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 upper 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 upper 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 upper A 1 and upper 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.1 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.2 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.”
