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
. 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
. 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
and
. 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.