Page:Grundgleichungen (Minkowski).djvu/57

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.^[1]

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)	$(x-x^{*})^{2}+(y-y^{*})^{2}+(z-z^{*})^{2}=(t-t^{*})^{2},\ t-t^{*}\geqq 0$

may be called a Ray-figure (Strahl-gebilde) of the space lime 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 B(x, y, z, t) be supposed to be fixed, the point B*(x*, y*, z*, z*) 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)	$mm^{*}\left({\frac {OA'}{B^{*}D^{*}}}\right)^{3}BD^{*}$

in the direction of BD* and another vector of suitable value

1. In a completely different manner than I do, H. Poincaré (Rend. Circ. Matem. Palermo, t. XXI (1906), p. 129) tried to harmonize Newton's law of attraction with the relativity postulate.