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.
So long as a body moves freely no force is appreciated by it. A falling aviator (neglecting air resistance) will not appreciate any gravitational force. He follows a natural track, thereby freeing himself from the force experienced in contact with matter. He acquires an accelerating motion with respect to an inertial system. By acquiring a particular accelerating motion an observer can annul any force experienced in any small region where the field of force can be considered constant.
Thus Einstein, interpreting the equality of gravitational and inertial mass, showed that the same quality manifests itself according to circumstances as “weight” or as inertia, and that all force is purely relative and may be treated as one phenomenon (an interruption in energy flow). This “Principle of Equivalence” shows that small portions of the World-Fabric, observed from a freely moving particle (free of force), could be treated as small portions of the World-Frame.[3]
If such observations were practicable, we could determine the Fabric curvature by referring point-event measurements to equation (1). We cannot observe from unique tracks but we can observe them from our restrained situation. Their importance is now apparent, because, by tracing them over a region, we are tracing something absolute in the Fabric—its geometrical character. We study this curvature by exploring separation-intervals on the tracks of freely moving bodies, relating these separation-intervals to actual measurements in terms of space and time components depending on the observer’s reference system. The law of curvature must be the law of gravitation. To illustrate the lines on which Einstein proceeded to survey the World-Fabric from the earth we will consider a similar but more simple problem—the survey of the sea-surface curvature from an airship. We study this curvature by exploring small distances on the tracks of ships (which we must suppose can only move uniformly on unique tracks—arcs of great circles), relating such distances to actual measurements in terms of length and breadth components depending on the observer’s reference system. This two-dimensional surface problem can be extended to the four-dimensional Fabric one.
We consider the surface to be covered by two arbitrarily drawn intersecting series of curves: curves in one series not intersecting each other, vide figure. This Gaussian system of coordinates is appropriate only when the smaller the surface considered, the more nearly it approximates to Euclidean conditions. It admits of defining any point on the surface by two numbers indicating the curves intersecting at that point. P is defined by
,
.
(very near P) is defined by
,
. The equation for the minute distance s between two adjacent points in such a system is given by the general formula
The g’s may be constants or functions of
,
. Their value is dependent on the observer’s reference system and on the geometrical character of the surface observed. The curves being arbitrary, the formula is appropriate for any reference system, or even if the observer does not know exactly what his reference system is. (The Fabric observer does not know what his space and time partitioning actually is because he is in a gravitational field). It is the g’s which disclose the geometry of an observer’s partitions, and their values also contain a reflection of the character of the region observed.
We find s by direct exploration with a moving ship (
is found by direct exploration with a freely moving particle);
,
are the observed length and breadth measurement differences which we have to relate to s. By making sufficient observations in a small area and referring them to the general formula we can find the values of the g’s for the observer’s particular reference system. Different values for g’s will be found if the observer changes his reference system, but there is a limitation to the values so obtainable owing to the part played by the surface itself, which is diffidently expressing its intrinsic geometrical character in the g’s in each observation.