Lorentz's Assumptions

In his book "The Theory of Electrons" (1909) Hendrik Lorentz wrote

This statement implies that Lorentz's approach was more fundamental, and therefore contained more meaningful physics, than the explicitly axiomatic approach of Einstein. However, a close examination of Lorentz's program reveals that he, no less than Einstein, simply postulated relativity. To understand what Lorentz actually did (and did not) accomplish, it's necessary to clearly understand the historical context and the conceptual issues involved.

Given any set of equations describing some class of physical phenomena with reference to a particular system of space and time coordinates, it may or may not be the case that the same equations apply equally well if the space and time coordinates of every event are transformed according to a certain rule. If such a transformation exists, then those equations (and the phenomena they describe) are said to be covariant with respect to that transformation. Furthermore, if those equations happen to be covariant with respect to a complete class of velocity transformations, then the phenomena are said to be relativistic with respect to those transformations. For example, Newton's laws of motion are relativistic, because they apply not only with respect to one particular system of coordinates x,t, but with respect to any system of coordinates x',t' related to the former system according to a complete set of velocity transformations of the form

From the time of Newton until the beginning of the 19th century many scientists imagined that all of physics might be reducible to Newtonian mechanics, or at least to phenomena that are covariant with respect to the same coordinate transformations as are Newton's laws, and therefore the relativity of Newtonian physics was regarded as complete, in the sense that velocity had no absolute significance, and each one of an infinite set of relatively moving coordinate systems, related by (1), was equally suitable for the description of all physical phenomena. This is called the principle of relativity, and it's important to recognize that it is just a hypothesis, similar to the principle of energy conservation. It is the result of a necessarily incomplete induction from our observations of physical phenomena, and it serves as a tremendously useful organizing principle, but only as long as it remains empirically viable. Admittedly we could regard complete relativity as a direct consequence of the principle of sufficient cause - within a conceptual framework of distinct entities moving in an empty void - but this is still a hypothetical proposition. The key point to recognize is that although we can easily derive the relativity of Newton's laws under the transformations (1), we cannot derive the correctness of Newton's laws, nor can we derive the complete relativity of physics from the presumptive relativity of the dynamics of material bodies.

By the end of the 19th century the phenomena of electromagnetism had become well-enough developed so that the behavior of the electromagnetic field - at least on a macroscopic level - could be described by a set of succinct equations, analogous to Newton's laws of motion for material objects. According to the principle of relativity (in the context of entities in an empty void) it was natural to expect that these new laws would be covariant with the laws of mechanics. It therefore was somewhat surprising when it turned out that the equations which describe the electromagnetic field are not covariant under the transformations (1). Apparently the principle of complete relativity was violated. On the other hand, if mechanics and electromagnetism are really not co-relativistic, it ought to be possible to detect the effects of an absolute velocity, whereas all attempts to detect such a thing failed. In other words, the principle of complete relativity of velocity continued to survive all empirical tests involving comparisons of the effects of velocity on electromagnetism and mechanics, despite the fact that the (supposed) equations governing these two classes of phenomena were not covariant with respect to the same set of velocity transformations.

At about this time, Lorentz derived the fact that although Maxwell's equations (taking the permissivity and permeability of the vacuum to be invariants) of the electromagnetic field are not covariant with respect to (1), they are covariant with respect to a complete set of velocity transformations, namely, those of the form

for a suitable choice of space and time units, where g = (1-v²)^-1/2. This was a very important realization, because if the equations of the electromagnetic field were not covariant with respect to any complete set of velocity transformations, then the principle of relativity could only have been salvaged by the existence of some underlying medium. The situation would have been analogous to finding a physical process in which energy is not conserved, leading us to seek for some previously undetected mode of energy. Of course, even recognizing the covariance of Maxwell's equations with respect to (2), the principle of relativity was still apparently violated because it still appeared that mechanics and electromagnetism were incompatible.

