Equation (1) shows that the geometry of the World-Frame referred to an inertial system is semi-Euclidean (hyperbolic), and that space and time measurements are relative to the observer’s inertial reference system. The equation shows that the World-Frame has a certain geometrical character which we distinguish as four-dimensional “flatness.” It is everywhere alike (homaloidal). Its flat character is shown by the straight line nature of the separation-interval and of the system to which it is most simply referred.
Thus we have found two absolute features in the World-Frame—(1) Its geometrical character—”flatness”; (2) The separation-interval—which can be expressed in terms of measurable variables called space and time partitions, this partitioning being dependent on the observer’s motion.
We are now in a position to explore the World-Fabric. Already we see that, studied under inertial conditions (free of force), it agrees with the World-Frame.
The World-Fabric
The General Theory of relativity is largely concerned with the investigation of the World-Fabric. Consider the World-Frame to be disturbed. We may regard this disturbance, which manifests itself in physical phenomena, as energy, or more correctly “action.”
When energy is thwarted in its natural flow, force is manifested, with which are associated non-uniform motions such as accelerations and rotations. This disturbed World-Frame we distinguish as the World-Fabric. It is found to have various non-Euclidean characters differing from the simple “flat” character of the World-Frame according to the degree of disturbance (action) in the region. Disturbance gives the fabric a geometrical character of “curvature”; the more considerable the disturbance, the greater the curvature. Thus an empty region (not containing energy, but under its influence) has less curvature than a region in which free energy abounds.
Our problem, after showing the relativity of force (especially gravitational force), is to determine the law underlying the fabric’s geometrical character; to ascertain how the degree of curvature is related to the energy influencing a region, and how the curvature of one region is linked by differential equations to that of neighboring regions. Such a law will be seen to be the law of gravitation.
We study the World-Fabric by considering tracks on which material particles and light-pulses progress; we find such tracks regulated and defined by the Fabric’s curvature, and not, as hitherto supposed, by attractive force inherent in matter. As a track is measurable by summing the separation-intervals between near-by point-events on it, all observers will agree which is the unique track between two distant point-events. Einstein postulates that freely progressing bodies will follow unique tracks, which are therefore called natural tracks (geodesics).
If material bodies are prevented from following natural tracks by contact with matter or other causes, the phenomenon of gravitational force is manifested relative to them. Whenever the natural flow of energy is interrupted force is born. For example, when the piston interrupts the flow of steam, or golf ball flow of club, force results—the interruption is mutual, and the force relative to both. Likewise when the earth interrupts the natural track of a particle (or observer) gravitational force is manifested relative to both.