The Danish Peace Academy
SCIENCE AND SOCIETY
Chapter 14 RELATIVITY - Albert Einstein
Albert Einstein was born in Ulm, Germany, in 1879. He was the son of middle-class, irreligious Jewish parents, who sent him to a Catholic school. Einstein was slow in learning to speak, and at first his parents feared that he might be retarded; but by the time he was eight, his grandfather could say in a letter: “Dear Albert has been back in school for a week. I just love that boy, because you cannot imagine how good and intelligent he has become.”
Remembering his boyhood, Einstein himself later wrote: “When I was 12, a little book dealing with Euclidian plane geometry came into my hands at the beginning of the school year. Here were assertions, as for example the intersection of the altitudes of a triangle in one point, which - though by no means self-evident - could nevertheless be proved with such certainty that any doubt appeared to be out of the question. The lucidity and certainty made an indescribable impression on me.”
When Albert Einstein was in his teens, the factory owned by his father and uncle began to encounter hard times. The two Einstein families moved to Italy, leaving Albert alone and miserable in Munich, where he was supposed to finish his course at the gymnasium. Einstein’s classmates had given him the nickname “Beidermeier”, which means something like “Honest John”; and his tactlessness in criticiz- ing authority soon got him into trouble. In Einstein’s words, what happened next was the following:
“When I was in the seventh grade at the Lutpold Gymnasium, I was summoned by my home-room teacher, who expressed the wish that I leave the school. To my remark that I had done nothing wrong, he replied only, ‘Your mere presence spoils the respect of the class for me’.”
Einstein left gymnasium without graduating, and followed his parents to Italy, where he spent a joyous and carefree year. He also decided to change his citizenship. “The over-emphasized military mentality of the German State was alien to me, even as a boy”, Einstein wrote later. “When my father moved to Italy, he took steps, at my request, to have me released from German citizenship, because I wanted to be a Swiss citizen.”
The financial circumstances of the Einstein family were now precarious, and it was clear that Albert would have to think seriously about a practical career. In 1896, he entered the famous Zürich Polytechnic Institute with the intention of becoming a teacher of mathematics and physics. However, his undisciplined and nonconformist attitudes again got him into trouble. His mathematics professor, Hermann Minkowski (1864-1909), considered Einstein to be a “lazy dog”; and his physics professor, Heinrich Weber, who originally had gone out of his way to help Einstein, said to him in anger and exasperation: “You’re a clever fellow, but you have one fault: You won’t let anyone tell you a thing! You won’t let anyone tell you a thing!”
Einstein missed most of his classes, and read only the subjects which interested him. He was interested most of all in Maxwell’s theory of electromagnetism, a subject which was too “modern” for Weber. There were two major examinations at the Zürich Polytechnic Institute, and Einstein would certainly have failed them had it not been for the help of his loyal friend, the mathematician Marcel Grossman.
Grossman was an excellent and conscientious student, who attended every class and took meticulous notes. With the help of these notes, Einstein managed to pass his examinations; but because he had alienated Weber and the other professors who could have helped him, he found himself completely unable to get a job. In a letter to Professor F. Ostwald on behalf of his son, Einstein’s father wrote: “My son is profoundly unhappy because of his present joblessness; and every day the idea becomes more firmly implanted in his mind that he is a failure, and will not be able to find the way back again.”
From this painful situation, Einstein was rescued (again!) by his friend Marcel Grossman, whose influential father obtained for Einstein a position at the Swiss Patent Office - Technical Expert (Third Class). Anchored at last in a safe, though humble, position, Einstein married one of his classmates, a Serbian girl named Mileva Maric. He learned to do his work at the Patent Office very efficiently; and he used the remainder of his time on his own calculations, hiding them guiltily in a drawer when footsteps approached.
In 1905, this Technical Expert (Third Class) astonished the world of science with five papers, written within a few weeks of each other, and published in the Annalen der Physik. Of these five papers, three were classics: One of these was the paper in which Einstein applied Planck’s quantum hypothesis to the photoelectric effect. The second paper discussed “Brownian motion”, the zig-zag motion of small particles suspended in a liquid and hit randomly by the molecules of the liquid.
This paper supplied a direct proof of the validity of atomic ideas and of Boltzmann’s kinetic theory.
