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.
XIX
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. [↑]
XX
EINSTEIN’S THEORY OF GRAVITATION
The Discussion of the General Theory and Its Most Important Application, from the Essay by
PROF. W. F. G. SWANN, UNIVERSITY OF MINNESOTA, MINNEAPOLIS
Newton’s great discovery regarding the motion of the planets consisted in his showing that these could all be summed up in the following statement: consider any planet in its relation to all particles in the universe. Write down, for the planet, in the line joining it to any particle, an acceleration proportional to the mass of the particle and to the inverse square of its distance from the planet. Then calculate the planet’s resultant acceleration by combining all the accelerations thus obtained.
We have here purposely avoided the use of the word “force,” for Newton’s law is complete as a practical statement of fact without it; and this word adds nothing to the law by way of enhancing its power in actual use. Nevertheless, the fact that the acceleration is made up as it were of non-interfering contributions from each particle in the line joining it to the planet strongly suggests to the mind something of the nature of an elastic pull for which the particle is responsible, and to which the planet’s departure from a straight-line motion is due. The mind likes to think of the elastic; ever since the time of Newton people have sought to devise some mechanism by which these pulls might be visualized as responsible for the phenomena in the same way as one pictures an elastic thread as controlling the motion of a stone which swings around at its end.
This search has been always without success; and now Einstein has found a rather different law which fits the facts better than Newton’s law. It is of such a type that it does not lend itself conveniently to expression in terms of force; the mind would gain nothing by trying to picture such forces as are necessary. It compensates for this, however, in being capable of visualization in terms of what is ultimately a much simpler concept.
In order to appreciate the fundamental ideas involved, suppose for a moment that gravitation could be annihilated, completely, and suppose I find myself upon this earth in empty space. You shall be seated at some point in space and shall watch my doings. If I am in the condition of mind of the people of the reign of King Henry VIII, I shall believe that the earth does not rotate. If I let go a stone, there being no gravity, I shall find that it flies away from me with an acceleration. You will know, however, that the stone really moves in a straight line with constant velocity, and that the apparent acceleration which I perceive is due to the earth’s rotation. If I have argued that acceleration is due to force, I shall say that the earth repels the stone, and shall try to find the law governing the variation of this force with distance. I may go farther, and try to imagine some reason for the force, some pushing action transmitted from the earth to the stone through a surrounding medium; and, you will pity me for all this wasted labor, and particularly for my attempt to find a mechanism to account for the force, since you know that if I would only accept your measurements all would appear so simple.
Let us probe this matter a little farther, however, from the stand-point of myself. I must believe in the reality of the force, since I have to be tied to my chair to prevent my departure from the earth. I might wonder how this field of force would affect the propagation of light, chemical action and so forth. For, even though I had discovered that, by using your measures, I could transform away the apparent effects of my field of force as far as concerned its power to hurl stones about, I could still regard this as a mathematical accident, and believe that the force was really there. Although I might suspect that the same transformation of view-point that would annul the field’s effect as regards the stones would also annul its effect as regards light, etc., I should not be sure of this, as you would be; and my conscience would hardly allow me to do more than look upon the assumption of complete equivalence between the apparent field and a change in the system of measurement as a hypothesis. I should be strongly tempted to make the hypothesis, however.
Now the question raised by Einstein is whether the force of gravity, which we experience as a very real thing, may be put upon a footing which is in some way analogous to that of the obviously fictitious centrifugal force cited above: whether gravitation may be regarded as a figment of our imagination engendered by the way in which we measure things. He found that it could be so regarded. He went still farther, and in his Principle of Equivalence, he postulated that the apparent effects of gravitation in all phenomena could be attributed to the same change in the system of our measurements that would account for the ordinary phenomena of gravitation. On the basis of this hypothesis he was able to deduce for subsequent experimental verification, the effects of gravitation on light. He did not limit himself to such simple changes in our measurements as were sufficient to serve the purpose of the problem of centrifugal force cited above; but, emboldened by the assumptions, in the older theory of relativity, of change in standards of length and time on account of motion, he went even farther than this, and considered the possibility of change of our measures due to mere proximity to matter.
