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

EINSTEIN'S THEORY OF RELATIVITY

A Simple Explanation of His Postulates and Their Consequences

BY T. ROYDS, KODAIKANAL OBSERVATORY, INDIA

Einstein’s theory of relativity seeks to represent to us the world as it really is instead of the world of appearances which may be deceiving us. When I was in town last week to buy 5 yards of calico I watched the draper very carefully as he measured the cloth to make sure I was not cheated. Yet experiment can demonstrate, and Einstein’s theory can explain, that the draper’s yardstick became longer or shorter according to the direction in which it was held. The length of the yardstick did not appear to me to change simply because everything else in the same direction, the store, the draper, the cloth, the retina of my eye, changed length in the same ratio. Einstein’s theory points out not only this, but every case where appearances are deceptive, and tries to show us the world of reality.

Einstein’s theory is based on the principle of relativity and before we try to follow his reasoning we must spend a little time in understanding what he means by “relativity” and in grasping how the idea arises. Suppose I wish to define my motion as I travel along in an automobile. I may be moving at the rate of 25 miles an hour relative to objects fixed on the roadside, but relative to a fellow-passenger I am not moving at all; relative to the sun I am moving with a speed of 18½ miles per second in an elliptical orbit, and again relative to the stars I am moving in the direction of the star Vega at a speed of 12 miles per second. Thus motion can only be defined relative to some object or point of reference. Now this is not satisfactory to the exact scientist. Scientists are not content with knowing, for example, that the temperature of boiling water is +100° C. relative to the temperature of freezing; they have set out to determine absolute temperatures and have found that water boils at 373° C. above absolute zero. Why should I not, therefore, determine the absolute motion of the automobile, not its motion relative to the road, earth, sun or stars, but relative to absolute rest?

Michelson and Morley set out in their famous experiment to measure the absolute velocity of their laboratory, which was, of course, fixed on the earth. The experiment consisted of timing two rays of light over two equal tracks at right angles to each other. When one track was situated in the direction of the earth’s motion they expected to get the same result as when two scullers of equal prowess are racing in a river, one up and down the stream and the other across and back; the winner will be the sculler rowing across the stream, as working out an example will convince. Even if the earth had been stationary at the time of one experiment, the earth’s motion round the sun would have been reversed 6 months later and would then have given double the effect. They found, however, that the two rays of light arrived always an exact dead heat. All experimenters who have tried since have arrived at the same result and found it impossible to detect absolute motion.

The principle of relativity has its foundation in fact on these failures to detect absolute motion. This principle states that the only motion we can ever know about is relative motion. If we devise an experiment which ought to reveal absolute motion, nature will enter into a conspiracy to defeat us. In the Michelson and Morley experiment the conspiracy was that the track in the direction of the earth’s absolute motion should contract its length by just so much as would allow the ray of light along it to arrive up to time.

We see, therefore, that according to the principle of relativity motion must always remain a relative term, in much the same way as vertical and horizontal, right and left, are relative terms having only meaning when referred to some observer. We do not expect to find an absolute vertical and are wise enough not to attempt it; in seeking to find absolute motion physicists were not so wise and only found themselves baffled.

The principle that all motion is relative now requires to be worked out to all its consequences, as has been done by Einstein, and we have his theory of relativity. Einstein conceives a world of four dimensions built up of the three dimensions of space, namely up and down, backwards and forwards, right and left, with time as the fourth dimension. This is an unusual conception to most of us, so let us simplify it into something which we can more easily picture but which will still allow us to grasp Einstein’s ideas. Let us confine ourselves for the present to events which happen on this sheet of paper, i.e., to space of two dimensions only and take time as our third dimension at right angles to the plane of the paper. We have thus built up a three dimensional world of space-time which is every bit as useful to us as a four dimensional representation so long as we only need study objects moving over the sheet of paper.

Suppose a fly is crawling over this sheet of paper and let us make a movie record of it. If we cut up the strip of movie film into the individual pictures and cement them together one above another in their proper order, we shall build up a solid block of film which will be a model of our simplified world of space-time and in which there will be a series of dots representing the motion of the fly over the paper. Just as I can state the exact position of an object in my room by defining its height above the floor, its distance from the north wall and its distance from the east wall, so we can reduce the positions of the dots to figures for use in calculations by measuring their distances from the three faces intersecting in the lines OX, OY, and OT, where OXAYTBCD represents the block of film. The mathematician would call the three lines OX, OY, OT the coordinate axes. Measuring all the dots in this way we shall obtain the motion of the fly relative to the coordinate axes OX, OY, OT. If we add a block OTDYEFGH of plain film we can use EX, EH, EF as coordinate axes and again obtain the motion of the fly relative to these new axes; or we

