The Physical side of Relativity

Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. The Physical side of Relativity

THE PHYSICAL SIDE OF RELATIVITY

The Immediate Contacts Between Einstein’s Theories and Current Physics and Astronomy

BY PROFESSOR WILLIAM H. PICKERING
HARVARD COLLEGE OBSERVATORY,
MANDEVILLE, JAMAICA

The Theory of Relativity will be treated first from the physical side, leaving the three astronomical tests to which it has been put to be discussed later. There is one astronomical fact however that must be mentioned in this connection, and this is the discovery of the aberration of light by Bradley in 1726. It is found that every star in the heavens apparently describes a small annual ellipse, whose major axis is 41″ in length. This Bradley showed to be due to a combination of the velocity of the earth in its orbit, and the velocity of light; and it is so explained in all the elementary text-books on astronomy. It implies a stationary ether through which the earth is moving. The importance of this statement will appear presently.

The subject is usually illustrated by supposing a [288]man to go out in a rainstorm carrying a vertical tube. If the rain is falling vertically, and the man stands still, the sides of the tube will not be wet, save by an occasional drop, but if the tube is moved, it must then be inclined forward in order to keep it dry. The angle of inclination, which corresponds to aberration, will depend on the relative velocity of the tube, corresponding to the earth, and the rain drops which correspond to the waves of light.

If three lines are dropped upon a point in space, each line being perpendicular to the plane containing the other two, we have what is known as a system of coordinates. Einstein’s original theory of relativity, which he now designates as the “special theory,” depends on two principles. The first is that “Every law of nature which holds good with respect to a coordinate system K must also hold good for any other system K′, provided that K and K′ are in uniform movement of translation.” The second principle is that “Light in a vacuum has a definite and constant velocity, independent of the velocity of its source.”

These two sentences may be considered as authoritative, being quoted in Einstein’s own words.1 The first of these principles need not greatly surprise us. The second is not well expressed, because it is ambiguous. He does not say how the first “velocity” is measured, whether relatively to the ether or relatively to the observer. In fact this is the very gist of the whole matter, as we shall presently see. In the case of sound the velocity is constant with regard to the medium, the air, in the case of light it is supposed to be constant with regard to the [289]observer. It reaches him with a constant velocity, no matter how he moves.

In order to understand this statement clearly let us consider the appended tabular diagram. On a calm day imagine a source of sound at S in line a. This may be either a gun or a bell. Imagine an observer 1,100 feet distant, located at O. The velocity of sound in air is 1,100 feet per second. This velocity we will take as unity, as indicated in the third column, and the velocity with which the sound reaches the observer is also 1, as shown in the fourth. It will reach him in a unit interval of 1 second, as shown in the fifth. If the bell is struck, it will give its normal pitch or frequency, which we will also call unity, in the sixth column.

Now imagine case b where the observer is on a train advancing toward S. When he is 1,100 feet distant, the gun is fired, but as he is advancing toward it, he hears it at O in rather less than a second, as shown in the fifth column. The velocity of the sound with regard to him is rather more than unity, as shown in the fourth column. If the bell is sounded, the pitch, that is the frequency, is raised, because he receives more sound waves per second than before.

In case c the observer is stationary, but the source of sound is receding. At a distance of 1,100 feet the gun is fired, and the observer hears it after an interval of just one second, as in case a. The velocities with regard to the observer and through the medium are also unity. If the bell is struck the pitch is lowered, since he receives fewer sound waves per second, the reverse of case b.

In case d imagine the source and the observer 1,100 feet apart, and advancing on the same train. When the gun is fired, the velocity of the sound waves will be greater with regard to the observer, and he will hear the sound in less than a second, as in case b. When the bell is struck it will have the normal pitch, the same as in case a.

We find therefore that for sound the velocity with regard to the medium is always unity, while the velocity with regard to the observer, and the interval elapsed, depend only on the motion of the observer himself, and are independent of the motion of the source. The frequency of the vibrations, on the other hand, depends only on the relative motion of the observer and the source, but is independent of their common motion in any direction. Further, it makes no difference whether the source and the observer are moving on a train, or whether they are stationary, and a uniform wind is blowing past them.