Recall that Lorentz took Maxwell's equations to be "the fundamental equations of the electromagnetic field" with respect to the inertial rest frame of the luminiferous ether. Needless to say, these equations were not logically derived from more fundamental principles, they were developed by a rational-inductive method whereby observed phenomena were analyzed into a small set of simple patterns, which were then formalized into mathematical expressions. Even the introduction of the displacement current was just a rational hypothesis. Admittedly the historical development of Maxwell's equations was guided to some extent by mechanistic analogies, but the mechanical world-view is itself a high-level conceptual framework based on an extensive set of abstract assumptions regarding dimensionality, space, time, plurality, persistent identities, motion, inertia, and various conservation laws and symmetries. Thus even if a completely successful mechanical model for the electromagnetic field existed, it would still be highly hypothetical.

Moreover, it was already clear by 1905 that Maxwell's equations are not fundamental, since the simple wave model of electromagnetic radiation leads to the ultra-violet catastrophe, and in general cannot account for the micro-structure of radiation, leading to such things as the photo-electric effect and other quantum phenomena. (Having just completed a paper on the photo-electric effect prior to starting his 1905 paper on special relativity, Einstein was very much aware that Maxwell's equations were not fundamental, and this influenced his choice of foundations on which to base his interpretation of electrodynamics. It's worth noting that although Lorentz derived the transformations (2) from the full set of Maxwell's equations (with the permissivity and permeability interpreted as invariants), these transformations actually follow from just one aspect of Maxwell's equations, namely, the invariance of the speed of light. Thus from the standpoint of logical economy, as well as to avoid any commitment to the fundamental correctness of Maxwell's equations, it is preferable to derive the Lorentz transformation from the minimum set of premises. Of course, having done this, it is still valuable to show that, as a matter of fact, Maxwell's equations are fully covariant with respect to these transformations.

To summarize the progress up to this point, Lorentz derived the general transformations (2) relating two systems of space and time coordinates such that if an electromagnetic field satisfies Maxwell's equations with respect to one of the systems, it also satisfies Maxwell's equations with respect to the other. Now, this in itself certainly does not constitute a derivation of the principle of relativity. To the contrary, the fact that (2) is different from (1) leads us to expect that the principle of relativity is violated, and that it ought to be possible to detect effects of absolute velocity, or, alternatively, to detect some underlying medium that accounts for the difference between (2) and (1). Lorentz knew that all attempts to detect an absolute velocity (or underlying medium) had failed, implying that the principle of complete relativity was intact, so something was wrong with the formulations of the laws of electromagnetism and/or the laws of mechanics.

Faced with this situation, Lorentz developed his "theorem of corresponding states", which asserts that all physical phenomena transform according to the transformation law for electrodynamics. This "theorem" is equivalent to the proposition that physics is, after all, completely relativistic. Since Lorentz presented this as a "theorem", it has sometimes misled people (including, to an extent, Lorentz himself) into thinking that he had actually derived relativity, and that, therefore, his approach was more fundamental or more constructive than Einstein's. However, an examination of Lorentz's "theorem" reveals that it was explicitly based on assumptions (in addition to the false assumption that Maxwell's equations are the fundamental equations of the electromagnetic field) which, taken together, are tantamount to the assumption of complete relativity. The key step occurs in �175 of The Theory of Electrons, in which Lorentz writes

Lorentz's equation [300] is simply the transformation law for electromagnetic forces, and his equations [305] give the relativistic expressions for the transverse and longitudinal masses of a particle. Lorentz has previously presented these expressions as

In other words, these are the assumptions required in order to make the theorem of corresponding states (i.e., the principle of relativity) true. Hence Lorentz simply postulates relativity, just as did Galileo and Einstein, and then backs out the conditions that must be satisfied by mechanical objects in order to make relativity true. Needless to say, if we assume these conditions, we can then easily prove the theorem, but this is tautological, because these conditions were simply defined as those necessary to make the theorem true. Not surprisingly, if someone just focuses on Lorentz's "proof", without paying attention to the assumptions on which it is based, he might be misled into thinking that Lorentz derived relativity from some more fundamental considerations. This arises from confusion over what Lorentz was actually doing. He was primarily deriving the velocity transformations with respect to which Maxwell's equations are covariant, after which he proceeded to determine how the equations of mechanics would need to be modified in order for them to be covariant with respect to these same transformations. He did not derive the necessity for mechanics to obey these revised laws, any more than Einstein or Newton did. He simply assumed it, and indeed he had no choice, because the laws of mechanics do not follow from the laws of electromagnetism. Why, then, does the myth persist (in some circles) that Lorentz somehow derived relativity?

