“If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy.”
I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.
If B*(x*, y*, z*, t*) be a solid (fester) space-time point, then the region of all those space-time points B (x, y, z, t), for which
(23) (x - x*)² + (y - y*)² + (z - z*)² = (t - t*)²
t - t* >= 0
may be called a “Ray-figure” (Strahl-gebilde) of the space time point B*.
A space-time line taken in any manner can be cut by this figure only at one particular point; this easily follows from the convexity of the figure on the one hand, and on the other hand from the fact that all directions of the space-time lines are only directions from B* towards to the concave side of the figure. Then B* may be called the light-point of B.
If in (23), the point (x y z t) be supposed to be fixed, the point (x* y* z* t*) be supposed to be variable, then the relation (23) would represent the locus of all the space-time points B*, which are light-points of B.
Let us conceive that a material point F of mass m may, owing to the presence of another material point F*, experience a moving force according to the following law. Let us picture to ourselves the space-time filaments of F and F* along with the principal lines of the filaments. Let BC be an infinitely small element of the principal line of F; further let B* be the light point of B, C* be the light point of C on the principal line of F*; so that OA′ is the radius vector of the hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is the point of intersection of line B*C* with the space normal to itself and passing through B. The moving force of the mass-point F in the space-time point B is now the space-time vector of the first kind which is normal to BC, and which is composed of the vectors
(24) mm*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of suitable value in direction of B*C*.