The Geometry of Gravitation
Let us consider a circular disc rotating with a uniform peripheral speed. According to the deductions from the “special theory” of relativity, an observer situated near the edge of this disc, but not rotating with it, will observe that units of length measured along the circumference of the disc are contracted. On the other hand, measurements along the diameter, which is at right angles to the direction of motion of the circumference, will show no contraction whatever, and, consequently the observer will find that the ratio of circumference to diameter has not the well known value 3.14159 … but exceeds this value, the difference being greater and greater as the peripheral speed approaches that of light. That is, the laws of ordinary geometry no longer hold true.
However, we know other cases in which the ordinary or Euclidean geometry is not applicable. Thus suppose that on the surface of a sphere we describe a series of concentric circles. Since the surface is curved, we are not surprised at finding that the circumference of any one of these circles is less than 3.14159 … times the distance across the circle as measured on the surface of the sphere. What this means, therefore, is that we cannot use Euclidean geometry to describe measurements on the surface of a sphere, and every schoolboy knows this from comparing Mercator’s projection of the earth’s surface with the actual representation on a globe.
When we come to think of it, the reason we realize all this is because our sense of three dimensions enables us to differentiate flat surfaces from those that are curved. Let us, however, imagine a two-dimensional being living on the surface of a large sphere. So long as his measurements are confined to relatively small areas he will find it possible to describe all his measurements in terms of Euclidean geometry. As, however, his area of operation increases he will begin to observe greater and greater discrepancies. Being unfamiliar with the existence of such a three-dimensional object as a sphere, and therefore not realizing that he is on the surface of one, our intelligent two-dimensional being will conclude that the disturbance in his geometry is due to the action of a force, and by means of plausible assumptions on the “law” of this force he will reconcile his observations with the laws of plane geometry.
Now since an acceleration in a gravitational field is identical with that due to centrifugal force produced by rotation, we concluded that the geometry in a gravitational field must also be non-Euclidean. That is, space in the neighborhood of matter is distorted or curved. The curvature of space bears the same relation to three dimensions that the curvature of a spherical surface bears to two dimensions, and that is why we do not perceive it, any more than the intelligent two-dimensional being would be aware of the distortion of his space (or surface). Furthermore, like this being, we have assumed the existence of a gravitational force to account for discrepancies in our geometrical measurements.
The identification in this manner of gravitational effects with geometrical curvature of space enables Einstein to derive a general law for the path of any particle in a gravitational field, with respect both to space and to time. Furthermore, the law expresses this motion in terms which are independent of the relative motion and position of the observer, and satisfies the condition that the fundamental laws of physics be equally valid for all observers. The solution of the problem involved the use of a new kind of higher calculus, elaborated by two Italian mathematicians, Ricci and Levi-Civita. The result is a law of motion which is extremely general in its validity.
For low velocities it approximates to Newton’s solution, and in the absence of a gravitational field it leads to the same conclusions as the special theory of relativity. There are three deductions from this law which have aroused a great deal of interest, and the confirmation of two of these by actual observation must be regarded as striking proof of Einstein’s theory.
XIII
AN INTRODUCTION TO RELATIVITY
A Treatment in Which the Mathematical Connections of Einstein’s Work are Brought Out More Strongly and More Successfully Than Usual in a Popular Explanation
BY HAROLD T. DAVIS, UNIVERSITY OF WISCONSIN, Madison, Wis.
One of the first questions which appears in philosophy is this: What is the great reality that underlies space and time and the phenomena of the physical universe? Kant, the philosopher, dismissed it as a subjective problem, affirming that space and time are “a priori” concepts beyond which we can say no more.
Then the world came upon some startling facts. In 1905 a paper appeared by Professor Albert Einstein which asserted that the explanation of certain remarkable discoveries in physics gave us a new conception of this strange four-dimensional manifold in which we live. Thus, the great difference between the space and time of philosophy and the new knowledge is the objective reality of the latter. It rests upon an amazing sequence of physical facts, and the generalized theory, which appeared several years later, founded as it is upon the abstruse differential calculus of Riemann, Christoffel, Ricci and Levi-Civita, emerges from its maze of formulas with the prediction of real phenomena to be sought for the in the world of facts.
We shall, therefore, approach the subject from this objective point of view. Let us go to the realm of actual physical events and see how the ideas of relativity gradually unfolded themselves from the first crude wonderings of science to the stately researches that first discovered the great ocean of ether and then penetrated in such a marvelous manner into some of its most mysterious properties.