His problem amounted to an attempt to find some way in which it was possible to conceive our scales and clocks as altered, relatively to some more fundamental set, so as to allow of the planetary motions being uniform and rectilinear with respect to these fundamental measures, although they appear as they do to us. If we allow our imaginations perfect freedom as to how the scales may be altered, we shall not balk at assuming alterations varying in any way we please, Einstein does, however, introduce restrictions for reasons which we will now discern.
If we imagine our whole universe, with its observers, planetary orbits, instruments, and everything else, embedded in a jelly, and then distort the jelly and contents in any way, the numbers at which our planetary orbits (or rather their telescopic images) intersect our scales will be unaltered. Moreover, we could vary, in any manner, the times at which all objects (including the clock hands) occupied their distorted positions, and the hand of some clock near the point where the planetary image crossed the scale would record for this occurrence the same dial reading as before. An inhabitant of this distorted universe would be absolutely unconscious of the change. Now the General Theory of Relativity which expresses itself in slightly varied forms, amounts to satisfying a certain philosophical craving of the mind, by asserting that the laws of nature which control our universe ought to be such that another universe like the above, whose inhabitants would be unconscious of their change, would also satisfy these laws, not merely from the standpoint of its own inhabitants, but also from the standpoint of our measurements. In other words, this second universe ought to appear possible to us as well as to its inhabitants.
Einstein decides to make his theory conform to this philosophical desire, and this greatly limits the modifications of clocks and scales which he permits himself for the purpose of representing gravitation. Further, if we express the alterations of the measures as functions of proximity to matter, velocity and so forth, our expressions for these alterations will include, as a particular case, that where matter is absent, although the scales and observer may still remain. Our alteration of the scales and clocks with velocity must thus revert, for this case, to that corresponding to the older theory of relativity, in order to avoid predicting that two observers, in uniform motion relative to each other in empty space, will measure different values for the velocity of light. In this way, the velocity of light comes to play a part in expressing the alterations of the measures.
Even with these restrictions, Einstein was able to do the equivalent of finding an alteration of scales and clocks in the presence of matter which would account for our finding that the planetary motions take place very nearly in accordance with Newton’s law. The new law has accounted with surprising accuracy for certain astronomical irregularities for which Newton’s law failed to account, and has predicted at least one previously unknown phenomenon which was immediately verified.
In conclusion, it may be of interest to state how the new law describes the motion of a particle in the vicinity of a body like the earth. The law amounts to stating that, if we measure a short distance, radially as regards the earth’s center, we must allow for the peculiarity of our units by dividing by
where r is the distance from the earth’s center, m the mass of the earth, c the velocity of light, and G the Newtonian gravitational constant. Tangential measurements require no correction, but intervals of time as measured by our clocks must be multiplied, for each particular place, by the above factor. Then, in terms of the corrected measures so obtained, the particle will be found to describe a straight line with constant velocity although, in terms of our actual measures, it appears to fall with an acceleration.
XXI
THE EQUIVALENCE HYPOTHESIS
The Discussion of This, With Its Difficulties and the Manner in Which Einstein Has Resolved Them, from the Essay by
PROF. E. N. DA C. ANDRADE, ORDNANCE COLLEGE, WOOLWICH, ENGLAND
Having shown that, of several systems all moving with reference to one another with uniform motion, no one is entitled to any preference over the others, and having deduced the laws for such systems, Einstein was confronted with a difficulty which had long been felt. A body rotating, which is a special case of an accelerated body, can be distinguished from one at rest, without looking outside it, by the existence of the so-called centrifugal forces.
This circumstance, which gives certain bodies an absolute or preferential motion, is unpalatable to the relativist; he would like there to be no difference as regards forces[1] between the case when the earth rotates with reference to outside bodies (the stars) considered as fixed, and the case when the earth is considered fixed and all outside bodies rotate around it. This point cannot be investigated by direct experiment; we can spin a top but we cannot keep a top at rest and spin the world round it, to see if the forces are same.
