Einstein on the Run

by Andrew Robinson

  QUANTUM THEORY

  Yet, despite the strangeness of its predictions, relativity was built on the mechanics of Newton modified by the electrodynamics of Maxwell, as Einstein was at pains to emphasise. Most modern physicists regard relativity theory as revolutionary, but Einstein himself did not, and reserved ‘revolutionary’ to describe his paper on the quantum theory. Ironically, although the quantum paper was published in April 1905 before his relativity paper in June 1905, his relativity paper does not refer to the quantum theory; the relativity paper treats electromagnetic radiation purely as a wave and never so much as hints that it might consist of particles or quanta of energy. Presumably Einstein recognised that one big new idea per paper would be indigestible enough for most physicists. Perhaps, too, his isolating of the two ideas in two separate papers reflected his own doubts about the quantum concept. Nevertheless, with these two papers he became the first physicist to accept what is today the orthodoxy in physics: light can behave both like a wave (in relativity theory) and like a particle (in quantum theory).

  Newton had been divided about the relative merits of waves and particles, on the whole favouring the latter in his ‘corpuscular’ theory of light, which dominated physics until convincing new evidence for the wave theory was discovered by Thomas Young (yet another English physicist admired by Einstein) soon after 1800. As for gravity, Newton had no idea at all as to how such a continuous influence might arise from discrete (in other words discontinuous) masses. Indeed, the debate about whether nature is fundamentally continuous or discontinuous runs through science – from the atomic theory of ancient Greece right up to the present day with its opposing concepts of analogue and digital, and the wave/particle ‘duality’ of subatomic entities like the electron. Russell is supposed to have asked: Is the world a bucket of molasses or a pail of sand? In mathematical terms, asked the physicist Rigden, ‘Is the world to be described geometrically as endless unbroken lines, or is it to be counted with the algebra of discrete numbers? Which best describes Nature – geometry or algebra?’

  Quantum theory, the modern corpuscular theory, was born with the new century, in 1900, as a result of the work of the physicist Max Planck, although it would remain in limbo until 1905, when Einstein’s paper would endow it with its true significance. Planck considered the energy of heat that had been measured emerging from a glowing cavity, termed a ‘black body’ because the hole leading to the cavity behaves almost as a perfect absorber and emitter of energy with no reflecting power (like a black surface). Planck tried to devise a theory to explain how the heat energy of a black body varied over different wavelengths and at different temperatures of the cavity. But he found that if he treated the heat as a continuous wave, this wave model did not agree with experiment. Only when he assumed that the energies of the ‘resonators’ (atoms) in the walls of the cavity that were absorbing and emitting heat were not continuous but could take only discrete values, did theory match experiment. Instead of continuous absorption and emission of energy, energy was exchanged between heat and atoms in packets or quanta. Moreover, the size of a quantum was proportional to the frequency of the resonator, which meant that high-frequency quanta carried more energy than low-frequency quanta. As a believer in nature as a continuum, and as an innately conservative man, Planck did not feel at all comfortable with what his calculation had told him, but in 1900 he reluctantly published his theoretical explanation of black-body radiation.

  Einstein was bolder than Planck. He was twenty years younger and had less stake than Planck in classical nineteenth-century physics. Probably encouraged by his disbelief in the ether, Einstein decided that it was not just the exchange of energy between heat/light and matter (i.e. absorption and emission) that was quantised – light itself was quantised. In his introduction to his April 1905 paper he radically stated: ‘According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localised in points in space, move without dividing, and can be absorbed or generated as a whole.’ Instead of moving particles, Einstein visualised a light beam as moving packets of energy. When this avant-garde concept was finally accepted by reluctant physicists in the 1920s, the packets were termed ‘photons’.

  Had there been no experimental support for Einstein’s assumption of quantised light, it would have met with an even more sceptical reaction than it in fact did. But fortunately there was at least some significant laboratory evidence. Though it was not detailed, Einstein audaciously interpreted the evidence with the quantum theory he had elaborated in the first part of his paper. The success of his theoretical explanation of the ‘photoelectric effect’ in his 1905 paper (which won him the Nobel prize) meant that light quanta could not be totally ignored, even if they were gravely distrusted.

  The photoelectric effect had been discovered by Hertz around 1888 while investigating electromagnetic waves. Hertz noticed that in a spark gap the spark gained in brightness when illuminated by ultraviolet (high-frequency) light. With the discovery of X-rays in 1895, and of the electron in 1897, followed by the experiments of Philipp Lenard (a former assistant to Hertz), it was soon accepted that high-frequency light could knock electrons out of the surface of a metal producing photoelectrons, so-called cathode rays. ‘I just read a wonderful paper by Lenard on the generation of cathode rays by ultraviolet light. Under the influence of this beautiful paper I am filled with such happiness and joy that I absolutely must share some of it with you,’ Einstein wrote to his fiancée Mileva in 1901. It may have been this paper by Lenard that started Einstein speculating on the quantised nature of light. For Lenard’s published data were in major contradiction with those expected from classical physics.

