No Hitching Posts

IN one of his autobiographical moments, Einstein has told of an incident that occurred when he was a child of five years. His father showed him a magnetic compass, offering it as an object of amusement. The dancing needle, never at rest, and yet persistently returning to its north-south direction, obeying some inscrutable order of nature as though subject to invisible pushes or pulls, fascinated the small boy. Nor was his interest only the casual response to a new toy. ‘It created in me a sensation which I consider to have been a dominant factor in my life up to the very present,’ Einstein declares.

This was his introduction to electromagnetism, that mysterious agency which not only swings the quivering compass needle, but also thrusts light through space, reflects it back and forth between mirrors, and so lays its swift beam along the crossarms of the Michelson-Morley interferometer for man to measure — if he can!

It seems historically right that the boy who was caught so early by the spell of electromagnetism should as a man be the one to untangle the skein of complexities that had grown out of electromagnetic attempts to measure the flight of the Earth through the ether. Einstein was an eight-year-old schoolboy in Munich when the Cleveland professors collaborated in their famous experiment. He was twentysix years old, employed as an examiner in the Swiss Patent Office at Berne, when he published his fundamental paper of 1905, ‘On the Electrodynamics of Moving Bodies.’ Under that technical title, and expressed for the most part in complicated algebraic symbols, was a view of nature so realistic that it seemed paradoxical — therefore, to many, impossible and unbelievable. But this new view, and particularly the generalization which Einstein gave it ten years later in his great synthesis, the General Theory of Relativity, has met successfully every experimental test. Relativity has explained phenomena which the classical physics could not account for, and it has predicted phenomena which were unknown but which when searched for have been found. ‘To accept it is not merely a pleasure,’ says F. S. C. Northrop, ‘but a necessity’ — a necessity of thought requisite to the intelligent interpretation of this strange hidden world which science now discovers is not the majestic machine of Newtonian kinematics, but something quite astonishingly different. How different we shall presently see.

I

Inasmuch as the theory of relativity is presented by its author in mathematical language, and in strictness of speaking cannot be expressed in any other, there is a certain presumption in every attempt to translate it into the vernacular. One might as well try to interpret Beethoven’s Fifth Symphony on a saxophone. Frankly we acknowledge that such profundities are communicable only by a full orchestra. But, lacking the virtuosity of an orchestra, we may venture to pick out a few chords on the piano, believing, as Einstein’s friend Moszkowski has suggested, that such efforts, even though vague and partial, ‘have a fruitful result if they succeed in focusing the attention of the reader or the hearer so that the connections, the leitmotivs, so to speak, of the doctrine, are at least suggested.’

Such is the plan of the present article: to recall the leitmotivs of relativity.

To begin with, Einstein is a realist. His theory rests on a physical fact. Physical facts are those natural effects which we verify by repeated observation and experiment. All the nineteenth-century attempts to measure the absolute motion of the Earth had been consistent in one respect: they showed an unfailing constancy in the velocity of light. Whether the measuring instrument was moving toward or away from the direction of the light, the velocity of the ray was the same in all experiments, as Michelson’s and Morley’s repeated measurements showed.

Then, said the realist, let us start with this observed physical fact. Since all the experiments agree on the result, let us assume that it is universally true. Let us assume that no matter how fast you move toward an oncoming ray of light, its relative velocity will be the same as it was when you were at rest — or, conversely, no matter how fast you move away from an oncoming ray of light, its rate of motion in your measuring device will show no change. The experiments give this result and no other result; since we are basing our science on experiment, we shall accept the only conclusion that experiment affords and build on it.

This acceptance seems reasonable, a logical consequence of the experimental method, but it involves difficulties. To illustrate: let us suppose that you are in an automobile traveling at thirty miles an hour, while another car is approaching at sixty miles an hour. As you pass on the road, what would you say is the relative speed of the other car? Sixty plus thirty, or ninety miles, of course— but that is quite contrary to what we are asked to believe of the velocity of light. If the two cars were abreast and moving in the same direction, the other car would pass you at the relative speed of sixty minus thirty, or thirty miles an hour — but light does not behave that way, according to the Michelson-Morley experiment. If the velocity of light is 186,000 miles a second, it is always that. If you move along with the light, your speed does not slow it down any. If the light is coming from a star directly ahead, and if the Earth is moving directly toward the light at many miles a second, that fact does not add anything to the relative velocity of the light. We may express the situation in the form of a simple equation: —

Velocity of light+Velocity of Earth= Velocity of light

This sounds suspiciously like nonsense — like saying, for example, that 1+2=1. But it is inherent in our postulate. If you take the velocity of light (186,000 miles a second in vacuo) as unity, and regard all other possible velocities on the Earth or in the skies as fractions thereof, Einstein’s hypothesis holds that, no matter how many of these fractions you add together, you will never exceed unity.

