Imagine a speedboat heading toward the opposite bank of a 400-yard river, whose flow carries it 300 yards downstream in the process. The water, of course, moves the boat. The speedboat winds up moving at an angle along the hypotenuse of a right triangle: 400 yards across and 300 yards downstream; it has traveled (because we’ve conveniently based the example on a Pythagorean triplet) a total of 500 yards in crossing the river. To end up directly across the river, it would have to point itself upstream while traveling by the same angle, and will in effect travel a longer distance (the hypotenuse of the above triangle) in crossing the bank directly. For the same reason – so the argument ran – the ether would move the light traveling in it, and the light would travel at a different speed when crossing the ether’s direction of motion.
In 1881 and 1887, two American physicists, Albert Michelson and Edward Morley, carried out an extremely sensitive experiment to detect what scientists were calling the ‘ether drift.’ Their instrument consisted of two ‘arms’, one pointing in the presumed direction of the ether’s motion and the other perpendicular to it, along which beams of light would travel back and forth. Mirrors were mounted on a bed of mercury, then rotated 90 degrees so that the light would be traveling in a different direction with respect to the ether’s motion. By bringing the beams together to create an interference pattern, Michelson and Morley would be able to detect any slight difference in their velocities. But the experiment failed to detect any such difference.
Physicists were baffled. Something was clearly wrong either with Newton’s or Maxwell’s equations.
At first they assumed the problem lay with Maxwell. He was the Johnny-come-lately. Maxwell’s equations had been around for only a few decades, while Newton’s laws had been around for two hundred years and successfully accounted for everything except a few disagreeable but minor discrepancies that there was little reason to think would not prove to be due to some experimental error or overlooked effect. Some of the brightest scientists of the day attempted to modify Maxwell’s equations to fit the Galileo transformations.3 But these equations proved remarkably resistant. They were embedded in an elaborate network of interrelated concepts, and any change in one rippled through the others with undesirable results.
As the nineteenth century drew to a close, many physicists interested in electrodynamics felt a deep dissatisfaction. There had to be an explanation – the constancy of the speed of light regardless of direction of motion had to be reconcilable with Maxwell and Newton – but none could be found. ‘The most incomprehensible thing about the world is that it is comprehensible’, Einstein once declared. The unstated corollary, is that, to a scientist, the most frustrating thing about the world is not being able to comprehend it.
Acts of Desperation
Dissatisfaction led to desperation. In 1889, Irish physicist George FitzGerald wrote a short, one-paragraph article – a mere five sentences, no equations – stating that ‘almost the only hypothesis’ that can reconcile the Michelson-Morley experiment with Maxwell and Newton ‘is that the length of material bodies changes, according as they are moving through the ether or against it by an amount depending on the square of the ratio of their velocities to that of light.’4 Suppose, FitzGerald thought, the arm of Michelson and Morley’s instrument pointing in the direction of motion shrank due to the impact of the ether on its molecules. If it shrank by just the right amount, it would ‘measure’ the light beam as going up and down in the direction of the ether at the same speed as it measured the light traversing the perpendicular arm. Still, the idea – objects shrink in size when moving at high speeds? – seemed too bizarre to take seriously.
Another desperate soul was Dutch theorist Hendrik Lorentz, who wrote to his friend Lord Rayleigh in 1892 about the predicament created by the Michelson-Morley experiment, ‘I am utterly at a loss to clear away this contradiction.’5 That year, he independently proposed the same idea that FitzGerald had, writing that ‘I can think of only one idea’ to explain the experiment, namely, that the ether causes some contraction effect in the length of a solid body. When he learned of FitzGerald’s idea and contacted him, FitzGerald was overjoyed to learn of a fellow champion of contraction, writing back that he had been ‘laughed at’ for his ideas.6 Lorentz then went on to work out in detail the set of transformations that would have to occur for this contraction to work. Lorentz found that time, too, would be affected. For while FitzGerald was only trying to save the Michelson-Morley experiment, which determined the light in two directions to be traveling at the same speed, Lorentz, more ambitiously, wanted to make sure that the speed of light stayed constant, and appeared the same to moving and nonmoving observers. To accomplish this, clocks would have to tick slower. He then produced a set of formulas now known as the Lorentz transformations, which gave compensations in length and time between stationary and moving systems that would preserve the possibility of the light moving at a constant speed in the ether as detected by the Michelson-Morley experiment, and thus the agreement between Maxwell and Newton. The compensation factor for both space and time was .7 Notice that when there is no relative motion (and v is 0), there is no correction. At low speeds, the correction is so small it would not be noticed. But the closer the object approached the speed of light, the larger the corrective factor grew – the more the object shrank in the direction of motion, and the slower clocks ticked. Meanwhile, however, most scientists continued to regard the idea as too strange to take seriously. But the fact that it was said at all shows the lengths to which scientists were prepared to go to save the ether.
