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Extending Classical Relativity

Problems with Electrodynamics

As the quotation at the beginning of this section says, it couldn't last. To begin with, the new science of electrostatics appeared to be an analog of gravitation, with the added feature that electrical charges could repel as well as attract. The equations for electrical force were of the same form as Newton's gravitational law, known to be covariant under Galilean transform, and it was expected that the same would apply. However, as the work of people like André-Marie Ampère, Michael Faraday, and Hans Christian Oersted progressed from electrostatics to electrodynamics, the study of electrical entities in motion, it became apparent that the situation was more complicated. Interactions between magnetic fields and electric charges produced forces acting in directions other than the straight connecting line between the sources, and which, unlike the case in gravitation and electrostatics, depended on the velocity of the charged body as well as its position. Since a velocity in one inertial frame can always be made zero in a different frame, this seemed to imply that under the classical transformations a force would exist in one that didn't exist in the other. And since force causes mass to accelerate, an acceleration could be produced in one frame but not in the other when the frames themselves were not accelerating relative to each other—which made no sense. The solution adopted initially was simply to exclude electrodynamics from the principle of classical relativity until the phenomena were better understood.

But things got worse, not better. James Clerk Maxwell's celebrated equations, developed in the period 1860–64, express concisely yet comprehensively the connection between electric and magnetic quantities that the various experiments up to that time had established, and the manner in which they affect each other across intervening space. (Actually, Wilhelm Weber and Neumann derived a version of the same laws somewhat earlier, but their work was considered suspect on grounds, later shown to be erroneous, that it violated the principle of conservation of energy, and it's Maxwell who is remembered.) In Maxwell's treatment, electrical and magnetic effects appear as aspects of a combined "electromagnetic field"—the concept of a field pervading the space around a charged or magnetized object having been introduced by Faraday—and it was by means of disturbances propagated through this field that electrically interacting objects influenced each other.

An electron is an example of a charged object. A moving charge constitutes an electric current, which gives rise to a magnetic field. An accelerating charge produces a changing magnetic field, which in turn creates an electric field, and the combined electromagnetic disturbance radiating out across space would produce forces on other charges that it encountered, setting them in motion—a bit like jiggling a floating cork up and down in the water and creating ripples that spread out and jiggle other corks floating some distance away. A way of achieving this would be by using a tuned electrical circuit to make electrons surge back and forth along an antenna wire, causing sympathetic charge movements (i.e., currents) in a receiving antenna, which of course is the basis of radio. Another example is light, where the frequencies involved are much higher, resulting from the transitions of electrons between orbits within atoms rather than oscillations in an electrical circuit.

 

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Maxwell's Constant Velocity

The difficulty that marred the comforting picture of science that had been coming together up until then was that the equations gave a velocity of propagation that depended only on the electrical properties of the medium through which the disturbance traveled, and was the same in every direction. In the absence of matter, i.e., in empty space, this came out at 300,000 kilometers per second and was designated by c, now known to be the velocity of light. But the appearance of this value in the laws of electromagnetism meant that the laws were not covariant under Galilean transforms between inertial frames. For under the transformation rules, in the same way that our airplane's velocity earlier would reduce to zero if measured in the bomb aimer's reference frame, or double if measured in the frame of another plane going the opposite way, the same constant velocity (depending only on electrical constants pertaining to the medium) couldn't be true in all of them. If Maxwell's equations were to be accepted, it seemed there could only exist one "absolute" frame of reference in which the laws took their standard, simplest form. Any frame moving with respect to it, even an inertial frame, would have to be considered "less privileged."

Putting it another way, the laws of electromagnetism, the classical Galilean transforms of space and time coordinates, and the principle of Newtonian relativity, were not compatible. Hence the elegance and aesthetic appeal that had been found to apply for mechanics didn't extend to the whole of science. The sense of completeness that science had been seeking for centuries seemed to have evaporated practically as soon as it was found. This was not very intellectually satisfying at all.