Let us take some velocities and see how this works out. The motion of the Earth in its orbit is about nineteen miles a second. The motion of the Sun and its planets with reference to the fixed stars is about twelve miles a second. The speed of rotation of the Milky Way is about a hundred and seventy-five miles a second. It is likely that our galaxy has in addition a motion of translation with reference to the outer galaxies, since all these outside systems appear to have such, so we may assume that there is an additional velocity here, say five hundred miles a second (which is a slow speed for a galaxy). While these various motions may be in different directions, and some may in fact neutralize others, yet it is possible that at some time the Earth will reach a point in its orbit coincident with the Sun’s reaching a point in its course through the Milky Way such that all these various motions are at one instant in the same general direction. Suppose, at this strategic moment, we flash a beam of light forward into space, thus adding to its motion these various cumulative velocities of our Earth. Common sense says that the light beam ought to be accelerated by a factor 19+12+175 + 500, or 706 miles a second. But the relativity equations admit no exception, the velocity of light cannot be accelerated, and our formula still stands: 186,000+706=186,000. You might imagine the light flashed from a beta particle shot out of the heart of a radium atom at a speed of 150,000 miles a second, but even this enormous addition of motion cannot affect the constancy of our standard. The result remains unchanged: 186,000 + 150,000=186,000. If the motion of the beta particle were in the opposite direction, and its velocity were subtracted instead of added, the result would be the same.

These arithmetical conclusions seem absurdities, and to hold to the postulate regardless of its apparent violation of common sense required a high degree of intellectual courage. Columbus employed something of the same valor when he resolved to accept the idea that ships could sail upside down on the other side of the Earth, a thing obviously contrary to common sense, but a consequence of his belief in the shape of the Earth. Evidence had convinced him that the Earth is a globe, and he acted on that assumption, trusting that his quest would resolve the absurdities. And so with Einstein. Having accepted an observed result as a physical fact, he resolutely held to it as the clue to reality, irrespective of the fictions that it might seem to involve.

‘This,’ says L. L. Whyte, in A Critique of Physics, ‘is the crucial step in the Special Theory of Relativity: the expression as postulates of facts given directly by experience, and the ruthless acceptance of their logical consequences.’

II

The postulate of light’s constancy of motion seems, at first glance, a slender clue on which to build a world. Here was this intricate complex of matter in space, huge aggregates of stuff called stars, huger aggregates of stars called galaxies, all in perpetual flight, but with no fixtures to refer their motions to, no landmarks to stake out the confines and designate the dimensions of the whole. Indeed, there were no confines. It was an infinite Universe; space was absolute, time was absolute, matter was absolute; infinity and eternity were ahead for every coursing star and every vibrant particle, and the forces that drove this mighty mechanism could accelerate velocities without limit, given time enough. That was the accepted world picture in 1905 when this unknown examiner of patents in Switzerland came forward with his idea of a world in which motion was restricted to the velocity of light as a maximum. It was the first limitation of the infinite.

But it was the only tangible result that had emerged from the MichelsonMorley experiments. So Einstein proceeded to ask whether it was possible to modify our fundamental ideas of space, time, and matter in such wise that the velocity of light would always register unity, irrespective of acceleration. The experiments had indicated the impossibility of measuring absolute motion. Might there not be an equal impossibility of detecting absolute space, time, and matter? What if space, time, and matter were not the same for all observers, but were relative entities that vary with the observer — just as Columbus found to be the case with those earlier absolutes, upness and downness?

