Poincare Contemplates Copernicus

 

 

In June of 1905 the theory of special relativity was described in two papers, one written by Henri Poincare, and the other by Albert Einstein. Poincare sent his paper “On the Dynamics of the Electron” to the Academy of Science in Paris on June 5, and Einstein submitted his paper “On the Electrodynamics of Moving Bodies” to the Annalen der Physik on June 30. Poincare followed up with a more detailed paper (now often referred to as the Palermo paper) in July, and Einstein followed with his paper “Does the Inertia of a Body Depend on Its Energy Content?” in September.

 

Admittedly it is controversial to claim that Poincare’s papers in the summer of 1905 described “special relativity”, since that name is now so closely associated with the particular point of view introduced by Einstein. Nevertheless, Poincare’s two 1905 papers, together with his previous writings, clearly described a theory of relativity, even if it was not identical to Einstein’s theory in all of its philosophical commitments. As early as 1898 Poincare had explicitly denied any justification for the idea of absolute simultaneity, and had proposed the operational definition of simultaneity based on light signals. By 1905 he had the full Lorentz transformations for electrodynamics, including the group property, and had corrected Lorentz’s expression for the current density. Moreover, he wrote

 

It appears that the impossibility of detecting 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.

 

He goes on to say that Lorentz had accounted for the negative results of Michelson and Morley by means of the Fitzgerald contraction, but that this by itself was not sufficient to ensure complete relativity, because the Fitzgerald contraction had been justified only for electromagnetic formations, not for whatever forces are responsible for holding atoms and sub-atomic particles together, nor for gravitational forces.  Indeed, Poincare shows that the entire inertia of a charged particle could be attributed to electromagnetic forces only if the charge was zero.

 

We must then admit that, in addition to electromagnetic forces, there are also non-electromagnetic forces or bonds. Therefore, we need to identify the conditions that these forces or these bonds must satisfy for electron  equilibrium to be undisturbed by the [Lorentz] transformation.

 

Thus the Lorentz-covariance of electromagnetism does not entitle us to assume that all of physics (particularly the mechanics of material objects) is Lorentz-covariant. Since Poincare did not know (and we still do not know today) the origin of inertial mass, he realized that it was necessary to simply infer the transformation properties of the forces responsible for inertial mass from the principle of relativity, i.e., to identify the conditions that these forces or bonds must satisfy for electron equilibrium to be Lorentz covariant. This had also been recognized by Lorentz. As Poincare noted

 

Lorentz judged it necessary to extend his hypothesis in such a way that the postulate remains valid in case there are forces of non-electromagnetic origin. According to Lorentz, all forces are affected by the Lorentz transformation … in the same way as electromagnetic forces.

 

In summary, Lorentz had shown two things, first, that the laws of electromagnetism (Maxwell’s equations) are covariant under Lorentz transformations, and second, somewhat tautologically, that if all physics (including the inertia of material bodies) is reducible to electromagnetism, then all physics is covariant under Lorentz transformations, and therefore the principle of relativity applies. Of course, this would entail a revision of the laws of mechanics, which had previously been thought to be covariant with respect to Galilean, not Lorentzian, transformations. However, no one knew how to show that mechanical inertia is reducible entirely to electromagnetic forces (and in fact we now know that only a small fraction of the masses of most particles is electromagnetic in origin), so it was not possible (and still is not possible today) to derive the principle of relativity in such a constructive way. What Lorentz and Poincare were obliged to do is to simply to assume the principle of relativity is valid (as all experiments indicated), and then to infer the consequences. One of these consequences is that, obviously, variations in all massless forces must propagate at the speed  of light.  At this point we begin to see one aspect of what Poincare has in mind when he talks about working out the consequences. He writes

 

If propagation of attraction [of the non-electromagnetic forces] occurs with the speed of light, it could not be a fortuitous accident. Rather, it must be because it is a function of the ether, and then we would have to try to penetrate the nature of this function, and to relate it to other fluid functions.