  With the wave theory of light, the more intense the light, the more energy it must have and the greater the number of electrons that should be ejected from the metal. This was observed by Lenard – yet only above a certain frequency of light. Below this frequency threshold, no matter how intense the light, it knocked out no electrons. Moreover, above the threshold, electron emission was observed even when the light was exceedingly weak. With the quantum theory, however, Einstein realised such behaviour was to be expected. One quantum of light (later called a photon) would knock out one electron, but only if the quantum carried enough energy to extract it from the surface of the metal. Since, as Planck had shown, the size of a quantum depended on its frequency, only quanta of a sufficiently high frequency or higher would knock out electrons – hence the existence of the threshold frequency. Moreover, even a very few quanta (a very weak intensity of light) would still eject a few electrons, provided that the quanta were above the threshold in frequency.

  So truly revolutionary was this discontinuous view of nature, which owed almost nothing to earlier physics, that light quanta took much more experimentation and a lot of fresh thinking to be accepted by other physicists. This happened only in the late 1920s. So we shall leave the quantum theory for now, and return to it much later, after following the next phase in Einstein’s struggle with relativity.

  GENERAL RELATIVITY

  In 1908, Einstein’s former mathematics professor at Zurich, Minkowski, reformulated relativity mathematically and introduced the new concept of ‘space-time’. He enthusiastically announced:

  The views of space and time which I wish to lay before you have sprung from the soil of experimental physics and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

  More prosaically, events in four-dimensional space-time are analogous to points in three-dimensional space. There is an analogy, too, between the interval separating events in space-time and the straight-line distance between points on a flat sheet of paper. The space-time interval is absolute, in other words its value does not change with the reference
frame used to compute it. In conventional space and time the stone falling from a uniformly moving train has two trajectories – straight down and parabolic – depending on whether it is observed from the train or from the embankment. But in the geometry of space-time it has only one trajectory, which Minkowski dubbed its ‘world line’.

  ‘Since the mathematicians pounced on the relativity theory I no longer understand it myself,’ Einstein apparently sighed on studying Minkowski’s treatment. As a physicist he was at this time somewhat ambivalent about pure mathematics. Even in his introduction to relativity for the general reader, he felt obliged to warn that mathematical talk of a ‘four-dimensional space-time continuum’ had nothing at all to do with the occult or with inducing ‘mysterious shuddering’. Yet, Einstein did admit that without Minkowski’s mathematics, the general theory of relativity might never have grown out of its infant state. ‘“Analytic” or “algebraically expressed” geometry is fundamental to modern theoretical physics, because of its ability to take the imagination way beyond everyday physical constraints,’ according to mathematician Robyn Arianrhod in Einstein’s Heroes: Imagining the World through the Language of Mathematics. ‘Newton used an early form of it (in his calculus) to visualise aspects of the mechanism that keeps “the stars in their courses”; Maxwell used it to imagine Faraday’s invisible fields; and Einstein used it to imagine the whole cosmos.’

  The following year, 1909, with the growing fame of relativity, Einstein’s academic career took off. After seven years he left the Patent Office in Bern to become a (non-tenured) professor of theoretical physics at the University of Zurich; was the guest of honour at the next annual meeting of German scientists in Salzburg; and received his first honorary degree in Geneva at the age of just thirty. In early 1911, he moved to Prague as a full professor, but stayed only sixteen months before moving back to Zurich in 1912, now as full professor of theoretical physics. While based in Prague, in late 1911 he attended the first Solvay Congress in Brussels and lectured about his quantum theory on terms of equality with the world’s greatest scientists: Lorentz, Planck and Poincaré – already known to him – as well as Marie Curie, Ernest Rutherford, Walther Nernst and others. Nernst’s student, Frederick Lindemann, secretary of the Congress – who as a future professor of physics at the University of Oxford would host Einstein in England – meeting him for the first time, recalled Einstein as ‘singularly simple, friendly and unpretentious. He was invariably ready to discuss physical questions, even with a mere post-graduate student, as I then was. . . . But his pre-eminence among the eighteen greatest theoretical physicists of the day, who were there assembled, was clear to any unprejudiced observer.’ Finally, in the spring of 1914, Einstein left Switzerland – while remaining a Swiss citizen – and arrived in Berlin where he was elected a member of the Prussian Academy – thereby reacquiring, in effect, German citizenship – on the understanding that he could devote his entire time to research.

  Solvay Congress in Brussels, Belgium, 1911: a gathering of the greatest scientists in the world at this time. Einstein, then only thirty-two years old, who gave the concluding address on his revolutionary quantum theory, stands second from right. Among the seated scientists are Walther Nernst (far left), Hendrik Lorentz (fourth from left), Marie Curie (second from right) and Henri Poincaré (far right). Those standing include Max Planck (second from left), Arnold Sommerfeld (fourth from left), Frederick Lindemann (fifth from left), James Jeans (fifth from right) and Ernest Rutherford (fourth from right).

