CONTENT
Written 1916
Translated: Robert W. Lawson (Authorised translation)


Preface
Part I: The Special Theory of Relativity 01. Physical Meaning of Geometrical Propositions
02. The System of Co-ordinates
03. Space and Time in Classical Mechanics
04. The Galileian System of Co-ordinates
05. The Principle of Relativity (in the Restricted Sense)
06. The Theorem of the Addition of Velocities employed in Classical Mechanics
07. The Apparent Incompatability of the Law of Propagation of Light with the Principle of Relativity
08. On the Idea of Time in Physics
09. The Relativity of Simultaneity
10. On the Relativity of the Conception of Distance
11. The Lorentz Transformation
12. The Behaviour of Measuring-Rods and Clocks in Motion
13. Theorem of the Addition of Velocities. The Experiment of Fizeau
14. The Hueristic Value of the Theory of Relativity
15. General Results of the Theory


16. Experience and the Special Theory of Relativity
17. Minkowski's Four-dimensial Space
Part II: The General Theory of Relativity 18. Special and General Principle of Relativity
19. The Gravitational Field
20. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity
21. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?
22. A Few Inferences from the General Principle of Relativity
23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
24. Euclidean and non-Euclidean Continuum
25. Gaussian Co-ordinates
26. The Space-Time Continuum of the Speical Theory of Relativity Considered as a Euclidean Continuum
27. The Space-Time Continuum of the General Theory of Realtivity is Not a Euclidean Continuum
28. Exact Formulation of the General Principle of Relativity
29. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
Part III: Considerations on the Universe as a Whole


30. Cosmological Difficulties of Newton's Theory
31. The Possibility of a "Finite" and yet "Unbounded" Universe
32. The Structure of Space According to the General Theory of Relativity
Appendices: 01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
03. The Experimental Confirmation of the General Theory of Relativity
04. The Structure of Space According to the General Theory of Relativity (sup. ch 32)
05. Relativity and the Problem of Space Note: The fifth appendix was added by Einstein at the time of the fifteenth re-printing of this book; and as a result is still under copyright restrictions so cannot be added without the permission of the publisher.


Preface
(December, 1916)
The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a "stepmotherly" fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for the trees. May the book bring some one a few happy hours of suggestive thought!
December, 1916
A. EINSTEIN
Introduction by Nigel Calder


INTRODUCTION “What really interests me is whether God had any choice in the creation of the world.” At the centenary of Albert Einstein’s birth, in 1979, I scripted that comment of his into a TV documentary about relativity. Some broadcasters expunged it from the soundtrack because they thought it was blasphemous. In truth it was a reverential remark, and the question remains a shrewd one. The relativity theories explained by Einstein in this book supply some of the basic rules that any properly functioning universe must obey. They make sure that atoms and natural forces will behave in the same way anywhere, across the great oceans of space and time. A supplementary rule helps to explain how stars and life can go on running for billions of years. When all the rules are known, will it turn out that only one possible kind of universe can be both self-consistent and con-genial for life? For any young would-be Einstein of today, the question is still on the table. And as the maestro commented on another occasion, to persevere with such difficult trains of thought requires feelings like those of “a religious person or a lover.” For you, the inquisitive reader, a sense of trying to read the mind of Godor to chat up Mother Nature, if you prefer—is an encouragement to accompany Einstein through the forest of tricky ideas contained in this slim volume. You will bear with him even when, from time to time, he uses a little high school mathematics to consolidate the reasoning. The payoff is worth the mental effort. Addressing you person-to-person, Einstein certainly wants viii introduction you to join him in his intellectual adventure. Gratuitous mystification and hero worship, which sometimes contaminate other people’s accounts of relativity, are absent here. As a writer, Einstein takes great pains to find examples and analogies to explain his points, but he pursues no prizes for his prose. He shares the opinion of the physicist Ludwig Boltzmann who said that, “Matters of elegance ought to be left to the tailor and to the cobbler.” When he wrote this book in German in 1916, Einstein’s name was scarcely known outside the physics institutes. He had just completed his masterpiece, the general theory of relativity. It provided a brand-new theory of gravity and it promised a new per-spective on the cosmos as a whole. He set out at once to share his excitement with as wide a public as possible. But


World War I was raging at the time and English-speaking countries scorned all things German. After the war, two British expeditions to the South Atlantic observed the total eclipse of the sun in May 1919. The astronomers photographed stars shifting in the sky, in a way that was said to support the Berlin professor’s outlandish ideas. Newton’s law of gravity was apparently out of date. Announced in London on November 6, 1919, the news made Einstein a celebrity overnight. “The typhoon of publicity crossed the Atlantic,” Ernest Rutherford noted. As the discoverer of the atomic nucleus, he was a rival for fame. Robert W. Lawson, a British physicist who had polished his knowledge of German while a prisoner of war in Austria, translated this book into English. He secured Einstein’s blessing for the book’s publication in 1920, and the eclipse results were included in an appendix. So why not engage in a little time travel? Imagine that it’s the aftermath of World War I. The usual method of longrange travel is still by railroad train. The U.S. government has prohibited alcohol, and bootleggers are admiring the newly developed Thompson submachine gun. Al Jolson and his song “Swanee” are all the rage. And that German chap’s account of how he upstaged the great Isaac Newton is here in your hands.
EINSTEIN’S FIRST THIRTY-SEVEN YEARS The transition from relative obscurity to a prominence unmatched by movie stars, signaled by the translation of this book, is also a moment to reflect on Einstein’s life till then. Born in Ulm in Germany in 1879, Albert was the son of an entrepreneurial electrician (seldom successful) and a musicallyminded mother. In the following year the family moved to Munich. Albert’s budding brain was enthused by a magnet given to him at the age of five, and by a book on Euclid’s “holy geometry” as he called it, when he was twelve. From sixteen onward, Einstein studied in Switzerland. He was brilliant and negligent by turns and often preoccupied with his own thoughts. In 1900, aged twentyone, he obtained a diploma as a science teacher from the Federal Institute of Technology in Zurich, but then had great difficulty securing a job. His private life was in a mess. Fellow student Mileva Maric


was pregnant with his child, Einstein’s parents opposed a marriage, and anyway he was broke. Not until the summer of 1902 did Einstein secure a permanent job, as a technical officer at the Swiss Patent Office. He married Mileva, and they settled in Bern. Their first child disappeared mysteriously—she was presumably adopted and/or died—but in 1904 the first of two sons was born. After a solid day’s work at the office, evaluating all sorts of inventions, Einstein would spend his spare time on fundamental physics, working at the table in the family apartment. The resulting papers went to the Annalen der Physik, which tolerantly published his ideas. He also wrote many review articles for that journal, on other people’s physics. Einstein was neither a competent experimenter nor a high-powered mathematician. His intuition about scientific concepts was unequaled, and when logic was on his side he would stick his neck far out, even when his conclusions ran counter to the received wisdom. A reinterpretation of the photoelectric effect was his first spectacular contribution, proving that light can behave as if it consists of particles, not waves. That was the subject of a paper written in March 1905, shortly before his twenty-sixth birthday. During the next few weeks Einstein completed two groundbreaking papers on molecular physics. In May he solved a puzzle about the speed of light that had taunted him for years, and before the end of June he had sent his first paper on relativity to Annalen der Physik. Three months later he followed it with a related paper on the equivalence of energy and mass. No wonder 1905 is called, in retrospect, Einstein’s miracle year. Yet the academic world was extremely slow to react. Not until 1909 was he able to give up his job at the Patent Office, on becoming an assistant professor at Zurich University. Thereafter his career took off, with a quick succession of posts in other universities culminating in a very prestigious appointment in Berlin, where he settled in 1914. All the while he was struggling to generalize his ideas about relativity. The special theory of 1905 dealt with conflicting views of the world that result from relative movements at a steady speed. If he could extend it to accelerated motions, a new theory of gravity would be in his grasp. He had the right ideas from 1907 onward, but the mathematics was so tricky that the general theory of relativity was not perfected until 1915.


