transformation. Just as in the case of relativistic length contraction, the physical content of these two relations is identical – and for the same reason: the passive coordinate systems are defined in terms of the active behavior of physical entities.

Of course, we are free to define any arbitrary coordinate system we like, and call it K′, without regard to any physical process, and then evaluate the “X contraction” of a given object by comparing its x and x′ projections in the K and K′ coordinate systems – but this has no physical significance. Ultimately our standard Cartesian coordinate systems are defined in terms of the behavior of ideal measuring rods under both translations and rotations. The physical symmetry of Euclidean geometry, represented by the invariance of X² + Y² (where X and Y are the magnitudes of the projections of a measuring rod onto the x and y axes), does not apply to arbitrary coordinate systems, it applies only to coordinate systems defined by that invariance. In other words, we map out a standard Cartesian coordinate system using solid measuring rods (or something equivalent). When we imagine translating or rotating a coordinate system, we are actually imagining rotating a system of ideal solid rods.

In this regard Poincare discussed (1903, Science and Hypothesis) the role of “solid bodies” in the empirical foundations of our conception of the geometry of space:

Among surrounding objects there are some which frequently experience displacements that may be corrected [restoring the original relations between the parts of the object and ourselves] simply by a correlative movement of our own body – namely, solid bodies. On the other hand, when the displacement of a body takes place with deformation, we can no longer by appropriate movements place our body in the same relative situation with respect to [the parts of] this body… It is only later that we learn how to decompose a deformable body into smaller elements such that each is displaced approximately according to the same laws as solid bodies. We thus distinguish deformations from other changes of state. Such a concept is very complex, even at this stage, and would not have been conceived at all had not the observation of solid bodies shown us before hand how to distinguish changes of position. If, then, there were no solid bodies in nature there would be no geometry.

These examples highlight the importance of keeping sight of the physical origin and significance of our standard coordinate systems, and not making the mistake of thinking they are somehow a priori concepts, devoid of physical meaning. Lack of clarity about this has often led to erroneous claims that length contraction refers only to passive transformations, and not to active transformations. People who make such claims seem to be motivated by a desire to deny the fact that a solid rod (for example) actually does tend to spatially contract (in terms of a fixed system of inertial coordinates) when it is set in motion. In a famous paper entitled “How to Teach Special Relativity” the physicist John Bell described this situation in terms of a thought experiment involving two spaceships connected by a string. If the spaceships begin to move at the same time, and accelerate always at the same rate in terms of the original coordinate system, the spatial distance between them will be constant in terms of that coordinate system. Bell asked several of his colleagues – who were presumably familiar with special relativity – what would happen to a string that was originally taut connecting the two spaceships, and he reports that most of them were initially resistant to the idea that the string would break. Bell’s scenario is actually quite trivial to understand, because length contraction implies that the string must spatially shrink (in terms of the original coordinate system) when set in motion, and the spaceships are preventing it from shrinking because they are maintaining a constant spatial distance. Hence the string must obviously be trying to contract, exerting forces on the spaceships trying to pull them together, and the string must eventually break (unless it is strong enough to actually pull the spaceships closer together).

To be fair, Bell also reports that, after thinking about it for awhile, his colleagues did eventually recognize that length contraction does indeed imply that the string will tend to contract and will eventually break. Nevertheless, it’s interesting that these well-educated physicists were initially so resistant to the idea that length contraction applies equally to passive and active transformations, in the sense that the physical forces responsible for the equilibrium configurations of physical entities transform, when those entities are set in motion, in such a way as to cause spatial contraction of the configurations. The reason for this attitude seems to be that many students of relativity are taught that it is an egregious error to say “objects shrink and clocks slow down when set in motion”. Such a statement is indeed problematic, because it implies that motion is absolute. However, the appropriate correction is to say (as we have explained above) that “objects spatially shrink and clocks slow down in terms of the inertial coordinates of their initial rest frame when they are set in motion relative to that frame”. This is a perfectly true statement. Unfortunately, rather than being given this correction, students are often taught to say that nothing happens to objects when they are set in motion, i.e., objects always maintain the same (tacitly, proper) length and clocks run always run at the same (tacitly, proper) rate. Length contraction and time dilation (so the unlucky student is taught) are to be understood only in terms of passive transformations. In other words, they are told that the only correct statement of length contraction is as the relation between the descriptions of an undisturbed physical configuration in terms of two different inertial coordinate systems. Moreover, they are often even given the impression that this “insight” is one of the main lessons of Einstein’s special relativity as an advance over Lorentzian relativity. In order to maintain this (frankly preposterous) position, if pressed, they are then compelled to argue that length contraction says nothing about how the lengths of solid rods change when their states of motion are changed. This, as we’ve seen, is flatly contradicted by the actual derivations of inertial coordinate systems and of the Lorentz transformation, which explicitly invoke the behaviors of measuring rods and clocks when changed from one state of motion to another.

Why, then, do so many students acquire the impression that derivations of the Lorentz transformation and the length contraction factor are somehow entirely independent of the behavior of equilibrium configurations of physical entities when placed in different states of motion? This is the misunderstanding that Bell lamented in his paper, and he had a valid point. I suspect that the reason that students of special relativity are so often misled about this is due to the continuing rear-guard battle with neo-Lorentzians over the soul of relativity. This is why such a seemingly innocuous and unobjectionable fact about relativity can, even today, arouse such heated responses.