The third paper was destined to establish Einstein’s reputation as one of the greatest physicists of all time. It was entitled On the Electrodynamics of Moving Bodies, and in this paper, Albert Einstein formulated his special theory of relativity.
The theory of relativity grew out of problems connected with Maxwell’s electromagnetic theory of light. Ever since the wavelike nature of light had first been demonstrated, it had been supposed that there must be some medium to carry the light waves, just as there must be some medium (for example air) to carry sound waves. A word was even invented for the medium which was supposed to carry electromagnetic waves: It was called the “ether”.
By analogy with sound, it was believed that the velocity of light would depend on the velocity of the observer relative to the “ether”.
However, all attempts to measure differences in the velocity of light in different directions had failed, including an especially sensitive experiment which was performed in America in 1887 by A.A. Michelson and E.W. Morley.
Even if the earth had, by a coincidence, been stationary with respect to the “ether” when Michelson and Morley first performed their experiment, they should have found an “ether wind” when they repeated their experiment half a year later, with the earth at the other side of its orbit. Strangely, the observed velocity of light seemed to be completely independent of the motion of the observer!
In his famous 1905 paper on relativity, Einstein made the negative result of the Michelson-Morley experiment the basis of a far-reaching principle: He asserted that no experiment whatever can tell us whether we are at rest or whether we are in a state of uniform motion. With this assumption, the Michelson-Morley experiment of course had to fail, and the measured velocity of light had to be independent of the motion of the observer.
Einstein’s Principle of Special Relativity had other extremely important consequences: He soon saw that if his principle were to hold, then Newtonian mechanics would have to be modified. In fact, Einstein’s Principle of Special Relativity required that all fundamental physical laws exhibit a symmetry between space and time. The three space dimensions, and a fourth dimension, ict, had to enter every fundamental physical law in a symmetrical way. (Here i is the square root of -1, c is the velocity of light, and t is time.)
When this symmetry requirement is fulfilled, a physical law is said to be “Lorentz-invariant” (in honor of the Dutch physicist H.A. Lorentz, who anticipated some of Einstein’s ideas). Today, we would express Einstein’s principle by saying that every fundamental physical law must be Lorentz-invariant (i.e. symmetrical in the space and time coordinates). The law will then be independent of the motion of the observer, provided that the observer is moving uniformly.
Einstein was able to show that, when properly expressed, Maxwell’s equations are already Lorentz-invariant; but Newton’s equations of motion have to be modified. When the needed modifications are made, Einstein found, then the mass of a moving particle appears to increase as it is accelerated. A particle can never be accelerated to a velocity greater than the velocity of light; it merely becomes heavier and heavier, the added energy being converted into mass.
From his 1905 theory, Einstein deduced his famous formula equating the energy of a system to its mass multiplied by the square of the velocity of light. As we shall see, his formula was soon used to explain the source of the energy produced by decaying uranium and radium; and eventually it led to the construction of the atomic bomb. Thus Einstein, a lifelong pacifist, who renounced his German citizenship as a protest against militarism, became instrumental in the construction of the most destructive weapon ever invented - a weapon which casts an ominous shadow over the future of humankind.
Just as Einstein was one of the first to take Planck’s quantum hypothesis seriously, so Planck was one of the first physicists to take Einstein’s relativity seriously. Another early enthusiast for relativity was Hermann Minkowski, Einstein’s former professor of mathematics. Although he once had characterized Einstein as a “lazy dog”, Minkowski now contributed importantly to the mathematical formalism of Einstein’s theory; and in 1907, he published the first book on relativity. In honor of Minkowski’s contributions to relativity, the 4-dimensional space-time continuum in which we live is sometimes called “Minkowski space”.
In 1908, Minkowski began a lecture to the Eightieth Congress of German Scientists and Physicians with the following words:
“ From now on, space by itself, and time by itself, are destined to sink completely into the shadows; and only a kind of union of both will retain an independent existence.”
Gradually, the importance of Einstein’s work began to be realized, and he was much sought after. He was first made Assistant Professor at the University of Zürich, then full Professor in Prague, then Professor at the Zürich Polytechnic Institute; and finally, in 1913, Planck and Nernst persuaded Einstein to become Director of Scientific Research at the Kaiser Wilhelm Institute in Berlin. He was at this post when the First World War broke out.
