Back to Contents

In the previous chapters I have shown you how to deduce the postulates of Relativity as theorems from the observation that we exist. At the beginning of the Twentieth Century, Albert Einstein was not so fortunate: he was obliged to devise the postulates by intuition. Before I show you how he used those postulates to deduce the features of the Lorentz Transformation, I want to show you how he may have conceived them and why it was important that he did so.

One of the great projects of Nineteenth Century physics was that of working out the nature of light. Though the prospect of that project's success was dim at the beginning of the century, by century's end physicists had succeeded well beyond their expectations, not only showing how light is related to electricity and magnetism, but also deducing and verifying experimentally the existence of related radiations. But the physics underlying a clearer understanding of light also revealed deeper, more subtle problems with physics as a whole. In spite of having the fundamental equations of electromagnetism in hand and in spite of performing sophisticated experiments suggested by those equations, physicists were unable to locate the foundations of the Universe. That failure gave physicists a serious problem, because the laws of electromagnetism make explicit reference to velocity in a way that implied to the physicists of the Nineteenth Century the existence of an experimentally verifiable state of absolute rest, a state that the best experiments of the day could not find.

Albert Einstein did not actually set out to solve that problem. By his own account, he had, at the age of sixteen (in 1895), devised a fantasy of flying alongside a ray of light. That fantasy was no mere daydream: he was using it to help him to imagine what the light in the ray would look like to him if he could pace it on its flight through space. He knew that light, as Maxwell had shown, consists of an electric field and a magnetic field, both so formed (specifically, in the shape of a wave) that when they move past an observer they appear to that observer to vibrate. The vibrations are important because a changing (i.e. vibrating) electric field generates a magnetic field (by Maxwell's addition to Ampere's law) and a changing magnetic field generates an electric field (by Faraday's law), so the moving fields in light regenerate each the other as the light propagates. But if you could travel with a ray of light, moving at the same speed at which it moves, the fields would not appear to you to vibrate, so would the light exist for you? Einstein reasoned that it would not, that for light to exist it had to move past all possible observers at a certain speed, and showed as a consequence (in 1905) that the Universe has no solid foundation.

That hypothetical foundation was called the Šther and it was a contemplation of light that led physicists to postulate its existence in the first place. In 1801 Thomas Young demonstrated conclusively that light is a wave-like phenomenon: he achieved that demonstration by showing that light displays both constructive and destructive interference, just as waves in a pan of water do. Reasoning from that demonstration and making an analogy with sound, which is a vibration of an elastic medium (i.e. air), physicists hypothesized that light is the vibration of some hugely dense, intensely stiff, superfluid medium, which they named after the celestial substance that the Ancient Greeks believed gave the sky its blue color, the Šther. In Maxwell's original work on electromagnetic theory in 1861 the Šther was given another use: Maxwell conceived electric and magnetic fields as strains induced in the Šther by the presence of electric charges and currents. In 1864, though, when he reworked his theory, he abandoned his use of the Šther, though he doesn't seem to have considered the consequences of doing so. And there were consequences.

In Newtonian dynamics the effect of a force is a change in a velocity. That means that in a calculation related to describing the path of a body affected by a force we are using the difference between the body's velocities measured before and after the application of the force, so it makes no difference in the calculation whether we use absolute velocities that are defined relative to some state of Absolute Rest that's the same for everyone or whether we use relative velocities that are defined relative to some convenient reference marker. But in the equations of electromagnetism velocity appears in the description of the cause of the force that electric currents exert and that force is an absolute quantity, so Maxwell's contemporaries were convinced that velocity must be absolute as well. They believed that all of the velocities used in the equations of electromagnetism had to be measured relative to a frame of Absolute Rest, which frame was occupied and marked by the Šther, the foundation stone of Reality, the Prime Meridian of velocity. Maxwell's abandoning the use of the Šther in his theorizing was tantamount to claiming that relative velocities can be used in the laws of electromagnetic force, but that claim couldn't actually be true to Reality, could it? Well, as always, there were experiments that would reveal the truth of the matter.