Just as he was beginning to apply it to the overall nature of the universe, he spared the time needed to write the present book. He was then age thirtyseven.
SPACE, TIME, AND LIGHT ALONG A RAILROAD Einstein starts his book by asking whether Euclid’s geometry is true. The Greek compendium about how lines and shapes relate to one another, on a flat surface, has been the bedrock of practical mathematics for two thousand years. Yet it is definitely correct only in abstract logic. In the real world, so Einstein warns us, its truth may turn out to be limited. This is a distant, ranging shot, and Section 1 may puzzle some readers, because Einstein does not return to Euclid’s vulnerability until halfway through the book. Its early appearance means that the author wants to shock us into thinking about things in his way. Euclid not necessarily correct? That was fighting talk in 1920. Nowadays scientists can directly test geometry across the solar system, because radar echoes from planets, and signals to and from spacecraft, can measure relative distances. The old rules about triangles, for example, don’t work exactly. Rather than complain that Euclid was wrong, the scientists now prefer to say that the empty space in the solar system is not perfectly “flat” in Euclid’s sense. The way gravity deforms space and time is what general relativity is all about. But first Einstein leads us into special relativity and the effects of motionspecial because it excludes accelerated move-ment. In Sections 2 to 4 he erects the traditional scaffolding of the physical world, which his theories are going to shake, namely the system of coordinates used for pinpointing events and tracing movements. There are three dimensions of space x, y, and z (meaning left-right, forward-back, and up-down) and one dimension of time t. Events can look very different to onlookers in different situa-tions, especially if they are moving relative to one another. In Section 3, Einstein introduces the railroad (the “railway” to the British translator) that sets the scene for his reasoning for many pages that follow. The first example of different viewpoints comes when I drop a stone from the window of a moving train. I see the stone go straight down from my hand to the ground. If you watch from the side of the track, you’ll see it following a curved (parabolic) path, because the stone inherits some sideways motion from the train.


In Section 4, Einstein equips the observers in the train and by the track with clocks, so that each has a complete coordinate system—a personal frame of reference in space and time. So far so obvious, but in Section 5 Einstein gives a preliminary hint that the world will get out of joint when light comes into the story. That is the meaning of his reference to developments in “electrodynamics and optics.” Approaching the tricky bit gradually, Einstein reasons that the laws of nature ought not to depend on who is watching. A raven flying in a straight line will appear to be going straight to an observer in a moving train, even though he’ll reckon the bird’s course and speed differently from a trackside observer. Similarly the laws of nature observed on the earth don’t change between winter and summer, even though our planet reverses its direction of travel around the sun as it hurries along in its orbit at 30 kilometers per second. A century after Einstein formulated these ideas, the absence of any seasonal variation in physical laws has been checked to the highest precision for which you could ever wish. Next, Einstein turns to the way speeds can be combined. In Section 6 a man walks forward along a moving train. An onlooker beside the track may reckon how fast the man is advancing by simply adding his speed of walking to the train’s velocity, but that will turn out to be an oversimplification. The first clue comes in Section 7, when Einstein imagines a beam of light being sent along a railroad embankment in the same direction as a train is traveling. He asks how fast the light goes in relation to the train. “Classical” ideas tempt you to think that the light must be going more slowly as judged from the train, because you should subtract the train’s speed. Not so, says Einstein. If a traveler on the train could measure the speed of the beam of light for himself, the result would be exactly the same as the speed of the same beam measured by someone stationary on the ground.
THE CONSTANT SPEED OF LIGHT How does Einstein know that the movement of a source of light, or of the detector that registers its arrival, has no effect on the speed of light as measured by any observer? When he developed the special theory of relativity in 1905, his convic-tion about this crucial point depended on his


intuition and on the theories of a Dutch physicist, Hendrik Lorentz. But in 1913, before he wrote this book, Einstein was rewarded with strong supporting evidence when a Dutch astronomer, Willem de Sitter, considered pairs of stars that orbit around each other. Sometimes a star swings towards the earth, as it circles its partner, and sometimes it’s receding on the other side of its partner. These phases of its orbit are distinguishable by shifts in frequency of the starlight, and they follow at regular intervals. If the light traveled faster when emitted by the star approaching the earth, it would overtake the light from the previous retreating phase and smear out the alternations. As astronomers can easily distinguish the comings and goings, de Sitter reasoned that the constancy of the speed of light was confirmed. Astronomy at invisible frequencies, which travel at the same speed as visible light, has refined de Sitter’s test. An X-ray star in another galaxy, the Small Magellanic Cloud, is orbiting around an unseen companion. It lies so far away that the slightest discrepancy in the light speed, due to the star’s own speed, would be detectable. Gamma ray bursts come from stupendous explosions that occur almost out to the limits of the observable universe. Even after taking billions of years to reach us, some bursts last for only a split second. That means there can be no difference at all in the speeds of emission from rapidly moving parts of the violently erupting source. The brevity of gamma-ray bursts now makes the constancy of light speed in empty space one of the surest facts in the whole of science. With this hindsight, Einstein’s conviction is correct. But the puzzle illustrated by the railroad train and the relative speed of the light beam “has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties.” The special theory of relativity is promised as the solution.
TIME BECOMES SLIPPERY Two lightning flashes are said to strike the railroad embankment at the same moment at different places, and Einstein spends several pages fretting about the meaning of “simultaneous” (Sections 8 and 9). After a mock dialogue with the reader, who supposedly defends old-fashioned thinking about the idea of time in physics, Einstein offers an exact definition. The lightning strokes are simultaneous if their light rays meet at the midpoint on the embankment between the places where they strike. But from the point


of view of a rider on a moving train, who happens to be midway between the points of impact when the lightning strikes, the flashes cannot be simultaneous. This observer will see the flash up ahead slightly before the one behind, because the train has moved him forward to meet its approaching rays. Once simultaneity becomes only a relative concept, time itself goes haywire. What the observer on the embankment considers to be one second is not one second for the person on the train. As a result (in Section 10) the speed of a person walking forward along the train, as judged from the embankment, turns out to be different from the speed of walking judged on the train itself. And the length of the train itself will appear different too. To make these crazyseeming propositions precise, Einstein brings in a mathematical device called the Lorentz transformation, named after the Dutch physicist mentioned earlier. Don’t worry if you can’t follow it all. Section 11 (reinforced by Ap-pendix 1) explains why the same beam of light has the same speed whether judged by a stationary or a moving observer. Time runs more slowly for the moving observer, to exactly the extent needed to secure the constancy of light speed. Similar mathematics in Section 12 tells you why no ordinary object could ever travel faster than light. It also shows that a measuring rod moving past you will appear shorter than when it is at rest; hence, the remark about the length of the train. Einstein’s way of putting it suggests a squeezing. In a more modern gloss, the rod, or the train, appears to be slightly rotated away from you as it passes, and so you see it foreshortened to the extent predicted by the formula. Einstein then returns to the question of how velocities are to be added together (Section 13). When this situation cropped up earlier, with a person walking forward along a moving train, the simpleminded answer was to add the man’s speed to the train’s speed. But in relativity the combined speed as gauged by a stationary watcher has to be reduced a little. With some satisfaction, Einstein revisits an experiment first devised by a French physicist, Armand Fizeau, in 1851. By measuring the speed of light in water flowing down a tube, it unwittingly tested Einstein’s formula for adding velocities. Repeated by Pieter Zeeman (yet another Dutchman, and a colleague of Lorentz), the experiment confirmed the formula’s accuracy to within 1 percent.


