Chapter VI: completes the preliminary course in the fundamentals of (9)

Evidently then if the distance in space and the interval in time separating two given events, such as the firing of a gun and the bursting of the shell, are measured by two observers in uniform relative motion, their estimates will not agree. Consider now the simple problem of measuring the distance between two points on an ordinary drawing-board. If we draw two perpendicular axes, we can define this distance by specifying the lengths of the projections on the two axes of the line joining the points. If we choose two different axes the projections will not be the same but will define the same length. Similarly, in a Euclidean four-space the distance between two points will be defined by the projections on the four axes, but if these axes be rotated slightly, the projections will be different, but will define the same length. Now, returning to the two observers just mentioned, it was noticed by Minkowski in 1908 that if the space measurements between the two events are split into the usual three components, and if the time measurements are multiplied by $\sqrt{-1}$, the difference between the two sets of measurements is exactly the same as would have occurred had these two events been points in a Euclidean fourspace, and two different observations made of their distance apart using two sets of axes inclined to each other. The velocity of light is made equal to 1 in this calculation by a suitable choice of units. This discovery threw a vivid light on the problem of space-time, showing that it is probably a true four-space of one negative dimension, a simple derivative of the much-discussed and now familiar Euclidean four-space.

Although this discovery gave a tremendous impetus to the progress of the theory, it is probable that it holds a deeper significance not yet revealed. It is probably a statement of the "stuff" of which the four-space is made, and perhaps also of how it is made; but the problem remains unsolved.

It thus becomes plain that our two observers are merely looking at the same thing from different viewpoints. Each has just as much right as the other to regard himself as being at rest in ordinary space (this is the postulate of the relativity of uniform motion) and to regard his time direction as a straight line in the four-space. The difference is merely that the two time axes are inclined to each other. If, however, one were moving with an acceleration with reference to the other his path in the four-space will appear curved to the other, though he himself, since he regards it as his time axis, will still assume it to be straight. If there is a body moving in what one observer sees to be a straight line, the other will, of course, in general see it as curved, and following the usual custom, since this body, without apparent reason, deviates from the straight path, will say there must be some force acting on it. Thus the curvature of his time axis, due to his accelerated motion, makes it appear that there is round him a field of force, which causes freely moving bodies to deviate from the straight path. Now if space-time is itself inherently curved it is not generally possible for any line in it to be straight any more than it is possible for any line on the surface of a sphere to be straight. Hence, all axes must be curved, and all observers, whatever their states of motion, must experience fields of force which are of the same nature as those due to motion only. The extra force experienced when a lift begins to rise is an example of force due to pure motion: gravitation is the similar force due to an inherent curvature of the four-space, and it was the postulate that these forces were similar that made possible Einstein's solution of the general problem of gravitation.

THE TIME DIAGRAM

The correlation of time with its geometrical analogue is of absorbing interest. Representing velocity by the common method of plotting a curve showing positions at various times and marking distances horizontally and times vertically, the velocity of light being 1, $MM'$ and $NN'$ will both represent this velocity. Since this is assumed to be the greatest velocity occurring in nature, all other possible velocities are represented by lines falling within the upper and lower V's. Now this diagram correctly represents two dimensions of Minkowski's Euclidean four-space so, transmuting to real but flat four-space by multiplying times by $\sqrt{-1}$, it is seen that there is a region outside which no effect can be propagated from $O$ since that would involve the existence of a velocity greater than that of light. This region represents the future of $O$. Similarly, $O$ can only be affected by events within the region derived from the downward-opening V, which therefore represents the past of $O$. The region between the two represents events which may be either simultaneous with $O$ or not, according to the velocity of the observer at $O$. Thus in this theory an event dictated by free-will, could affect points in its "future" region, but not in any other, which agrees with experience and shows that the theory is not essentially "determinist." If "free-will" is really free, the future is not yet determined, and the fourspace must be in some way formed by the will as time progresses.

The trains of thought inspired by Einstein's postulates have already carried us to a pinnacle of knowledge unprecedented in the history of man. On every hand, as we look out upon the universe from our new and lofty standpoint, unexpected and enthralling vistas open up before us, and we find ourselves confronting nature with an insight such as no man has ever before dared aspire to.