While many other German intellectuals produced manifestos justifying Germany’s invasion of Belgium, Einstein dared to write and sign an anti-war manifesto. Einstein’s manifesto appealed for cooperation and understanding among the scholars of Europe for the sake of the future; and it proposed the eventual establishment of a League of Europeans. During the war, Einstein remained in Berlin, doing whatever he could for the cause of peace, burying himself unhappily in his work, and trying to forget the agony of Europe, whose civilization was dying in a rain of shells, machine-gun bullets, and poison gas.
The work into which Einstein threw himself during this period was an extension of his theory of relativity. He already had modified Newton’s equations of motion so that they exhibited the space-time symmetry required by his Principle of Special Relativity. However, Newton’s law of gravitation remained a problem. Obviously it had to be modi- fied, since it was not Lorentz-invariant; but how should it be changed?
What principles could Einstein use in his search for a more correct law of gravitation? Certainly whatever new law he found would have to give results very close to Newton’s law, since Newton’s theory could predict the motions of the planets with almost perfect accuracy. This was the deep problem with which he struggled.
In 1907, Einstein had found one of the principles which was to guide him - the Principle of Equivalence of inertial and gravitational mass. After turning Newton’s theory over and over in his mind, Einstein realized that Newton had used mass in two distinct ways: His laws of motion stated that the force acting on a body is equal to the mass of the body multiplied by its acceleration; but according to Newton, the gravitational force on a body is also proportional to its mass.
In Newton’s theory, gravitational mass, by a coincidence, is equal to inertial mass; and this holds for all bodies. Einstein wondered - can the equality between the two kinds of mass be a coincidence? Why not make a theory in which they necessarily have to be the same? He then imagined an experimenter inside a box, unable to see anything outside it. If the box is on the surface of the earth, the person inside it will feel the pull of the earth’s gravitational field. If the experimenter drops an object, it will fall to the floor with an acceleration of 32 feet per second per second. Now suppose that the box is taken out into empty space, far away from strong gravitational fields, and accelerated by exactly 32 feet per second per second. Will the enclosed experimenter be able to tell the difference between these two situations? Certainly no difference can be detected by dropping an object, since in the accelerated box, the object will fall to the floor in exactly the same way as before.
With this “thought experiment” in mind, Einstein formulated a general Principle of Equivalence: He asserted that no experiment whatever can tell an observer enclosed in a small box whether the box is being accelerated, or whether it is in a gravitational field. According to this principle, gravitation and acceleration are locally equivalent, or, to say the same thing in different words, gravitational mass and inertial mass are equivalent.
Einstein soon realized that his Principle of Equivalence implied that a ray of light must be bent by a gravitational field. This conclusion followed because, to an observer in an accelerated frame, a light beam which would appear straight to a stationary observer, must necessarily appear very slightly curved. If the Principle of Equivalence held, then the same slight bending of the light ray would be observed by an experimenter in a stationary frame in a gravitational field. Another consequence of the Principle of Equivalence was that a light wave propagating upwards in a gravitational field should be very slightly shifted to the red. This followed because in an accelerated frame, the wave crests would be slightly farther apart than they normally would be, and the same must then be true for a stationary frame in a gravitational field. It seemed to Einstein that it ought to be possible to test experimentally both the gravitational bending of a light ray and the gravitational red shift.
This seemed promising; but how was Einstein to proceed from the Principle of Equivalence to a Lorentz-invariant formulation of the law of gravitation? Perhaps the theory ought to be modeled after Maxwell’s electromagnetic theory, which was a field theory, rather than an “action at a distance” theory. Part of the trouble with Newton’s law of gravitation was that it allowed a signal to be propagated instantaneously, contrary to the Principle of Special Relativity. A field theory of gravitation might cure this defect, but how was Einstein to find such a theory? There seemed to be no way.