It seemed, one scientist would say later, that ‘all the forces of Nature had entered on a conspiracy’ with the goal of ‘preventing us from measuring or even detecting our motion through the ether.’8
The consternation mounted. Great scientists began reaching for fantastic ideas. In 1898, French mathematician Henri Poincaré toyed with the idea of giving up absolute time in favor of ‘local time’, and soon tried to use it to explain the puzzle regarding the speed of light in ether. In a public lecture at the 1904 World’s Fair in St. Louis, Poincaré remarked almost whimsically, ‘Perhaps we should construct a whole new mechanics, of which we only succeed in catching a glimpse…in which the velocity of light would become an impassible limit.’9
Thus the problem that Lord Kelvin had called ‘Cloud No. 1’ obscuring the ‘beauty and clearness’ of nineteenth-century dynamical theory was only getting more and more difficult. The year after Poincaré spoke, in 1905, all the fantastic ideas that had been cited to try to banish it – the contraction of space and time at high velocities, the nonexistence of absolute space and time, and the speed of light as an absolute upper limit – were shown, in one form or another, to be true.
Enter Einstein
How had Einstein, at that time still a patent clerk, come to take up this particular problem? The same way everyone else did: dissatisfaction.
Years later, Einstein wrote to a friend that, while only about 5 or 6 weeks elapsed between his conception of the idea of the special theory of relativity and a finished paper about it, ‘the arguments and building blocks were being prepared over a period of years.’10
The earliest argument emerged in late 1895 or early 1896, when Einstein was sixteen. It came in the form of what he would call a ‘childlike thought-experiment.’ (The adjective ‘childlike’, in the sense of pure and direct, was often applied to Einstein.) What, the youth asked himself, would happen if he were traveling at the speed of light, and looked over at a light beam riding next to him?11 Newton said it could happen, Maxwell said it couldn’t.
This simple puzzle – can you or can’t you catch up to a light wave – had to have an answer, but none could be fashioned from the existing tools of physics. The puzzle focused the dissatisfaction of young Einstein, providing him with the mixture of bewilderment and curiosity needed for him to start fashioning the arguments and building blocks.
Einstein brooded for years. ‘I must confess’, he told a frien
d later, ‘that at the very beginning, when the Special Theory of Relativity began to germinate in me, I was visited by all sorts of nervous conflicts. When young, I used to go away for weeks in a state of confusion, as one who at that time had yet to overcome the stage of stupefaction in his first encounter with such questions.’12 One day in 1905 he went to visit his close friend and patent office colleague Michele Besso, poured out the details of his ‘battle’ with the problem, and departed. But in the process of laying out the problem, Einstein found the solution. The next day, he dropped in on Besso again and greeted him with the words, ‘Thank you. I’ve completely solved the problem.’13
The result was ‘On the Electrodynamics of Moving Bodies’, one of the most famous and momentous scientific papers ever written, sent to the journal Annalen der Physik in June 1905. Despite the angst behind the genesis of the paper, it follows a simple yet powerful logic – ‘a deep, almost childlike freshness of approach’14 – that is relatively easy to understand.