Such questions give a hint of the method of reasoning that, expressed in the precise phrasing of mathematics and freed of the anthropomorphic implications of common sense, produced the theory of relativity. If you grant the principle of the invariability of the velocity of light, then all the rest follows logically. By working from this principle it can be shown that as the motion of a planet, rocket, airplane, or other moving body increases, the measurement of the dimensions of the body and of all its material objects changes at a rate which is proportional to the acceleration. As the velocity of the moving body quickens, its spacial distances shrink, time moves more slowly, and matter grows more massive — and these changes become more pronounced as the velocity approaches that of light itself.

1. I have said that matter grows more massive. If the Earth could be gradually speeded up, weights would gradually increase. If the Earth’s motion could be accelerated to the velocity of 161,000 miles a second, every pound on it or in it would weigh two pounds, and with increasing acceleration the masses would yet more rapidly increase until at the velocity of light each terrestrial mass would be infinitely great. This consequence follows from the principle of inertia, since mass (or inertia) is the resistance that a material body offers to a change of motion. If the resistance remained the same irrespective of motion, then any acceleration able to increase the velocity at all would by continued application go on increasing it indefinitely. In other words, it would then be possible to produce a velocity greater than that of light, and indeed there would be no limit. But we have accepted and are building on the postulate that nothing can exceed in motion the velocity of light, therefore we must assume an inertia or resistance which will harmonize with this concept. It can be shown that if the motion of an accelerated body causes its mass to change in the ratio determined by Einstein, then a resistance to change is set up such that it increases rapidly as the velocity of light is approached, and at that velocity, 186,000 miles a second, becomes an infinite resistance making impossible any further acceleration.

Thus, the fact that the velocity of light is a constant, and that no body can ever move faster than that fixed speed, requires as a consequence this curious behavior of matter. The measurement of material particles shot out of radium, at speeds ranging up to 185,000 miles a second, beautifully confirms this law of relativity. By their deflection between the poles of a powerful magnet, the swift particles can be weighed in flight — and it has been found that the faster they move, the greater is their mass.

2. I have said that time passes more slowly. This is a consequence of this mass variation. On a planet moving at a higher velocity, the masses of the pendulums, balance wheels, and other time-keeping mechanisms are increased in the manner suggested in item 1 above; therefore, as heavier masses they must move more slowly. As the more massive bob swings more slowly, lengthening the duration between ticks, the hands of the clock turn more slowly. A person on the planet would be unaware of this slowing down of time, for, since all clocks are equally affected, there would be no master clock by which to check deviations. To learn the right time from another planet would involve the transfer of a light signal, and, since light has its finite velocity, some time would be required in getting the light signal communicated, thus introducing an element of uncertainty and invalidating any attempt to fix the standard of one system by that of another. In other words, the idea of simultaneity loses its sense.

Perhaps some hardheaded ‘rationalist’ will insist that this is pushing our thesis too far. Even if we cannot synchronize the time between two systems, may there not be two events occurring simultaneously? Quite independent of our ability to clock them, may not exact coincidences really occur? Relativity must answer, No. ‘If such questions are allowed,’ writes D. E. Richmond, in The Dilemma of Modern Physics, ‘one gets into no end of trouble. In conjunction with the theory of relativity there has therefore grown up a philosophy of science which states that questions are physically meaningless unless they admit of experimental answer.’

3. I have said that spacial distances shrink. Yardsticks contract in the direction of the accelerated motion of the planet or other body; but as all other things on the planet similarly contract, including the observer, the shrinkage will be imperceptible — except to some outsider moving at a different velocity.

One consequence of this close relatedness of space and time, and their relativity to motion, is the fact that when a velocity is accelerated by another velocity, the two never add up to their common-sense arithmetical sum. Thus, if we see a railway train traveling at 60 miles an hour, and in the train a man is walking forward at 5 miles an hour, the velocity of the man with reference to the ground is not 60+5 as we should expect from classical physics. No, it is something less than 65 by a fraction in such a proportion to the velocity of light that at the latter velocity it would disappear. All accelerations are inexorably tied to this constant, this ‘first discovered law of the new physics,’ in a ratio such that its rule can never be violated.