Case C represents Einstein’s statement, as confirmed by Majorana’s experiment. It does not differ from case c for sound. Case D is more complex, but accepting the statement above that the velocity is constant with regard to the observer, we see that the velocity through the medium must be less, and that the interval elapsed will be constant, as in case B. Could we use the brighter stars and planets as sources of light, several of these cases could be further tested.

This brings us at once to statements that contradict our common sense. For instance, Jeans says “no matter what the velocity of the observer is, the light surface, as observed by that observer, is invariably a sphere having that observer as center.”3 That is to say the light surface, or wave front, is a contracting, [292]not an expanding, sphere. This, if confirmed, would go a long way toward making our universe a subjective rather than an objective phenomenon. Again imagine a flash of light, such as an explosion, to occur when an observer is in a given position. It makes no difference how the observer may move while the light is approaching him, whether several miles forward or backward, the light will reach him in exactly the same time, as is shown by Michelson’s experiment. Or if two observers are at the same spot when the explosion occurs, and one moves forward, and the other backward, they will both see the explosion at exactly the same instant.

This sounds ridiculous, but not only is it what Jeans says, but it is the logical interpretation of Einstein’s second principle, if Einstein means by velocity, velocity with regard to the observer. If he means velocity with regard to the medium, then the case is exactly the same as that of sound in air, and Michelson’s experiment as well as the Maxwell-Lorentz theory of light are contradicted. This theory is now universally accepted, and Michelson’s experiment has been carefully repeated by other observers, and fully confirmed. This is the very heart of the relativity question.

If we state the matter objectively it comes to this. The velocity of light with regard to the ether is a variable quantity, depending merely on where the observer chooses to go. As Eddington well says, “these relations to the ether have no effect on the phenomena and can be disregarded—a step which appears to divest the ether of the last remnants of substantiality.”4[293]

The only way of avoiding this apparent absurdity seems to be to consider that the ether moves with the earth. Michelson’s result would then be fully explained. Of course this can only be true for a few miles above the earth’s surface. Beyond that the ether must either be stationary or move with the sun. The velocity of light with regard to the ether would then be a constant, just as the velocity of sound is constant with regard to the air. This would contradict Einstein’s second principle as it is generally understood. The trouble with this suggestion is that it fails to account for aberration, which, as already explained, appears to require that the earth should be moving through the ether. To meet this emergency would involve some modification of the undulatory theory of light, which apparently would not be impossible, but has not yet been made.

In 1915 Einstein brought out an extension of his first principle. This he calls the “general theory of relativity.” It states that in our choice of coordinate systems we “should not be limited in any way so far as their state of motion is concerned.”1 This leads to the three astronomical consequences mentioned later in this paper, two of which have been more or less confirmed, and the third practically contradicted as far as quantitative measures are concerned.5

That portion of the mass of a body due to its electrical charge can be readily shown experimentally to vary with the velocity of the body. Einstein has shown the same to be true of the normal mass, as is illustrated in the advance of the perihelion of the orbit of Mercury. He has also pointed out that gravitation, inertia and centrifugal force are all closely related, and obey similar laws. Thus if we rise from the earth with accelerated velocity, we apparently increase our weight. Again if the velocity of rotation of the earth on its axis should be increased, our weight would be diminished. These [295]facts are suggestive when we come to consider the ultimate cause of gravitation.

Another fact which must be rather startling to the older school of scientists is that momentum is no longer simply mv, mass times velocity, but that the velocity of light c, comes into the question, and the formula for momentum now assumes the form of

For ordinary velocities this correction is extremely small, but it has been shown to be necessary, both theoretically and experimentally, when dealing with the high velocities with which we are now familiar.