 

This passage suggests that Poincare is still wedded to the idea of the ether, and even to a fluidic ether (if the word fluid has been translated accurately), but in other writings Poincare had already questioned whether the ether really exists as a substantial entity. It seems that he sometimes used the word to refer to the vacuum, without any commitment to a substantial medium. In any case, the passage also shows that Poincare would not be satisfied with the simple hypothesis that all the varied phenomena of nature happen to be covariant with respect to Lorentz transformations with the same characteristic constant c. In other words, he was not prepared to accept this as simply a fortuitous accident. Thus he believed that Lorentz covariance, if universal, must be due to the nature of the context (i.e., the ether) within which those forces exist. He goes on to say

 

We cannot be content with a simple juxtaposition of formulas that agree with each other by good fortune alone; these formulas must, in a manner of speaking, interpenetrate. The mind will be satisfied only when it believes it has perceived the reason for this agreement, and the belief is strong enough to entertain the illusion that it could have been predicted.

 

Just three years later, Minkowski delivered his famous lecture “Space and Time”, in which he described what we now call the Minkowski structure of spacetime based on Einstein’s approach to relativity, with it’s non-positive-definite metric and the group of transformations that he called Gc, as opposed to the group of transformations in Galilean G in spacetime. Minkowski might almost have been replying to Poincare statement above when he wrote

 

…since Gc is mathematically more intelligible than G, it looks as though the thought might have struck some mathematician, fancy free, that after all, as a matter of fact, natural phenomena do not possess an invariance with the group G, but rather with the group Gc, c being finite and determinate, but in ordinary units of measure extremely great.  Such a premonition would have been an extraordinary triumph for pure mathematics.

 

Thus Minkowski believes he has “perceived the reason for this agreement [i.e., the Lorentz covariance of all phenomena], and the belief is strong enough to entertain the illusion that it could have been predicted”. Interestingly, Poincare himself seems to have had a premonition of his own. In the Palermo paper he continued

 

But the question may be viewed from a different perspective, better shown via an analogy. Let us imagine a pre-Copernican astronomer who reflects on Ptolemy’s system; he will notice that for all the planets, one of two circles - epicycle or deferent - is traversed in the same time. This fact cannot be due to chance, and consequently between all the planets there is a mysterious link we can only guess at.  Copernicus, however, destroys this apparent link by a simple change in the coordinate axes that were considered fixed. Each planet now describes a single circle, and orbital periods become independent… 

 

It is possible that something analogous is taking place here. If we were to admit the postulate of relativity, we would find the same number in the law of gravitation and the laws of electromagnetism - the speed of light - and we would find it again in all other forces of any origin whatsoever. This state of affairs may be explained in one of two ways: either everything in the universe would be of electromagnetic origin, or this aspect - shared, as it were, by all physical phenomena - would be a mere epi-phenomenon, something due to our methods of measurement.  

 

How do we go about measuring?  The first response will be: we transport solid objects considered to be rigid, one on top of the other. But that is no longer true in the current theory if we admit the Lorentzian contraction. In this theory, two lengths are equal, by definition, if they are traversed by light in equal times.  Perhaps if we were to abandon this definition Lorentz’s theory would be as fully over-thrown as was Ptolemy’s system by Copernicus’s intervention.

 

In a sense, this is precisely how Lorentz’s theory was “fully over-thrown” by Einstein. In fact, the correspondence is more than just an analogy, because the innovation of both Copernicus and Einstein was nothing other than the adoption of inertia as the basis of measurement.  From a strictly kinematic standpoint we can just as well regard the Earth or the Sun as the “center” of the solar system, but the Sun is (nearly) the inertial center, and the success of the Copernican approach over Ptolemy was ultimately due to the greater simplicity and clarity that can be achieved when things are described in terms of an inertial coordinate system. Likewise Einstein advocated the use of inertial coordinate systems, which Lorentz regarded merely as effective coordinates. Einstein pointed out that these “effective coordinates” were precisely the coordinates in terms of which inertia is homogeneous and isotropic, which is the definition of an inertial coordinate system.  So, rather than adhering to the earthcentric coordinate system of Ptolemy and Lorentz as the one true rest frame, Copernicus and Einstein argued for the greater clarity and simplicity of inertial coordinates as the best context within which to organize our knowledge.