MASS AND ENERGY When this book was written, practicable speeds of motion were too slow, and clocks and rulers too imprecise, to test many predictions of special relativity directly. Undismayed, Einstein went on to make some sweeping inferences from his theory. His assertion, in Section 14, that any general law of nature must be consistent with special relativity, reconfirms his early requirement that the laws of nature cannot depend on who is looking. Einstein’s own most famous law is that mass and energy are equivalent. In Section 15 he introduces the idea by way of another remarkable prediction of special relativity, namely that the mass of a body increases when it travels at high speed. In classical physics it gains in energy of motion. In relativity that kinetic energy makes itself felt as additional mass. Radiant energy absorbed by a body also increases its mass. Indeed the total mass becomes a measure of its total energy. But in this reckoning the body starts with inherent energy even when at rest. It is a huge amount, given by the body’s rest mass rest mass multiplied by the square of the speed of light, or E=mc2. Concentrations of energy available in 1916 were too small in relation to experimental masses for this equivalence of mass and energy to be tested. And in Section 16, when Einstein trawls for evidence in support of special relativity as a whole, the haul is meager—just small deviations from classical expectations in experiments with electrons, and a favorable gloss that he can put on the failure to detect any difference in the speed of light in two directions at right angles, in a nineteenth-century experiment. Modern particle accelerators confirm Einstein’s predictions more directly. They prolong the life of unstable subatomic particles by achieving speeds that stretch time for them. The masses of accelerated particles increase to the point where new matter can be created from them. E= mc2 accounts for the long-lasting power of the sun and the stars, and appears as the nuclear energy that power engineers and bombmakers have learned how to tap. To trace all the consequences of special relativity now verified by scientists would be to recapitulate much of the physics and astronomy of the past one hundred years. Perhaps the crowning glory is antimatter, predicted by Paul Dirac in England when he applied special relativity to the theory of subatomic particles.


Antimatter is now known to shower down from the sky above us, and when a particle meets its antiparticle, both of them disappear in a burst of radiant energy, exactly in accordance with E=mc2.
ONWARD TO GENERAL RELATIVITY “The non-mathematician is seized by a mysterious shuddering when he hears of ‘four-dimensional’ things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more commonplace statement than that the world in which we live is a four-dimensional space-time continuum.” Einstein makes this remark at the start of Section 17, which is a preamble about geometry for the transition from special relativity to general relativity. Figuring prominently is the mathematician Hermann Minkowski, who had called Einstein “a lazy dog” when teaching him in Zurich. Minkowski invented a mathematical trick that treats time as if it were just an extra dimension of space. Appendix 2 has a little more on this subject. Without this method, the juggling with space-time whereby Einstein revolutionized the theory of gravity would have been much more difficult. The different meanings of special and general relativity appear in Section 18. The democratic principle that Einstein is pursuing requires the laws of nature to remain the same regardless of how the observer is moving. While the special theory compares different views of the world due to uniform motion, as in the railroad train moving steadily along the track, the general theory removes that restriction and allows for all kinds of movement, including acceleration and rotation. Some puzzles about gravity are set out in Section 19. Whatever the “intermediary medium” is, which pulls a stone down when we drop it, it acts equally on any other object. Everything falls with the same acceleration, if there’s no air resistance. And isn’t it odd that the force of gravity acting on a body is propor-tional to its mass—exactly the same quality that crops up when you gauge the body’s resistance to acceleration, its inertia? These features find an explanation in Section 20, when gravity is seen to be very like any other accelerating system. Einstein invites us to visualize a man living in a big chest that’s drifting in empty space. He must tie himself to the floor if he is not to float about. The modern reader has seen videos of astronauts drifting weightlessly in their spaceships, but Einstein has to picture it for himself.


Unable to invoke a space rocket to propel the box, he imagines a “being” pulling on a rope attached to the lid of the chest and imparting a steady acceleration. The man in the chest can then think himself at home on the earth. He no longer tends to float, and any object he releases will fall to the floor. The steady acceleration through empty space will feel to him just like gravity. What’s more, the simulation fully accounts for the equal effect of gravity on all objects, and for the equality of inertial and gravitational mass. A similar situation prevails for a person on a train when the brakes are applied hard. He can say, if he wishes, that he is jerked forward by a shortlived gravitational field. It also slows down the embankment (and the planet in general) that were rushing past him while he sat stationary in his own frame of reference. By this time the reader may think that the reasoning is quaint, but in Section 21 it’s clear that Einstein is in earnest. He is simply stressing that, in general relativity, no point of view can take preference over any other.
GRAVITY BENDS LIGHT The similarity between gravity and any other acceleration means that light must be affected by gravity like any other substance. This crucial ingredient of Einstein’s theory first appears in Section 22. There he predicts that the light from a star grazing the sun will be deflected, so that it will change its apparent position in the sky by 1.7 seconds of arc (roughly one twothousandths of a degree). At the time of an eclipse, he says, stars seen beyond the sun ought to appear shifted outwards from the sun to that sort of extent, compared with their normal positions in the sky. This was the prediction that made Einstein famous. He was doubly lucky. When wrestling with his early ideas about general relativity, in 1911, he published a wrong answer for the deflection of starlight—half the correct result. The outbreak of World War I prevented astronomers from testing Einstein’s prediction at the total eclipse of 1914, before he came up with the right number in 1915. The second stroke of luck was that the British astronomer Arthur Eddington, who led the effort to test it at the eclipse of 1919, was predisposed to believe Einstein’s theory. Looking for starshifts of less than a millimeter on the photographic plates, Eddington’s team put aside several plates that gave “wrong” results, and picked ’n’ mixed the rest until the


average was about right (see Appendix 3b). It was pretty sloppy science, yet Eddington let the message ring out around the world: “Newton’s theory of gravity is dead—long live Einstein’s!” Fortunately, light bending to the extent required by general relativity has been amply verified since then. Radio waves are invisible light, and astronomers used widely spaced radio telescopes in accurate observations of Quasar 3C279, which regularly passes behind the sun. A European starmapping satellite called Hipparcos (1989–93) detected the deflection of starlight even from stars lying far from the sun’s direction in the sky. Hipparcos scientists were able to verify Einstein’s theory to an accuracy of 1/10 of 1 percent. Gravitational lenses lying far away in the universe give an up-to-date demonstration of light bending in the Einsteinian fash-ion. The gravity of a cluster of galaxies, and of invisible dark matter associated with it, acts as an untidy magnifier. It en-hances the view of even more distant galaxies by magnifying them, albeit with distortion and multiple images.
COMING CLEAN ABOUT THE SPEED OF LIGHT An ordinary lens works by delaying the light passing through it, because light travels more slowly through glass than through air. The same is true of gravitational lenses. Einstein comments in Section 22 that, against all expectations from special relativity, the deflection of light by gravity implies a change in the speed of light in the sun’s vicinity. “A curvature of rays of light can only take place when the velocity of propagation of light varies with position.” What a pity that remark was not printed in italics in Einstein’s book, or painted on balloons for all to see! Researchers and teachers ignored it for half a century, until radar echoes from Venus and Mercury in the late 1960s turned changes in the speed of light into an observed fact. Radar pulses sent out from the Haystack observatory in Massachusetts were clearly delayed whenever the planets were on the far side of the sun, as seen from the earth. The radio waves (a variety of light) slowed down as they passed the sun on their outward and return journey. Even in the 1970s it was hard to get more than the most grudging admission from experts on relativity that gravity slows down light, although Einstein himself was unabashed about it sixty years earlier. Undue emphasis on the constancy of light speed made general relativity unnecessarily opaque to students and the


general public for several decades. Just come clean, and admit that light dawdles a little near a massive object like the sun, and Einstein’s theory of gravity is far easier to understand. The experts were not being entirely perverse. One of the glories of general relativity is that the speed of light does indeed remain the same, provided you measure it on the spot. If you could station a heat-resistant spacecraft beside the sun to gauge Haystack’s radar pulses whizzing past, they would seem to be traveling at just the usual speed of light. The reason is that time, too, runs more slowly in the spacecraft, under the influence of the sun’s strong gravity. It’s only the distant observer, with a faster clock, who notices the slowdown. In special relativity, you’ll remember, different rates of time on the train and on the embankment enabled the observers to get the same answer for the speed of light. In general relativity, too, changes in clockrates always keep the speed the same, as measured locally. That’s how Einstein ensures that natural laws hold good everywhere. Despite the effects of stronger gravity, atoms, particles, and radiant energy on the sun interact according to exactly the same laws as on the earth. To say so is to run ahead of the chain of explanation in this book. These retrospective hints may nevertheless give you a sense of destination, as Einstein approaches “a serious difficulty” that “lies at the heart of things” and “lays no small claims on the patience and on the power of abstraction of the reader.” Also helpful, perhaps, is to note that the next step in the argument matches an idea illustrated in the science fiction movie 2001: A Space Odyssey, where a large space station simulates normal gravity by centrifugal force. The station rotates at an appropriate speed and the astronauts walk around a floor at the rim, with their feet pointing outwards, away from the center.
SLITHERING IN SPACE-TIME The usual picture of gravity is turned inside out in Section 23, by putting an observer on a disc that is rotating. He feels a force pushing him outward, and like the man in the accelerated box he is authorized by general relativity to call it gravity—a peculiar kind of gravity that becomes stronger the farther you are from the center. The man is also moving relative to the center of the disc. Recalling the effects of relative motion between a train and the track, Einstein notes that the man’s clock will run more slowly than a clock at the center of the disc.


