# Chapter 17 Special Relativity

Should the local power grid be run using alternating or direct current? On the one hand, it is easier to store direct current in batteries. On the other hand, it is easier to step up, or down, the voltage of alternating current. While a large company may have the means to invest in both standards, a small company must choose one, or the other. The power engineering firm, Einstein & Cie, was one such company.

Founded in 1880 by the brothers Jakob and Hermann Einstein, Einstein & Cie built electric generators in Munich. Their business took off throughout the 1880s as DC generators remained in high demand. Unfortunately, these successful years also coincided with the invention of practical closed-coil transformers and AC induction motors. By the mid 1890s, the city of Munich had switched to AC power, and Einstein & Cie went bankrupt. Jacob and Hermann moved their families to two different Italian cities to found new firms, while Hermann’s teenage son, Albert, stayed in Munich supposedly to finish his studies at the prestigious Luitpold Gymnasium. This did not last long, as Albert found the formal instruction uninteresting and soon joined his parents in Italy a few months later, without earning his diploma. This gave him much more freedom to follow his interests, and after about a year he moved to Switzerland to continue his studies—but this time in mathematical physics.

After earning a degree, but still working on his doctoral thesis, Albert could not find a teaching position. He did, however, use his technical background to secure a well-paying job working in a Swiss patent office. During this time, Einstein appeared to be at his most happy and productive. His patent office job was not too demanding, which allowed him to devote thought to questions of natural philosophy. At the same time, Einstein was a member of a small book group that read current works on the philosophy of science. Among these works was Henri Poincaré’s book, Science and Hypothesis, which outlined many of the inconsistencies in theoretical physics at that time.

Einstein’s miracle year came in 1905 when he published four papers that forever altered physics. Einstein’s first paper made the radical proposal that light is not a wave after all, but rather behaves as if it were a particle. This made little sense to other physicists of the day, especially since it appeared to contradict everything nineteenth century scientists had learned about interference. Yet Einstein made testable predictions in this paper, and their later experimental confirmation by Robert A. Millikan would pull down the edifice of Maxwell’s wave theory of light. Einstein’s second 1905 paper gave a statistical proof that atoms and molecules really exist based on an analysis of Brownian motion, and it was this paper that would become his doctoral dissertation on the determination of Avogadro’s number and the size of molecules. Einstein’s third and fourth papers are the subject of this chapter.

Albert Einstein’s paper On The Electrodynamics of Moving Bodies answered the question that dogged theoretical physics throughout the late nineteenth century: “What is the nature of the aether?” Einstein’s solution was simple: there is no aether. It simply does not exist. Instead, he asserted, Galileo’s principle of relativity (but not his transformation equations) is equally true for electrodynamics and mechanics.

The two fundamental postulates that Einstein presented as the foundation of his theory are:

1. The laws governing the changes of the state of any physical system do not depend on which one of two coordinate systems in uniform translational motion relative to one another these changes are referred to.
2. Every ray of light moves in the co-ordinate system “at rest” with the definite velocity V independent of whether this ray of light is emitted by a body at rest or in motion. [1]

The first postulate holds that electrodynamics, like mechanics, ought to conform to the principle of relativity. There is no privileged inertial frame, such as the rest frame of the aether, to which all laws of electrodynamics must be referred. All inertial frames, i.e. frames moving uniformly with respect to one another, are equally correct for describing electrodynamics as well as mechanics. Far from overthrowing Galileo or Newton, Einstein was extending the principle of relativity to all of physics. No violation of this principle has ever been found.

Einstein’s second postulate is our Fundamental Law 15.1, which asserts that light propagates in a vacuum at the same speed (V=c) regardless of the motion of the source or the observer. At the time, it was simply an inspired guess suggested by the form of Maxwell’s wave equation (see Discussion 15.3).

This assumption would, at first sight, appear to conflict with the principle of relativity. If a light source and an observer are in uniform relative motion with a constant velocity, v, then the Galilean transformations demand that the speed of light measured by the observer is not simply equal to c, but instead equals c+v or c-v.  Yet, this is exactly what Einstein’s second postulate denies!  If Einstein’s assertions are true, then the second postulate must be reconciled with the principle of relativity and a new set of transformations must supplant the Galilean transformations. As we mentioned in Chapter 15, such a set of transformations, called the Lorentz-FitzGerald transformations, does indeed exist and we will see how these both preserve Maxwell’s theory and allow Einstein’s two postulates to coexist.

