Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of USD5,000

THE PRACTICAL SIGNIFICANCE OF RELATIVITY

The Best Discussion of the Special Theory Among All the Competing Essays

BY PROFESSOR HENRY NORRIS RUSSELL, PRINCETON UNIVERSITY

Can a small child catch a baseball moving sixty miles an hour without getting hurt? We should probably answer “No”—but suppose that the boy and his father were sitting side by side in an express train, and the ball was tossed lightly from one to the other. Then there would be no trouble about it, whether the train was standing still, or going at full speed. Only the relative motion of ball and boy would count.

This every-day experience is a good illustration of the much discussed Principle of Relativity, in its simplest form. If there were no jolting, the motion of the train, straight ahead at a uniform speed, would have no effect at all upon the relative motions of objects inside it, nor on the forces required to produce or change these motions. Indeed, the motion of the earth in its orbit, which is free from all jar, but a thousand times faster, does not influence even the most delicate apparatus. We are quite unconscious of it, and would not know that the earth was moving, if we could not see other bodies outside it. This sort of relativity has been recognized for more than two centuries and lies at the bottom of all our ordinary dynamical reasoning, upon which both science and engineering are based.

But there are other things in nature besides moving bodies,—above all, light, which is intimately related to electricity and magnetism, and can travel through empty space, between the stars. It moves at the enormous speed of 186,000 miles per second, and behaves exactly like a series of vibrations or “waves.” We naturally think of it as travelling through some medium, and call this thing, which carries the light, the “ether.”

Can we tell whether we are moving through this ether, even though all parts of our apparatus move together, and at the same rate? Suppose that we have two mirrors, M and N, at equal distances, d, from a point O, but in directions at right angles to one another, and send out a flash of light from O. If everything is at rest, the reflected flashes will evidently come back to O at the same instant, and the elapsed time will be 2 d slash c seconds if c is the velocity of light.

But suppose that O, M, and N are fastened to a rigid frame work, and all moving in the direction upper O upper M, with velocity V. The light which goes from O toward M, at the speed c, will overtake it with the difference of their speeds, c minus v, taking d slash left-parenthesis c minus v right-parenthesis seconds to reach M. On the way back, O will be advancing to meet it, and the return trip will occupy d slash left-parenthesis c plus v right-parenthesis seconds. The elapsed time for the round trip comes out 2 c d slash left-parenthesis c squared minus v squared right-parenthesis seconds, which is longer than when the system was at rest—the loss of time in the “stern chase” exceeding the saving on the return.

The light which is reflected from N has a different history. When it starts, O and N have certain positions in the ether, upper O 1 and upper N 1. By the time it reaches the mirror, this is at upper N 2, and O is at upper O 2, and when it returns, it finds O at upper O 3. The distances for the outward and inward journeys are now equal, but (as is obvious from the figure), each of them is greater than d, or upper O 2 upper N 2, and the time for the round trip will be correspondingly increased. A simple calculation shows that it is 2 d slash StartRoot c squared minus v squared EndRoot.

The increase above the time required when the system was at rest is less in this case than the preceding. Hence, if the apparatus is moving through the ether, the flashes reflected from M and N will not return at the same instant.

For such velocities as are attainable—even the 18 miles per second of the earth in its orbit—the difference is less than a hundred-millionth of the elapsed time. Nevertheless, Michelson and Morley tried to detect it in their famous experiment.

A beam of light was allowed to fall obliquely upon a clear glass mirror (placed at O in the diagram) which reflected part of it toward the mirror, M, and let the rest pass through to the mirror N. By reuniting the beams after their round trips, it was possible to tell whether one had gained upon the other by even a small fraction of the time of vibration of a single light wave. The apparatus was so sensitive that the predicted difference, though amounting to less than a millionth part of a billionth of a second, could easily have been measured; but they actually found no difference at all—though the earth is certainly in motion.

Other optical experiments, more intricate, and even more delicate, were attempted, with the same object of detecting the motion of the earth through the ether; and they all failed.

The Special Theory and Its Surprising Consequences

It was upon these facts that Einstein based his original, or “special” theory of Relativity. He assumed boldly that the universe is so constituted that uniform straight-ahead motion of an observer and all his apparatus will not produce any difference whatever in the result of any physical process or experiment of any kind. Granting this, it follows that if all objects in the visible universe were moving uniformly together in any direction, no matter how fast, we could not find this out at all. We cannot determine whether the universe, as a whole, is at rest or in motion, and may as well make one guess as another. Only the relative motions of its parts can be detected or studied.