What’s more, if he puts a measuring rod along the edge of the disc in the direction of movement, to start measuring the circumference, the rod will be shorter than it would be at the center. On the other hand, the rod is not shortened when pointing toward the center of the disc, to measure the diameter. As a result, the circumference of the disc will seem to be greater than the diameter multiplied by (pi, 3.14 . . . ) which would be the case if the disc were at rest. When the effective length of a measuring rod can change, Euclid’s geometry for flat surfaces no longer works. General relativity needs a suppler frame of reference, which Einstein sets up in Sections 24 to 28. First he imagines a rectangular grid of rods laid out on a marble slab, which goes askew if you heat part of the slab and some of the rods expand in length. The squares of the grid are no longer square. Not to worry. Carl Friedrich Gauss, a German mathemati-cian and physicist who flourished in the early nineteenth century, devised a system of coordinates in which the grid can be crooked and the lines curved. And just as Minkowski added time to the three dimensions of space in a rectangular system, Einstein adds supple time to Gauss’s supple system for describing space. Then, like a child in a floppy climbing-frame, he has a framework of space-time in which to play with his general theory of relativity. Einstein imagines his four-dimensional world to be inhabited by slithery creatures—he calls them molluscs—that can move about and change shape ad lib. General relativity requires that all molluscs should have “equal right and equal success” in formulating the laws of nature. This invertebrate democracy might seem like a recipe for total confusion. Instead it imposes such strict legislation on the universe that the distortions of space-time due to the influences of massive bodies provide a precise and novel theory of gravity.
IMPROVING ON NEWTON The mathematics that Einstein used to tame his supple space-time is too abstruse for the wide readership he aims for in this book. He contents himself with summarizing some key results, in Section 29. First, Newton’s law of gravity, in which the gravitational force between two bodies is inversely proportional to the square of the distance between them, springs ready-made from the molluscs’ weird world. Unlike Newton, Einstein can explain what is happening. Masses deform space, with the result that


other masses follow curved tracks—as when the sun forces the planets to orbit around it. Moreover, Newton’s law of gravity is only approximately correct. Deviations become evident where gravity is strong, and they show how Einstein’s theory improves on Newton’s. One, already mentioned, is the extent of the bending of starlight when it passes near the sun. Another improvement concerns the misbehavior of the planet Mercury, first noticed by the French astronomer Urbain Leverrier in 1865, which finds a ready explanation in Einstein’s theory of gravity. The planet’s elliptical orbit around the sun gradually swivels because of interactions with other planets, but this “precession” is greater than predicted by Newton’s theory. Searches for an unknown planet that might explain the discrepancy were unavailing. The explanation is slightly stronger gravity near the sun, provided by Einstein’s theory. All planetary orbits are affected but Mercury’s the most because it is closest to the sun (see also Appendix 3a). Radar observations of Mercury later confirmed that the swiveling matches Einstein’s theory to a high degree of accuracy. More spectacular in this regard is a pulsating radio star, or pulsar, discovered in 1974. It goes very closely around and around a silent companion, on an orbit that swivels far more rapidly than Mercury’s. In a double pulsar reported in 2004, the effect is even greater. These systems are also seen to be shedding energy, supposedly by radiating gravitational waves that Einstein predicted in 1916. The third innovation from general relativity described in this book is nowadays known to scientists as the gravitational red-shift. Einstein mentions it only briefly in Section 29 but gives more detail in Appendix 3c, where he writes: “An atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated.” As a symptom of gravity’s amazing power to slow down time, the characteristic light emitted by atoms and molecules—their spectral lines —will appear to distant observers to have lower frequencies in strong gravity than on the earth or in empty space. They will be shifted towards the red end of the spectrum. Verification of the gravitational red-shift came in 1924. Walter Adams in California discovered that Sirius B is a very dense star, the first white dwarf ever identified. He reported that some emissions from hydrogen atoms showed marked reductions in


frequency as required by Einstein’s theory. Much as with the 1919 eclipse story, historians of science question the reliability of Adams’s result. The light from Sirius B was contaminated by light from the much brighter Sirius A. Never mind. The gravitational red-shift is now observed routinely in many astronomical objects including the sun. Even more convincingly for non astronomers, the effect of gravity in slowing time is demonstrated directly with atomic clocks. They run faster in high-flying aircraft than they do on the ground.
A COSMOLOGICAL SKETCH Part III of the book is entitled “Considerations on the Universe as a Whole.” It is very brief, and lest it should disappoint any-one familiar with Einstein’s contributions to cosmology, be aware that when he wrote this book for the general public in 1916, his ideas were still maturing. A key scientific paper, “Cosmological Considerations on the General Theory of Relativity,” did not appear until the following year. Astronomical knowledge of the time, primitive by contemporary standards, misleads Einstein badly. Like Newton, he imagines stars scattered through the immensity of space and moving about only slowly. He is troubled, as Newton was, by the problem that gravity will tend to drag all the stars together. This outcome might be avoided if the stars were very evenly scattered, which was Newton’s own suggestion. By Einstein’s young days, astronomers knew very well that the stars of the Milky Way are not at all uniform in their distribution. To avoid having all the stars fall together in a heap, the German astronomer Hugo von Seeliger suggested that gravity must weaken at long ranges, more rapidly than prescribed by Newton’s inverse square law. This is the main theme of Section 30. Einstein indicates that he might welcome such an idea if only there were a logical reason for it. Another approach to the problem of the collapsing starfield appears in Section 31. Einstein proposes that cosmic space may be folded back upon itself. He invites us to share the worldview of flat beings living on what they perceive as a flat, two-dimensional surface, but which is in fact a sphere of large but finite size. There is no boundary to the flat creatures’ universe. If they traveled far enough they would come back to their starting point, on a great circle. Without having to make a world tour, the flat beings can figure out what


kind of universe they are living in, and even measure its diameter, by discovering subtle discrepancies between the predictions of Euclid for truly flat surfaces, and what they find in practice. Similarly, you can imagine a super-geometry in which our own three-dimensional space is so folded that it is “finite” yet “unbounded.” Such a universe is congenial for general relativity and promises a way of preventing the stars falling together. This is Einstein’s assertion in the very brief Section 32 that, except for the appendices, brings his book to a close. The story ends abruptly with a cursory description of one possible form for the universe, which Einstein happens to like. The reader is left with a strong sense of unfinished business. To pursue in any detail here the dramatic cosmology that unfolded in the twentieth century would be too lengthy a departure from the main subjects of this book. Yet not to mention akin to Seeliger’s idea of gravity weakening at long ranges. To stop the stars falling together, adjusted the strength of gravity as required. Without , a universe is unstable and it must be either imploding or expanding, which was contrary to the myopic impressions of early twentieth-century astronomy. The discovery in the 1920s of the great cosmic expansion, in which the Milky Way is just one of many galaxies and the spaces between clusters of galaxies grow rapidly, made Einstein think which it apparently began. Yet at the end of the twentieth century the expansion of the universe turned out to be accelerating and Einstein’s has come back in triumph in the driving seat of the cosmos. And what about Euclid, with whom this book starts and finishes? Einstein wanted astronomers to emulate the flat beings of his folded twodimensional universe and discover the overall geometry of the real world. If the universe were not “flat” in Euclid’s ideal sense, it should act as a lens, and very distant objects should appear magnified or shrunk. The most distant observable objects—clumps of hot gas that existed soon after the Big Bang—are now mapped by radio microwaves and they look neither bigger nor smaller than expected. In the geometers’ heaven, it’s a draw. Einstein trumps Euclid in the distorted space-time surrounding planets, stars, and galaxies, but the geometry of the universe at large still conforms very well to what that old Greek taught his students in Alexandria 2,300 years ago.