But all these transformations, it must be remembered, are apparent only to observers outside the moving system and at rest relative to it. Indeed, the transformations have their basis in the fact that the measured values of one system (or frame of reference) are referred to those of another in a different state of motion. If an airplane of thirty feet length and ten tons weight should take to the air, and attain a speed of 161,000 miles a second, its transformations would not be apparent to its pilot, though to an observer on the ground they would be conspicuous — assuming, of course, superhuman sensory powers to match the highly imaginary airplane speed. To the man on the ground the plane would appear to be but fifteen feet long, the hands of the clock carried in the plane would appear to be moving only half as fast as those on the ground, and the weight of the plane as ‘sensed’ by our imaginary gravitational detector would register twenty tons instead of ten. While none of these changes would be known to the occupant of the plane, — since all his measuring devices have changed in exactly the same ratio, — he would behold equally strange transformations down on the ground. For, from the point of view of the aviator, appearances are the same as though it were the Earth that was sweeping past at 161,000 miles per second while his plane was at rest. Therefore, the aviation field would seem half as long as it was when he took off, the clock over the hangar entrance would seem to be moving half as fast as normal, and objects on the ground would seem double weight.

Which measurer is right — the pilot on the moving plane, or the observer on the stationary Earth? Both, answers the relativist, for who is to say which is the true length, the true time, or the true mass of anything? An observer on the Sun would see that the Earth is not stationary, and consequently would see transformations not apparent to either the pilot or the groundsman; while an observer on Arcturus would see that the Sun too is in motion, dragging the Earth and the airplane along its path at an added velocity; therefore, all measurements of objects on Sun, Earth, and airplane would be still different for the eye of Arcturus. Since there is no hitching post anywhere, no observer at rest who can be said to have the measuring rod or the clock, we must discard the notion of space, time, and matter as absolutes. So far as we can apprehend them by our measuring devices, each of these entities is relative, relative to the motion of the observer — therefore, in its ultimate definition, a derivative of motion. Thus that universal quality which Heraclitus twentyfive centuries ago attributed to nature when he said, ’All things flow,’ is now revealed as the moulder, the transformer, and in a sense the creator, of the things.

‘This idea of the derivative character of matter, space, and time lies at the heart of the modern principle of relativity,’ says Herbert Dingle, in Relativity for All. ‘It deserves particular emphasis, for, if it is at once grasped, the greater part of the difficulty of the subject disappears. It is the event that is the immediate entity of perception; nature is the sum total of events, and every instrument of thought that our minds employ can be traced back to its ultimate origin in events. Two observers of nature see, not necessarily the same matter, but the same events, because events finally constitute the external physical world.’

III

But we see the external world as lumps of matter and waves of radiation, these stars, planets, solid rocks, green growing things, and the other objects, animate and inanimate, of nature. Are they events? The perception of them is, and that is all we really know. If your eye touches a star as truly as your hand touches this table, then the event of seeing has equal validity with the event of touching, and every glance at the heavens is as precisely an event as every handclasp, every birth, death, railway collision, or earthquake. Of the handclasp, birth, death, accident, or earthquake, we know two details if we know the event at all directly: (1) the place of the occurrence, and (2) the time of the occurrence. The two are inseparable; we cannot experience space except at a time, nor can we experience time except at a place.

All measurements have this same dual character. When we say that an hour has passed, we mean that the pointer hand of a clock or watch has described a certain distance, a measurement of space on a dial. That is our hour, and whether we measure time as the Naval Observatory does by recording the transit of stars across the hairline of a meridian-circle telescope, or as the ploughboy does by estimating the position of the Sun in the sky, it is a distance measurement. Similarly with measurements of space. From New York to Chicago is 17 hours by express train, and the hours of time are as valid a rating of the distance as the miles of space which separate the two cities.

Thus, the interval between two events, as well as the events themselves, is a closely knit fabric of space and time, and every determination of the interval involves both time measurement and space measurement. One cannot make the trip from New York to Chicago, or from the Sun to Arcturus, without reckoning a time difference as well as a space difference; and likewise one cannot run his eye from one end of the measuring rod to the other without rating (unconsciously perhaps) the time lapse between the two events as well as the space distance.

These ideas were not included by Einstein in his original treatment of relativity, but were first suggested the same year by Henri Poincaré, the French mathematician; but it was their elaboration and formalization by the Göttingen professor, Herman Minkowski, that finally won the attention of men who think on these things. In 1908, Minkowski gave Einstein’s theory a mathematical restatement. He pointed out that the new ideas of motion could be portrayed very elegantly by accepting this inseparability of space and time as a physical fact, and treating time as a fourth dimension coördinate with the three dimensions of space. ‘From this hour,’ said Minkowski, ‘space of itself and time of itself sink to mere shadows, and only a kind of blend of the two retains an independent existence.’