In considering the problem of how to devise laws which should make all rotations relative, Einstein conceived the brilliant yet simple idea that gravitation could be brought into the scheme as an acceleration effect, since both ordinary accelerational forces and gravitational forces are proportional to the same thing, the mass of a body. The impossibility of separating the two kinds of effect can be easily seen by considering the starting of an elevator. When the elevator is quickly accelerated upwards we feel a downward pull, just as if the gravitational pull had been increased, and if the acceleration continued to be uniform, bodies tested with a spring balance would all weigh more in the elevator than they did on firm ground. In a similar way the whole of the gravitational pull may be considered to be an accelerational effect, the difficulty being to devise laws of motion which will give the effects that we find by actual observation.
But it is obvious that we cannot, by ordinary mechanics, consider the earth as being accelerated in all directions, which we should have to do, apparently, to account for the fact that the gravitational pull is always toward the center. [It is obvious that we cannot explain gravitation by assuming that the earth’s surface is continually moving outward with an accelerated velocity.][227] So Einstein found that, as long as we treat the problem by Euclid’s geometry, we cannot reach a satisfactory solution. But he found that to the four-dimensional space made up of the three ordinary dimensions of space, together with the time-dimension which we have already mentioned in discussing the special theory, may be attributed a peculiar geometry, the nature of which departs more and more from Euclidean geometry as we approach a gravitational body, and the net result of which is to make possible the universal correspondence of gravitation and acceleration.
This modification of the geometry of space is often spoken of as the “curvature of space,” an expression which is puzzling, especially as the space which is “curved” is four-dimensional time-space. But we can get an idea of what is meant by considering figures, triangles say, drawn on the surface, of a sphere. These triangles, although drawn on a surface, will not have the same properties as triangles drawn on flat paper—their three angles will not together equal right angles. They will be non-Euclidean. This is only a rough analogy, but we can see that the curvature of the surface causes a departure from Euclidean geometry for plane figures, and consequently the departure from Euclidean laws extended to four dimensions may be referred to as caused by “curvature of space.”
It is difficult to imagine a lump of matter affecting the geometry of the space round it. Once more we must use a rough illustration. Imagine a very hot body, and that, knowing nothing of its properties, we have to measure up the space round it with metal measuring-rods. The nearer we are to the body, the longer the rods will become, owing to the expansion of the metal. When we measure out a square, one side of which is nearer the body than the opposite side, its angles will not be right angles. If we knew nothing of the laws of heat we should say that the body had made the space round it non-Euclidean.
Einstein found, then, that by taking the properties of space, as given by measurement, to be modified in the neighborhood of masses of matter, he could devise general laws according to which gravitational effects would be produced, and there would be no absolute rotation. All forces will be the same whether a body rotates with everything outside it fixed, or the body is fixed, and everything rotates round it. All motion is then relative, and the theory is one of “general relativity.” The velocity of light is, however, no longer constant, and its path is not a straight line, if it is passing near gravitating matter. This does not contradict the special theory, which did not allow for gravitation. Rather, the special theory is a particular case to which the generalised theory reduces when there is no matter about, just as the Newtonian dynamics is a special case of the special theory, which we obtain when all velocities are small compared to that of light.
[1] There is, on any view, no difference as regards observation of position only.—Author. [↑]
XXII
THE GENERAL THEORY
Fragments of Particular Merit on This Phase of the Subject
BY VARIOUS CONTRIBUTORS
When Dorothy was carried by the cyclone from her home in Kansas to the land of Oz, together with her uncle’s house and her little dog Toto, she neglected to lower the trap door over the hole in the floor which formerly led to the cyclone cellar and Toto stepped through. Dorothy rushed to the opening expecting to see him dashed onto the rocks below but found him floating just below the floor. She drew him back into the room and closed the trap.