From these troubles Albert Einstein was rescued (a third time!) by his staunch friend Marcel Grossman. By this time, Grossman had become a professor of mathematics in Zürich, after having written a doctoral dissertation on tensor analysis and non-Euclidian geometry - the very things that Einstein needed. The year was 1912, and Einstein had just returned to Zürich as Professor of Physics at the Polytechnic Institute. For two years, Einstein and Grossman worked together; and by the time Einstein left for Berlin in 1914, the way was clear. With Grossman’s help, Einstein saw that the gravitational field could be expressed as a curvature of the 4-dimensional space-time continuum.
The mathematical methods appropriate for describing the curvature of a many-dimensional space had already been developed in the early 19th century by Nickolai Ivanovich Lobachevski (1793-1856), Karl Friedrich Gauss (1777-1855) and Bernard Riemann (1826-1866). As an example of a curved space, we might think of the 2-dimensional space formed by the surface of a sphere. The geometry of figures drawn on a sphere is non-Euclidian: Parallel lines meet, and the angles of a triangle add up to more than 180 degrees. Non-Euclidian spaces of higher dimension are hard to visualize, but they can be treated mathematically.
Gauss and Riemann had introduced a “metric tensor” which contained all the necessary information about the curvature of a non- Euclidian space; and Einstein saw that this metric tensor could be used to express the gravitational field. The orbits of the planets became “geodesics” in curved space. A geodesic is the shortest distance between two points, but in the curved space-time continuum of Einstein’s theory, the geodesics were not straight lines.
By 1915, working by himself in Berlin, Einstein was able to show that the simplest theory of this form yielded Newton’s law of gravitation as a first approximation, and in a higher approximation, it gave the correct movement of the perihelion of the orbit of Mercury. It had long been known that Mercury’s point of closest approach to the sun (its perihelion) drifted slowly forward at the rate of between 40 and 50 seconds of arc per century. Einstein calculated that the change of Mercury’s perihelion each century should be 43 seconds of arc. In January, 1916, he wrote to his friend Paul Ehrenfest:
“Imagine my joy at the feasibility of the general covariance, and at the result that the equations yield the correct perihelion of mercury. I was beside myself with ecstasy for days.”
In 1919, a British expedition, headed by Sir Arthur Eddington, sailed to a small island off the coast of West Africa. Their purpose was to test Einstein’s prediction of the bending of light in a gravitational field by observing stars close to the sun during a total eclipse. The observed bending agreed exactly with Einstein’s predictions; and as a result he became world-famous.
The general public was fascinated by relativity, in spite of the abstruseness of the theory (or perhaps because of it). Einstein, the absentminded professor, with long, uncombed hair, became a symbol of science. The world was tired of war, and wanted something else to think about.
Einstein met President Harding, Winston Churchill and Charlie Chaplin; and he was invited to lunch by the Archbishop of Canterbury. Although adulated elsewhere, he was soon attacked in Germany. Many Germans, looking for an excuse for the defeat of their nation, blamed it on the pacifists and Jews; and Einstein was both these things.
The mass defect
Albert Einstein’s famous relativistic formula, relating energy to mass, soon yielded an understanding of the enormous amounts of energy released in radioactive decay. Marie and Pierre Curie had noticed that radium maintains itself at a temperature higher than its surroundings. Their measurements and calculations showed that a gram of radium produces roughly 100 gram-calories of heat per hour.
This did not seem like much energy until Rutherford found that radium has a half-life of about 1,000 years. In other words, after a thousand years, a gram of radium will still be producing heat, its radioactivity only reduced to one-half its original value. During a thousand years, a gram of radium produces about a million kilocalories - an enormous amount of energy in relation to the tiny size of its source!
Where did this huge amount of energy come from? Conservation of energy was one of the most basic principles of physics. Would it have to be abandoned?
The source of the almost-unbelievable amounts of energy released in radioactive decay could be understood through Einstein’s formula equating the energy of a system to its mass multiplied by the square of the velocity of light, and through accurate measurements of atomic weights. Einstein’s formula asserted that mass and energy are equivalent. It was realized that in radioactive decay, neither mass nor energy is conserved, but only a quantity more general than both, of which mass and energy are particular forms.