One attempt at a way out, the "ballistic theory," hypothesized the speed of light (from now on taken as representing electromagnetic radiation in general) as constant with respect to the source. Its speed as measured in other frames would then appear greater or less in the same way as that of bullets fired from a moving airplane. Such a notion was incompatible with a field theory of light, in which disturbances propagate at a characteristic rate that has nothing to do with the movement of their sources, and was reminiscent of the corpuscular theory that interference experiments were thought to have laid to rest. But it was consistent with the relativity principle: Light speed would transform from one inertial frame, that of the source, to any other just like the velocity of a regular material body.

However, observations ruled it out. In binary star systems, for example, where one star is approaching and the other receding, the light emitted would arrive at different times, resulting in distortions that should have been unmistakable but which were not observed. A series of laboratory experiments 65 also told against a ballistic explanation. The decisive one was probably one with revolving mirrors conducted by A. A. Michelson in 1913, which also effectively negated an ingenious suggestion that lenses and mirrors might reradiate incident light at velocity c with respect to themselves—a possibility that the more orthodox experiments hadn't taken into account.

Another thought was that whenever light was transmitted through a material medium, this medium provided the local privileged frame in which c applied. Within the atmosphere of the Earth, therefore, the speed of light should be constant with respect to the Earth-centered frame. But this runs into logical problems. For suppose that light were to go from one medium into another moving relative to the first. The speeds in the two domains are different, each being determined by the type of medium and their relative motion. Now imagine that the two media are progressively rarified to the point of becoming a vacuum. The interaction between matter and radiation would become less and less, shown as a steady reduction of such effects as refraction and scattering to the point of vanishing, but the sudden jump in velocity would still remain without apparent cause, which is surely untenable.

Once again, experimental evidence proved negative. For one thing, there was the phenomenon of stellar aberration, known since James Bradley's report to Newton's friend Edmond Halley, in 1728. Bradley found that in the course of a year the apparent position of a distant star describes an ellipse around a fixed point denoting where it "really" is. The effect results from the Earth's velocity in its orbit around the Sun, which makes it necessary to offset the telescope angle slightly from the correct direction to the star in order to allow for the telescope's forward movement while the light is traveling down its length. It's the same as having to tilt an umbrella when running, and the vertically falling rain appears to be coming down at a slant. If the incoming light were swept along with the atmosphere as it entered (analogous to the rain cloud moving with us), the effect wouldn't be observed. This was greeted by some as vindicating the corpuscular theory, but it turns out that the same result can be derived from wave considerations too, although not as simply. And in similar vein, experiments such as that of Armand Fizeau (1851), which measured the speed of light through fast-flowing liquid in a pipe, and Sir George Airy (1871), who repeated Bradley's experiment using a telescope filled with water and showed aberration didn't arise in the telescope tube, demonstrated that the velocity of light in a moving medium could not be obtained by simple addition in the way of airplanes and machine-gun bullets or as a consequence of being dragged by the medium.

Relativity is able to provide interpretations of these results—indeed, the theory would have had a short life if it couldn't. But the claim that relativity is thereby "proved" isn't justified. As the Dutch astronomer M. Hoek showed as early as 1868, attempts at using a moving material medium to measure a change in the velocity of light are defeated by the effect of refraction, which cancels out the effects of the motion. 66

Michelson, Morely, and the Ether That Wasn't

These factors suggested that the speed of light was independent of the motion of the radiation source and of the transmitting medium. It seemed, then, that the only recourse was to abandon the relativity principle and conclude that there was after all a privileged, universal, inertial reference frame in which the speed of light was the same in all directions as the simplest form of the laws required, and that the laws derived in all other frames would show a departure from this ideal. The Earth itself cannot be this privileged frame, since it is under constant gravitational acceleration by the Sun (circular motion, even at constant speed, involves a continual change of direction, which constitutes an acceleration) and thus is not an inertial frame. And even if at some point its motion coincided with the privileged frame, six months later its orbit would have carried it around to a point where it was moving with double its orbital speed with respect to it. In any case, whichever inertial frame was the privileged one, sensitive enough measurements of the speed of light in orthogonal directions in space, continued over six months, should be capable of detecting the Earth's motion with respect to it.

