The Intervals of Space-Time

The A B C of Relativity by Bertrand Russells, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. VII. INTERVALS IN SPACE-TIME

The special theory of relativity, which we have been considering hitherto, solved completely a certain definite problem: to account for the experimental fact that, when two bodies are in uniform relative motion, all the laws of physics, both those of ordinary dynamics and those connected with electricity and magnetism, are exactly the same for the two bodies. “Uniform” motion, here, means motion in a straight line with constant velocity. But although one problem was solved by the special theory, another was immediately suggested: what if the motion of the two bodies is not uniform? Suppose, for instance, that one is the earth while the other is a falling stone. The stone has an accelerated motion: it is continually falling faster and faster. Nothing in the special theory enables us to say that the laws of physical phenomena will be the same for an observer on the stone as for one on the earth. This is [Pg 92]particularly awkward, as the earth itself is, in an extended sense, a falling body: It has at every moment an acceleration[4] towards the sun, which makes it go round the sun instead of moving in a straight line. As our knowledge of physics is derived from experiments on the earth, we cannot rest satisfied with a theory in which the observer is supposed to have no acceleration. The general theory of relativity removes this restriction, and allows the observer to be moving in any way, straight or crooked, uniformly or with an acceleration. In the course of removing the restriction, Einstein was led to his new law of gravitation, which we shall consider presently. The work was extraordinarily difficult, and occupied him for ten years. The special theory dates from 1905, the general theory from 1915.

It is obvious from experiences with which we are all familiar that an accelerated motion is much more difficult to deal with than a uniform one. When you are in a train which is traveling steadily, the motion is not noticeable so long as you do not look out of the window; but when the brakes are applied suddenly you are precipitated forwards, [Pg 93]and you become aware that something is happening without having to notice anything outside the train. Similarly in a lift everything seems ordinary while it is moving steadily, but at starting and stopping, when its motion is accelerated, you have odd sensations in the pit of the stomach. (We call a motion “accelerated” when it is getting slower as well as when it is getting quicker; when it is getting slower the acceleration is negative.) The same thing applies to dropping a weight in the cabin of a ship. So long as the ship is moving uniformly, the weight will behave, relatively to the cabin, just as if the ship were at rest: if it starts from the middle of the ceiling, it will hit the middle of the floor. But if there is an acceleration everything is changed. If the boat is increasing its speed very rapidly, the weight will seem to an observer in the cabin to fall in a curve directed towards the stern; if the speed is being rapidly diminished, the curve will be directed towards the bow. All these facts are familiar, and they led Galileo and Newton to regard an accelerated motion as something radically different, in its own nature, from a uniform motion. But this distinction could only be maintained by regarding motion as absolute, not relative. If all motion is relative, [Pg 94]the earth is accelerated relatively to the lift just as truly as the lift relatively to the earth. Yet the people on the ground have no sensations in the pits of their stomachs when the lift starts to go up. This illustrates the difficulty of our problem. In fact, though few physicists in modern times have believed in absolute motion, the technique of mathematical physics still embodied Newton’s belief in it, and a revolution in method was required to obtain a technique free from this assumption. This revolution was accomplished in Einstein’s general theory of relativity.

It is somewhat optional where we begin in explaining the new ideas which Einstein introduced, but perhaps we shall do best by taking the conception of “interval.” This conception, as it appears in the special theory of relativity, is already a generalization of the traditional notion of distance in space and time; but it is necessary to generalize it still further. However, it is necessary first to explain a certain amount of history, and for this purpose we must go back as far as Pythagoras.

Pythagoras, like many of the greatest characters in history, perhaps [Pg 95]never existed: he is a semi-mythical character, who combined mathematics and priestcraft in uncertain proportions. I shall, however, assume that he existed, and that he discovered the theorem attributed to him. He was roughly a contemporary of Confucius and Buddha; he founded a religious sect, which thought it wicked to eat beans, and a school of mathematicians, who took a particular interest in right-angled triangles. The theorem of Pythagoras (the forty-seventh proposition of Euclid) states that the sum of the squares on the two shorter sides of a right-angled triangle is equal to the square on the side opposite the right angle. No proposition in the whole of mathematics has had such a distinguished history. We all learned to “prove” it in youth. It is true that the “proof” proved nothing, and that the only way to prove it is by experiment. It is also the case that the proposition is not quite true—it is only approximately true. But everything in geometry, and subsequently in physics, has been derived from it by successive generalizations. The latest of these generalizations is the general theory of relativity.