Recall that Lorentz had labored for decades to account for each detailed aspect of physical laws (sometimes modifying those laws when necessary) to account for the persistent appearance of relativity. One of his great accomplishments was showing that Maxwell’s equations actually are invariant when expressed in terms of a class of (what he regarded as fictitious) coordinate systems. If everything in nature was ultimately electromagnetic in origin (which it isn’t), and if Maxwell’s equations had unlimited validity (which they don’t), it would follow that all physical phenomena are relativistic in terms of these coordinate systems. However, Lorentz and Poincare already knew that the purely electromagnetic view of the world was not viable, and to account for the complete relativity of our experience it is necessary to assume that all forces – including the then unknown forces responsible for the stability of atoms, and even the inertial forces – must be invariant in terms of the very same systems of “fictitious” coordinates that had been inferred from Maxwell’s equations. Thus Lorentz and Poincare were compelled to simply assume the Lorentz invariance of all relevant attributes of physical phenomena, including mechanical inertia. Thus they did not arrive at their version of relativity constructively. Admittedly Lorentz performed a great deal of constructive labor, but ultimately he found it necessary (as did Poincare very explicitly) to simply take the empirical fact of relativity itself as a premise, and “back out” what the laws of physics must be in order for relativity to hold good. This of course is exactly what Einstein did as well. And of course it was already known from Lorentz’s work on Maxwell’s equations that the speed of light is c in terms of every one of the coordinate systems related by Lorentz transformations. The adoption of the relativity principle by Lorentz and Poincare was not a constructive step, any more than it was for Einstein. For all of them it was simply a principle inferred from experience. As Poincare wrote in 1905

It appears that this impossibility to detect the absolute motion of the Earth by experiment may be a general law of nature; we are naturally inclined to admit this law, which we will call the Postulate of Relativity and admit without restriction. Whether or not this postulate, which up to now agrees with experiment, may later be corroborated or disproved by experiments of greater precision, it is interesting in any case to ascertain its consequences.

One could argue that Lorentz arrived at the light principle in a more constructive way, since he deduced it from Maxwell’s equations, but by 1905 Einstein already knew that Maxwell’s equations cannot claim unlimited validity, because they do not correctly represent the behavior of electromagnetic radiation on the most fundamental level (i.e., light quanta). Hence it is misleading to suggest that Lorentz established the light principle constructively. At best he inferred it from the general phenomenological success of Maxwell’s equations – which of course is precisely what Einstein did.

It’s undeniably true – and perfectly understandable, considering his decades of labor – that Lorentz himself tended to view his approach to relativity as more “constructive”, even though he candidly acknowledged that he eventually found it necessary for further progress to simply assume the very same principles that Einstein explicitly took as the foundations of special relativity. Ultimately the only operative feature of Lorentz’s “constructivism” was his reservations about the validity of Lorentz invariance itself. We see the same lack of commitment in Poincare’s quote above, where he highlights the possibility that the principle of relativity may someday be disproved by experiments of greater precision. Lorentz viewed the Lorentz invariance of electromagnetism, and of the molecular forces, and of inertia, etc, as essentially independent facts. He argued that, when examining new phenomena, we should not simply assume from the start that they will be Lorentz invariant. He always kept in view the possibility – even the hope – that relativity would fail, because only such failure would vindicate his reservations.

In contrast, Einstein’s view is unequivocally that the principles of special relativity are universal, which enables him to consolidate them into the very terms of description (i.e., the spacetime coordinates). Indeed Lorentz himself said

Einstein may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.

In other writings, including private correspondence with Einstein, Lorentz admitted that (as we’ve discussed elsewhere) his preference for absolute time was based on his quasi-religious belief in a universal spirit that perceives everything in each instant of absolute time. But of course he also admitted that this went outside the bounds of physics.

This historical background may explain (at least in part) why so many accounts of special relativity seem determined to deny or obscure the fact that relativistic length contraction actually does entail the spatial shrinking of equilibrium configurations in terms of their original rest frame coordinates when the configurations are set in motion relative to those coordinates. (Oddly, such accounts are usually quite happy to accept that clocks – or twins – with changing states of motion experience the expected time dilation, even as they deny that measuring rods with changing states of motion undergo the expected length contraction.) In their zeal to represent all relativistic effects as purely kinematic, modern accounts of special relativity often fail to describe the physical (dynamical) basis for inertial coordinate systems, which then compels them to portray length contraction (and time dilation) as relevant purely and exclusively to passive transformations. (We’ve discussed elsewhere the unfortunate confusions over the terms kinematic and dynamic in discussions of the foundations of relativity.) They evidently fear that any acknowledgement of the relevance of length contraction to active transformations, and indeed any acknowledgement of the active physical basis of the so-called passive transformations, might lead students to a less simplistic (and less superficial) understanding of the foundations of special relativity. The situation is complicated by the fact that many individuals who make it their business to point out the dynamical basis of inertial coordinate systems are actually neo-Lorentzians (as was Bell, to some extent), and as such they are confused about the degree to which Lorentz’s work was actually “constructive”, and they mistakenly think a weaker, less economical, and less falsifiable conceptual framework like Lorentz’s is superior to a stronger, more economical, and more falsifiable conceptual framework like Einstein’s. To fight against these rather silly neo-Lorentzian beliefs, many people go to the opposite extreme, and try to deny any and every statement that a neo-Lorentzian makes, even though some of their statements are obviously correct – not to mention entirely consistent with the Einsteinian view.

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