The quantitative verification of the equivalence of mass and energy depended on very accurate measurements of atomic weights. Until 1912, the atomic weights of the elements were a puzzle. For some elements, the weights were very nearly integral multiples of the atomic weight of hydrogen, in units of which carbon was found to have an atomic weight almost exactly equal to 12, while nitrogen, oxygen and sodium were respectively 14, 16 and 23. This almost exact numerical correspondence made the English chemist, William Prout (1785-1850), propose that hydrogen might be the fundamental building-block of nature, and that atoms of all elements might be built up out of hydrogen. Prout’s hypothesis was destined to be killed several times, and revived several times. It was soon discovered that many elements have atomic weights which are not even nearly integral multiples of the weight of hydrogen. This discovery killed Prout’s hypothesis for the first time. However, through their studies of radioactive decay, Rutherford and Soddy discovered isotopes; and isotopes revived Prout’s hypothesis.
Rutherford and Soddy demonstrated that in the decay of uranium to its final product, lead, a whole chain of intermediates is involved, all of them radioactive, and each one changing spontaneously to the next. But what elements could these intermediate links of the decay chain be? After all, among the known elements, only uranium, polonium, radium, actinium and thorium were radioactive - and one could show that these elements could not represent all the intermediates of the Rutherford-Soddy decay chain.
In 1912, in Rutherford’s Manchester laboratory, a young chemist named Georg von Hevesy was trying to separate by chemical means two radioactive decay products known to be different from each other because their half-lives were different. But no matter what he tried, von Hevesy could not separate them. All chemical methods failed. Hevesy discussed his troubles with Niels Bohr, who suggested that the two decay products might be atoms with the same nuclear charges, but different atomic weights. Since the number of electrons was determined by the nuclear charge, and since the chemical properties were determined by the number of electrons, it would be impossible to separate the two decay products by chemical means. They were, in fact, different varieties of the same element.
The same idea occurred simultaneously and independently to Frederick Soddy. In the autumn of 1912, he published a detailed paper explaining the concept, and introducing the word “isotope”. Each chemical element, Soddy explained, is a mixture of isotopes. For those elements whose atomic weight is nearly an integral multiple of the atomic weight of hydrogen, a single isotope dominates the mixture. All the isotopes of a given element have the same nuclear charge (atomic number) and the same number of electrons; but two different isotopes of the same element have different atomic weights and different nuclear properties, some isotopes being radioactive, while others are stable. When a nucleus emits a beta-particle (a high-speed electron carrying one unit of negative charge, but very little mass), the weight of the nucleus is almost unchanged, but its charge increases by one unit. Therefore beta-decay produces a product which is one place higher in the periodic table than its parent.
In alpha-decay, on the other hand, a helium ion, with two units of positive charge, and four units of mass, is thrown out of the decaying nucleus. Therefore, in alpha-decay, the product is two places lower in the periodic table, and four atomic mass units lighter than the parent atom.
The concept of isotopes allowed Frederick Soddy to identify clearly all the intermediate links in the decay chains which he and Rutherford had studied; and he later received the Nobel Prize in Chemistry for his work. Georg von Hevesy became the first scientist to use radioactive isotopes as tracers in biochemistry; and he also received the Nobel Prize in Chemistry.
Meanwhile, at the Cavendish Laboratory in Cambridge, J.J. Thomson and his student, Francis Aston (1877-1945), developed a “massspectrograph” - an instrument which could separate isotopes from one another by accelerating them with both electric and magnetic fields. In Aston’s hands, the mass spectrograph became a precision instrument. Using it, he could not only separate isotopes from one another - he could also measure their masses very accurately. He found these masses to be almost exactly integral multiples of the mass of a hydrogen atom, but not quite! There was always a little mass missing!
The explanation for the missing mass - the mass defect - was found through Prout’s hypothesis (newly revived) and Einstein’s formula relating mass to energy. The nucleus of an atom was visualized as being composed of hydrogen nuclei (protons) and electrons bound tightly together. The mass defect, through Einstein’s formula, was equivalent to the energy which would be needed to separate these elementary particles. By observing the mass defects of isotopes, one could calculate their binding energies; and from these, the vast amounts of energy available for release through nuclear transmutation could also be calculated.
For the first time, humans realized the enormous power which was potentially available in the atomic nucleus.
Suggestions for further reading
1. Paul Arthur Schlipp (editor), Albert Einstein:
Philosopher-Scientist, Open Court Publishing Co., Lasalle Illinois