3.6 The End of My Latin
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Leaving the old, both worlds at once they view |
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That stand upon the threshold of the new. |
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Edmund Waller, 1686 |
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In his book "The Theory of Electrons" (1909) Hendrik Lorentz wrote |
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Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field. |
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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 useful to review the fundamental conceptual issues that he faced. |
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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 |
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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. |
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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. |
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for a suitable choice of space and time units, where γ = (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. |
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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. |
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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. |
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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. |
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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 |
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We are now prepared for a theorem concerning corresponding states of electromagnetic vibration, similar to that of §162, but of a wider scope. To the assumptions already introduced, I shall add two new ones, namely (1) that the elastic forces which govern the vibratory motions of the electrons are subjected to the relation [300], and (2) that the longitudinal and transverse masses m' and m" of the electrons differ from the mass m0 which they have when at rest in the way indicated by [305]. |
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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 |
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...the assumptions required for the establishment of the theorem, that the systems S and S0 can be the seat of molecular motions of such a kind that, in both, the effective coordinates of the molecules are the same function of the effective time. |
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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? |
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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. |
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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. |
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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. |
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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. |
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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’s 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. Einstein’s achievement in special relativity was essentially to show that Lorentz’s results (and more) actually follow unavoidably from just a small subset of his assumptions, and that these can be consistently interpreted as primitive aspects of space and time. |
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The first published reference to Einstein's special theory of relativity appeared in a short note by Walter Kaufmann reporting on his experimental results involving the deflection of electrons in an electromagnetic field. Kaufmann's work was intended as an experimentum crucis for distinguishing between the three leading theories of the electron, those of Abraham, Bucherer, and Lorentz. In his note of 30 November 1905, Kaufmann wrote |
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In addition there is to be mentioned a recent publication of Mr. A. Einstein on the theory of electrodynamics which leads to results which are formally identical with those of Lorentz's theory. I anticipate right away the general result of the measurements to be described in the following: the results are not compatible with the Lorentz-Einstein fundamental assumptions. |
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Kaufmann's results were originally accepted by most physicists as favoring the Abraham theory, but gradually people began to have doubts. Although the results disagreed with the Lorentz-Einstein model, the agreement with Abraham's theory was not particularly good either. This troubled Planck, so he conducted a careful analysis of Kaufmann's experiment and his analysis of the two competing theories. It was an interesting example of scientific "detective work" by Planck. |
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Kaufmann in 1905 had measured nine characteristic deflections d1,d2,..,d9 for electrons passing though nine different field strengths. Then he had computed the nine values that would be predicted by Abraham's theory, and the nine values that would be predicted by Lorentz-Einstein. However, in order to derive the "predictions" from the theories for his particular experimental setup he needed to include an attenuation factor "k" on the electric field strength. This factor is actually quite a complicated function of the geometry of the plates and coils used to establish the electric field. Kaufamnn selected a particular value of "k" that he thought would be reasonable. |
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Now, both the Abraham and the Lorentz-Einstein theory predicted the electron's velocity could never exceed c, but Planck noticed that Kaufmann's choice of k implied a velocity greater than c for at least one of the data points, and therefore was actually inconsistent with both theories. This caused Planck to suspect that perhaps Kaufmann's assumed value of k was wrong. Unfortunately the complexity of the experimental setup made it impossible to give a firm determination of the attenuation factor from first principles, but Planck was nevertheless able to extract some useful information from Kaufmann's data. |
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Planck took the nine data points and "backed out" the values of k that would be necessary to make them agree with Abraham's theory. Then he did the same for the Lorentz-Einstein theory. All these values of k were well within the range of plausibility (given the uncertainty in the experimental setup), so nothing definite could be concluded, but Planck noted that the nine k-values necessary to match the Lorentz-Einstein theory to the measurements were all nearly equal, whereas the nine k-values necessary to match Abraham showed more variation. From this, one might actually infer a slight tilt in favor of the Lorentz-Einstein theory, simply by virtue of the greater consistency of k values. |
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Naturally this inconclusive state of affairs led people to try to think of an experiment that would be more definitive. In 1908 Bucherer performed a variation of Kaufmann's experiment, but with an experimental setup taking Planck's analysis into account, so that uncertainty in the value of k basically "cancels out". Bucherer's results showed clear agreement with the Lorentz-Einstein theory and disagreed with the Abraham theory. Additional and more refined experiments were subsequently performed, and by 1916 it was clear that the experimental evidence did in fact support what Kaufmann had called "the Lorentz-Einstein fundamental assumptions". |
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Incidentally, it's fascinating to compare the reactions of Lorentz, Poincare, and Einstein to Kaufmann's results. Lorentz was ready to abandon his entire model (and life's work) since it evidently conflicted with this one experiment. As he wrote to Poincare in 1906, the length contraction hypothesis was crucial for the coherence of his entire theoretical framework, and yet |
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Unfortunately my hypothesis of the flattening of electrons is in contradiction with Kaufmann's results, and I must abandon it. I am, therefore, at the end of my Latin. |
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Poincare agreed that, in view of Kaufmann's results "the entire theory may well be threatened". It wasn't until the announcement of Bucherer's results that Lorentz regained confidence in his own theoretical model. Interestingly, he later cited those results as one of the main reasons for his eventual acquiescence with the relativity principle, noting that if Lorentz-covariance is actually as comprehensive as these experimental results show it to be, then the ether concept is entirely devoid of heuristic content. (On the other hand, he did continue to maintain that there were some benefits in viewing things from the standpoint of absolute space and time, even if we are not at present able to discern such things.) |
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Einstein's reaction to Kaufmann's apparently devastating results was quite different. In a review article on relativity theory in 1907, Einstein acknowledged that his theory was in conflict with Kaufmann's experimental results, and he could find nothing wrong with either Kaufmann's experiment or his analysis, which seemed to indicate in favor of Abraham's theory over relativity. Nevertheless, the young patent examiner continued |
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It will be possible to decide whether the foundations of the relativity theory correspond with the facts only if a great variety of observations is at hand... In my opinion, both [the alternative theories of Abraham and Bucherer] have rather slight probability, because their fundamental assumptions concerning the mass of moving electrons are not explainable in terms of theoretical systems which embrace a greater complex of phenomena. A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended is its area of applicability. |
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This is a remarkable defense of a scientific theory against apparent experimental falsification. While not directly challenging the conflict between experiment and theory, Einstein nevertheless maintained that we should regard relativity as most likely correct, essentially on the basis of it's scope and conceptual simplicity. Oddly enough, when later confronted with similar attempts to justify other people's theories, Einstein was fond of saying that "a theory should be as simple as the facts allow - but no simpler". Yet here we find him serenely confident that the "facts" rather than his theory will ultimately be overturned, which turned out to be the case. This sublime confidence in the correctness of certain fundamental ideas was a characteristic of Einstein throughout his career. When asked what he would have done if the eclipse observations had disagreed with the prediction of general relativity for the bending of light, Einstein replied "Then I would have felt sorry for the dear lord, because the theory is correct." |
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