‘It is well known’, Einstein begins, ‘that Maxwell’s electrodynamics – as usually understood at present – when applied to moving bodies, leads to asymmetries [eccentric results] that do not seem to attach to the phenomena [that is, they seem to be an artifact of our theories rather than a part of the world].’15 He gives examples, and says that these, ‘and the failure of attempts to detect a motion of the earth relative to the ‘light medium’, ‘ lead to the postulate that there is no such thing as ‘absolute rest.’ He calls this conjecture ‘the principle of relativity’, and says that he will combine it with the postulate that in empty space ‘light is always propagated with a definite velocity V which is independent of the state of motion of the emitting body.’
Thus Einstein framed his paper around the logical requirement of reconciling the two key principles: relativity and the constancy of the speed of light. These are ‘seemingly incompatible’, Einstein says. Only seemingly. For he develops a reconciliation in the rest of the paper, claiming that, based on logic alone, he can produce a ‘simple and consistent electrodynamics of moving bodies’ with no need for the supposition of an ether or for an absolute rest frame. What would it take for observers on two different inertial frames to see light travel at the same speed? Einstein determines that it would require the same contraction factor for length in the direction of motion and time that Lorentz proposed.
But while Lorentz had based his work (as had FitzGerald) on the assumption that the ether existed and that the contraction was real (due to the effect of ether on molecular forces), Einstein based his only on the assumption of the validity of the principles of relativity and of the constancy of light. That is, while Lorentz and FitzGerald got their results by trying to save the ether, Einstein arrived at the same result by getting rid of it. As scientists said at the time, ‘There is no conspiracy of concealment, because there is nothing to conceal.’ Or as Feynman liked to say, a universal conspiracy is a law of nature.
Einstein refers to the contraction factor as ‘ß’ in his paper. Its deduction is most easily and frequently presented as a Pythagorean problem. Suppose two inertial reference frames, A and B, are moving at velocity v with respect to each other. In A, a beam of light is sent from a source, perpendicularly to the direction of motion, to bounce back off a mirror at a distance d away from the source. From the point of view of someone on A, the light simply travels a distance 2d. But to someone on B, for whom A – source, mirror, and all – is gliding past at a velocity v, the light travels a longer path; we’ll call it 2d’. Half of this path, d’, is the hypotenuse of a right-angled triangle whose other sides are d and vt’/2. Thus (d’)2 = (d)2 + (vt’/2)2. Yet according to the second principle, the light has the same speed, c, covering the same distance in the same time, seen from B as it does seen from A. That is, V (the symbol Einstein is using for the speed of light) is equal to 2d / t in A and to 2d’/t’ in B. How can that happen? Only if the distance and time of objects in A are shorter in A as seen from B. By how much? By just the amount that d is shorter than d’; that is, d/d’ or t/t’, or the contraction factor ß. If V = 2d/t, then d = Vt/2; and if V = 2d’/t’, then d’ = Vt’/2. Substituting in the Pythagorean equation gives us ß (or the contraction factor t/t’ we are seeking) = .
This, the seminal paper of what would become known as the ‘special theory of relativity’ (a usage that Einstein began in 1915, to distinguish it from his then-new ‘general theory of relativity’), was published in September 26, 1905. It introduced some radical changes in notions of space and time. A paper with such fundamental implications, however – especially one put together in such a short time by someone working feverishly on so many things – was bound to have more consequences than its author could foresee while composing it. One struck him almost immediately. Sometime in fall 1905 he wrote to his friend Conrad Habicht,
A consequence of the study on electrodynamics did cross my mind. Namely, the relativity principle, in association with Maxwell’s fundamental equations, requires that the mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable reduction of mass would have to take place in the case of radium. The consideration is amusing and seductive; but for all I know, God Almighty might be laughing at the whole matter and might have been leading me around by the nose.16
Being led ‘around by the nose’ – reminiscent of how Meno’s slave must have felt, learning something which appears to be true, yet which also must be further explored.