GENERAL RELATIVITY STILL THRIVES To say much about the rest of Einstein’s life story in this introduction might break the mental link that I have tried to fashion with the relatively young Einstein who wrote the book. He would divorce Mileva and marry his cousin Elsa. The 1921 No-bel Prize would come his way—not for relativity but for his interpretation of the photoelectric effect. Later he would fall out with his fellow physicists over the interpretation of the quantum theory. In 1932, as a Jewish refugee from the Nazis, he would find sanctuary in Princeton, New Jersey, and live there until his death in 1955. Much more relevant to the reader is the subsequent career of Einstein’s cleverest brainchild, general relativity. Despite many decades of efforts by experimenters and theorists to prove it imperfect, it still holds sway. Oftrepeated promises of a superior quantum theory of gravity remain only a speculation. A great theory should make surprising predictions that can be verified by observation, and go on to take unexpected discoveries in its stride. General relativity has performed supremely well on both counts. In addition to several successful tests already described, a huge effort is now going into the direct detection of gravitational waves, which should squeeze and stretch space as they pass by. Failure to find them would be surprising because, as mentioned, the behavior of orbiting pulsars makes sense only if they are radiating gravitational waves. Another prediction currently under test with a satellite is that the earth should drag space-time around it as it rotates. The finest example of explaining the unexpected came with the discovery of quasars in 1963. These compact sources of radiation in the hearts of some galaxies were far too powerful to rely on the nuclear energy that lights the stars. Ready to hand was an awful possibility implicit in general relativity. A massive object might collapse into a black hole, which would then be capable of squeezing huge amounts of energy out of any stars or gas falling into it. The idea of black holes won acceptance only gradually. Observations established the compactness of the quasars and the presence of material feeding their hearty appetites. Not until 1994 did direct confirmation of the reality of black holes come in results from the Japanese satellite ASCA. Variations in the wavelength of X-ray emissions from iron atoms in a stormy galaxy made a pattern predicted for material orbiting closely around


a black hole. And a loss of energy by individual X-ray particles showed time slowing down in the intense gravity near the black hole, just as general relativity requires.
Suggestions for Further Reading
The bibliography concerning Einstein is huge. Here are a few well-regarded books, written or edited from a modern perspective.
Calder, Nigel. Einstein’s Universe: The Layperson’s Guide. New York: Penguin, 2005. This is an updated edition of a book first published in 1979. It covers much of the same ground as the present classic by Einstein himself, but with all the bonuses of scientific hindsight.
Overbye, Dennis. Einstein in Love. New York: Viking Penguin, 2000. Drawing on many unpublished letters, this book tells of Einstein’s romances with his first wife, Mileva, and with his second, Elsa, which took place during his most creative years.
Schilpp, Paul Arthur (editor and translator). Albert Einstein: PhilosopherScientist. La Salle, Ill.: Open Court, 1982. Of special interest here are Einstein’s own “Autobiographical Notes” written in 1946, where he told how his ideas developed but avoided saying much about his “merely personal” life.
White, Michael, and Gribbin, John. Einstein: A Life in Science. New York: Plume Books, 2005. First published in 1993, this book skillfully interweaves the personal, public, and scientific strands of Einstein’s whole life, including his persistent misgivings about the quantum theory.


Part I. The Special Theory of
Relativity


01- Physical Meaning of
Geometrical Propositions
K = co-ordinate system
x, y = two-dimensional co-ordinates
x, y, z = three-dimensional co-ordinates
x, y, z, t = four-dimensional co-ordinates
t = time
I = distance
v = velocity F = force
G = gravitational field
Part I: The Special Theory of Relativity In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration. Geometry sets out form certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of


these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves. It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation. If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically
rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the


"truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses. Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.
Footnotes 01
1) It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for the present purpose.




02-The System of Co-ordinates


On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a " distance " (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry ; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement
of length. 1) Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place
specification " Times Square, New York," [A] I arrive at the following result. The earth is the rigid body to which the specification of place refers; " Times Square, New York," is a well-defined point, to which a name has
been assigned, and with which the event coincides in space.2) This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Times Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed. (a) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by. the completed rigid body. (b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.


(c) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud. From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry. In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available ; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be
sought in accordance with the above considerations. 3) We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances;" the "distance" being represented physically by means of the convention of two marks on a rigid body.
Footnotes 02


1) Here we have assumed that there is nothing left over i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.
[A] Einstein used "Potsdamer Platz, Berlin" in the original text. In the authorised translation this was supplemented with "Tranfalgar Square, London". We have changed this to "Times Square, New York", as this is the most well known/identifiable location to English speakers in the present day. [Note by the janitor.]
2) It is not necessary here to investigate further the significance of the expression "coincidence in space." This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.
3) A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book.


03-Space and Time in Classical
Mechanics


The purpose of mechanics is to describe how bodies change their position in space with "time." I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins. It is not clear what is to be understood here by "position" and "space." I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space" ? From the considerations of the previous section the answer is self-evident. In the first place we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of " body of reference " we insert " system of co-ordinates," which is a useful idea for mathematical description, we are in a position to say : The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing
trajectory (lit. "path-curve" 1)), but only a trajectory relative to a particular body of reference. In order to have a complete description of the motion, we must specify how the body alters its position with time ; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction ; the man at the railway-carriage window is holding one of them, and the man on


the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.


Footnotes 03
1) That is, a curve along which the body moves.


04-The Galileian System of Co
ordinates


As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of coordinates, permissible in mechanics, which can be used in mechanical description. The visible fixed stars are bodies for which the law of inertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of coordinates relative to which the fixed stars do not move in a circle. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a " Galileian system of coordinates." The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.


05-The Principle of Relativity (in
the restricted sense)
In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation ("uniform" because it is of constant velocity and direction, " translation " because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say : If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co
ordinate system K1 provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding section, it follows that:
If K is a Galileian co-ordinate system. then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform
motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.
We advance a step farther in our generalisation when we express the tenet
thus: If, relative to K, K1 is a uniformly moving co-ordinate system devoid
of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense). As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more


recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity became ripe for discussion, and it did not appear impossible that the answer to this question might be in the negative. Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the principle of relativity. Even though classical mechanics does not supply us with a sufficiently broad basis for the theoretical presentation of all physical phenomena, still we must grant it a considerable measure of " truth," since it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful. The principle of relativity must therefore apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable. We now proceed to the second argument, to which, moreover, we shall return later. If the principle of relativity (in the restricted sense) does not
hold, then the Galileian co-ordinate systems K, K1, K2, etc., which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. In this case we should be constrained to believe that natural laws are capable of being formulated in a particularly simple manner, and of course only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K0) of
a particular state of motion as our body of reference. We should then be justified (because of its merits for the description of natural phenomena) in calling this system " absolutely at rest," and all other Galileian systems K " in motion." If, for instance, our embankment were the system K0 then our
railway carriage would be a system K, relative to which less simple laws would hold than with respect to K0. This diminished simplicity would be
due to the fact that the carriage K would be in motion (i.e."really")with respect to K0. In the general laws of nature which have been formulated
with reference to K, the magnitude and direction of the velocity of the carriage would necessarily play a part. We should expect, for instance, that the note emitted by an organpipe placed with its axis parallel to the


direction of travel would be different from that emitted if the axis of the pipe were placed perpendicular to this direction. Now in virtue of its motion in an orbit round the sun, our earth is comparable with a railway carriage travelling with a velocity of about 30 kilometres per second. If the principle of relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth. For owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year. However,
the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is very powerful argument in favour of the principle of relativity.


06-The Theorem of the Addition of
Velocities Employed in Classical
Mechanics


Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly or, in other words, with what velocity W does the man advance relative to the embankment during the process ? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total be covers the distance W=v+w relative to the embankment in the second considered. We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained ; in other words, the law that we have just written down does not hold in reality. For the time being, however, we shall assume its correctness.