Alas for those who preferred to believe in the existence of the Šther, the Universe seems somewhat perversely to be designed specifically to hide the existence of the Šther from physicists conducting clever experiments. And the experiments were quite clever. In 1887 Albert Abraham Michaelson (1852-1931) and Edward Williams Morley (1838-1923) attempted to measure Earth's motion through the Šther with an interferometer, a device that splits light into two beams, makes those two beams follow two different paths, and then brings them together so that they will create an interference pattern. They hypothesized that light traveling across the direction of the Šther's flow through their laboratory would move at a speed different from the speed at which light would travel in the direction with or against the Štherial flow. To test that hypothesis, they passed light of a single frequency through a tilted beam splitter (a partly transparent mirror that reflects half the light striking it and allows the other half to pass through it) that sent the two resulting rays down paths oriented perpendicular to each other. Reflected off mirrors at the ends of the paths, the rays came back through the beam splitter and were projected onto a small screen in such a way that they interfered with each other; that is, the electromagnetic fields in the rays so canceled or augmented each other that the resulting light formed a pattern of alternating dark and bright bands on the screen. Those bands were the clever part of Michaelson and Morley's experiment.

In accordance with their hypothesis, Michaelson and Morley figured that the time that the light required to traverse one of the paths in their interferometer would change as the orientation of the path was changed from being parallel to the direction of the Šther wind to being perpendicular to it. They didn't have the means to measure the traversal times directly, so they designed their experiment to measure them indirectly. That's why they split one beam of light into two beams that then traversed paths oriented at a right angle to each other. They had in mind the idea that as the apparatus was rotated (and it was mounted on a granite slab floating on a pool of mercury to enable it to be rotated smoothly) the traversal time along one path would increase slightly and the traversal time along the other path would decrease slightly. Those changes would change the way in which the two rays interfered with each other on the screen and thus make the pattern of dark and bright bands appear to shift sideways. Measurement of that shift, correlated with the orientation of the apparatus, would enable Michaelson and Morley to calculate the speed and direction at which the hypothetical Šther wind blew through their laboratory. So they switched on their apparatus, took their data, and discovered that the speed of the Šther through their apparatus was precisely zero. It was always zero. Whenever they made their measurements, regardless of time of day or time of year, it was always zero. And they did conduct their experiment at different times of the year, just in case Earth periodically slips into some Štherial doldrum. But Earth changes its velocity by sixty kilometers per second every six months, so a zero result throughout the year was a truly strange result to obtain.

In 1892 the Irish physicist George F. Fitzgerald (1851-1901) offered a suitably strange hypothesis to explain Michaelson and Morley's result. He reasoned that if electromagnetic fields are strains in the Šther, as Maxwell had initially suggested, then when the Šther blows through a body it might alter the electromagnetic fields that hold the body's atoms together; specifically, he hypothesized that the body would be made to shrink in the direction parallel to the direction in which the Šther was moving. When he used that idea in its mathematical form to analyze Michaelson and Morley's experiment, he found that the presumed shrinkage of the interferometer would, as the apparatus was rotated, cancel the difference between the changes in the times that the two rays took to traverse their paths through the apparatus, thereby producing the null result.

Three years later the Dutch physicist Hendrik Antoon Lorentz (1853-1928) elaborated Fitzgerald's hypothesis, in part by noting that a clock moving through the Šther would tell a time different from the time told by a clock at rest in the Šther. He summed up his elaboration of Fitzgerald's work in four equations that we call the Lorentz Transformation, though the interpretation that we give those equations differs remarkably from the one that Lorentz and his contemporaries gave them. And, although we no longer believe that it is caused by the Šther wind, we call the shrinkage of objects due to their relative motions the Lorentz-Fitzgerald contraction in remembrance of the two men's contribution to Relativity.

As disappointing as their results were, Michaelson and Morley did not pronounce the last word on clever experiments to find the Šther. If the Lorentz-Fitzgerald contraction prevented interferometers from detecting the effects of the moving Šther, then someone would have to devise an experiment that did not depend upon the length of their apparatus. In 1903 the obscure team of Trouton and Noble did just that. They were inspired by the notion that an electric current imposes upon the Šther a stress that strains the Šther into manifesting a magnetic field. In order to be the source of such a stress, the current must consist of electric charges moving through the Šther, so an electric charge that's stationary in the laboratory will nonetheless generate a magnetic field if the Šther blows over it. To test that hypothesis, Trouton and Noble attached two flat metal plates to each other face to face with a narrow gap between them, suspended them from a thin quartz fiber, and put equal and opposite electric charges on the plates. If their hypothesis was correct, the magnetic forces generated by the Šther wind blowing through the apparatus would exert a torque that would act to turn the plates parallel to the direction in which the Šther was moving. The apparatus and its successors were all sensitive enough to detect the expected torque, but regardless of how the apparati were turned or when the experiments were performed, the measured torque was always zero. And in this case there was no analogue of the Lorentz-Fitzgerald contraction to provide theoretical cover for the null result.