The theorem of Pythagoras was itself, in all probability, a [Pg 96]generalization of an Egyptian rule of thumb. In Egypt, it had been known for ages that a triangle whose sides are 3, 4, and 5 units of length is a right-angled triangle; the Egyptians used this knowledge practically in measuring their fields. Now if the sides of a triangle are 3, 4, and 5 inches, the squares on these sides will contain respectively 9, 16, and 25 square inches; and 9 and 16 added together make 25. Three times three is written “3²”; four times four, “4²”; five times five, “5².” So that we have

3² + 4² = 5².

It is supposed that Pythagoras noticed this fact, after he had learned from the Egyptians that a triangle whose sides are 3, 4 and 5 has a right angle. He found that this could be generalized, and so arrived at his famous theorem: In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides.

Similarly in three dimensions: if you take a right-angled solid block, [Pg 97]the square on the diagonal (the dotted line in the figure) is equal to the sum of the squares on the three sides.

This is as far as the ancients got in this matter.

The next step of importance is due to Descartes, who made the theorem of Pythagoras the basis of his method of analytical geometry. Suppose you wish to map out systematically all the places on a plain—we will suppose it small enough to make it possible to ignore the fact that the earth is round. We will suppose that you live in the middle of the plain. One of the simplest ways of describing the position of a place is to say: starting from my house, go first such and such a distance east, then such and such a distance north (or it may be west in the first case, and south in the second). This tells you exactly where the place is. In the rectangular cities of America, it is the natural method to adopt: in New York you will be told to go so many blocks east (or west) and then so many blocks north (or south). The distance you have to go east is called x, and the distance you have to [Pg 98]go north is called y. (If you have to go west, x is negative; if you have to go south, y is negative.) Let O be your starting point (the “origin”); let OM be the distance you go east, and MP the distance you go north. How far are you from home in a direct line when you reach P? The theorem of Pythagoras gives the answer. The square on OP is the sum of the squares on OM and MP. If OM is four miles, and MP is three miles, OP is 5 miles. If OM is 12 miles and MP is 5 miles, OP is 13 miles, because 12² + 5² = 13². So that if you adopt Descartes’ method of mapping, the theorem of Pythagoras is essential in giving you the distance from place to place. In three dimensions the thing is exactly analogous. Suppose that, instead of wanting merely to fix positions on the plain, you want to fix stations for captive balloons above it, you will then have to add a third quantity, the height at which the balloon is to be. If you call the height z, and if r is the direct distance from O to the balloon, you will have

r² = x² + y² + z²,

and from this you can calculate r when you know x, y, and z. For example, if you can get to the balloon by [Pg 99]going 12 miles east, 4 miles north, and then 3 miles up, your distance from the balloon in a straight line is 13 miles, because

12 × 12 = 144,4 × 4 = 16,3 × 3 = 9,144 + 16 + 9 = 169 = 13 × 13.

But now suppose that, instead of taking a small piece of the earth’s surface which can be regarded as flat, you consider making a map of the world. An accurate map of the world on flat paper is impossible. A globe can be accurate, in the sense that everything is produced to scale, but a flat map cannot be. I am not talking of practical difficulties, I am talking of a theoretical impossibility. For example: the northern halves of the meridian of Greenwich and the ninetieth meridian of west longitude, together with the piece of the equator between them, make a triangle whose sides are all equal and whose angles are all right angles. On a flat surface, a triangle of that sort would be impossible. On the other hand, it is possible to make a square on a flat surface, but on a sphere it is impossible. Suppose you try on the earth: walk 100 miles west, then 100 miles north, then 100 miles east, then 100 miles south. You might think this would make a square, but it wouldn’t, because you would not at the end have come back to [Pg 100]your starting point. If you have time, you may convince yourself of this by experiment. If not, you can easily see that it must be so. When you are nearer the pole, 100 miles takes you through more longitude than when you are nearer the equator, so that in doing your 100 miles east (if you are in the northern hemisphere) you get to a point further east than that from which you started. As you walk due south after this, you remain further east than your starting point, and end up at a different place from that in which you began. Suppose, to take another illustration, that you start on the equator 4,000 miles east of the Greenwich meridian; you travel till you reach the meridian, then you travel northwards along it for 4,000 miles, through Greenwich and up to the neighborhood of the Shetland Islands; then you travel eastward for 4,000 miles, and then 4,000 miles south. This will take you to the equator at a point 4,000 miles further east than the point from which you started.