07-The Apparent Incompatibility of
the Law of Propagation of Light
with the Principle of Relativity
There is hardly a simpler law in physics than that according to which light is propagated in empty space. Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c= 300,000 km./sec. At all events we know with great exactness that this velocity is the same for all colours, because if this were not the case, the minimum of emission would not be observed simultaneously for different colours during the eclipse of a fixed star by its dark neighbour. By means of similar considerations based on observa- tions of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light. The assumption that this velocity of propagation is dependent on the direction "in space" is in itself improbable. In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties? Let us consider how these difficulties arise. Of course we must refer the process of the propagation of light (and indeed every other process) to a rigid reference-body (co-ordinate system). As such a system let us again choose our embankment. We shall imagine the air above it to have been removed. If a ray of light be sent along the embankment, we see from the above that the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that


we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity w of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we have w = c-v. The velocity of propagation ot a ray of light relative to the carriage thus comes cut smaller than c. But this result comes into conflict with the principle of relativity set forth in Section V. For, like every other general law of nature, the law of the transmission of light in vacuo [in vacuum] must, according to the principle of relativity, be the same for the railway carriage as reference-body as when the rails are the body of reference. But, from our above consideration, this would appear to be impossible. If every ray of light is propagated relative to the embankment with the velocity c, then for this reason it would appear that another law of propagation of light must necessarily hold with respect to the carriage — a result contradictory to the principle of relativity. In view of this dilemma there appears to be nothing else for it than to abandon either the principle of relativity or the simple law of the propagation of light in vacuo. Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple. The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable to the principle of relativity. The development of theoretical physics shows, however, that we cannot pursue this course. The epochmaking theoretical investigations of H. A. Lorentz on the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain leads conclusively to a theory of electromagnetic phenomena, of which the law of the constancy of the velocity of light in vacuo is a necessary consequence. Prominent theoretical physicists were theref ore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which were contradictory to this principle. At this juncture the theory of relativity entered the arena. As a result of an analysis of the physical conceptions of time and space, it became evident that in realily there is not the least incompatibilitiy between the principle of


relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically rigid theory could be arrived at. This theory has been called the special theory of relativity to distinguish it from the extended theory, with which we shall deal later. In the following pages we shall present the fundamental ideas of the special theory of relativity.


08-On the Idea of Time in Physics


Lightning has struck the rails on our railway embankment at two places A and B far distant from each other. I make the additional assertion that these two lightning flashes occurred simultaneously. If I ask you whether there is sense in this statement, you will answer my question with a decided "Yes." But if I now approach you with the request to explain to me the sense of the statement more precisely, you find after some consideration that the answer to this question is not so easy as it appears at first sight. After some time perhaps the following answer would occur to you: "The significance of the statement is clear in itself and needs no further explanation; of course it would require some consideration if I were to be commissioned to determine by observations whether in the actual case the two events took place simultaneously or not." I cannot be satisfied with this answer for the following reason. Supposing that as a result of ingenious considerations an able meteorologist were to discover that the lightning must always strike the places A and B simultaneously, then we should be faced with the task of testing whether or not this theoretical result is in accordance with the reality. We encounter the same difficulty with all physical statements in which the conception " simultaneous " plays a part. The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case. We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.) After thinking the matter over for some time you then offer the following suggestion with which to test simultaneity. By measuring along the rails, the connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an
arrangement (e.g. two mirrors inclined at 900) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.


I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel constrained to raise the following objection: "Your definition would certainly be right, if only I knew that the light by means of which the observer at M perceives the lightning flashes travels along the length A M with the same velocity as along the length B M. But an examination of this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle." After further consideration you cast a somewhat disdainful glance at me and rightly so — and you declare: "I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, that in every real case it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand is indisputable. That light requires the same time to traverse the path A M as for the path B M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity." It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the
body of reference 1) (here the railway embankment). We are thus led also to a definition of " time " in physics. For this purpose we suppose that clocks of identical construction are placed at the points A, B and C of the railway line (co-ordinate system) and that they are set in such a manner that the positions of their pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the " time " of an event the reading (position of the hands) of that one of these clocks which is in the immediate vicinity (in space) of the event. In this manner a time-value is associated with every event which is essentially capable of observation. This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference-body are set in such a manner that a


particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position, of the pointers of the other clock, then identical " settings " are always simultaneous (in the sense of the above definition).


Footnotes
1) We suppose further, that, when three events A, B and C occur in different places in such a manner that A is simultaneous with B and B is simultaneous with C (simultaneous in the sense of the above definition), then the criterion for the simultaneity of the pair of events A, C is also satisfied. This assumption is a physical hypothesis about the the of propagation of light: it must certainly be fulfilled if we are to maintain the law of the constancy of the velocity of light in vacuo.




09-The Relativity of Simulatneity


Up to now our considerations have been referred to a particular body of reference, which we have styled a " railway embankment." We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig 1. People travelling in this train will with a vantage view the train as a rigid reference-body (co-ordinate system); they regard all events in
reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises : Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A B of the embankment. But the events A and B also
correspond to positions A and B on the train. Let M1 be the mid-point of the distance A B on the travelling train. Just when the flashes (as judged
from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves towards the right in the diagram with the
velocity v of the train. If an observer sitting in the position M1 in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him


simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result: Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time ; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event. Now before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. But we have just seen that this assumption is incompatible with the most natural definition of simultaneity; if we discard this assumption, then the conflict between the law of the propagation of light in vacuo and the principle of relativity (developed in Section 7) disappears. We were led to that conflict by the considerations of Section 6, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each second of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance w relative to the railway line in a time which is equal to one second as judged from the embankment. Moreover, the considerations of Section 6 are based on yet a second assumption, which, in the light of a strict consideration, appears to be arbitrary, although it was always tacitly made even before the introduction of the theory of relativity.




10-On the Relativity of the
Conception of Distance


Let us consider two particular points on the train 1) travelling along the embankment with the velocity v, and inquire as to their distance apart. We already know that it is necessary to have a body of reference for the measurement of a distance, with respect to which body the distance can be measured up. It is the simplest plan to use the train itself as reference-body (co-ordinate system). An observer in the train measures the interval by marking off his measuring-rod in a straight line (e.g. along the floor of the carriage) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often the rod has to be laid down is the required distance. It is a different matter when the distance has to be judged from the railway
line. Here the following method suggests itself. If we call A1 and B1 the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the embankment. In the first place we require to determine the points A and B of the embankment which
are just being passed by the two points A1 and B1 at a particular time t judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section 8. The distance between these points A and B is then measured by repeated application of thee measuring-rod along the embankment. A priori it is by no means certain that this last measurement will supply us with the same result as the first. Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself. This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section 6. Namely, if the man in the carriage covers the distance w in a unit of time measured from the train, — then this distance — as measured from the embankment — is not necessarily also equal to w.


Footnotes
1)e.g. the middle of the first and of the hundredth carriage.




11-The Lorentz Transformation


The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section 7) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: (1) The time-interval (time) between two events is independent of the condition of motion of the body of reference. (2) The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference. If we drop these hypotheses, then the dilemma of Section 7 disappears, because the theorem of the addition of velocities derived in Section 6 becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of Section 6 in order to remove the apparent disagreement between these two fundamental results of experience? This question leads to a general one. In the discussion of Section 6 we have to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment ? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity ? In other words : Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train ? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another. Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in Section 2 we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework.Fig.2


Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as " co-ordinate planes " (" co-ordinate system "). A co-ordinate system K then corresponds to the embankment, and a co-ordinate system K' to the train. An event, wherever it may have taken place, would be fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate planes,
and with regard to time by a time value t. Relative to K1, the same event
would be fixed in respect of space and time by corresponding values x1, y1,
z1, t1, which of course are not identical with x, y, z, t. It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements.
Obviously our problem can be exactly formulated in the following manner.
What are the values x1, y1, z1, t1, of an event with respect to K1, when the magnitudes x, y, z, t, of the same event with respect to K are given ? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every
ray) with respect to K and K1. For the relative orientation in space of the co


ordinate systems indicated in the diagram (Fig. 2), this problem is solved by means of the equations :
y1 = y y z1 = z
This system of equations is known as the " Lorentz transformation." 1) If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:
x1 = x - vt ; y1 = y ; z1 = z ; t1 = t
This system of equations is often termed the " Galilei transformation." The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light c in the latter transformation. Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body K and for the reference-body
K1. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation
x = ct,
i.e. with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t involves a relation
between x1 and t1. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:


from which, by division, the expression
x1 = ct1
immediately follows. If referred to the system K1, the propagation of light takes place according to this equation. We thus see that the velocity of
transmission relative to the reference-body K1 is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.