Other than the Michelson-Morley experiment, there was scant direct experimental evidence for Einstein’s postulate of the universality of c.  However, his theory as a whole did explain such diverse phenomena as the Doppler effect, Fresnel’s drag coefficient, and stellar aberration.

Einstein’s second postulate has been confirmed many times since, which have put tiny upper limits on the value of a hypothetical $k$ parameter $\left( k=\tfrac{\left| {c}'-c \right|}{v} \right).$  These experiments include: de Sitter’s 1913 observations[2] of the light received from binary stars,[3] a 1964 pion decay measurement, $\left( k<1\,\times {{10}^{-4}} \right)$,[4] and a 1977 observation of x-ray sources in binary star systems $\left( k<2\,\times {{10}^{-9}} \right)$.[5] We may, therefore, be confident that the speed of light is independent of the observer’s reference frame.

Rather than focusing on the aether, which Einstein discarded as an unnecessary abstraction, his special theory of relativity focuses instead on what can, at least in principle, be observed. Consider this key passage from Einstein’s introduction to his original paper:

Like every other electrodynamics, the theory to be developed is based on the kinematics of the rigid body, since assertions of each and any theory concern the relations between rigid bodies (coordinate systems), clocks, and electromagnetic processes.[6]

This need to base theory on measurement led Einstein to ask what we mean by saying two events are simultaneous, and to demand an operational definition of simultaneity: one that stresses the use of clocks, rigid measuring rods, and light signals. In this manner, the old assumption of an absolute time that flows evenly and universally was revealed to be nothing more than metaphysical prejudice.

The aether, which had been retained from an overly mechanistic model of electromagnetic interactions, also proved unnecessary. Over and over, a predicted effect of such an aether had been contradicted by experiment, and every time this happened scientists resorted to more exotic refinements of the aether model. With no compelling evidence that it even existed, the aether—like the notion of absolute time—seemed to arise from an uncritical acceptance of common sense. The aether was a superfluous notion at best, Einstein argued, and the science of electricity and magnetism could dispense with it.

Even matter’s very existence, at least as something distinct from energy, came into doubt with Einstein’s relativity. Einstein’s fourth 1905 paper addressed the question of relativistic dynamics and the equivalence of mass and potential energy.

We begin this chapter with Einstein’s postulates, and use a series of thought experiments in order to illustrate the theory of special relativity and relativistic electrodynamics. In Section 17.1, we not only show that measurements of length and time depend on the observer’s frame of reference, but that the order of events can also be relative. This interdependence of space and time naturally leads to denoting the time and location of an event with a single 4-vector, making the Lorentz-FitzGerald transformations mathematically similar to a rotation in spacetime.

In Section 17.2, we reproduce our most fundamental thought experiments from early in this text—but with a completely relativistic approach. We begin with a third axiom, that charge is both conserved and independent of reference frame. Just as we derived the electrodynamic Galilean transformations by considering moving rods, capacitors, and inductors, we now derive the corresponding electrodynamic Lorentz-FitzGerald transformations. Moreover, we also use simple moving capacitors and inductors to derive the relativistic energy and momentum relations, including the famous equation ${{\mathbb{E}}_{0}}=m{{c}^{2}}$.

Finally, in Section 17.3, we introduce Einstein summation notation that is common in both special and general relativity. This leads to a discussion of polar and axial vectors, and covariant and contravariant tensors. It is in this form that Maxwell’s set of equations most elegantly displays the interdependence between the electric and magnetic fields, and the unique nature of light as a wave without a medium.

[1] From “On the Electrodynamics of Moving Bodies,” by Albert Einstein (1905), translated by Anna Beck, ©1989 by the Hebrew University of Jerusalem.

[2] W. de Sitter, “On the constancy of the velocity of Light,” Proceedings of the Royal Netherlands Academy of Arts and Science, 16 (1913), pp. 395-396.

[3] J. G. Fox pointed out in the 1960s that light emitted by binary stars must pass through surrounding gases making de Sitter’s conclusion premature because his measurements would be independent of the original motion of the binary stars (Am. J. Phys. 30 (1962), p. 297 and J. Opt. Soc. 57 (1967), p. 967).

[4] T. Alveger, et al. Phys. Letters, 12 (1964), 260.

[5] K. Brecher, “Is the speed of light independent of the velocity of the source?”, Physical Review Letters, 39 (17) (1977), pp. 1051–1054.

[6] A. Einstein, “On the Electrodynamics of Moving Bodies,” Annalen der Physik 17 (1905), 891-921, Translated by Anna Beck, in The Collected Papers of Albert Einstein, Volume 2, English Translation, ©1989 by the Hebrew University of Jerusalem