In a sense, what we have just been saying is not quite fair, because, except on the equator, traveling due east is not the shortest route from a place to another place due east of it. A ship traveling (say) [Pg 101]from New York to Lisbon, which is nearly due east, will start by going a certain distance northward. It will sail on a “great circle,” that is to say, a circle whose centre is the centre of the earth. This is the nearest approach to a straight line that can be drawn on the surface of the earth. Meridians of longitude are great circles, and so is the equator, but the other parallels of latitude are not. We ought, therefore, to have supposed that, when you reach the Shetland Islands, you travel 4,000 miles, not due east, but along a great circle which lands you at a point due east of the Shetland Islands. This, however, only reinforces our conclusion: you will end at a point even further east of your starting point than before.

What are the differences between the geometry on a sphere and the geometry on a plane? If you make a triangle on the earth, whose sides are great circles, you will not find that the angles of the triangle add up to two right angles: they will add up to rather more. The amount by which they exceed two right angles is proportional to the size of the triangle. On a small triangle such as you could make with strings on your lawn, or even on a triangle formed by three ships which can [Pg 102]just see each other, the angles will add up to so little more than two right angles that you will not be able to detect the difference. But if you take the triangle made by the equator, the Greenwich meridian, and the ninetieth meridian, the angles add up to three right angles. And you can get triangles in which the angles add up to anything up to six right angles. All this you could discover by measurements on the surface of the earth, without having to take account of anything in the rest of space.

The theorem of Pythagoras also will fail for distances on a sphere. From the point of view of a traveler bound to the earth, the distance between two places is their great circle distance, that is to say, the shortest journey that a man can make without leaving the surface of the earth. Now suppose you take three bits of great circles which make a triangle, and suppose one of them is at right angles to another—to be definite, let one be the equator and one a bit of the meridian of Greenwich going northward from the equator. Suppose you go 3,000 miles along the equator, and then 4,000 miles due north; how far will you be from your starting point, estimating the distance along a great circle? [Pg 103]If you were on a plane, your distance would be 5,000 miles, as we saw before. In fact, however, your great circle distance will be considerably less than this. In a right-angled triangle on a sphere, the square on the side opposite the right angle is less than the sum of the squares on the other two sides.

These differences between the geometry on a sphere and the geometry on a plane are intrinsic differences; that is to say, they enable you to find out whether the surface on which you live is like a plane or like a sphere, without requiring that you should take account of anything outside the surface. Such considerations led to the next step of importance in our subject, which was taken by Gauss, who flourished a hundred years ago. He studied the theory of surfaces, and showed how to develop it by means of measurements on the surfaces themselves, without going outside them. In order to fix the position of a point in space, we need three measurements; but in order to fix the position of a point on a surface we need only two—for example, a point on the earth’s surface is fixed when we know its latitude and longitude.[Pg 104]

Now Gauss found that, whatever system of measurement you adopt, and whatever the nature of the surface, there is always a way of calculating the distance between two not very distant points of the surface, when you know the quantities which fix their positions. The formula for the distance is a generalization of the formula of Pythagoras: it tells you the square of the distance in terms of the squares of the differences between the measure quantities which fix the points, and also the product of these two quantities. When you know this formula, you can discover all the intrinsic properties of the surface, that is to say, all those which do not depend upon its relations to points outside the surface. You can discover, for example, whether the angles of a triangle add up to two right angles, or more, or less, or more in some cases and less in others.

But when we speak of a “triangle,” we must explain what we mean, because on most surfaces there are no straight lines. On a sphere, we shall replace straight lines by great circles, which are the nearest possible approach to straight lines. In general, we shall take, instead of straight lines, the lines that give the shortest route on [Pg 105]the surface from place to place. Such lines are called “geodesics.” On the earth, the geodesics are great circles. In general, they are the shortest way of traveling from point to point if you are unable to leave the surface. They take the place of straight lines in the intrinsic geometry of a surface. When we inquire whether the angles of a triangle add up to two right angles or not, we mean to speak of a triangle whose sides are geodesics. And when we speak of the distance between two points, we mean the distance along a geodesic.