can add block after block so as to keep the axes moving. We can conceive of other changes of axes. The operator making the movie record might have taken the fly for the hero of the piece and moved the camera about so as to keep the fly more or less central in the picture; or he might, by turning the handle first fast and then slow and by moving the camera, have made the fly appear to be doing stunts. Moving the camera would change the axes of x and y, and turning the handle at different speeds would change the axis of time. Again, we might change the axes by pushing the block out of shape or by distorting it into a state of strain. Whatever change of axes we make, any dot in the block of film will signify a coincidence of the fly with a certain point of the paper at a certain time, and the series of dots will, in every case, be a representation of the motion of the fly. Maybe the representation will be a distorted one, but who is to say which is the absolutely undistorted representation? The principle of relativity which we laid down before says that no one set of coordinates will give the absolute motion of the fly, so that one set is as good as another. The principle that all motion is relative means, therefore, that no matter how we change our coordinates of space-time, the laws of motion which we deduce must be the same for all changes.

To use an analogy, the sculptured head of Shakespeare on my table may appear to have hollow cheeks when I admit light from the east window only, or to have sunken eyes with light from the skylight in the roof, but the true shape of the head remains the same in all lights.

Hence, if with reference to two consecutive dots in our block of film a mathematical quantity can be found which will not change no matter how we changes our axes of coordinates, that quantity must be an expression of the true law of motion of the fly between the two points of the paper and the two times represented by these two dots. Einstein has worked out such a quantity remaining constant for all changes of coordinates of the four dimensional world of space-time.

In passing we may notice a feature of Einstein’s world of space-time which we shall doubtless find it difficult to conceive, namely, that there is no essential difference between a time and a distance in space. Since one set of coordinates is as good as another, we can transform time into space and space into time according as we choose our axes. For example if we change OX, OT, the axes of x and time in Fig. 2, into OX', OT' by a simple rotation, the new time represented by OT' consists partly of OA in the old time and partly of OB in the old x direction. Referring to our block of movie film again, it means that although I might separate the block into space and time by slicing it into the original pictures, I can just as readily slice it in any direction I choose and still get individual pictures representing the motion of the fly but with, of course, new time and space. So whilst I may be believe that a liner has travelled 3,000 miles in 4 days, an observer on a star who knows nothing of my particular axes in space-time may say, with equal truth, that it went 2,000 miles in 7 days. Thus, time and space are not two separate identities in Einstein’s view; there only exists a world of four dimensions which we can split up into time and space as we choose.

Let us see now how Einstein explains gravitation. When a body is not acted on by any forces (except gravitation) the quantity which remains constant for all changes of coordinates implies that the body will follow that path in the space of an outside observer which takes the least time. It is an observed fact that one body attracts another by gravitation; that is, the path of one body is bent from its course by the presence of another. Now we can bend the path of the fly in our block of film by straining the block in some way. Suppose, therefore, that I strain the world so as to bend the path of a body exactly as the gravitation due to some other body bends it; i.e., by a change of coordinates I have obtained the same effect as that produced by gravitation. Einstein’s theory, therefore, explains gravitation as a distortion of the world of space-time due to the presence of matter. Suppose first that a body is moving with no other bodies near; according to Einstein it will take the path in space which requires the least time, i.e., a straight line as agrees with our experience. If now the world be strained by the presence of another body or by a change of coordinates it will still pursue the path of least time, but this path is now distorted from the straight line, just as in a similar way the path on a globe requiring the least time to travel follows a great circle. So, on Einstein’s view of gravitation, the earth moves in an elliptical path around the sun not because a force is acting on it, but because the world of space-time is so distorted by the presence of the sun that the path of least time through space is the elliptical path observed. There is, therefore, no need to introduce any idea of “force” of gravitation. Einstein’s theory explains gravitation only in the sense that he has explained it away as a force of nature and makes it a property of space-time, namely, a distortion not different from an appropriate change of coordinates. He does not, however, explain how or why a body can distort space-time. It is noteworthy that whilst the law of gravitation and the law of uniform motion in a straight line when no force is acting were separate and independent laws under Newton, Einstein finds one explanation for both under the principle of relativity.1

1 The balance of Dr. Royds’ essay is given to a discussion of the phenomena of Mercury’s perihelial advance, the deflection of light under the gravitational field of the sun, and the shift in spectral lines, in connection with which alone Einstein’s theory makes predictions which are sufficiently at variance with those of Newtonian science to be of value in checking up the theory observationally. In the interest of space conservation and in the presence of Dr. Pickering’s very complete discussion of these matters we omit Dr. Royds’ statement.—Editor.?

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