 

The difference between Copernicus and Einstein is that the former argued for the use of inertial measures of space, whereas the latter argued for the use of inertial measures of time as well. We can see from the preceding quote that on some level Poincare recognized that a Copernican shift in point of view might be close at hand (actually it was being composed in Berne even as Poincare was writing his paper in Paris), accounting for the apparently coincidental Lorentz-covariance of all physical phenomena as an inevitable consequence of the fundamental measures of space and time. However, when he commented on what aspect of our measures might be re-evaluated he spoke only about spatial measures, neglecting to mention his own brilliant insights regarding the operational meaning of our measures of time. Instead, Poincare seems to hint that the inertial measure of space (which he associated with light-based metrics) might be abandoned. As it turned out, the shift in viewpoint that overthrew Lorentz’s model retained the inertial measure of space, and merely combined it with the inertial measure of time.

 

Perhaps the most striking difference between Poincare and Einstein in their treatments of relativity in 1905 is that, despite his obvious mastery of the quantitative implications, Poincare was far more cautious and tentative. Indeed he concluded the preface of the Palermo paper by almost apologizing for publishing “these few partial results” at the very moment when experiments with cathode rays (presumably Kaufmann’s) “seem to threaten the entire theory”. Aristotle said the mark of an educated mind is that it can entertain an idea without accepting it. Poincare certainly had an educated mind, and he was exploring the implications of what he called the postulate of relativity, while at the same time reserving judgment as to whether this postulate “may later be corroborated or disproved by experiments of greater precision”. We should also point out that the second half of the Palermo paper concerns relativistic gravitation, and we now know that the simple concept of relativity that both Poincare and Einstein envisaged in 1905 was not consistent with the phenomena of gravitation. This realization came to Einstein only in 1907, but we can see that Poincare was already concerned about it (and rightly so) in 1905. Poincare’s quickness and depth prevented him from unambiguously embracing (special) relativity as a principle, rather than just a postulate. In fairness, it must be admitted that his reservations were justified (at least with regard to gravity, although not with regard to the cathode ray results). In addition, Poincare made the important point that gravitational effects ought to propagate at the speed c in a relativistic theory, and explained why this was not ruled out by Laplace’s well-known argument.

 

During the remaining seven years of Poincare’s life (before his untimely death in 1912) he continued to write on the subject of relativity, but never mentioned the contributions of Einstein. This is understandable, considering how fully Poincare must have felt he had anticipated all the major ideas in Einstein’s approach to the subject, and how in some respects he had actually surpassed Einstein’s development of the subject in their simultaneous papers of 1905. It’s also possible that Poincare made note of the lack of references to himself in Einstein’s main relativity papers. There is, however, one exception to this mutual failure to acknowledge each other. In 1906 Einstein published a paper in the Annalen der Physik entitled “The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy”, in which he presents a more general derivation of the equivalence between inertia and energy than the one he had given in September of the previous year. He says in the introduction

 

Although the simple formal considerations that have to be carried out to prove this statement are in the main already contained in a work by H. Poincare [in Lorentz-Festschrift (1900)], for the sake of clarity I shall not base myself upon that work.

 

Aside from this one acknowledgement, Einstein never mentioned Poincare’s contribution to relativity until years later. Remarkably, Pais tells us (in his book “Subtle is the Lord”) that in the early 1950s he asked Einstein what he had thought of Poincare’s Palermo paper, and Einstein replied that he had not read it. This strikes me as amazing, almost as if Poincare had lived another half century after 1905 and never read Einstein’s paper. One would think that sheer idle curiosity, if nothing else, would have motivated Einstein to read the Palermo paper at some point. Furthermore, in a letter to David Hilbert in 1919, Einstein wrote about

 

…the hypothesis of a cosmic pressure (already taken similarly into account by Poincare, in order to make the electron comprehensible).

 

It was in the Palermo paper that Poincare discussed this “cosmic pressure” necessary to hold an electron’s charge together, so the above comment seems to suggest that Einstein actually had read the paper, contrary to what he told Pais. In any case, Pais asked Einstein if he would like to read it, and Einstein said yes, so Pais loaned him his “second-hand exemplar of the Gauthier-Villar reprint”. It was never returned, and after Einstein’s death Helen Dukas told Pais she couldn’t find it among his possessions. It’s hard to know what to make of this, but Pais notes that Einstein in 1953 (two years before his death) wrote to the organizers of an upcoming celebration of the 50th anniversary of special relativity, saying his health would not allow him to attend, and adding

 

Hopefully on this occasion the contributions of H. A. Lorentz and H. Poincare will also be suitably appreciated.

 

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