Footnotes
1) A simple derivation of the Lorentz transformation is given in Appendix I.




12-The Behaviour of Measuring
Rods and Clocks in Motion


Place a metre-rod in the x1-axis of K1 in such a manner that one end (the
beginning) coincides with the point x1=0 whilst the other end (the end of
the rod) coincides with the point x1=I. What is the length of the metre-rod relatively to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K. By means of the first equation of the Lorentz transformation the values of these two points at the time t = 0 can be shown to be
the distance between the points being . But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its
length with a velocity v is of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity v=c we should
have , and for stiII greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body. Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these became meaningless if we choose values of v greater than c. If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to K, then we should have found that the length of the rod as judged
from K1 would have been ; this is quite in accordance with the principle of relativity which forms the basis of our considerations.


A Priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galileian transformation we should not have obtained a contraction of the rod as a consequence of its motion. Let us now consider a seconds-clock which is permanently situated at the
origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks : t=0 and
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but
seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.


13-Theorem of the Addition of
Velocities. The Experiment of
Fizeau


Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment. In Section 6 we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics- This theorem can also be deduced readily from the Galilei transformation (Section 11). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system
K1 in accordance with the equation
x1 = wt1 By means of the first and fourth equations of the Galilei transformation we
can express x1 and t1 in terms of x and t, and we then obtain x = (v + w)t This equation expresses nothing else than the law of motion of the point with reference to the system K (of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section 6, W=v+w A) But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation
x1 = wt1 B)
we must then express x1and t1 in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation
which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we


are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity v ? In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube. It is clear that we have the problem of Section 6 again before us. The tube plays the part of the railway embankment or of the co-ordinate system K,
the liquid plays the part of the carriage or of the co-ordinate system K1, and finally, the light plays the part of the
Fig. 3
man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by W, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts.
Experiment1) decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of


flow v on the propagation of light is represented by formula (B) to within one per cent. Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of MaxwellLorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of each other, on which electrodynamics was built.


Footnotes
1) Fizeau found , where is the index of refraction of the liquid. On the other hand, owing to the
smallness of as compared with I,
we can replace (B) in the first place by , or to the same order of approximation by
, which agrees with Fizeau's result.


14-The Heuristic Value of the
Theory of Relativity


Our train of thought in the foregoing pages can be epitomised in the following manner. Experience has led to the conviction that, on the one hand, the principle of relativity holds true and that on the other hand the velocity of transmission of light in vacuo has to be considered equal to a constant c. By uniting these two postulates we obtained the law of transformation for the rectangular co-ordinates x, y, z and the time t of the events which constitute the processes of nature. In this connection we did not obtain the Galilei transformation, but, differing from classical mechanics, the Lorentz transformation.
The law of transmission of light, the acceptance of which is justified by our actual knowledge, played an important part in this process of thought. Once in possession of the Lorentz transformation, however, we can combine this with the principle of relativity, and sum up the theory thus: Every general law of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space-time variables x, y, z, t of the original coordinate system K, we introduce new
space-time variables x1, y1, z1, t1 of a co-ordinate system K1. In this connection the relation between the ordinary and the accented magnitudes is given by the Lorentz transformation. Or in brief : General laws of nature are co-variant with respect to Lorentz transformations. This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental assumptions of the theory would have been disproved. Let us now examine what general results the latter theory has hitherto evinced.


15-General Results of the Theory


It is clear from our previous considerations that the (special) theory of relativity has grown out of electrodynamics and optics. In these fields it has not appreciably altered the predictions of theory, but it has considerably simplified the theoretical structure, i.e. the derivation of laws, and — what is incomparably more important — it has considerably reduced the number of independent hypothese forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour. Classical mechanics required to be modified before it could come into line with the demands of the special theory of relativity. For the main part, however, this modification affects only the laws for rapid motions, in which the velocities of matter v are not very small as compared with the velocity of light. We have experience of such rapid motions only in the case of electrons and ions; for other motions the variations from the laws of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer given by the wellknown expression
but by the expression
This expression approaches infinity as the velocity v approaches the velocity of light c. The velocity must therefore always remain less than c, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain


When is small compared with unity, the third of these terms is always small in comparison with the second,
which last is alone considered in classical mechanics. The first term mc2 does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass; depends on the velocity. We shall speak of its essential significance later. The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the canservation of energy and the law of the conservation of mass these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it. The principle of relativity requires that the law of the concervation of energy should hold not only with reference to a co-ordinate system K, but
also with respect to every co-ordinate system K1 which is in a state of uniform motion of translation relative to K, or, briefly, relative to every " Galileian " system of co-ordinates. In contrast to classical mechanics; the Lorentz transformation is the deciding factor in the transition from one such system to another. By means of comparatively simple considerations we are led to draw the following conclusion from these premises, in conjunction with the fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity v, which absorbs 1) an amount of energy E0 in the form of
radiation without suffering an alteration in velocity in the process, has, as a consequence, its energy increased by an amount
In consideration of the expression given above for the kinetic energy of the body, the required energy of the body comes out to be


Thus the body has the same energy as a body of mass
moving with the velocity v. Hence we can say: If a body takes up an amount of energy E0, then its inertial mass increases by an amount
the inertial mass of a body is not a constant but varies according to the change in the energy of the body. The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form
we see that the term mc2, which has hitherto attracted our attention, is
nothing else than the energy possessed by the body 2) before it absorbed the energy E0.
A direct comparison of this relation with experiment is not possible at the present time (1920; see Note, p. 48), owing to the fact that the changes in energy E0 to which we can Subject a system are not large enough to make
themselves perceptible as a change in the inertial mass of the system.
is too small in comparison with the mass m, which was present before the alteration of the energy. It is owing to this circumstance that classical mechanics was able to establish successfully the conservation of mass as a law of independent validity. Let me add a final remark of a fundamental nature. The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium)


of the type of Newton's law of gravitation. According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental role in this theory. In Part II we shall see in what way this result becomes modified in the general theory of relativity.


Footnotes
1)E0 is the energy taken up, as judged from a co-ordinate system moving
with the body.
2) As judged from a co-ordinate system moving with the body.
[Note] The equation E = mc2 has been thoroughly proved time and again since this time.


16-Experience and the Special
Theory of Relativity


To what extent is the special theory of relativity supported by experience ? This question is not easily answered for the reason already mentioned in connection with the fundamental experiment of Fizeau. The special theory of relativity has crystallised out from the Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of experience which support the electromagnetic theory also support the theory of relativity. As being of particular importance, I mention here the fact that the theory of relativity enables us to predict the effects produced on the light reaching us from the fixed stars. These results are obtained in an exceedingly simple manner, and the effects indicated, which are due to the relative motion of the earth with reference to those fixed stars are found to be in accord with experience. We refer to the yearly movement of the apparent position of the fixed stars resulting from the motion of the earth round the sun (aberration), and to the influence of the radial components of the relative motions of the fixed stars with respect to the earth on the colour of the light reaching us from them. The latter effect manifests itself in a slight displacement of the spectral lines of the light transmitted to us from a fixed star, as compared with the position of the same spectral lines when they are produced by a terrestrial source of light (Doppler principle). The experimental arguments in favour of the Maxwell-Lorentz theory, which are at the same time arguments in favour of the theory of relativity, are too numerous to be set forth here. In reality they limit the theoretical possibilities to such an extent, that no other theory than that of Maxwell and Lorentz has been able to hold its own when tested by experience. But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself — i.e. without making use of the theory of relativity — appears extraneous. It is known that cathode rays and the so-called ß-rays emitted by radioactive substances consist of negatively electrified particles (electrons) of very small inertia and large velocity. By examining the deflection of these rays under the influence of electric and magnetic fields, we can study the law of motion of these particles very exactly. In the theoretical treatment of these electrons, we are faced with the difficulty that electrodynamic theory of itself is unable to give an account of their nature. For since electrical masses of one sign repel each other, the


