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It is a well-known fact that Maxwell's theory of electromagnetism is the only piece of nineteenth-century physics that was unchanged under Einstein's Special Relativity. In fact it is possible to derive Special Relativity from Maxwell's equations, as in this paper. But doing so seems to require the proper mindset to even consider doing so; it is not immediately obvious that Maxwell's equations lead to SR, without the support of late nineteenth-century physics like the famous Michelson-Morley experiment. Could Maxwell himself have done so? The answer appears to be yes. Here is an analysis I came across that attempts to show that if Maxwell had done one simple thing more, which he had done with other work but for some reason failed to do in this case, he would have been inevitably led to Special Relativity. He also talks, in true AH.com fashion, about some of the impact this advance would have had on nineteenth-century culture.

Maxwell’s Theory of Relativity

...what would it mean to us if James Clerk Maxwell had discovered the Special Theory of Relativity in 1861, forty-four years before Einstein did in Real History? Of course, we say, such a thing couldn’t have possibly happened. After all, wasn’t it Maxwell who hypothesized the luminiferous æther, whose motion relative to Earth the Michelson-Morley and Trouton-Noble experiments were specifically designed to detect and measure? And wasn’t it the failure of those experiments to find any hint of an ætherial effect that led George FitzGerald and Hendrik Lorentz to propose the contraction of lengths, one of the conceptual stepping stones that supported Einstein when he made his great intuitive leap into Relativity? Therefore, wasn’t it Maxwell’s theory that had to be overcome before Relativity could be discovered and thus wasn’t it effectively impossible for Maxwell to discover Relativity? Well, not exactly.

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Though he gave it a special plausibility, we have little doubt that Maxwell did not take the æther as seriously as others did. He abandoned the concept altogether in his 1864 paper, in which he first presented the famous eight equations (condensed to four in modern textbooks) that sum up all of electromagnetic theory.

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However useful or funny he may have found it, Maxwell most likely employed the æther for Michael Faraday’s sake. The two men were good friends and Maxwell had a knack for converting the most sophisticated mathematical ideas of physics into concepts that Faraday could think with comfortably. In a letter that he wrote to Maxwell in 1857 Faraday reminded him of that special ability and urged him to use it, when he had completed his works, to translate the "hieroglyphics" of higher mathematics into imagery more easily accessible to minds unskilled in mathematics. If, indeed, such urgings prompted Maxwell to employ the æther concept, then History might easily have followed a different path.

One of the tasks that Maxwell had accomplished early on was a demonstration that theories based on mathematical descriptions of Faraday’s lines of force are fully equivalent to theories drawn from Newtonian concepts of centers of force exerting actions at a distance. He thus used the older concepts as a test to confirm the validity of the newer concepts, thereby providing us an example of what we now call the correspondence principle. In the light of that effort, his failure to make a similar confirmatory test of his theory of electromagnetic waves looks strange. The test is easy to make and if Maxwell had made it, he would have run into an intriguing problem.

After deducing the existence and properties of electromagnetic waves from manipulation of his equations of abstract mathematics, Maxwell could have confirmed his discovery by direct application of the lines-of-force model. Mathematically the test would have involved translating Maxwell’s Equations from their conventional form as a set of differential equations into an equivalent set of integral equations, a minor exercise for Maxwell. Conceptually the test consists of imagining an experimenter observing a field of magnetic lines passing through a series of galvanometers. The lines move in a direction perpendicular to the direction in which they point, so, in accordance with Faraday’s law, they will generate an electric field whose intensity stands in proportion to both the magnetic field’s intensity and its velocity. That electric field will be manifested in all of the galvanometers as the magnetic field passes through them, leading the experimenter to conclude that the electric field travels with the magnetic field. But Maxwell’s version of Ampere’s law says that a moving electric field will generate a magnetic field, one that, in this case, will act to support the original magnetic field.