Many interpreted this universal frame as the hypothetical "ether" that had been speculated about long before Maxwell's electromagnetic theory, when experiments began revealing the wave nature of light. If light consisted of waves, it seemed there needed to be something present to be doing the "waving"—analogous to the water that carries ocean waves, the air that conducts sound waves, and so on. The eighteenth to early nineteenth centuries saw great progress in the development of mathematics that dealt with deformation and stresses in continuous solids, and early notions of the ether sought an interpretation in mechanical terms. It was visualized as a substance pervading all space, being highly rigid in order to propagate waves at such enormous velocity, yet tenuous enough not to impede the motions of planets. Maxwell's investigations began with models of fields impressed upon a mechanical ether, but the analogy proved cumbersome and he subsequently dispensed with it to regard the field itself as the underlying physical reality. Nevertheless, that didn't rule out the possibility that an "ether" of some peculiar nature might still exist. Perhaps, some concluded, the universal frame was none other than that within which the ether was at rest. So detection of motion with respect to it could be thought of as measuring the "ether wind" created by the Earth's passage through it in its movement through space.

The famous experiment that put this to the test, repeated and refined in innumerable forms since, was performed in 1887 by Albert Michelson and Edward Morley. The principle, essentially, was the same as comparing the round-trip times for a swimmer first crossing a river and back, in each case having to aim upstream of the destination in order to compensate for the current, and second covering the same distance against the current and then returning with it. The times will not be the same, and from the differences the speed of the current can be calculated. The outcome was one of the most famous null results in history. No motion through an ether was detected. No preferred inertial reference frame could be identified that singled itself out from all the others in any way.

So now we have a conundrum. The elaborate experimental attempts to detect a preferred reference frame indicated an acceptance that the relativity principle might have to be abandoned for electromagnetism. But the experimental results failed to identify the absolute reference frame that this willingness allowed. The laws of electromagnetism themselves had proved strikingly successful in predicting the existence of propagating waves, their velocity and other quantities, and appeared to be on solid ground. And yet an incompatibility existed in that they were not covariant under the classical transforms of space and time coordinates between inertial frames. The only thing left to question, therefore, was the process involving the transformations themselves.

Lorentz's Transforms for Electromagnetics

Around the turn of the twentieth century the Dutch theoretical physicist Hendrick Lorentz followed the path of seeking alternative transformation laws that would do for electromagnetics what the classical transforms had done for mechanics. Two assumptions that few people would question were implicit in the form of the Galilean transforms: (1) that observers in all frames will measure time the same, as if by some universal clock that ticks the same everywhere; and (2) while the space coordinates assigned to points on a rigid body such as a measuring rod might differ, the distance between them would not. In other words, time intervals and lengths were invariant.

In the Lorentz Transforms, as they came to be called, this was no longer so. Time intervals and lengths measured by an observer in one inertial frame, when transformed to another frame, needed to be modified by a factor that depended on the relative motion between them. Lorentz's system retained the notion of an absolute frame in which the ether is at rest. But the new transforms resulted in distances being reduced in the direction of motion relative to it, and it was this fact which, through an unfortunate coincidence of effects, made detection of the motion unobservable. As a matter of fact, an actual physical shrinkage of precisely this form—the "Fitzgerald Contraction"—had been proposed to explain the Michelson-Morley result as due to a shortening of the interferometer arms in the affected direction. Some textbook writers are of the opinion that Lorentz himself took the contractions as real; others, that he used them simply as mathematical formalisms, symbolizing, as it were, some fictitious realm of space and time that applied to electromagnetic phenomena. I don't claim to know what Lorentz thought. But here was a system which acknowledged a preferred frame as required by Maxwell's equations (defined by the constancy of c), yet at the same time observed the relativity that the optical experiment seemed to demand. Okay, maybe things were a bit messy in that a different system applied to mechanics. But everything more or less worked, and maybe that was just the way things from now on would have to be.

Except that somebody called Albert Einstein wasn't happy with it.

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