negative electrical masses constituting the electron would necessarily be scattered under the influence of their mutual repulsions, unless there are forces of another kind operating between them, the nature of which has
hitherto remained obscure to us.1) If we now assume that the relative distances between the electrical masses constituting the electron remain unchanged during the motion of the electron (rigid connection in the sense of classical mechanics), we arrive at a law of motion of the electron which does not agree with experience. Guided by purely formal points of view, H. A. Lorentz was the first to introduce the hypothesis that the form of the electron experiences a contraction in the direction of motion in consequence of that motion. the contracted length being proportional to the expression
This, hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that particular law of motion which has been confirmed with great precision in recent years. The theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. We arrived at a similar conclusion in Section 13 in connection with the experiment of Fizeau, the result of which is foretold by the theory of relativity without the necessity of drawing on hypotheses as to the physical nature of the liquid. The second class of facts to which we have alluded has reference to the question whether or not the motion of the earth in space can be made perceptible in terrestrial experiments. We have already remarked in Section 5 that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galileian transformation for changing over from one body of reference to another. Now assuming that the Maxwell-Lorentz equations hold for a reference-body K, we then find that they do not hold
for a reference-body K1 moving uniformly with respect to K, if we assume that the relations of the Galileian transformation exist between the co


ordinates of K and K1. It thus appears that, of all Galileian co-ordinate systems, one (K) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding K as at rest with respect to a hypothetical æther of space. On the other hand, all coordinate
systems K1 moving relatively to K were to be regarded as in motion with
respect to the æther. To this motion of K1 against the æther ("æther-drift "
relative to K1) were attributed the more complicated laws which were
supposed to hold relative to K1. Strictly speaking, such an æther-drift ought also to be assumed relative to the earth, and for a long time the efforts of physicists were devoted to attempts to detect the existence of an æther-drift at the earth's surface. In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. A ray of light requires a perfectly definite time T to pass from one mirror to the other and back again, if the whole system be at rest with respect to the
æther. It is found by calculation, however, that a slightly different time T1 is required for this process, if the body, together with the mirrors, be moving relatively to the æther. And yet another point: it is shown by calculation that
for a given velocity v with reference to the æther, this time T1 is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the differeace in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a " specially favoured " (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can


be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses ; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.


Footnotes
1) The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces.


17-Minkowski's Four-Dimensional
Space


The non-mathematician is seized by a mysterious shuddering when he hears of "four-dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum. Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (coordinales) x, y, z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by coordinates such as x1, y1, z1, which may be as near as we choose to the
respective values of the co-ordinates x, y, z, of the first point. In virtue of the latter property we speak of a " continuum," and owing to the fact that there are three co-ordinates we speak of it as being " three-dimensional." Similarly, the world of physical phenomena which was briefly called " world " by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z, and a time coordinate, the time value t. The" world" is in this sense also a continuum; for to every event there are as many "neighbouring" events (realised or at least thinkable) as we care to choose, the co-ordinates x1, y1, z1, t1 of which
differ by an indefinitely small amount from those of the event x, y, z, t originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent role, as compared with the space coordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the
Galileian transformation (t1 = t) The four-dimensional mode of consideration of the "world" is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation:


Moreover, according to this equation the time difference Δt1 of two events
with respect to K1 does not in general vanish, even when the time
difference Δt1 of the same events with reference to K vanishes. Pure " space-distance " of two events with respect to K results in " time-distance " of the same events with respect to K. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical
space.1) In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure. These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes. Minkowski's work is doubtless difficult of access to anyone inexperienced in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall leave it here at present, and revert to it only towards the end of Part 2.


Footnotes
1) Cf. the somewhat more detailed discussion in Appendix II.




Part II. The general theory of
relativity


18-Special and General Principle of
Relativity


The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. Let as once more analyse its meaning carefully. It was at all times clear that, from the point of view of the idea it conveys to us, every motion must be considered only as a relative motion. Returning to the illustration we have frequently used of the embankment and the railway carriage, we can express the fact of the motion here taking place in the following two forms, both of which are equally justifiable : (a) The carriage is in motion relative to the embankment,
(b) The embankment is in motion relative to the carriage. In (a) the embankment, in (b) the carriage, serves as the body of reference in our statement of the motion taking place. If it is simply a question of detecting or of describing the motion involved, it is in principle immaterial to what reference-body we refer the motion. As already mentioned, this is self-evident, but it must not be confused with the much more comprehensive statement called "the principle of relativity," which we have taken as the basis of our investigations. The principle we have made use of not only maintains that we may equally well choose the carriage or the embankment as our reference-body for the description of any event (for this, too, is self-evident). Our principle rather asserts what follows : If we formulate the general laws of nature as they are obtained from experience, by making use of (a) the embankment as reference-body,
(b) the railway carriage as reference-body, then these general laws of nature (e.g. the laws of mechanics or the law of the propagation of light in vacuo) have exactly the same form in both cases. This can also be expressed as follows : For the physical description of
natural processes, neither of the reference bodies K, K1 is unique (lit. " specially marked out ") as compared with the other. Unlike the first, this latter statement need not of necessity hold a priori; it is not contained in the conceptions of " motion" and " reference-body " and derivable from them; only experience can decide as to its correctness or incorrectness.


Up to the present, however, we have by no means maintained the equivalence of all bodies of reference K in connection with the formulation of natural laws. Our course was more on the following Iines. In the first place, we started out from the assumption that there exists a reference-body K, whose condition of motion is such that the Galileian law holds with respect to it : A particle left to itself and sufficiently far removed from all other particles moves uniformly in a straight line. With reference to K (Galileian reference-body) the laws of nature were to be as simple as
possible. But in addition to K, all bodies of reference K1 should be given preference in this sense, and they should be exactly equivalent to K for the formulation of natural laws, provided that they are in a state of uniform rectilinear and non-rotary motion with respect to K ; all these bodies of reference are to be regarded as Galileian reference-bodies. The validity of the principle of relativity was assumed only for these reference-bodies, but not for others (e.g. those possessing motion of a different kind). In this sense we speak of the special principle of relativity, or special theory of relativity. In contrast to this we wish to understand by the "general principle of
relativity" the following statement : All bodies of reference K, K1, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion. But before proceeding farther, it ought to be pointed out that this formulation must be replaced later by a more abstract one, for reasons which will become evident at a later stage. Since the introduction of the special principle of relativity has been justified, every intellect which strives after generalisation must feel the temptation to venture the step towards the general principle of relativity. But a simple and apparently quite reliable consideration seems to suggest that, for the present at any rate, there is little hope of success in such an attempt; Let us imagine ourselves transferred to our old friend the railway carriage, which is travelling at a uniform rate. As long as it is moving uniformly, the occupant of the carriage is not sensible of its motion, and it is for this reason that he can without reluctance interpret the facts of the case as indicating that the carriage is at rest, but the embankment in motion. Moreover, according to the special principle of relativity, this interpretation is quite justified also from a physical point of view.


If the motion of the carriage is now changed into a non-uniform motion, as for instance by a powerful application of the brakes, then the occupant of the carriage experiences a correspondingly powerful jerk forwards. The retarded motion is manifested in the mechanical behaviour of bodies relative to the person in the railway carriage. The mechanical behaviour is different from that of the case previously considered, and for this reason it would appear to be impossible that the same mechanical laws hold relatively to the non-uniformly moving carriage, as hold with reference to the carriage when at rest or in uniform motion. At all events it is clear that the Galileian law does not hold with respect to the non-uniformly moving carriage. Because of this, we feel compelled at the present juncture to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relatvity. But in what follows we shall soon see that this conclusion cannot be maintained.