Einstein's Masterwork

by John Gribbin

  This brings us to the idea of reference frames. A reference frame is the place you make measurements from, like the laboratory in the previous example. We know that the lab is actually being carried along with the Earth’s motion, but we can treat it as if it were at rest. All the laws of mechanics (Newton’s laws) work perfectly in the lab. Those same laws also work perfectly in a frame of reference moving at a constant velocity relative to the lab – for example, in an aircraft flying at a smooth and steady speed, or, in the example Einstein always favoured, in a train rolling smoothly along a track at a constant speed in a straight line (that is, at constant velocity). Such frames, in which Newton’s laws work perfectly, are now called ‘inertial frames’.

  The first person to spell most this out, long before the days of trains and planes, was Galileo Galilei, in the 17th century; his insights were part of the foundations of Newton’s work. As far as the physical behaviour of matter and Newton’s laws are concerned, by 1905 it had been known for hundreds of years that there is no difference in how things behave if your frame of reference is a lab in an immovable building or another lab in another frame of reference moving steadily at a constant velocity relative to the first lab. Newton himself believed that there must be an absolute ‘standard of rest’ in the Universe against which all motion could be measured, and this tied in with the idea of the ether. But no experiment involving Newton’s laws could ever detect motion relative to this hypothetical absolute rest frame.

  Now, in 1905, Einstein had realised that no experiment involving Maxwell’s equations could ever detect motion relative to this hypothetical absolute rest frame, either. So there was no need for the ether. Referring to the way electricity is generated by the relative motion of a wire and a magnet, he continued:

  Examples of this sort, together with the unsuccessful attempts to detect a motion of the earth relative to the ‘light medium’, lead to the conjecture that not only the phenomena of mechanics but also those of electrodynamics have no properties that correspond to the concept of absolute rest. Rather, the same laws of mechanics and optics will be valid for all coordinate systems in which the equations of mechanics hold.

  In this context, ‘coordinate system’ means the same as ‘reference frame’. Einstein is saying that all of the laws of physics, both for mechanics and electrodynamics, are the same in any reference frame moving at constant velocity relative to any other reference frame in which those laws apply. There is no special reference frame which can be regarded as at rest in an absolute sense. He called this his first postulate, which, echoing Poincaré, he named the ‘principle of relativity’.

  The second postulate is even simpler, and comes straight from Maxwell’s equations:

  Light always propagates in empty space with a definite velocity V that is independent of the state of motion of the emitting body.

  Einstein based this postulate on Maxwell’s equations, not on the experiments that tried to detect the effect of the Earth’s motion on the measured speed of light. As he realised, all that could be said about those experiments was that they had ‘failed to detect’ any movement of the Earth relative to the hypothetical ether, and failed to measure any change in the speed of light. But there could still be effects too small for the experimenters to have measured. The postulate, though, was precise and unequivocal – as were the results he obtained from these two disarmingly simple assumptions.

  The other key ingredient of the Special Theory is a direct consequence of the second postulate. For objects moving at constant velocity relative to one another, each object can be regarded as carrying its own reference frame (coordinate system) along with it. Einstein’s theory had to deal with the relationship between coordinate systems, clocks and electromagnetic processes. And the first thing he had to do to develop this theory was to spell out the way the second postulate affects our ideas about time.

  In everyday life, we all have an idea in our heads that it is the same ‘time’ everywhere at once. But what does this really mean? If some master clock in London sends out a time signal by radio at noon, and I have a clock which receives that radio signal and automatically sets itself to noon, my clock will actually be a fraction of a second behind the clock in London, because it takes radio waves, travelling at the speed of light, a certain amount of time to reach my clock. As long as I don’t move my clock, and I know how far it is to London, I can get round this problem by building in an allowance for the time taken by the radio waves to travel to the clock. Since light travels at a constant speed, the only completely accurate way to do this is to send a light signal to London and back, time how long the return journey takes and divide by two to work out the difference. But what if this process is watched by a moving observer?

  The first postulate says that this observer is entitled to regard himself as stationary, with the clocks moving past him at a constant velocity. He will see the calibrating light pulse from the clock head off to London and bounce back, but the return journey will be a different distance in his frame of reference because the whole moving coordinate system will have shifted forward while the light was on its journey. So the ‘stationary’ observer and the ‘moving’ observer do not agree on the distance between the clock and London, and they do not agree on what ‘the same time’ means. In other words, measurements of both time and space are relative – they depend on how the person who makes them (the observer) is moving, relative to the things they are measuring. Remember, also, that the second postulate says that each observer will see all light pulses, whether they originate in his own frame of reference or the other one, moving at the same speed. Putting the appropriate mathematics into all of this led Einstein to find a system of equations that could be used to transform the measurements made by one observer into the equivalent measurements in any other coordinate frame moving at constant velocity relative to that observer.o These coordinate transformations are exactly the same as the equations found by Lorentz a year earlier, although Einstein had not read Lorentz’s 1904 paper in 1905. But there is a huge difference in the interpretation of those equations.