Einstein mailed a three-page paper outlining this consequence, entitled ‘Does the Inertia of a Body Depend Upon its Energy Content?’ to the Annalen the day after his relativity paper appeared, and it was published later that year. As historian of science John Rigden, among others, has pointed out, this paper does not break new ground, and simply draws a consequence that was logically implicit in the previous paper, and easily could have been its final section. If it had, Rigden says, ‘it would have made a spectacular conclusion.’17
Einstein opens the ‘Energy Content’ paper in a disarmingly modest key, ‘The results of an electrodynamic investigation published by me recently in this journal lead to a very interesting conclusion.’ He reaches the conclusion via the following example. Suppose an object (an atom, say) of mass m at rest in reference frame A emits two beams of light – thus, it expends energy – in opposite directions. Let’s say the total amount of energy lost is L (as in the previous paper, Einstein uses the now-unfamiliar notation of L for energy and V for the speed of light), so each light beam carries away the energy L/2. An observer on A sees the object as having no net change in kinetic energy; the atom is standing still, has shed some of the energy it had in an excited state, and continues to have the same mass that it was originally stamped with. But an observer in B, for whom A is moving, sees something different. The forward-moving light beam has more momentum than the backward one, meaning that the atom has had a net change – a decrease – in kinetic energy. This can happen only if the atom’s velocity or its mass decreases. But its velocity is the same; in the rest frame, there is no recoil. The only other possibility is that, from the perspective of the frame in which the atom is moving, the mass has decreased. The atom has not gained any mass from the point of view of its rest frame; its ‘inertial mass’ is the same. But its mass from the point of view of the laboratory, which views it as a moving object, changes. By how much? Applying the tools of the previous paper, Einstein finds that the conversion factor, once more, is ß.
Einstein continues – again, using the unfamiliar notation of L for energy and V for the speed of light – as follows:
If a body releases the energy L in the form of radiation, its mass decreases by L/V2. Since obviously here it is inessential that the energy withdrawn from the body happens to turn into energy of radiation rather than into some other kind of energy we are led to the more general conclusion: The mass of a body is a measure of its energy content.18
This is the first appearance in print of the idea eventuall
y to become famous as E = mc2. It is not presented explicitly in the form of an equation, and it is not in its familiar symbols. However, the startling, even revolutionary mass-energy concept is fully articulated. The concept transformed some of the most fundamental notions of how the universe is assembled. It put together two things long thought to be utterly different: energy, whose conservation principle was a crowning achievement of nineteenth-century physics, and mass, whose conservation principle was a crowning achievement of eighteenth-century science.19 They can change into each other.
It also revolutionized the requirements for objectivity. On the Newtonian stage, energy and mass remained the same when observed from different inertial frames; on Einstein’s, they remain virtually the same at low speeds, but changes take place the closer the speed of the frames get to that of light. What is objective – really out there – is what changes in length and clock time by this amount when witnessed from another, sufficiently fast, inertial frame.20
Over the next several years, Einstein referred to this result several times, though again in the form of descriptions or in his original symbols, and not yet in his now-famous version. In a footnote to a 1906 paper, for instance, Einstein wrote that ‘the principle of the constancy of mass is a special case of the energy principle.’21 Early in 1907, in another Annalen article, he refers to energy as e, mass as the Greek letter μ, and the speed of light as V, and he uses the equation
Here the famous equation – energy is equal to the mass times the speed of light squared – appears with the corrective factor ß, which takes into account the effect when the body is in motion. That is, let’s take a piece of matter such as an electron. At rest, every electron has the same mass; it’s as if nature stamped that mass on each electron when it was created. Whenever that electron is weighed in its own reference frame, it always has that mass. Now suppose we look at it from another reference frame, in which the electron is moving. If E = mc2, and c is a constant, then m and e have to vary in exactly the same way as the energy increases. The electron’s inertial mass – its mass from its own rest frame – does not change. But its mass as you measure it in the laboratory, where you see it as moving, changes. And ß, the compensation factor, is the transformation that tells you what to multiply by to get the rest mass. Leaving out that compensating factor gives you what he calls in a footnote the ‘simplifying stipulation μV2 = ε0.’22
A Brief Guide to the Great Equations Page 16