At speeds achievable in a Nineteenth Century laboratory the amount of support that a magnetic field would obtain from its induced electric field is insignificant. Thus the field must gain support from a material source, such as a bar magnet or a current-carrying wire. But of course we know a speed, the speed of light, at which the induced electric field has sufficient strength and moves fast enough to generate full support for its magnetic field. Fields traveling past the experimenter at that speed can fly free of all material sources. In fact, according to Maxwell’s theory, free-flying fields can travel only at the speed of light. If they travel slower, they will undersupport each other and decay and if they travel faster, they will oversupport each other and grow: in either case they would violate the conservation of energy theorem.

When Maxwell deduced the electromagnetic wave equation from combining the differential equations of Faraday’s law and Ampere’s law, the speed at which the waves propagate appeared as a mathematical abstraction that then needed an associated physical interpretation.

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...the "c" in Maxwell’s wave equation does not represent velocity relative to a medium, as many physicists believed at the time. The lines-of-force derivation outlined above removes the ambiguity from the interpretation: in it the speed has been specifically described as the speed at which the fields move past an experimental apparatus. Maxwell would have understood that the fields must conform to the requirement that they fly at the same constant speed past any apparatus and that understanding seems to lead directly to a logical contradiction.

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Imagine, as Maxwell would have done, that two experimenters have set up arrays of galvanometers to measure the field of a free-flying electromagnetic wave. One experimenter has set up his apparatus on a flatcar that workers have integrated into an express train and the other has set up his apparatus on the platform of a small station, a minor whistle stop for which the express won’t even slow down. An assistant so generates the wave that it flies along the track without spreading; thus, when it passes through the station experimenter’s apparatus at the speed of light, all of the galvanometers in the array will measure the same value for the amplitude of the induced electric field. Given that fact and the knowledge that the train moves away from the wave’s source, we can form a preliminary expectation that the wave will fly through the train experimenter’s apparatus at a speed slightly less than the speed of light and that, therefore, the fields in the wave won’t fully support each other: we should expect the wave to display a progressively diminishing electric field on successive galvanometers in the array. Fulfillment of that expectation would create a contradiction in which the train experimenter reports that the wave faded as it flew, being extinguished by the defect in its velocity, while the station experimenter maintains that the wave flew along the track without diminution.

Confronted with that dilemma and unable to deny the validity of the laws that create it, Maxwell would have to question his interpretation of the laws. He would almost certainly begin by reaffirming Formal Logic’s law of the excluded middle, which stands as the basis for our belief in the non-contrariness of Nature’s laws. That reaffirmation might take the form of an axiom worded to have obvious application to the two railroad experimenters:

1. The character of any phenomenon has the same form for all observers, regardless of how the traits of that character may be distorted by the uniform motion of the observers relative to each other and relative to the objects associated with the phenomenon.

We can sum up the character of the phenomenon under consideration as "an electromagnetic field flying free of any entanglement in material sources, the fields so supporting each other that they can travel together without diminution". Having reaffirmed that proposition, Maxwell would then proceed to resolve the contradiction between the railroad experimenters by stating a second axiom:

2. Any electromagnetic wave will always pass observers at the same speed, regardless of how the observers move relative to each other and relative to the source of the wave.

We recognize those axioms simply as reworded versions of the two postulates that Einstein presented as the foundations of Special Relativity in his 1905 paper "On the Electrodynamics of Moving Bodies":

1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.

2. Any ray of light moves in the ‘stationary’ system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.


Thus Maxwell could have come to the foundations of Special Relativity, though at the expense of exchanging one contradiction for another. He would now have two observers in relative motion seeing the same thing, the electromagnetic wave, pass each of them at the same speed.

The above paper then continues with speculation on some of the effects of a mid-nineteenth-century theory of Special Relativity on culture, but strangely does not discuss its effects on science. So what effects do you think this earlier breakthrough would have? (Other than Maxwell being even more famous than he already is -- with SR added to his existing work on electromagnetism and thermodynamics, I have no doubt that he would be considered to be at least as great as Newton.)
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