  Lorentz had found a system of equations that worked, but which were solely based on the need to explain the experimental failure to measure the motion of the Earth relative to the ether. But he still thought in terms of the ether, and there was no underlying principle shoring up his equations. This is a little like the way Planck came up with an equation for the black-body curve that fitted the curve but had no foundation in terms of an underlying principle. By contrast, as he had with the black-body radiation, Einstein started out from first principles and proved that the world really must work in accordance with the equations.

  The best example of how different Einstein’s view of the world was even to that of Lorentz and Poincaré is that he understood the importance of the symmetry in what we still call the Lorentz transformations. Poincaré had noticed that the transformations are symmetrical. If one observer sees a moving object shrunk in the direction of its motion, then the symmetry implies that an observer riding with the moving object sees his own frame of reference as perfectly normal, but the first observer and everything else in his coordinate system shrunk. Poincaré dismissed this as a quirk of the equations, with no physical significance – it made no sense to him to turn his world view around and think of the ether being ‘shrunk’ as it moved past us. But Einstein, who had no need of the ether, saw the symmetry in the transformations as a fundamental truth of profound physical significance.

  Einstein proved that an observer in one inertial frame would perceive objects in a different inertial frame shrunk in the direction of their motion relative to him, and he would see clocks in the other inertial frame running more slowly than clocks in his own inertial frame. An observer in the other inertial frame would see the mirror image of this – he would see the first observer’s clocks running slow, and the first observer’s rulers and other equipment (and, indeed, the first observer) shrunk. All of this has now been confirmed by experiments. The e
ffects are very small unless the relative velocities involved are a sizeable fraction of the speed of light (which is why we don’t notice them in everyday life and they are not common sense), but particles are routinely observed travelling at such speeds in ‘atom smashing’ machines like those at CERN, in Geneva, and Fermilab, in Chicago. The Special Theory has been proved accurate time and time again. It also makes one further prediction, which Einstein himself didn’t spot in June 1905, but which he quickly realised and wrote about in another paper, which was a kind of footnote to the paper on the Special Theory. That realization led to the most famous equation in science – although, disappointingly, the equation does not appear in its familiar form in the paper itself.

  In the summer of 1905, Einstein wrote a letter to Conrad Habicht in which he said:

  One more consequence of the paper on electro-dynamics has also occurred to me. The principle of relativity, in conjunction with Maxwell’s equations, requires that mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable decrease in mass should occur in the case of radium. The argument is amusing and seductive; but for all I know the Lord might be laughing over it and leading me around by the nose.11

  Einstein mentions radium because this archetypal radioactive element emits energy in the form of radiation and heat all the time. The origin of this energy had been a puzzle for science ever since the discovery of radium by Marie and Pierre Curie at the end of the 1890s; Einstein’s discovery implied that the matter radium was made of was slowly being converted into energy, so that the radium itself would gradually lose mass. There had previously been suggestions that electromagnetic energy might be associated with mass, and even that the electron’s mass might be completely attributed to its electromagnetic field. But Einstein was suggesting something different: that all matter had an energy equivalent, an energy that might in principle, be liberated; and he calculated a precise value for this energy.

  His little paper pointing this out (just three pages long in its printed form) was received by the Annalen der Physik on 27 September. It was published before the end of the year, but in the next volume of the Annalen (volume 18) from the volume containing the three great papers on Brownian motion, light quanta and the Special Theory (volume 17, now a valuable collectors’ item). Still using V to denote the velocity of light, in his own words Einstein concluded:

  If a body emits the energy L in the form of radiation, its mass decreases by L/V2. Here it is obviously inessential that the energy taken from the body turns into radiant energy, so we are led to the more general conclusion:

  The mass of a body is a measure of its energy content; if the energy changes by L, the mass changes in the same sense by L/9 × 1020 if the energy is measured in ergs and the mass in grams.

  In those units, the speed of light is 3 × 1010 cm per second, and 9 × 1020 is the speed of light squared. Putting E for the energy rather than L, c for the speed of light rather than V, m for mass, and re-arranging the equation slightly to have the energy, rather than the mass, on the left hand side, the equation Einstein discovered becomes:

  E = mc2

  The only equation that everybody knows.

  Like all of the results from the papers written by Einstein in his annus mirabilis, this prediction has since been amply confirmed by experiment – not least the awesome ‘experiment’ of the nuclear bomb. It is now understood that the conversion of mass into energy provides the energy source which keeps the Sun and stars shining, and is therefore the ultimate source of the energy on which life on Earth depends. Which makes this little paper just about the most important ‘footnote’ in scientific history.

  As I hope I have made clear, however, all of the work that Einstein produced in 1905 was of its time. The statistical ideas which underpinned the doctoral thesis and the work on Brownian motion were a significant step forward, but still part of the mainstream of the investigation of atoms and molecules. The light quantum paper jumped off from the work of Max Planck, and also used statistical ideas from thermodynamics. Even the paper on the Special Theory – although strikingly different in its foundations from the approach used by Lorentz and Poincaré – came up with the same transformation equations, and it is not unlikely that something similar to the Special Theory would soon have emerged from the line of thinking pioneered by Poincaré himself. In isolation, each contribution was something that an individual physicist at the height of his powers might have been proud of, as the biggest achievement of his career – even the relatively mundane PhD paper, since to many physicists a PhD is the greatest achievement of their academic lives.