  At this point, the relativity theory of 1905 began to be known as ‘special’ relativity, to distinguish it from the later, more general theory, following Einstein’s own terminology introduced in 1915. Of course the ‘general’ theory subsumes the ‘special’ theory, indeed it reduces to the special theory under conditions of uniform motion with constant velocity (as with the example of the train and the falling stone). In such an idealised universe, without gravity, special relativity alone is sufficient. But in the real physical universe, which is pervaded by gravity and accelerations due to gravity as well as various other kinds of forces, there is no such thing as absolutely uniform motion, only approximations to it, so we need the more general theory.

  Einstein’s aim was to make his 1905 relativity theory valid for all moving coordinate systems. Then, as he noted ironically, there would be an end to the violent disputes that had racked human thought since Copernicus, because ‘The two sentences, “the Sun is at rest and the Earth moves,” or “the Sun moves and the Earth is at rest,” would simply mean two different conventions concerning two different coordinate systems.’ In 1905, he had done away with Newton’s concepts of absolute space and absolute time. Now, using the concept of space-time introduced by Minkowski and radically developed by Einstein with the help of his mathematician friend Grossmann, Einstein would devise a more sophisticated theory which would also do away with gra-vity’s inexplicable instantaneous action at a distance, while at the same time retaining Newton’s laws of motion and his inverse-square law of gravitational attraction as a first approximation to physical reality.

  The initial inkling of how to generalise relativity struck Einstein in 1907, and it is a moment reminiscent of Newton’s contemplation of the falling apple, though trickier to comprehend. ‘I was sitting on a chair in my Patent Office in Bern. Suddenly a thought struck me: if a man falls freely, he would not feel his weight,’ Einstein later recalled. In other words, if you were to jump off a rooftop or better still a high cliff, you would not feel gravity. ‘I was taken aback. The simple thought experiment made a deep impression on me. It was what led me to the theory of gravity.’ He called this ‘the happiest thought of my life’.

  To drive home the point, he imagined that as you fall, you let go of some rocks from your hand. What happens to them? They fall at the same rate as you, side by side. If you were to concentrate only on the rocks (admittedly difficult!), you would not be able to tell if they were falling to the ground. An observer on the ground would see you and the rocks accelerating together for a smash, but to you the rocks, relative to your reference frame, would appear to be ‘at rest’.

  Or imagine being inside a moving lift while standing on a weight scale. As the lift descends, the faster it accelerates, the less you will feel your weight and the lighter will be the weight reading on the scale. If the lift cable were to snap and the lift to go into free fall, your weight according to the scales would be zero. Then gravity would not exist for you in your immediate vicinity. In other words, the existence of gravity is relative to acceleration.

  From such thinking, which became intensive only after he moved to Prague in 1911, Einstein restated a venerable idea that has become known as his ‘equivalence principle’ – the idea that gravity and acceleration are, in a certain sense, equivalent. It encompasses the fact, first observed by Galileo Galilei, that gravity accelerates all bodies equally. In more scientific language, inertial mass (as defined by Newton’s second law of motion) equals gravitational mass (as defined by gravity). Newton had simply assumed this equivalence as self-evident in formulating his gravitational equation, but Einstein felt that by understanding the physical reason for the equivalence he could gain insight into how to include gravity in relativity theory. Modern physicists have different ways to state the equivalence principle. For example, it is ‘the idea that the physics in an accelerated laboratory is equivalent to that in a uniform gravitational field’, according to Tony Hey and Patrick Walters.

  For the next few years Einstein became obsessed with thoughts of accelerating closed boxes. On a Swiss Alpine hike with Marie Curie, her two daughters and their governess in the summer of 1913, Einstein toiled along crevasses and up steep rocks without seeing either, stopping periodically to discuss science. Once, Eve Curie remembered with amusement, Einstein seized her mother’s arm and burst out: ‘You understand, what I need to know is exactly what happens in a lift when it falls into emptiness.’ At a packed lecture in Vienna the following month,
he entertained an audience of scientists by asking them to imagine two physicists awakening from a drugged sleep to find themselves standing in a closed box with opaque walls but with all their instruments. They would be unable to discover, he said, whether their box was at rest in the Earth’s gravity or was being uniformly accelerated upwards through empty space (in which gravity is taken to be negligible) by some mysterious external force.

  In a similar example, Einstein imagined a small hole in the wall of a lift which is being accelerated upwards by an external force. A light ray enters the lift through the hole. The ray travels to the opposite wall of the lift. But as it does so, the lift moves upwards. The ray therefore meets the opposite wall at a point a little below its point of entrance. For an observer outside the lift, there is no difficulty: the lift is accelerating upwards and so the light ray is bent downwards into a slight curve. (Had the lift been moving uniformly, the ray would have appeared to travel in a straight line.) But for an observer inside the lift who believes that the lift is at rest and that it is gravity that is acting on the lift, the curved ray poses a problem. How can a ray of light be affected by gravity? Well, said Einstein, it must be: ‘A beam of light carries energy and energy has mass’ – as shown in his famous equation E = mc2, derived from his 1905 theory of relativity. ‘But every inertial mass is attracted by the gravitational field, as inertial and gravitational masses are equivalent. A beam of light will bend in a gravitational field exactly as a body would if thrown horizontally with a velocity equal to that of light.’