The author of the chronicle of Dorothy’s adventures explains that the same force which held up the house held up Toto but this explanation is not necessary. Dorothy was now floating through space and house and dog were subject to the same forces of gravitation which gave them identical motions. Dorothy must have pushed the dog down onto the floor and in doing so must herself have floated to the ceiling whence she might have pushed herself back to the floor. In fact gravitation was apparently suspended and Dorothy was in a position to have tried certain experiments which Einstein has never tried because he was never in Dorothy’s unique position.][188]
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The Principle of Equivalence, of which Einstein’s suspended cage experiment is the usual illustration, and upon which the generalized theory of relativity is built, is thus stated by Prof. Eddington: “A gravitational field of force is precisely equivalent to an artificial field of force, so that in any small region it is impossible, by any conceivable experiment, to distinguish between them. In other words, force is purely relative.”
This may be otherwise stated by going back to our idea of a four-dimensional world, the points of which represent the positions and times of events. If we mark in such a space-time the successive positions of an object we get a line, or curve, which represents the whole history of the object, inasmuch as it shows us the position of the object at every time. The reader may imagine that all events happen in one plane, so that only two perpendicular dimensions are needed to fix positions in space, with a third perpendicular dimension for time. He may then conceive, if he may not picture, an analogous process for four-dimensional space-time. These lines, “tracks of objects through space-time,” were called by Minkowski “world-lines.” We may now say that all the events we observe are the intersections of world-lines. The temperature at noon was 70°. This means that if I plot the world-line of the top of the mercury column and the world-line of a certain mark on the glass they intersect in a certain point of space-time. All that we know are intersections of these world-lines. Suppose now we have a large number of them drawn in our four-dimensional world, satisfying all known intersections, and let us suppose the whole embedded in a jelly. We may distort this jelly in any way, changing our coordinates as we please, but we shall neither destroy nor create intersections of world-lines. It may be proved that a change from one system of reference, to which observations are referred, to any other system, moving in any way with respect to the first system, may be pictured as a distortion of the four-dimensional jelly. The laws of nature, therefore, being laws that describe intersections, must be expressible in a form independent of the reference system chosen.
From these postulates, Einstein was able to show such a formulation possible. His law may be stated very simply:—All bodies move through space-time in the straightest possible tracks.
The fact that an easy non-mathematical explanation can not be given, of how this law is reached, or of just why the straightest track of Mercury through space-time will give us an ellipse in space after we have split space-time up into space and time, is no valid objection to the theory. Newton’s law that bodies attract with a force proportional to their masses and inversely proportional to the square of the distance is simple, but no one has ever given an easy non-mathematical proof of how that law requires the path of Mercury to be an ellipse, with the sun at a focus, instead of some other curve.][182]
* * *
One of the grave difficulties we have in gaining a satisfactory comprehension of Einstein’s conceptions, is that they do not readily relate themselves to our modes of geometrical thought. Within limits we may choose our own geometry, but it may be at the cost of unwieldy complication. If we think with Newton in terms of Euclidean geometry and consider the earth as revolving around the sun, the motions of our solar system can be stated in comparatively simple terms. If, on the other hand, we should persist in stating them, as Ptolemy would have done, from the earth as a relatively stationary center, our formulas will become complicated beyond ready comprehension. For this reason it is much simpler in applying the theory of relativity, and in considering and describing what actually happens in the physical universe, to use geometrical conceptions to which the actual conditions can be easily related. We find such an instrument in non-Euclidean geometry, wherein space will appear as though it were projected from a slightly concave mirror. It is in this sense that some speak of space as curved. The analogy is so suggestive it tempts one to linger over it. Unless there were material objects within the range of the mirror, its conformation would be immaterial; the thought of the space which the mirror, as it were, circumscribes, is dependent upon the presence of such material objects. The lines of light and of all other movement will not be quite “straight” from the view-point of Euclidean geometry. A line drawn in a universe of such a nature must inevitably return upon itself. Nothing therefore, can ever pass out of this unlimitedly great but yet finite cosmos. But even now, since our imaginary mirror is only very slightly concave, it follows that for limited regions like the earth or even the solar system, our conception of geometry may well be rectilinear and Euclidean. Newton’s law of gravitation will be quite accurate with only a theoretical modification drawn from the theory of relativity.][82]
