Einstein's Masterwork

by John Gribbin

  This was impossible to explain on the wave model of light. But it was utterly simple to explain using Einstein’s ‘heuristic principle’. Planck’s equation implied that electromagnetic radiation only existed in little packets of energy, quanta with E = h. For a particular wavelength (or frequency) of radiation, every quantum has the same energy. So energy could be handed over to electrons only in quanta that size. As Einstein put it:

  The simplest conception is that a light quantum transfers its entire energy to a single electron.

  In which case, as long as was the same, every electron would receive the same amount of energy and would rush away from the metal surface with the same velocity. If you turned the brightness of the light up, there would be more quanta, but they would still each have the same energy h, so there would be more ejected electrons, but each still moving with the same velocity. The only way to make the electrons move faster would be to use a different wavelength of light, with a bigger value of the frequency (which means a shorter wavelength, further into the ultraviolet).

  This was very nearly what Lenard had found, but not quite. The trouble was, the experiments involved were still extremely difficult, and although Lenard had found that shorter-wavelength ultraviolet light did produce ejected electrons with more energy, he couldn’t measure the energy precisely. The results weren’t good enough to say exactly how much extra energy the electrons got for a particular change in wavelength. Einstein’s calculations did predict a very precise relationship, but it was more precise than the experimental results. His equation agreed with the experimental data, but within the range of uncertainty allowed by the data it was conceivable (if unlikely) that some other equation might work just as well. So all he could say in 1905 was that:

  As far as I can tell, this conception of the photo-electric effect does not contradict its properties as observed by Mr Lenard.

  With no absolute experimental confirmation of the validity of Einstein’s calculations, the idea of light quanta (especially ‘particles’ of light that still somehow had wavelengths associated with them) was so shocking to physicists in 1905 that Einstein’s paper was largely ignored for years. The only person who really took much notice of it was an American experimental physicist, Robert Millikan, who was so infuriated when he heard about it that he promptly set out to try to prove Einstein was wrong. But it’s an indication of how little attention was paid to Einstein’s photoelectric paper when it was published that Millikan, who was already working on the photoelectric effect and investigating other properties of electrons in 1905, didn’t even learn about the paper for several years.

  Millikan worked at the University of Chicago and was 37 in 1905, eleven years older than Einstein and ten years younger than Planck. But it was only in 1912, when he was 44, that he began his determined effort to measure the properties of electrons ejected by the photoelectric effect accurately enough to test Einstein’s predictions and – he firmly expected – to prove Einstein was wrong. In a classic example of the scientific method at work, the sceptical Millikan actually found that the relationship between the energy of the ejected electrons and the wavelength of the radiation involved exactly matched Einstein’s predictions. But even then, he could not at first accept the reasoning behind Einstein’s prediction. When he announced the results of four years of intensive research into the problem in 1916, he said:

  The Einstein equation accurately represents the energy of electron emission under irradiation with light [but] the physical theory upon which the equation is based [is] totally unreasonable.

  Nevertheless, he admitted that his results, combined with Einstein’s equation, provided ‘the most direct and most striking evidence so far obtained for the reality of Planck’s h’.9

  The Nobel Committee were no less cautious when they awarded Einstein the Physics Prize in 1922 (it was actually the 1921 Prize, held over for a year; Millikan received the Prize in 1923). The citation noted the work of Millikan in proving Einstein’s prediction right, but referred only to ‘the discovery of the law of the photo-electric effect’ (in other words, the equation tested by Millikan) and avoided mentioning the physical model on which the equation was based. But as it happened, just a year later, in 1923, new experiments involving the interaction between electromagnetic radiation and electrons finally established the reality of light quanta, which were then given the name ‘photons’ by the American chemist Gilbert Lewis in 1926.

  Although it is not strictly relevant to our story, it’s amusing to see how Millikan rewrote his own history of these events as the reality of photons became more and more firmly established. Having entirely dismissed the physical theory on which the equation was based in 1916, in 1949 he wrote in an article in the journal Reviews of Modern Physics that: ‘I spent ten years of my life testing that 1905 equation of Einstein’s and contrary to all my expectations, I was compelled in 1915 to assert its unambiguous verification in spite of its unreasonableness.’ In 1951, two years before he died, he wrote in his autobiography that: ‘I think it is correct to say that the Einstein view of light quanta, shooting through space in the form of localised light pulses, or, as we now call them, photons, had practically no convinced adherents prior to about 1915, by which time convincing experimental proof had been found.’ No mention here that even Millikan himself had still not been convinced of the reality of light quanta in 1915!

  As I have mentioned, part of the problem of convincing scientists that light quanta were real was the enormous success of the wave model of light, and in particular Maxwell’s equations. At the end of the 19th century, it seemed quite clear that particles were particles and waves were waves. When cathode rays were discovered, nobody knew if they were waves or particles until J.J. Thomson devised the experiments which proved that they were particles. Then, they could be neatly labelled and the possibility that they might be waves forgotten. It was equally natural to assume that light could only be one thing or the other. It wasn’t until well into the 1920s that physicists began to come to terms with the uncomfortable truth that it was possible for light to somehow be both a wave and a particle, and to realise that the everyday laws of common sense do not apply on the very small scale of entities such as photons and electrons. As we shall see, Einstein also played a key part in these discoveries.