Do you see what I mean about the Universe appearing to be so designed that the Šther eludes the grasp of clever experimentalists? Apparently none of our experiments can manipulate the laws of physics in a way that will reveal the existence or the motion of the Šther. What that means, though, is that in our theorizing about those laws of physics we can ignore the Šther, pretend that it doesn't exist, and yet enjoy the confidence that our theories won't go wrong as a result. But how can we ignore the Šther in the laws of electromagnetic force? The velocities that appear in those laws must be measured relative to something. If that something is not the Šther, then what is it? According to Einstein, it's everyone's favorite inertial frame - their own.

Einstein first presented his theory of Special Relativity in the 1905 Sep 26 issue of Annalen der Physik, in a paper titled "On the Electrodynamics of Moving Bodies". In that paper he showed that the laws of electromagnetic force are not invalidated if the velocities used in the calculations are referred to arbitrarily chosen inertial frames of reference. He showed that the calculation of the electromagnetic forces acting among several bodies observed from one inertial frame can be translated into a correct description of the forces as they would be measured by an observer in another frame, the translation being made with formulae based upon the Lorentz Transformation. His first move was, of course, to deduce the Lorentz Transformation from two postulates; that is, from two statements asserted as if they were axioms, even though they both lack the axiom's necessary property of being self-evident. The first of these postulates, as translated into English from the original German, is

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

Putting that statement another way, we have

COSMIC THEOREM 1: The results of any given experiment or observation and the laws of physics derived from them are the same for all experimenters or observers, regardless of how those experimenters or observers are positioned, oriented, or moving relative to each other.

That postulate, the Principle of Relativity, did not originate with Einstein. The Italian mathematician and natural philosopher Galileo Galilei (1564-1642) first described the principle in 1633 in his book "Dialogue on the Two Chief World Systems". He described the principle indirectly by stating that certain phenomena (such as the flight of insects, the fall of objects, and the motion of a man jumping about as though playing hopscotch) observed aboard a ship sailing across a smooth sea would appear no different from the same phenomena observed on land. Later that same century (in 1687) Isaac Newton (1642-1725) offered much the same description in his Principia.

Einstein's version of the principle, the statement that the laws of physics must be expressed in the same mathematical form, regardless of which inertial frame of coordinates is chosen as the frame of reference, is equivalent to claiming that there is no experiment that will reveal which frame is that of absolute rest. Thus did Einstein indirectly dismiss the Šther as being of any relevance to his theory. But if all inertial frames are perfectly equivalent to each other, then the physical constants (the proportionality factors that appear in the equations of physics) must also be the same for all observers. In particular, the electric permittivity of vacuum and the magnetic permeability of vacuum must be the same in all inertial frames. Because the product of those two numbers is inversely proportional to the square of the speed at which electromagnetic waves propagate through vacuum, we are led to Einstein's second postulate, the one that makes Special Relativity truly special and that marks the place in the theory where Einstein's stroke of genius struck hardest. That postulate is

"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."

We can restate that postulate in a perfectly equivalent form that refers to the scientists who observe the ray rather than to the motion or nonmotion of the ray's source. We have, then

COSMIC THEOREM 2: Any phenomenon that moves at the same speed at which the boundary of space moves passes all observers at that same speed, regardless of how those observers are positioned, oriented, or moving relative to each other.

The "determined velocity c" to which Einstein referred is just the speed of light, which we hypothesized is the same as the speed at which the boundary of space seems to recede from us and it has the value 299,792.458 kilometers per second or 186,234.709 miles per second.

With those two postulates in mind, Einstein performed a series of imaginary experiments, involving fast-moving trains, in which experiments he worked out the features of the Lorentz Transformation before going on to work out the dynamical laws that govern the interactions among electromagnetic fields and moving bodies. Now I'm going to guide you through a repetition of those thought experiments.


Back to Contents