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One of the grave difficulties we have in gaining a satisfactory comprehension of Einstein’s conceptions, is that they do not readily relate themselves to our modes of geometrical thought. Within limits we may choose our own geometry, but it may be at the cost of unwieldy complication. If we think with Newton in terms of Euclidean geometry and consider the earth as revolving around the sun, the motions of our solar system can be stated in comparatively simple terms. If, on the other hand, we should persist in stating them, as Ptolemy would have done, from the earth as a relatively stationary center, our formulas will become complicated beyond ready comprehension. For this reason it is much simpler in applying the theory of relativity, and in considering and describing what actually happens in the physical universe, to use geometrical conceptions to which the actual conditions can be easily related. We find such an instrument in non-Euclidean geometry, wherein space will appear as though it were projected from a slightly concave mirror. It is in this sense that some speak of space as curved. The analogy is so suggestive it tempts one to linger over it. Unless there were material objects within the range of the mirror, its conformation would be immaterial; the thought of the space which the mirror, as it were, circumscribes, is dependent upon the presence of such material objects. The lines of light and of all other movement will not be quite “straight” from the view-point of Euclidean geometry. A line drawn in a universe of such a nature must inevitably return upon itself. Nothing therefore, can ever pass out of this unlimitedly great but yet finite cosmos. But even now, since our imaginary mirror is only very slightly concave, it follows that for limited regions like the earth or even the solar system, our conception of geometry may well be rectilinear and Euclidean. Newton’s law of gravitation will be quite accurate with only a theoretical modification drawn from the theory of relativity.][82]
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The way in which a curvature of space might appear to us as a force is made plainer by an example. Suppose that in a certain room a marble dropped anywhere on the floor always rolled to the center of the room; suppose the same thing happened to a baseball, a billiard ball, and a tennis ball. These results could be explained in two ways; we might assume that a mysterious force of attraction existed at the center of the floor, which affected all kinds of balls alike; or we might assume that the floor was curved. We naturally prefer the latter explanation. But when we find that in the neighborhood of a large material body all other bodies move toward it in exactly the same manner, regardless of their nature or their condition, we are accustomed to postulate a mysterious attractive force (gravitation); Einstein, on the contrary, adopts the other alternative, that the space around the body is curved.][223]
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In the ordinary “analytical geometry,” the position and motion of all the points considered is referred to a rigid “body” or “frame of reference.” This usually consists of an imaginary room of suitable size. The position of any point is then given by three numbers, i.e., its distances from one side wall and from the back wall and its height above the floor. These three numbers can only give one point, every other point having at least one number different. In four-dimensional geometry a fourth wall may be vaguely imagined as perpendicular to all three walls, and a fourth number added, giving the distance of the “point” from this wall also. Since “rigid” bodies do not exist in gravitational fields the “frame of reference” must be “non-rigid.” The frame of reference in the Gaussian system need not be rigid, it can be of any shape and moving in any manner, in fact a kind of jelly. A “point” or “event” in the four dimensioned world is still given by four numbers but these numbers do not represent distances from anywhere; all that is necessary is that no two events shall have exactly the same four numbers to represent them, and that two events which are very close together shall be represented by numbers which differ only slightly from one another. This system assumes so little that it will be seen to be very wide in its scope; although to the ordinary mind, what is gained in scope seems more than that lost in concreteness. This does not concern the mathematician, however, and by using this system he gains his object, proving that the general laws of nature remain the same when expressed in any Gaussian coordinate system whatever.][220]
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Einstein enunciates a general principle that it is possible to find a transformation of coordinate axes which is exactly equivalent to any force, and in particular one which is equivalent to the force of gravitation. That is by concentrating our attention on the transformation which is a purely mathematical operation we can afford to neglect the force completely. To get a better idea of this principle of equivalence as it is called, let us consider a relatively simple example (which actually has nothing to do with gravitation, but which will serve to make our notions clearer.) A person on the earth unconsciously refers all his experiences, i.e., the motions of the objects around him to a set of axes fixed in the earth on which he stands. However, we know that the earth is rotating about its axis, and his axes of reference are also rotating with respect to the space about him. From the point of view of general relativity it is exactly because we do refer motions on the surface of the earth to axes rotating with the earth that we experience the so-called centrifugal force of the earth’s rotation, with which everyone is familiar. If we could find it convenient to transform from moving axes to fixed axes, the force would vanish, since it is exactly equivalent to the transformation from one set of axes to the other. However, we find it unnatural to refer daily experiences to axes that are not placed where we happen to be, and so we prefer to take the force and rotating axes instead of no force and fixed axes.][272]
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We seem to have a direct experience of force in our muscular sensations. By pushing or pulling we can set bodies in motion. It is natural to assume, that something similar occurs, when Nature set bodies in motion. But is this not a relic of animism? The savage and the ancients peopled all the woods and skies with Gods and demons, who carries on the activities of nature by their own bodily efforts. Today we have dispossessed the demons, but the ghost of a muscular pull still holds the planets in place.][141]