* * *
The way in which a curvature of space might appear to us as a force is made plainer by an example. Suppose that in a certain room a marble dropped anywhere on the floor always rolled to the center of the room; suppose the same thing happened to a baseball, a billiard ball, and a tennis ball. These results could be explained in two ways; we might assume that a mysterious force of attraction existed at the center of the floor, which affected all kinds of balls alike; or we might assume that the floor was curved. We naturally prefer the latter explanation. But when we find that in the neighborhood of a large material body all other bodies move toward it in exactly the same manner, regardless of their nature or their condition, we are accustomed to postulate a mysterious attractive force (gravitation); Einstein, on the contrary, adopts the other alternative, that the space around the body is curved.][223]
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In the ordinary “analytical geometry,” the position and motion of all the points considered is referred to a rigid “body” or “frame of reference.” This usually consists of an imaginary room of suitable size. The position of any point is then given by three numbers, i.e., its distances from one side wall and from the back wall and its height above the floor. These three numbers can only give one point, every other point having at least one number different. In four-dimensional geometry a fourth wall may be vaguely imagined as perpendicular to all three walls, and a fourth number added, giving the distance of the “point” from this wall also. Since “rigid” bodies do not exist in gravitational fields the “frame of reference” must be “non-rigid.” The frame of reference in the Gaussian system need not be rigid, it can be of any shape and moving in any manner, in fact a kind of jelly. A “point” or “event” in the four dimensioned world is still given by four numbers but these numbers do not represent distances from anywhere; all that is necessary is that no two events shall have exactly the same four numbers to represent them, and that two events which are very close together shall be represented by numbers which differ only slightly from one another. This system assumes so little that it will be seen to be very wide in its scope; although to the ordinary mind, what is gained in scope seems more than that lost in concreteness. This does not concern the mathematician, however, and by using this system he gains his object, proving that the general laws of nature remain the same when expressed in any Gaussian coordinate system whatever.][220]
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Einstein enunciates a general principle that it is possible to find a transformation of coordinate axes which is exactly equivalent to any force, and in particular one which is equivalent to the force of gravitation. That is by concentrating our attention on the transformation which is a purely mathematical operation we can afford to neglect the force completely. To get a better idea of this principle of equivalence as it is called, let us consider a relatively simple example (which actually has nothing to do with gravitation, but which will serve to make our notions clearer.) A person on the earth unconsciously refers all his experiences, i.e., the motions of the objects around him to a set of axes fixed in the earth on which he stands. However, we know that the earth is rotating about its axis, and his axes of reference are also rotating with respect to the space about him. From the point of view of general relativity it is exactly because we do refer motions on the surface of the earth to axes rotating with the earth that we experience the so-called centrifugal force of the earth’s rotation, with which everyone is familiar. If we could find it convenient to transform from moving axes to fixed axes, the force would vanish, since it is exactly equivalent to the transformation from one set of axes to the other. However, we find it unnatural to refer daily experiences to axes that are not placed where we happen to be, and so we prefer to take the force and rotating axes instead of no force and fixed axes.][272]
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We seem to have a direct experience of force in our muscular sensations. By pushing or pulling we can set bodies in motion. It is natural to assume, that something similar occurs, when Nature set bodies in motion. But is this not a relic of animism? The savage and the ancients peopled all the woods and skies with Gods and demons, who carries on the activities of nature by their own bodily efforts. Today we have dispossessed the demons, but the ghost of a muscular pull still holds the planets in place.][141]
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The general theory is an extension of the special theory which enables the law of gravitation to be deduced. Not in Newton’s form, it is true, but in a better form, that is, one that accounts for two important facts otherwise not explained. But it is a far more general theory that indicated above. It is a complete study of the relations between laws expressed by means of any four coordinates (of which three space and one time is a special case), and the same laws expressed in the four coordinates of a system having any motion whatever with respect to the first system. By restricting this general study in accordance with certain postulates about the nature of the universe we live in, we arrive at a number of conclusions which fit more closely with observed facts that the conclusions drawn from Newton’s theory.][221]