The theory of relativity is so widespread in its application that several other theories have become more or less intimately combined with it, for which Einstein is in no way responsible. One of these is known as the Fitzgerald-Lorentz theory, that all bodies are subject to a contraction in the direction of their motions through space. This was first suggested in order to explain the Michelson-Morley experiment, but has proved inadequate to do so, particularly when the observer is receding from the source. This contraction is expressed by the same factor used in the denominator of the revised expression for momentum, given above. Again the quantity c is so enormous, that even for large bodies at planetary velocities the contraction amounts to very little. Thus the earth moving at a speed of eighteen miles per second in its orbit, is flattened only 1/200,000,000, or 2.5 inches. On the other [296]hand for high velocities of many thousand miles per second, such as we have become familiar with in the case of the radioactive substances, the flattening is a very considerable fraction of the diameter of the moving body, one-half or more, and in the case of the corpuscles of light, if that theory were adopted, this flattening becomes equal to the diameter, and their thickness is reduced to zero.

When we view Einstein’s theories from the astronomical standpoint, the earliest fact bearing on relativity that we need consider was the discovery of aberration, by Bradley, in 1726, as seen above. In 1872 Airy observed the star γ Draconis through a telescope filled with water. Since the velocity of light is less in water than in air, we should naturally expect to find the aberration appreciably increased. It was found, on the other hand, however, to be unaffected.

In 1887 the results of the famous Michelson-Morley experiment were published.7 In this experiment the velocity of light was measured in various directions with regard to the motion of the earth in its orbit. If the ether were stationary, and the earth moving through it, different velocities should be obtained in different directions. Such was not the case however, and the experiment indicated that the ether moved with the earth. It thus flatly contradicted the conclusions founded on aberration.

Einstein’s Special Theory of Relativity, of 1905, as we have seen, resolves this contradiction. But as we shall presently see, it is the General Theory, of 1915, that leads to astronomical applications of broad scope. It indicates, for instance, that there is [297]no essential difference between gravitation and inertia. This idea may be crudely illustrated by our feelings of increased weight when an elevator starts rapidly upwards. A man while falling freely in space ceases to feel the pull of gravitation.

But we must not as yet conceive of the theory of relativity as a universally accepted and unquestioned truth of science. Eddington is its leading English exponent, and he is supported by such men as Jeans, Larmor, and Jeffreys. On the other hand, the theory has been severely criticised by Lodge, Fowler, Silberstein, and Sampson. Few American scientists have expressed any opinions in print on the subject, and the recent eclipse observations, to which we shall refer later, are to be repeated with more suitable instruments for verification in 1922, in the hope of obtaining more accurate and accordant results.8

An appurtenance of the Einstein theories which bears much the same relation to them as does the Lorentz-Fitzgerald contraction, mentioned above, is the idea, first clearly stated by Minkowski, that time is a kind of space—a fourth dimension. This the reader will doubtless find to be the most difficult portion of the theory to picture in his own mind. It is entirely unsupported by experiment or observation, necessarily so, and is based wholly on mathematical and philosophical conceptions. Our distinction between space and time seems to be that the direction in which we progress without effort is time; the other directions, in which we have to make an exertion to move ourselves, or in which we are carried, are space. How many dimensions empty space may have, we really have no means of knowing, because [298]we can neither see nor feel it. Matter we know has three, length, breadth, and thickness, also that it lies remote from us in three corresponding directions. These facts may have given us the erroneous impression that space too has only three dimensions. Now it is claimed that time is a fourth, and that there are also others.

In order to illustrate this, Eddington asks us to imagine a movie film taken of a man or of any moving object. Let the separate pictures be cut apart and piled on one another. This would form a sort of pictorial history of the individual for a brief interval in his life, in the form of a cube. If we attempt to pick it up, it falls apart, thus clearly showing the difference between time and space. But suppose it now all glued together in one solid cube, so that it is no easier to cut a section in one direction than in another. That is Minkowski’s idea of space and time, and further, that the direction in which we should cut it depends merely on the velocity with which we are moving through space. I should cut it parallel to the films, but a man on a rapidly moving star, in order to separate it into space and time, would cut it in an inclined direction. That is a thing which may be true, but it is one which we believe no mortal man can clearly picture to himself.