It is completely unthinkable that this theory can ever be swept aside. Apart from experimental verifications which, in point of fact, lend it the strongest support, no one could work through the theory without feeling that here, in truth, the inner workings of the universe were laid bare before him. The harmony with nature is far too complete for any doubt to arise of its truth.

XVI

THE QUEST OF THE ABSOLUTE

Modern Developments in Theoretical Physics, and the Climax Supplied by Einstein

BY DR. FRANCIS D. MURNAGHAN, JOHNS HOPKINS UNIVERSITY BALTIMORE

We shall discuss the more important aspects of the theory popularly known as the "Einstein Theory of Gravitation" and shall try to show clearly that this theory is a natural outcome of ideas long held by physicists in general. These ideas are:

(a) The impossibility of "action at a distance;" in other words we find an instinctive repugnance to admit that one body can affect another, remote from it, instantaneously and without the existence of an intervening medium.

(b) The independence of natural, i.e., physical, laws of their mathematical mode of expression. Thus, when an equation is written down as the expression of a physical law it must be satisfied, no matter what units we choose in order to measure the quantities occurring in the equation. As our physics teacher used to say "the expression of the law must have in every term the same dimensions." More than this the choice of the quantities used to express the law--if there be a choice open--must have no effect on its correctness. As we were told--"all physical laws are capable of expression as relations between vectors or else as relations between magnitudes of the same dimensions." We shall hope to make this clearer in its proper place in the essay, as its obvious generalization is Einstein's cardinal principle of relativity.

The measurements which an experimental physicist makes are always the expression of a coincidence of two points in space at the same time. If we ask such an experimenter what he means by a point in space he tells us that, for him, the term has no meaning until he has a material body with reference to which he can locate the point by measurements; in general it requires three measurements and he expresses this by saying that space has three dimensions. He measures his distance, as a rule, parallel to three mutually perpendicular lines fixed in the material body--a Cartesian reference-frame so-called. So that a "point in space" is equivalent to a given material reference-frame and three numbers or coordinates. If, for any reason, we prefer to use a new material reference-frame the coordinates or measurements will change and, if we know the relative positions of the two material reference-frames, there is a definite relation between the two sets of three coordinates which is termed a transformation of coordinates. But which particular material reference-frame shall we use? The first choice would, we think, be that attached to the earth. But, even yet, we are in doubt as there are numberless Cartesian frameworks attached to the earth (as to any material body) and it is here that our idea (b) begins to function. We say it must be immaterial which of these Cartesian frames we use. In each frame a vector has three components and when we change from one frame to another the components change in such a way that if two vectors have their three components equal in one framework they will be equal in any other attached to the same material system. So our idea (b), which says that our physical equations must be vector equations, is equivalent to saying that the choice of the framework attached to any given material body can have no effect on the mode of expression of a natural law.

Shall we carry over our idea (b) to answer the next question: "To which material body shall we attach our framework?" To this question Newton gave one answer and Einstein another. We shall first consider Newton's position and then we may hope to see clearly where the new theory diverges from the classical or Newtonian mechanics. Newton's answer was that there is a particular material frame with reference to which the laws of mechanics have a remarkably simple form commonly known as "Newton's laws of motion" and so it is preferable to use this framework which is called an absolute frame.

What is the essential peculiarity of an absolute frame? Newton was essentially an empiricist of Bacon's school and he observed the following facts. Let us suppose we have a framework of reference attached to the earth. Then a small particle of matter under the gravitational influence of surrounding bodies, including the earth, takes on a certain acceleration $A_1$. Now suppose the surrounding bodies removed (since we cannot remove the earth we shall have to view the experiment as an abstraction), and another set introduced; the particle, being again at its original position, will begin to move with an acceleration $A_2$. If both sets of surrounding bodies are present simultaneously the particle begins to move with an acceleration which is approximately but not quite the sum of $A_1$ and $A_2$. Newton postulated there there is a certain absolute reference frame in which the approximation would be an equality; and so the acceleration, relative to the material frame, furnishes a convenient measure of the effect of the surrounding bodies--which effect we call their gravitational force. Notice that if the effect of the surrounding bodies is small the acceleration is small and so we obtain as a limiting case, Newton's law of inertia which says that a body subject to no forces has no acceleration; a law which, as Poincaré justly observed, can never be subjected to experimental justification. The natural questions then arise: which is the absolute and privileged reference-frame and how must the simple laws be modified when we use a frame more convenient for us--one attached to the earth let us say? The absolute frame is one attached to the fixed stars; and to the absolute or real force defined as above, we must add certain terms, usually called centrifugal forces. These are referred to as fictitious forces because, as it is explained, they are due to the motion of the reference-frame with respect to the absolute frame and in no way depend on the distribution of the surrounding bodies. Gravitational force and centrifugal forces have in common the remarkable property that they depend in no way on the material of the attracted body nor on its chemical state; they act on all matter and are in this way different from other forces met with in nature, such as magnetic or electric forces. Further Newton found that he could predict the facts of observation accurately on the hypothesis that two small particles of matter attracted each other, in the direction of the line joining them, with a force varying inversely as the square of the distance between them. This law is an "action at a distance" law and so is opposed to the idea (a).