This blend of the two ‘shadows’ is the world of space-time. It is within this space-time continuum that motions produce the events which we experience as the phenomena of nature.

Every event takes place at a point of space which the mathematician finds it convenient to designate dimensionally by three numbers (the space coordinates, x, y, z), and at a moment of time (the time coördinate, t). These four numbers define a world-point or world-element, and the sum total of the world-elements is the world itself, the Universe.

But within space-time there is something. Minkowski prefers not to call it matter or electricity, as these terms are already barnacled with meanings and implications. Call it substance, and call every atom and light wave and lesser or larger particle a ‘substantial point.’ But the substantial points are not stationary. Everything is in motion, so every substantial point is continually changing its position. At a given moment a given atom coincides with the world-point x, y, z, t, but at the next moment it coincides with a different world-point x’, y’, z’, t’ — just as the train going from New York to Chicago is continually changing its position. If Albany at 4 P. M. on November 1, 1935, is x, y, z, t, then Schenectady at 4.29 the same day is x’, y’, z’, t’, and so on. The worldpoints, corresponding to the railway stations, are stationary; the substantial points, corresponding to the train, are moving. Just as the train travels its track from one station to the next, so the atom travels its path through space-time, describing a continuous track which Minkowski called its world-line.

Every substantial point has its world-line, and history is made — that is, events occur — when world-lines cross. When an astronomer observes a star, the event is the result of the intersection of one world-line (that described by the light ray sent out by the star) with another world-line (that described by the retina of the observer’s eye). When a sleeper is awakened by the nightingale’s song, the phenomenon is likewise an effect of a crossing of world-lines — the world-line of the sound wave encounters the world-line of the eardrum. And so with everything that is seen, heard, felt, tasted, smelled, or in any other way sensed. It matters not whether the encounter is personally willed, as in the case of the star observation, or accidental, as was the nightingale’s song; it is the intersection of world-lines that makes the events of life, all the phenomena of nature, from stars to fireflies, from observing a solar eclipse to the reading of this word. All devices for taking measurements— clocks, thermometers, speedometers, electroscopes, spectrographs, thermopiles — are simply arrangements for recording or registering the intersections of world-lines. A complex of atoms, such as the human body is, threads an invisible cable of world-lines through space-time; every moment of life is a multiplex of encounters with other world-lines.

And so with those vaster complexes, the stars. Their motions, pulsations, occupations, and catastrophes must also be wrought of this strange interweaving of space and time, threaded with world-lines innumerable, endlessly crossing and recrossing, in the mysterious manifold of destiny.

IV

This revelation of the world as a four-dimensional medium, fabricated of three parts space and one part time, brought to the doctrine of relativity a new and powerful illumination. It provided the natural mathematical foundation for the novel conception of space, time, and matter as derivatives of motion. But it did more. It pointed the way to an extension of the principle to embrace a wider range of phenomena, and gave the lead which Einstein now followed to his great synthesis.

His first achievement, that published in his original paper of 1905, is known as the Special Theory of Relativity because its conclusions are specialized and apply only to uniform motion in a straight line. But it is obvious that the world has not restricted its kinematics to uniform linear motion: we have only to note the rotations and revolutions of stars, planets, and other massive bodies to see that straight-line motion is rare. A fundamental theory of nature must include every kind of motion.

Moreover — or, perhaps one should say, consequently — the Special Theory had left untouched the concept of physical forces. This concept, you will remember, had been invented by Newton to account for the acceleration of bodies. The forces of gravitation, however, are not any less mysterious than the propelling angels which they supplanted; in fact, the forces might be called the angels under another name; and it was clear that any generalization of relativity to include rotation and revolution must also take into account the law of gravitation and its underlying postulates.

One of these postulates is the idea of action at a distance. Newton assumed that gravitation propagates its influence at infinite speed—that the attraction between two bodies is instantaneously operative. But the Special Theory of Relativity holds that nothing can be accelerated to a velocity greater than that of light.