If I wish to give a complete dimensional description of myself in my four dimensions, I must give my length, my breadth, and my thickness, ever since I came into being, and also the course I have traversed through space since that time. This latter distance will be expressed in terms of a unit whose length is 186,000 miles, the distance traversed by light in one second. The distance which I travel through space annually is enormous, and very complex as to direction. It involves not merely my own motions as I cross the room, or take a train or steamer, but also those due to the rotation of the earth on its axis, its revolution round the sun, and the motion of the latter through the heavens. In general I travel, or in other words increase my length in the fourth dimension, by over 4,000 units [300]a year. The fourth dimension accordingly, if this view is accepted, is simply a distance like the other three, and perfectly easy to understand.

We now come to the three actual tests by which the theory has been tried. The planets as is well known revolve about the sun in ellipses, with the sun in one of the foci. That is to say, the sun is not in the center, but a little on one side of it. The end of the ellipse where the planet comes nearest to the sun is called the perihelion, and here the planet is moving most rapidly. The other end is called the aphelion, and here the motion is slowest. According to Newton’s theory of gravitation, if a spherical sun possesses a single planet or companion, its orbit will be permanently fixed in space unless perturbed by some other body. If a second planet exist, it will cause the perihelion of the first slowly to advance. According to Einstein the mass of a planet depends in part on its velocity. It will therefore be less at aphelion where it is moving slowly than at perihelion where it is moving rapidly, consequently in addition to the Newtonian attraction we have another one which increases as we approach the sun. The effect of this will be to cause the perihelion of the orbit to advance, whether there is a second planet or not.

Among the larger planets Mercury has the most eccentric orbit, and it also moves most rapidly, so that it is particularly well adapted to test the relativity theory. The observed advance of its perihelion is 574″ per century, instead of the theoretical figure 532″, due to the other planets—a difference of 42″.11 This has long been a puzzling discrepancy between observation and the law of gravitation. [301]Prior to Einstein, attempts were made to eliminate it by assuming a certain oblateness of the solar disk. If the equatorial diameter exceeded the polar by only 0″.5 the whole advance would be accounted for, but not only has this ellipticity failed of detection, but if it existed, it should produce a very noticeable and inadmissible change in the inclination of Mercury’s orbit, amounting to about 3″ per century, as has been demonstrated by both Herzer and Newcomb.12

Einstein from computations alone, without introducing any new constants or hypotheses whatever, showed, if the theory of relativity be accepted, that the sun should produce an acceleration of 43″ per century, thus entirely accounting for the observed discrepancy, far within the limits of accuracy of the observations. The only other planet whose orbit has a large eccentricity, and that is suitable for investigation, is the planet Mars. Here the discrepancy between observation and theory is very slight, only 4″, and a portion of that may be due to the attraction of the asteroids. This deviation is so slight that it may well be due entirely to accidental errors of observation, but however that may be, Einstein’s theory reduces it to 2″.7.

This all seems very satisfactory and complete, but the trouble with it is that the coincidence for Mercury is rather too good. It is based on the assumption that the sun is a perfect sphere, and that the density of its surface is uniform from the equator to the poles. This would doubtless be true if the sun did not revolve on its axis. In point of fact it does revolve, in a period in general of about 26 [302]days. Consequently an object on its equator must experience a certain amount of centrifugal force. Therefore if its surface were of uniform density the shape of the sun would be an oblate spheroid.

It can be readily shown that the theoretical excess of the equatorial over the polar diameter, due to the centrifugal force, should amount to only 0″.04, an amount which could hardly be detected by observation, and might readily be concealed by a slight excess of equatorial over polar density. Any reasonable excess of density at the center would diminish this result but slightly. The molecular weight of the central material13 is probably about 2. This computed equatorial excess is one-twelfth of the amount necessary to cause the observed advance, and should therefore cause an advance of the perihelion of about 3″.5 per century, reducing the difference between the observed advance and that caused by gravitation to 38″.5. According to Einstein the advance due to relativity should be, as we saw, 43″, a discrepancy of 4″.5 per century, or 10 per cent. Jeffreys has remarked that any discrepancy such as 10″ “would be fatal to a theory such as Einstein’s, which contains no arbitrary constituent capable of adjustment to suit empirical facts.”14 It must be pointed out here however, that so far as known, this small correction to the motion of Mercury’s perihelion has not previously been suggested, so that there has been no opportunity hitherto for its criticism by others.