The next step in our generalizing process is rather difficult: it is the transition to non-Euclidean geometry. We live in a world in which space has three dimensions, and our empirical knowledge of space is based upon measurement of small distances and of angles. (When I speak of small distances, I mean distances that are small compared to those in astronomy; all distances on the earth are small in this sense.) It was formerly thought that we could be sure à priori that space is Euclidean—for instance, that the angles of a triangle add up to two right angles. But it came to be recognized that we could not prove this by reasoning; if it was to be known, it must be known as the result of [Pg 106]measurements. Before Einstein, it was thought that measurements confirm Euclidean geometry within the limits of exactitude attainable; now this is no longer thought. It is still true that we can, by what may be called a natural artifice, cause Euclidean geometry to seem true throughout a small region, such as the earth; but in explaining gravitation Einstein is led to the view that over large regions where there is matter we cannot regard space as Euclidean. The reasons for this will concern us later. What concerns us now is the way in which non-Euclidean geometry results from a generalization of the work of Gauss.

There is no reason why we should not have the same circumstances in three-dimensional space as we have, for example, on the surface of a sphere. It might happen that the angles of a triangle would always add up to more than two right angles, and that the excess would be proportional to the size of the triangle. It might happen that the distance between two points would be given by a formula analogous to what we have on the surface of a sphere, but involving three quantities instead of two. Whether this does happen or not, can only [Pg 107]be discovered by actual measurements. There are an infinite number of such possibilities.

This line of argument was developed by Riemann, in his dissertation “On the hypotheses which underlie geometry” (1854), which applied Gauss’s work on surfaces to different kinds of three-dimensional spaces. He showed that all the essential characteristics of a kind of space could be deduced from the formula for small distances. He assumed that, from the small distances in three given directions which would together carry you from one point to another not far from it, the distances between the two points could be calculated. For instance, if you know that you can get from one point to another by first moving a certain distance east, then a certain distance north, and finally a certain distance straight up in the air, you are to be able to calculate the distance from the one point to the other. And the rule for the calculation is to be an extension of the theorem of Pythagoras, in the sense that you arrive at the square of the required distance by adding together multiples of the squares of the component distances, together possibly with multiples of their products. From certain characteristics in the formula, you can tell what sort of [Pg 108]space you have to deal with. These characteristics do not depend upon the particular method you have adopted for determining the positions of points.

In order to arrive at what we want for the theory of relativity, we now have one more generalization to make: we have to substitute the “interval” between events for the distance between points. This takes us to space-time. We have already seen that, in the special theory of relativity, the square of the interval is found by subtracting the square of the distance between the events from the square of the distance that light would travel in the time between them. In the general theory, we do not assume this special form of interval, except at a great distance from matter. Elsewhere, we assume to begin with a general form, like that which Riemann used for distances. Moreover, like Riemann, Einstein only assumes his formula for neighboring events, that is to say, events which have only a small interval between them. What goes beyond these initial assumptions depends upon observation of the actual motion of bodies, in ways which we shall explain in later chapters.[Pg 109]

We may now sum up and re-state the process we have been describing. In three dimensions, the position of a point relatively to a fixed point (the “origin”) can be determined by assigning three quantities (“co-ordinates”). For example, the position of a balloon relatively to your house is fixed if you know that you will reach it by going first a given distance due east, then another given distance due north, then a third given distance straight up. When, as in this case, the three co-ordinates are three distances all at right angles to each other, which, taken successively, transport you from the origin to the point in question, the square of the direct distance to the point in question is got by adding up the squares of the three co-ordinates. In all cases, whether in Euclidean or in non-Euclidean spaces, it is got by adding multiples of the squares and products of the co-ordinates according to an assignable rule. The co-ordinates may be any quantities which fix the position of a point, provided that neighboring points must have neighboring quantities for their co-ordinates. In the general theory of relativity, we add a fourth co-ordinate to give the time, and our formula gives “interval” instead of spatial distance; moreover we [Pg 110]assume the accuracy of our formula for small distances only. We assume further that, at great distances from matter, the formula approximates more and more closely to the formula for interval which is used in the special theory.

We are now at last in a position to tackle Einstein’s theory of gravitation.