We have tacitly supposed that the space in which we make our measurements is that made familiar to us by the study of Euclid's elements. The characteristic property of this space is that stated by the theorem of Pythagoras that the distance between two points is found by extracting the square root of the sum of the squares of the differences of the Cartesian coordinates of the two points. Mathematicians have long recognized the possibility of other types of space and Einstein has followed their lead. He abandons the empiricist method and when asked what he means by a point in space replies that to him a point in space is equivalent to four numbers how obtained it is unnecessary to know a priori; in certain special cases they may be the three Cartesian coordinates of the experimenter (measured with reference to a definite material framework) together with the time. Accordingly he says his space is of four dimensions. Between any two "points" we may insert a sequence of sets of four numbers, varying continuously from the first set to the second, thus forming what we call a curve joining the two points. Now we define the "length" of this curve in a manner which involves all the points on it and stipulate that this length has a physical reality, i.e., according to our idea (b) its value is independent of the particular choice of coordinates we make in describing the space. Among all the joining curves there will be one with the property of having the smallest length; this is called a geodesic and corresponds to the straight line in Euclidean space. We must now, for lack of an a priori description of the actual significance of our coordinates, extend the idea of vector so that we may speak of the components of a vector no matter what our coordinates may actually signify. In this way are introduced what are known as tensors; if two tensors are equal, i.e., have all their components equal, in any one set of coordinates they are equal in any other and the fundamental demand of the new physics is that all physical equations which are not merely the expression of equality of magnitudes must state the equality of tensors. In this way no one system of coordinates is privileged above any other and the laws of physics are expressed in a form independent of the actual coordinates chosen; they are written, as we may say, in an absolute form.

THE GRAVITATIONAL HYPOTHESIS

Einstein flatly denies Newton's hypothesis that there is an absolute system (and, indeed, many others before him had found it difficult to admit that so insignificant a part of the universe as our fixed star system should have such a privileged position as that accorded to it in the Newtonian Mechanics). In any system, he says, we have no reason to distinguish between the so-called real gravitational force and the so-called fictitious centrifugal forces--if we wish so to express it gravitational force is fictitious force. [10] A particle moving in the neighborhood of material bodies moves according to a law of inertia--a physical law expressible, therefore, in a manner quite independent of the choice of coordinates. The law of inertia is that a particle left to itself moves along the geodesics or shortest lines in the space. If the particle is remote from other bodies the space has the Euclidean character and we have Newton's law of inertia; otherwise the particle is in a space of a non-Euclidean character (the space being always the four-dimensional space) and the path of the particle is along a geodesic in that space. Einstein, in order to make the theory more concrete, makes a certain stipulation as to the nature of the gravitational space which stipulation is expressed, as are all physical laws, by means of a tensor equation--and this is sometimes called his law of gravitation.

Perhaps it will be well, in exemplification, to explain why light rays, which pass close to the sun, should be bent according to the new theory. It is assumed that light rays travel along certain geodesics known as minimal geodesics. The sun has an intense gravitational field near it--or, as we now say, the departure of the four-dimensional space from the Euclidean is very marked for points near the sun--but for points so remote as the earth this departure is so small as to be negligible. Hence the form of the geodesics near the sun is different from that near the earth. If the space surrounding the sun were Euclidean the actual paths of the light rays would appear different from geodesics or straight-lines. Hence Einstein speaks of the curvature of the light rays due to the gravitational field of the sun; but we must not be misled by a phrase. Light always travels along geodesics (or straight lines--the only definition we have of a straight line is that it is a geodesic); but, owing to the "distortion" of the space they traverse, due to the sun, these geodesics reach us with a direction different from that they would have if they did not pass through the markedly non-Euclidean space near the sun.