This seems simple and easy enough to understand. But the consequences which follow from it are extraordinary, and at first acquaintance seem almost absurd.

In the first place, if an observer measures the velocity of light, he must always get the same result, no matter how fast he and his apparatus are moving, or in what direction (so long as the motion is uniform and rectilinear). This sounds harmless; but let us go back to the Michelson-Morley experiment where the light came back in exactly the same time from the two mirrors. If the observer supposes himself to be at rest, he will say that the distances upper O upper M and upper O upper N were equal. But if he fancies that the whole universe is moving in the direction upper O upper M, he will conclude that M is nearer to O than N is—for if they were equidistant, the round-trip would take longer in the first case, as we have proved. If once more he fancies that the universe is moving in the direction upper O upper N, he will conclude that N is nearer to O than M is. His answer to the question which of the two distances, upper O upper M or upper O upper N, is the greater will therefore depend on his assumption about the motion of the universe as a whole.

Similar complications arise in the measurement of time. Suppose that we have two observers, A and B, provided with clocks which run with perfect uniformity, and mirrors to reflect light signals to one another. At noon exactly by his clock, A sends a flash of light towards B. B sees it come in at 12:01 by his clock. The flash reflected from B’s mirror reaches A at 12:02 by A’s clock. They communicate these observations to one another.

If A and B regard themselves as being at rest, they will agree that the light took as long to go out as it did to come back, and therefore that it reached B at just 12:01 by A’s clock, and that the two clocks are synchronized. But they may, if they please, suppose that they (and the whole universe) are moving in the direction from A towards B, with half the speed of light. They will then say that the light had a “stern chase” to reach B, and took three times as long to go out as to come back. This means that it got to B at 1½ minutes past noon by A’s clock, and that B’s clock is slow compared with A’s. If they should assume that they were moving with the same speed in the opposite direction, they would conclude that B’s clock is half a minute fast.

Hence their answer to the question whether two events at different places happen at the same time, or at different times, will depend on their assumption about the motion of the universe as a whole.

Once more, let us suppose that A and B, with their clocks and mirrors, are in relative motion, with half the speed of light, and pass one another at noon by both clocks. At 12:02 by A’s clock, he sends a flash of light, which reaches B at 12:04 by his clock, is reflected, and gets back to A’s clock at 12:06. They signal these results to each other, and sit down to work them out. A thinks that he is at rest, and B moving. He therefore concludes that the light had the same distance to go out as to return to him and took two seconds each way, reaching B at 12:04 by A’s clock, and that the two clocks, which agreed then, as well as at noon, are running at the same rate.

B, on the contrary, thinks that he is at rest and A in motion. He then concludes that A was much nearer when he sent out the flash than when he got it back, and that the light had three times as far to travel on the return journey. This means that it was 12:03 by A’s clock at the instant when the light reached B and B’s clock read 12:04. Hence A’s clock is running slow, compared with B’s.

Hence the answer to the question whether two intervals of time, measured by observers who are in motion relative to one another, are of the same or of different durations, depends upon their assumptions about the motion of the universe as a whole.

Now we must remember that one assumption about the motion of the universe as a whole is exactly as good—or bad—as another. No possible experiment can distinguish between them. Hence on the Principle of Relativity, we have left no absolute measurement of time or space. Whether two distances in different directions are to be called equal or not—whether two events in different places are to be called simultaneous or not—depends on our arbitrary choice of such an assumption, or “frame of reference.” All the various schemes of measurement corresponding to these assumptions will, when applied to any imaginable experiment, predict exactly the same phenomena. But, in certain important cases, these predictions differ from those of the old familiar theory, and, every time that such experiments have been tried, the result has agreed with the new theory, and not with the old.

We are therefore driven to accept the theory of relativity, strange as it is, as being more nearly “true to nature” than our older ideas. Fortunately, the difference between the results of the two become important only when we assume that the whole visible universe is moving together much faster than any of its parts are moving relatively to one another. Unless we make such an unwarranted assumption, the differences are so small that it takes the most ingenious and precise experiments to reveal them.

The Generalization

Not content with all this, Einstein proceeded, a few years ago, to develop a “general” theory of relativity, which includes the effects of gravitation.

To make this idea clear, let us imagine two observers, each, with his measuring instruments, in a large and perfectly impervious box, which forms his “closed system.”

The first observer, with his box and its contents, alone in space, is entirely at rest.