One of the eclipse photographs

The arrows pointing to the star-images have been inserted by hand; and the star-images themselves have had to be materially strengthened in order to make them show in the engraving at all.

Photograph submitted by Dr. Alexander McAdie, Harvard University, by courtesy of the Royal Observatory, Greenwich.

It was due largely to the success with Mercury that it was decided to put the relativity theory to another test. According to the Newtonian theory, [303]as stated by Newton himself, corpuscles as well as planets have mass, and must therefore be attracted by the sun. According to Einstein, owing to their high velocity, this attraction must be twice as great as it would be according to the theory of gravitation. If the ray of light proceeding from a star were to pass nearly tangent to the sun’s limb it should be deflected 0″.87 according to Newton. According to the theory of relativity it should be deflected 1″.75. Stars of course cannot usually be observed near the sun. It is therefore necessary to take advantage of a total solar eclipse, when the sun is completely hidden by the moon, in order to secure these observations.

Two expeditions, one to Africa, and one to South America, observed successfully the total eclipse of May 29, 1919. The former was located on the Island of Principe in the Gulf of Guinea. The latter was located at Sobral, Brazil. Their equipment and results are shown in the following table, where the successive columns give the location, the aperture in inches of the telescopes employed, their focus in feet, the number of plates secured, the number of stars measured, their mean deduced deflection from their true positions by the attraction of the sun, and the deviations from the theoretical results.15 In the first and last line of the table shown herewith, this

deviation is taken from Einstein’s computed value of [304]1″.75. In the second line the difference shown is from the value required by the Newtonian theory, 0″.87. The results obtained with this telescope were rejected however, although they were much the most numerous, because it was found that for some reason, supposed to be the heating of the mirror by the sun before the eclipse, the star images were slightly out of focus, and were therefore considered unreliable. The results with the two other telescopes were not very accordant, but the 4-inch had the longer focus, secured the greater number of plates, and showed the greater number of stars. The results obtained with it therefore appear to have been the more reliable. They differ from Einstein’s prediction by 13 per cent. In future expeditions to test this question, the mirror in front of the telescope will be eliminated.

We now come to the final test which has been applied to Einstein’s theory. Einstein showed that in the intense gravitational field of the sun, the theory of relativity required that all of the spectrum lines should be shifted slightly toward the red end. The shift however is exceedingly small, and can only be detected and measured with the most powerful modern instruments. Moreover only certain lines can be used, because owing to varying pressure in the solar atmosphere, which affects many lines, as well as to rapid motion in the line of sight, which may affect all of them, still larger displacements are liable to occur.

The reference numbers in the above text have nothing to do with the numbers used in other parts of this volume to acknowledge the work of the various contestants; they refer to Dr. Pickering’s sources, as follows:

1 Journ. Brit. Astron. Assoc., 1919, 30, 76.

2 Comptes Rendus, 165, 424, and 167, 71.

3 Monthly Notices R. A. S., 1919, 80, 104.

4 Monthly Notices R. A. S., 1917, 77, 379.

5 Astro-Physical Journal, 1917, 46, 249. Journ. Brit. Astron. Assoc., 1920, 30, 276.

6 Monthly Notices, R. A. S., 1917, 77, 377.

7 Amer. Journ. Sci., 34, 333.

8 Monthly Notices, R. A. S., 1920, 80, 628.

9 The Observatory 1920, April. From an Oxford Note Book.

10 Monthly Notices, R. A. S., 1917. 78, 3 De Sitter, 1919, 80, 121, Jeans, 80, 146 Jeffreys.

11 “Gravitation and the Principle of Relativity,” Eddington. Royal Institution of Great Britain, 1918.

12 Journ. Brit. Astron. Assoc., 1920, 30, 125.

13 “The Interior of a Star,” Eddington. Scientia, 1918, 23, 15.

14 Monthly Notices, R. A. S., 1919, 80, 138.

15 Monthly Notices. R. A. S., 1920, 80, 415. Journ. Brit. Astron. Assoc., 1919, 30, 46.

16 Astro-Physical Journ., 1917, 46, 249.

17 Journ. Brit. Astron. Assoc., 1920, 30, 276.

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