  This ‘wave-particle duality’ lies at the heart of quantum physics, and it is now well established that just as light (which was formerly thought of as a wave) behaves like a stream of particles under some circumstances, so electrons (and other entities that were formerly thought of as particles) have a dual nature and behave under some circumstances like waves. Electrons can even be made to interfere with one another in a variation on Young’s experiment.i Einstein was the first person to understand that light could behave as a wave under some circumstances (as in Young’s experiment) and like a particle in other circumstances (as in the photoelectric effect). This flexibility of approach allowed him to keep faith with the aspects of the wave model that worked – notably Maxwell’s equations – even while he was rejecting the wave model in situations where it did not apply. With the photoelectric paper submitted for publication in mid-March 1905, Einstein’s long fascination with light was about to bear fruit in an even more spectacular way – once he had finished writing up the paper that would become his PhD thesis and his paper on Brownian motion.

  The special one

  The last of the four great papers of Einstein’s annus mirabilis emerged from his fertile brain soon after he had submitted his paper on Brownian motion to the Annalen der Physik. The breakthrough was triggered, he later recalled, by a discussion with his old friend Michele Besso, sometime in the middle of May.j An intense burst of activity over the next six weeks saw the key paper on the Special Theory of Relativity delivered to the Annalen der Physik on 30 June (after Mileva had carefully checked Einstein’s calculations for slips, a mundane task which didn’t even earn her an acknowledgement in the paper). It was published at the end of September, in the
same week that the editor of the Annalen received a second paper on the subject, in which Einstein spelled out the famous relationship between mass and energy.k

  Curiously to modern eyes, this key paper about the nature of space and time is actually titled ‘On the Electrodynamics of Moving Bodies’. This reflects the importance of light – an electromagnetic entity – in Einstein’s theory, but also highlights the way in which the puzzle of relative motion had developed in the 1890s. Following the success of Maxwell’s equations, physicists in the last quarter of the 19th century were convinced that light was a form of vibration in the ether, and various experiments were carried out to try to measure the motion of the Earth through the ether. If light travels at a certain fixed speed through the ether (as Maxwell’s equations seemed to imply), and the Earth is moving in the same direction, then you would expect from everyday experience of how speeds add up that the speed of that light relative to the Earth would be less than the speed of that light through the ether. Conversely, if the Earth were running head on into a light beam travelling through the ether, then you would expect the speed of the light beam measured in the experiments to be equal to its speed through the ether plus the speed of the Earth through the ether.

  Making such measurements proved extremely difficult (chiefly because the speed of light is so big, 300,000 kilometres per second), but the predictions encouraged experimenters to develop new techniques in the 1880s and 1890s which were accurate enough to measure the calculated effects. But even when these experiments became sophisticated enough to take account of things like the Earth’s movement around the Sun, and its daily rotation on its own axis, they always measured the same velocity for light, whether it was moving in the same direction as the Earth, in the opposite direction to the Earth, or at any angle across the line of the Earth’s motion.

  The first person to take these results seriously and try to find an explanation for what was going on (rather than just assuming the experimenters were making a mistake) was George Fitzgerald, who was professor of Natural and Experimental Philosophy at Trinity College, Dublin. In 1889 he wrote a paper, which he sent to the American journal Science, in which he pointed out that the experimental results could be explained if the experimental apparatus (and everything else) shrank slightly in the direction of its motion through the ether. The experimental apparatus involved is much more complicated than a simple ruler, but in effect he said that if your ruler shrank by a tiny amount then the time taken for light to whizz past the ruler from one end to the other would be a little less, and you would measure a different speed than if the ruler had not shrunk. ‘Paper’, though, is perhaps too grand a word for Fitzgerald’s squib, which contained no mathematical calculations and consisted only of a single paragraph. It was clear that in order for all experiments to always measure the same speed for light, the shrinking had to obey a precise mathematical formula, which Fitzgerald did not spell out. This shows, for example, that in order to make a metre-long ruler shrink to 99 centimetres (that is, a reduction in length of just 1 per cent), it would have to be moving at one seventh of the speed of light, 43,000 kilometres per second.

  Fitzgerald had no detailed physical explanation for why objects should shrink in this way, and his colleagues in Dublin laughed at the idea. Although the paper was published in Science, nobody took any notice.l So when the Dutch physicist Hendrik Lorentz came up with a similar idea in 1892, and the appropriate mathematical equation to describe the shrinking effect, he didn’t know about the similarity to Fitzgerald’s earlier work until this was pointed out by the British physicist Oliver Lodge.