The second observer, with his box and its contents, is, it may be imagined, near the earth or the sun or some star, and falling freely under the influence of its gravitation.

This second box and its contents, including the observer, will then fall under the gravitational force, that is, get up an ever-increasing speed, but at exactly the same rate, so that there will be no tendency for their relative positions to be altered.

According to Newton’s principles, this will make not the slightest difference in the motions of the physical objects comprising the system or their attractions on one another, so that no dynamical experiment can distinguish between the condition of the freely falling observer in the second box and the observer at rest in the first.

But once more the question arises: What could be done by an optical experiment?

Einstein assumed that the principle of relativity still applied in this case, so that it would be impossible to distinguish between the conditions of the observers in the two boxes by any optical experiment.

It can easily be seen that it follows from this new generalized relativity that light cannot travel in a straight line in a gravitational field.

Imagine that the first observer sets up three slits, all in a straight line. A ray of light which passes through the first and second will obviously pass exactly through the third.

Suppose the observer in the freely falling system attempts the same experiment, having his slits P, Q, R, equally spaced, and placing them at right angles to the direction in which he is falling. When the light passes through P, the slits will be in certain position upper P 1 upper Q 1 upper R 1 (Figure). By the time it reaches Q, they will have fallen to a lower level, upper P 2 upper Q 2 upper R 2, and when it reaches R, they will be still lower, upper P 3 upper Q 3 upper R 3. The times which the light takes to move from P to Q and Q to R will be the same: but, since the system is falling ever faster and faster the distance upper R 2 upper R 3 will be greater than upper Q 1 upper Q 2. Hence, if the light which has passed through P and Q moves in a straight line, it will strike above R, as is illustrated by the straight line in the figure. But, on Einstein’s assumption, the light must go through the third slit, as it would do in the system at rest, and must therefore move in a curved line, like the curved line in the figure, and bend downward in the direction of the gravitational force.

The Tests

Calculation shows that the deviation of light by the moon or planets would be too small to detect. But for a ray which had passed near the sun, the deflection comes out 1.7, which the modern astronomer regards as a large quantity, easy to measure. Observations to test this can be made only at a total eclipse, when we can photograph stars near the sun, on a nearly dark sky. A very fine chance came in May, 1919, and two English expeditions were sent to Brazil and the African coast. These photographs were measured with extreme care, and they show that the stars actually appear to be shifted, in almost exactly the way predicted by Einstein’s theory.

Another consequence of “general relativity” is that Newton’s law of gravitation needs a minute correction. This is so small that there is but a single case in which it can be tested. On Newton’s theory, the line joining the sun to the nearest point upon a planet’s orbit (its perihelion) should remain fixed in direction, (barring certain effects of the attraction of the other planets, which can be allowed for). On Einstein’s theory it should move slowly forward. It has been known for years that the perihelion of Mercury was actually moving forward, and all explanations had failed. But Einstein’s theory not only predicts the direction of the motion, but exactly the observed amount.

Einstein also predicts that the lines of any element in the solar spectrum should be slightly shifted towards the red, as compared with those produced in our laboratories. Different observers have investigated this, and so far they disagree. The trouble is that there are several other influences which may shift the lines, such as pressure in the sun’s atmosphere, motion of currents on the sun’s surface, etc., and it is very hard to disentangle this Gordian knot. At present, the results of these observations can neither be counted for or against the theory, while those in the other two cases are decisively favorable.

The mathematical expression of this general relativity is intricate and difficult. Mathematicians—who are used to conceptions which are unfamiliar, if not incomprehensible, to most of us—find that these expressions may be described (to the trained student) in terms of space of four dimensions and of the non-Euclidean geometry. We therefore hear such phrases as “time as a sort of fourth dimension,” “curvature of space” and others. But these are simply attempts—not altogether successful—to put mathematical relationships into ordinary language, instead of algebraic equations.

More important to the general reader are the physical bearings of the new theory, and these are far easier to understand.

Various assumptions which we may make about the motion of the universe as a whole, though they do not influence the observed facts of nature, will lead us to different ways of interpreting our observations as measurements of space and time.

Theoretically, one of these assumptions is as good as any other. Hence we no longer believe in absolute space and time. This is of great interest philosophically. Practically, it is unimportant, for, unless our choice of an assumption is very wild, our conclusions and measurements will agree substantially with those which are already familiar.

Finally, the “general” relativity shows that gravitation and electro-magnetic phenomena—(including light) do not form two independent sides of nature, as we once supposed, but influence one another (though slightly) and are parts of one greater whole.

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