The consideration of the fundamental four-dimensional space as being non-Euclidean where matter is present gives a possibility of an answer to the world old question: Is space finite or infinite? Is time eternal or finite? The fascinating possibility arises that the space may be like the two-dimensional surface of a sphere which to a limited experience seems infinite in extent and flat or Euclidean in character. A new Columbus now asks us to consider other possibilities in which we should have a finite universe--finite not only as to space measurement but as to time (for the space may be such that all of the four coordinates of its points are bounded in magnitude). However, although Einstein speaks of the possibility of a finite universe, we do not, personally, think his argument convincing. Points on a sphere may be located by the Cartesian coordinates of their stereographic projections on the equatorial plane and these coordinates, which might well be those actually measured, are not bounded.

THE SPECIAL RELATIVITY THEORY

In our account of the Einstein theory we have not followed its historical order of development for two reasons. Firstly, the earlier Special Relativity Theory properly belongs to a school of thought diametrically opposed to that furnishing the "General Theory of Relativity" and, secondly, the latter cannot be obtained from the former by the process of generalization as commonly understood. Einstein, when proposing the earlier theory, adopted the position of the empiricist so that to him the phrase, a point in space, had no meaning without a material framework of reference in which to measure space distances. When he came to investigate what is meant by time and when he asked the question "what is meant by the statement that two remote events are simultaneous?" it became evident that some mode of communication between the two places is necessary; the mode adopted was that by means of light-signals. The fundamental hypothesis was then made that the velocity of such signals is independent of the velocity of their source (some hypothesis is necessary if we wish to compare the time associated with events, when one material reference-system is used, and the corresponding time when another in motion relative to the first is adopted). It develops that time and space measurements are inextricably interwoven; there is no such thing as the length of a body or the duration of an event but rather these are relative to the reference-system. [11] Minkowski introduced the idea of the space of events--of four dimensions--but this space was supposed Euclidean like the three-dimensional space of his predecessors. To Einstein belongs the credit of taking from this representation a purely formal mathematical character and of insisting that the "real" space--whose distances have a physical significance--is the four-dimensional space. But we cannot insist too strongly on the fact that in the gravitational space of the general theory there is no postulate of the constancy of velocity of a light-signal and accordingly no method of assigning a time to events corresponding to that adopted in the special theory. In this latter theory attention was confined to material systems moving with uniform velocity with respect to each other and it developed that the velocity of light was the ultimate velocity faster than which no system could move--a result surprising and a priori rather repugnant. It is merely a consequence of our mode of comparing times of events; if some other method--thought transference, let us say--were possible the velocity of this would be the "limiting velocity."

In conclusion we should remark that the postulated equivalence of "gravitational" and "centrifugal" forces demands that anything possessed of inertia will be acted upon by a gravitational field and this leads to a possible identification of matter and energy. Further our guiding idea (a) will prompt us to say, following the example of Faraday in his electrical researches, that the geodesics of a gravitational space have a physical existence as distinct from a mere mathematical one. The four-dimensional space we may call the ether, and so restore this bearer of physical forces to the position it lost when, as a three-dimensional idea in the Special Relativity Theory, it had to bear an identical relation to a multitude of relatively moving material systems. The reason for our seemingly paradoxical title for an essay on Relativity will be clear when it is remembered that in the new theory we consider those space-time properties which are absolute or devoid of reference to any particular material reference-frame. Nevertheless, although the general characteristics of the theory are thus described, without reference to experiment, when the theory is to be tested it is necessary to state what the four coordinates discussed actually are--how they are determined by measurement. It is our opinion that much remains to be done to place this portion of the subject on a satisfactory basis. For example, in the derivation of the nature of the gravitational space, surrounding a single attracting body, most of the accounts use Cartesian coordinates as if the space were Euclidean and step from these to polar coordinates by the formulæ familiar in Euclidean geometry. But these details are, perhaps, like matters of elegance, if we shall be allowed to give Einstein's quotation from Boltzmann, to be left to the "tailor and the cobbler."