To answer this question, we need to examine Lorentz's derivation of the theorem of corresponding states in greater detail. First, Lorentz justified the contraction of material objects in the direction of motion (with respect to the ether frame) on the basis of his "molecular force hypothesis", which asserts that the forces responsible for maintaining stable configurations of matter transform according to the electromagnetic law. This can only be regarded as a pure assumption, rather than a conclusion from electromagnetism, for the simple reason that the molecular forces are necessarily not electromagnetic, at least not in the Maxwellian sense. Maxwell's equations are linear, and it is not possible to construct bound states from any superposition of linear solutions. Hence Lorentz's molecular force hypothesis cannot legitimately be inferred from electromagnetism. It is a sheer hypothesis, amounting to the simple assumption that all intrinsic mechanical aspects of material entities are covariant with electromagnetism.

Second, and even more importantly, Lorentz justifies the applicability of the "effective coordinates" for the laws of mechanics of material objects by assuming that the inertial masses (both transverse and longitudinal) of material objects transform in the same way as do the "electromagnetic masses" of a charged particle arising from self-reaction. Admittedly it was once hoped that all inertial mass could be attributed to electromagnetic self-reaction effects, which would have provided some constructive basis for Lorentz's assumption, but we now know that only a very small fraction of the effective mass of an electron is due to the electromagnetic field. Again, it is simply not possible to account for bound states of matter in terms of Maxwellian electromagnetism, so it does not logically follow that the mechanics of material objects are covariant with respect to (2) simply because the electromagnetic field is covariant with respect to (2). Of course, we can hypothesize that this is case, but this is simply the hypothesis of complete physical relativity.

Thus Lorentz did not in any way derive the fact that the laws of mechanics are covariant with respect to the same transformations as are the laws of electromagnetism. He simply observed that if we assume they are (and if we assume every other physical effect, even those presently unknown to us, is likewise covariant), then we get complete physical relativity - but this is tautological. If all the laws of physics are covariant with respect to a single set of velocity transformations (whether they are of the form (1) or (2) or any other), then by definition physics is completely relativistic. The doubts about relativity that arose in the 19th century were due to the apparent fact that the laws of mechanics and the laws of electromagnetism were not covariant with respect to the same set of velocity transformations. Obviously if we simply assume that they are covariant with respect to the same transformations, then the disparity is resolved, but it's important to recognize that this represents just the assumption - not a derivation - of the principle of relativity.

An alternative approach to preserving the principle of relativity would be to assume that electromagnetism and mechanics are actually both covariant with respect to the velocity transformations (1). This would necessitate modifications of Maxwell's equations, and indeed this was the basis for Ritz's emission theory. However, the modifications that Ritz proposed eventually led to conflict with observation, because according to the relativity based on (1) speeds are strictly additive and there is no finite upper bound on the speed of energy propagation. The failure of emission theories illustrates the important fact that there are two verifiable aspects of relativistic physics. The first is the principle of relativity itself, but this principle does not fully determine the observable characteristics of phenomena, because there is more than one possible relativistic pattern, and these patterns are observationally distinguishable. This is why relativistic physics is founded on two distinct premises, one being the principle of relativity, and the other being some empirical proposition sufficient to identify the particular pattern of relativity (Euclidean, Galilean, Lorentzian) that applies. Lorentz theorem of corresponding states represents the second of these premises, whereas the first is simply assumed, consistent with the apparent relativity of all observable phenomena.

For more on the distinction between constructive theories versus principle theories, see Lorentz to Minkowski - Constructing the Principles. For a discussion of how the Lorentzian approach accounts for relativistic effects, see Why Was Michelson Surprised? For another related discussion, see Conditions for Relativity.

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