  What made Einstein so special, and the annus mirabilis so miraculous, was that all four pieces of work were produced by the same young man, working outside the mainstream of scientific life, in his spare time, while holding down a demanding job at the patent office which required his attendance there six days a week for eight hours a day. Not to mention the inevitable disturbance caused by a year-old baby back at his apartment. And it wasn’t as if it really took him a year to do all this. Although, admittedly, a lot of prior thought had gone in to all this work, the four great papers were actually written between March and June 1905, at a rate of one a month, and the E = mc2 paper was finished just three months later, at the end of September.

  It would take a little while for the importance of all this to sink in, and for the true genius of Einstein to be appreciated. Indeed, he even stayed at the patent office (partly by choice) for another four years before at last becoming a university professor at the age of 30. He would also continue to make major contributions to physics until he was in his forties, a remarkably long time for any theoretical physicist. The rest of his life, and his place in history, would be forever coloured by that outburst of creativity in 1905; but it would be ten years before, in 1915, he produced something really special (in the everyday sense of the term!): a theory that was very much not of its time, and which in all probability would not otherwise have emerged for decades, if at all in the form that we know it. Even for Einstein, though, the road to the General Theory was, as we shall see, far from straightforward.

  Footnotes

  a The story that in later life he gave up wearing socks so as not to have the bother of finding clean ones to wear is true, and typical.

  b This number is called Avogadro’s ‘constant’ today, but we shall stick with the name familiar in Einstein’s day.

  c Remember that what matters is not so much the number on the front, 0.5 or 2.1, as the agreement of the number of ‘powers of ten’ in the exponent, 23.

  d You sometimes see slightly different versions of the titles of Einstein’s papers. The titles and quotes from Einstein’s papers that are used here, are taken from Stachel, which is the most accessible source.

  e All of this works for other solutions as well, of course, but we shall stick with sugar and water because that was the example Einstein used in his dissertation.

  f A white-hot object looks white because the peak of its spectrum is in the middle of the rainbow of colours; so it radiates all the colours, which combine to make white light. If it were even hotter, it would look blue, as some stars do.

  g His equation was later refined by James Jeans and became known as the Rayleigh-Jeans law.

  h Incidentally, this is exactly what astronomers do now ‘see’, using sensitive detectors called charge-coupled devices, when they point their telescopes towards the faintest and most distant objects in the Universe. They can literally count the photons arriving one by one.

  i Deliciously, while J.J. Thomson received the Nobel Prize for ‘proving’ that electrons are particles, his son George received the Nobel Prize for ‘proving’ that electrons are waves. They were both right.

  j The genesis of the Special Theory was described by Einstein in a lecture in Japan, in 1922; the lecture was reprinted in Physics Today in August 1982.

  k Of course, the ‘Special Theory’ paper was not known by that name at the ti
me; Einstein introduced the name in 1915, to distinguish it from his General Theory of Relativity. But I will use the name, since, as with the ‘Brownian motion’ paper, we have the benefit of hindsight. I emphasise that ‘Special’ in this context means the theory is a ‘special case’ dealing only with objects travelling at constant velocities; the General Theory deals with accelerations as well. But I reiterate that it is always ‘Special Theory of Relativity;’ there is no such thing as the ‘theory of special relativity,’ since it is the theory that is ‘special’ not the relativity!

  l Indeed, for some time after his paper had been published on the other side of the Atlantic, Fitzgerald himself didn’t know that it had appeared in print.

  m Again, with a different meaning from what ‘ion’ means to a scientist today.

  n Shortly before he died, Einstein told his biographer Carl Seelig that in 1905 he knew about Lorentz’s work of 1895, but not about his later work or Poincaré’s contributions. This may be an exaggeration, but he probably had not actually read Lorentz’s 1904 paper, since the Proceedings of the Amsterdam Academy were not exactly easy to get hold of in Bern.

  o I do not have space to go through the argument in detail here; the easiest way to understand what is going on is provided by Lewis Epstein in his book Relativity Visualized.

  3

  The Long and Winding Road

  The geometry of relativity; Moving on; In the shadow of a giant; On the move; First steps; What Einstein should have known; The masterwork

  The work Einstein published in his annus mirabilis didn’t immediately set the scientific world on fire, but it was noticed and drew him into correspondence with a widening circle of physicists, many of whom were astonished to find that he was a junior patent officer and not a professor at the University of Bern. In May 1906, he wrote in a letter to Lenard that: ‘My papers are meeting with much acknowledgement and are giving rise to further investigations. Prof. Planck [Berlin] wrote me about it recently.’1 The paper Planck was particularly interested in was not, however, the one on light quanta, about which he had reservations, but the Special Theory of Relativity, of which he was an early and enthusiastic champion.