Another challenge to gravitation was involved in Einstein’s picture of the relativity of space, time, and matter. According to Newton, the attraction between two bodies operates with a force which is proportional to the product of the masses divided by the square of the distance. But when masses and distance change with the relative motion of the observer, who is to say what are the masses and the distance?

Besides, there was something unaccountable in the perfect indifference of the force of gravitation to other influences. You can pound a body into whatever form you wish, smash it and grind it to powder, electrify it, magnetize it, heat it, evaporate it, do with it what you will, and none of these changes will affect its gravitational response. This is not true of the other forces of nature. If you heat a magnet, it loses its magnetism. If you touch an electrified body with a grounded copper wire, its charge disappears. You can screen off heat, magnetism, electricity, light, radio waves; even the penetrating gamma rays of radium are stopped by a few sheets of lead — but nothing will stop or modify the force of gravitation. It acts as if it were everywhere, instantaneously, and unconditionally operative. There is no other force like it.

Perhaps it is not a force? Einstein asked that searching question. Suppose a race of people lived on a disk under an enclosing dome which shut out the rest of the world. Suppose the disk and dome began slowly to rotate; the inhabitants would be unable to detect the smooth motion, for they would have no outside point of reference by which to gauge it. But presently, as they ventured away from the centre of the disk, they would feel a strange influence tending to pull them outward; and the farther they wandered from the centre, the more pronounced would be this influence. They might develop a theory of a gravitational force in the wall of the dome, or outside, that was attracting them, and it would be possible to derive a mathematical law to account for the behavior which we know as the familiar centrifugal effect.

But who, when he speaks of centrifugal force, really pictures a force? When an object is flung off a whirling wheel, we attribute the centrifugal effect to its inertia. The object simply tends to continue in the direction in which it is moving — that is, in a straight line tangent to the curve of the wheel. It is taking the path of least resistance, the course which will cause the least change in its motion, therefore the least opposition to its inertia. This inertia, according to an ancient law of physics, is a direct index to the mass of the body; it is also called the inertial mass, and determines the degree of force necessary to give the body an acceleration. But the body also has its gravitational mass, an index to its gravitational force. All this is familiar Newtonian mechanics, but to many a hypercritical physicist it had long seemed strange that the inertial mass and the gravitational mass of any given body always came out equal.

Perhaps they are the same thing, ventured Einstein. By rotating the disk under the dome we create an artificial gravitational field, though we know it is really an effect of inertia. May it not be that the gravitational fields of Earth, Moon, Sun, and stars are equally fictitious, being also effects of inertia? Is it not possible that certain natural tracks are the paths of least resistance for the movement of inertial masses? Indeed, are not these tracks the world-lines which ponderable bodies describe as they move through Minkowski’s world of space-time?

This principle of the equivalence of inertia and gravitation was announced by Einstein in 1911, six years after his publication of the Special Theory, and its attainment marks the first step toward the generalization which came four years later. Einstein had long since given up his post in the Swiss Patent Office, and while serving as professor at various universities, first at Zurich, then at Prague, then Zurich again, and finally at Berlin, worked persistently on this problem of gravitation. His collaborator was Marcel Grossman, an able mathematician and friend of student days. ‘Our final results appear almost self-evident,’ said Einstein in an address at the University of Glasgow in 1933, ‘ but the years of searching in the dark for a truth that one feels but cannot express, the intense desire, and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are known only to him who has himself experienced them.’

Einstein and his associate had settled in Berlin early in 1914, in those electric months that, preceded the World War; and while the military machine was moving into action on half a dozen European fronts the gentle professor within the four walls of his study room was winning a far more adventurous and lasting, if less sanguinary, victory. The difficulties involved, the mental struggles that had to be undergone, the trials and failures and retrials and eventual success, are concisely suggested in the Glasgow address.

V

In brief, his solution involved the adoption of a new kind of geometry — new only in the sense that its application was novel, for the metric itself had been provided fifty years before by the Göttingen professor Bernhard Reimann.¹ Einstein discovered that the geometry of Euclid — the familiar geometry of common usage, in which a straight line is the shortest distance between two points, and in which the angles of a triangle always add up to two right angles — could not be used as a description of the world of space-time.