  The contraction formula became known as the Lorentz-Fitzgerald contraction, which seems a little unfair both in terms of the chronology and alphabetically. There was, though, a sound reason why it became known as Lorentz-Fitzgerald contraction, not Fitzgerald-Lorentz contraction. Unlike Fitzgerald, Lorentz developed this mathematical equation alongside a physical picture of what might be going on to make moving objects shrink.

  In the early 1890s, Lorentz, who was born in 1853, was professor of Theoretical Physics at the University of Leiden. He had developed a theory of electrodynamics which he was already calling the ‘electron theory’ – although, confusingly from our point of view, this did not involve the particles now known as electrons. He suggested that all matter is made up of electrically charged particles, some with positive charge and some with negative charge, held together by electromagnetic forces. In 1892, he simply referred to these entities as ‘charged particles’, but in 1895 he referred to them as ‘ions’.m It was only in 1899, two years after the identification of cathode rays as streams of negatively charged particles, that he started calling these particles ‘electrons’. The name stuck, even though the ‘electron theory’ did not last.

  Like Fitzgerald, Lorentz argued that the experimental observations of the constancy of the speed of light could be explained if moving objects shrank in the direction they were moving, and he came up with the appropriate formula for the contraction. Also like Fitzgerald, he assumed that the cause was the motion of the object relative to the ether, which provided a standard frame of reference against which all motion could, in principle, be measured. But he went further by suggesting that the reason why objects shrank in this way was because of a physical effect of the ether on the moving objects. Specifically, he suggested that there was an electric force which was caused by the motion, which had the effect of squeezing the charged particles of which the moving object was made. Lorentz took these ideas much further than Fitzgerald (not least because Fitzgerald died in 1901, at the early age of 49) and came up with a complete theory that he published in a Dutch journal, which was not very widely read, in 1904. As well as describing how the length of a moving object was related to its motion relative to the ether, this work raised the idea of ‘relative time’ and the idea of synchronising clocks by using light signals – which was also, as we shall see, a central feature of Einstein’s work.

  In all of this, Lorentz was encouraged by the French mathematician Henri Poincaré, a year younger than Lorentz, who publicised the ideas and became interested in the mathematical foundations of the equations involved. Indeed, the first appearance of the term ‘relativity principle’ was in a lecture Poincaré gave at the World Exhibition in St Louis in 1904. It was also Poincaré who first used the term ‘Lorentz transformations’ to describe the whole package of equations that Lorentz had derived in his 1904 paper.

  Einstein had followed at least some of these developments, and was well aware of the fact that all experiments showed no effect of the motion of the Earth on the measured speed of light, even though he does not seem to have been particularly familiar with the details of all the experiments. He later said that he had spent seven years puzzling over the electrodynamics of moving bodies before the breakthrough in 1905, and this is borne out by a letter he wrote to Mileva in August 1899, in which he said:

  I’m more and more convinced that the electrodynamics of moving bodies as it is presented today doesn’t correspond to reality, and that it will be possible to present it in a simpler way. The introduction of the term ‘ether’ into theories of electricity has led to the conception of a medium whose motion can be described without, I believe, being able to ascribe physical meaning to it.10

  This is where Einstein would make his dramatic breakthrough in 1905. He did away with the ether. Instead of saying that what matters is motion relative to the ether, he said that what matters is how two objects move relative to each other, and that there is no absolute standard of rest against which motion can be measured.

  There’s another especially intriguing feature of the paper on what became known as the Special Theory. It contains no references at all to any earlier work, not even that of Lorentz, or the experiments involving the speed of light. Instead, it starts from first principles and Maxwell’s equations to build a logical, consistent mathematical structure which leads inevitably to the conclusions about the nature of space and time. By
structuring his paper in this way, Einstein is clearly proclaiming to the world that he has discovered a fundamental, absolute truth about the nature of the Universe, to rank with such fundamental mathematical truths as Pythagoras’ theorem concerning the lengths of the sides of right-angled triangles. It does not depend on experiments or theoretical models, it is part of the very fabric of the way the world works.n

  In this spirit, Einstein starts with just two facts about the world – what the mathematicians would call postulates or axioms – and constructs the whole edifice of his theory by building upward from those foundations. The first postulate comes straight from the world of the practical application of electromagnetism in dynamos and electric motors, the industry where Einstein’s father had worked for so long. The 19th-century boom in this industry was based on the work of Michael Faraday, who discovered in 1831 that when a conducting wire moves in a magnetic field, an electric current flows in the wire. Or rather, using modern terminology, he found that when a wire moves relative to a magnetic field an electric current flows in the wire. It doesn’t matter whether the magnet is fixed in place in the laboratory and the wire moves past it, or whether the wire is fixed in place in the laboratory and the magnet moves past it. Either way, an electric current flows in the wire. As Einstein put it in the opening paragraph of his paper on the Special Theory:

  The observable phenomenon here depends only on the relative motion of conductor and magnet.

  And this leaves no role for the ether, since the observed phenomenon is not affected by the motion of either the magnet or the wire relative to the ether. If, for example, both the wire and the magnet move alongside each other in the same direction (any direction!) and at the same speed (any speed!), there is no current in the wire.