XVII

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 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 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.

Velocity
Source in to Interval Frequency Observer
Medium Observer

Air

a S 1 1 1 1 O b S 1 1+ 1- 1+ O c S 1 1 1 1- O d S 1 1+ 1- 1 O

Ether

A S 1 1 1 1 O B S 1- 1 1 1+ O C S 1 1 1 1- O D S 1 - 1 1 1 O

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.

In the case of light waves we shall find a very different state of affairs, although the rules for frequency are the same as they are for sound. In case A we have the normal conditions, where both the source and observers are stationary. In case B we have a representation of the Michelson-Morley experiment as supplemented by that of Majorana,2 where the source is stationary and the observer advances. Unlike the case of sound, the interval elapsed, as shown by the experiment, is now the same as in case A, and since the distance to the observer is less, the velocity of light with respect to the ether must also be less than unity. Since the observer is advancing against the light, this will permit the velocity of light with regard to the observer to remain unity, in conformity with the second principle of relativity. Compare with case b for sound. As Jeans expresses it, "The velocity of light in all directions is the same, whatever the motion of the observer."3 That is to say it appears to be the same to him, however he moves.

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, 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

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

As is well known the kinetic energy of a moving body may be expressed as $e = 1/2mv^2$, but if the body is charged electrically, the fraction becomes $1/2(m + m')v^2$, where $m'$ is a quantity dependent on the square of the electrical charge. That is to say, we have the normal mass of the body, and also what we may call its electrical mass. If when in this condition a portion of the mass is electrical, the question at once occurs to us, why may not the whole mass be electrical, in other words, a form of energy? Although this has not been satisfactorily proved hitherto, yet such is the general belief among physicists. As Einstein puts it "inert mass is nothing else than latent energy."1 The same idea is sometimes expressed as "the mass of ordinary matter is due to the electromagnetic energy of its ultimate particles, and electromagnetic energy wherever found must possess mass, i.e., inertia."6 If that is so, since a ray of light on the undulatory theory is a form of electromagnetic energy, it too must possess mass. Since all mass with which we are familiar is subject to the attraction of gravitation, it seemed likely that a ray of light would be bent out of its course in passing near the sun, and this as we have seen was proved to be true at the recent solar eclipse.

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 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

$$\frac{m v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

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 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 g 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 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 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.

On the other hand Turner has recently made a very interesting point,9 namely, that the fourth dimension as actually treated by the mathematicians is not time itself, but time multiplied by a constant--the velocity of light.10 Without affecting the astronomical proofs of relativity at all, this simplifies our conceptions enormously. In ordinary everyday life time and space cannot be identical, any more than a yard can be identical with a quart. On what is known to physicists as the centimeter-gram-second system, distance is represented by $l$, mass by $m$, and time by $t$. Velocity is then distance divided by time, $l/t$, or as we say in English units, so many feet per second, and the fourth dimension may be expressed as time multiplied by velocity, $t \times l/t = l$. That is to say, it is simply distance, just like the other three dimensions. To say that time is the fourth dimension from this point of view, appears to us just as ridiculous as it would be to attempt to measure the velocity of a train in quarts. It is quite correct, however, although unusual, to speak of a given train as moving at a speed of 10 quarts per square inch per second, $l^3/l^2 t = l/t$. This would be equivalent to a velocity of 33 miles per hour.

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 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. 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 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.

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, 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

Location Aperture Focus Plates Stars Defl. Dev.

Principe 13 11 2 5 1''.60 -0''.15
Sobral 13 11 19 12 0 .93 (+0 .06)
,, 4 19 8 7 1 .98 +0 .23

deviation is taken from Einstein's computed value of 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.

According to the theory of relativity the displacement of the lines should be $+0.0080 A$. St. John at Mt. Wilson found a displacement for the cyanogen lines of only $+0.0018 A$.16 Evershed at Kodaikanal found +0.0060 at the north pole of the sun, and +0.0080 at the south pole. These latter values however were only for the stronger lines. The weaker lines give much smaller shifts, as do those of calcium and magnesium.17 According to Einstein all lines should give nearly the same shift, an amount proportional to the wave length. It therefore appears that we must conclude by saying that Einstein's theory of relativity has been partially, but not completely, verified.

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.