By its name geometry means earth-measurement, and by geometry we ordinarily mean the system first set forth by the Greek mathematician Euclid, which has been taught in our schools ever since, more than 2000 years. But the theory of relativity shows that only in regions almost infinitesimally small is Euclidean geometry applicable. On the surface of a tennis court the sum of the angles of a triangle will always equal two right angles, and here too the Pythagorean theorem of the square on the hypotenuse will always hold. But on the surface of a sphere these rules break down. San Francisco and Yokohama both lie close to the 35th parallel of latitude, and on a map this parallel appears as the straight line between the two. But no shipmaster, wishing to make the shortest voyage between the two ports, would follow that parallel. No, he would sail a course which describes the arc of a great circle sweeping northward toward the Alaskan islands and then southward to Japan. This curved course is the geodesic — the actual shortest distance between the two points on the globe.

On the two-dimensional surface of the tennis court, a straight line is the geodesic. On the three-dimensional surface of the Earth, the arc of a great circle is the geodesic. In the four-dimensional continuum of space-time, the orbit of a planet or star is the geodesic.

The analogy cannot be pressed further, for the four-dimensional form is not picturable, but in general we may say that just as the surface of the Earth is curved into a shape which determines its geodesics, so is space-time curved. It is curved by the matter or masses which it contains.

The idea may be suggested by the effect of a magnet. If you bring a powerful magnet into a room in which there is a compass, the needle will instantly respond. The space surrounding the magnet shows new properties, for if you hold a sheet of paper above the poles of the magnet and sprinkle the paper with iron filings, the filings will arrange themselves in a symmetrical pattern of lines indicating the magnetic field. The effect manifests itself only in the proximity of a magnet; and the larger the magnet, the more powerful and far-reaching is its field. Now, instead of attributing the response of the filings and needle to some inexplicable force within the magnet, we might find that there was a distortion of the space surrounding the magnet and see the behavior of the filings and needle as a natural conformity to this space peculiarity.

This supposition is not offered to explain magnetism, but only to illustrate the new conception of gravitation. There are no forces. There is simply matter in motion through space-time, and the matter itself is described as singularities in space-time. As these singularities — the stars and planets, for example — move through the allsurrounding medium, they affect it. Space-time, whose normal undisturbed tendency is to smooth out into unrelieved flatness, crumples up around the Sun into vast invisible and unpicturable curvature, and the planets moving with the inertia of their masses simply swing into the tracks ordained for them by the curvature. Nearer the Sun the curvature would be so acute as to spiral them into its depths; farther from the Sun the curvature flattens and at great distances it would approximate a straight line. ‘Gravitation,’ says the British physicist, E. T. Whittaker, ‘simply represents a continual effort of the universe to straighten itself out.’

We have spoken of these orbits of planets and stars as geodesics, and have likened them to the geodesics on the tennis court and on the surface of a sphere. Curiously, though, a geodesic in space-time is not the shortest distance between two points, but the longest distance that a body can cover in a given time. It can be shown that the orbit of the Moon is that track in which it can traverse the space in the shortest time — and so with all the planetary orbits. Nature, it seems, is lazy. One of its fundamentals is the ‘Principle of Least Work.’ To squeeze the most travel out of the least time would seem to be a rather consistent ingenuity in a world where motion appears to be the primary rule of being.

The General Theory of Relativity brings us a picture of the cosmos in which space by itself and time by itself have indeed ceased to exist, and the blend of the two has become a supple theatre for events. The theatre continually changes with the events which it stages. Not only is space-time moulded and transformed at every point by the matter and motions which it contains, but the very rules of the geometry by which its manifold is measured are shown to be relative, and the relations are expressed in purely physical terms. Gravitation is a consequence of this geometrical relativity. Light rays travel, not necessarily in the straight lines of Euclidean space, but in the geodesics determined by these peculiarities. Particles affect other particles, planets affect other planets, not through any mysterious attraction, but through the inevitable distortions of space-time which they produce.

All things, by immortal power,
Near or far,
Hiddenly
To each other linked are,
That thou canst not stir a flower
Without troubling of a star.

Everything affects everything. That is the meaning of the new law of gravitation.

1. Even earlier than Reimann’s work is that of the Russian mathematician, Nicholas Ivanovitch Lobatchewsky, who was the first to reject Euclid’s well-known postulate of parallel lines. It was, however, Reimann’s particular form of the new geometry that Einstein adopted. — AUTHOR↩