Page:Grundgleichungen (Minkowski).djvu/48

which represents a hyperboloidal shell, contains the space-time points $A(x,\ y,\ z,\ t=0,\ 0,\ 0,\ 1)$, and all points A', which after a Lorentz-transformation enter into the newly introduced system of reference as $(x',\ y',\ z',\ t'=0,\ 0,\ 0,\ 1)$.

The direction of a radius vector OA' drawn from to the point A' of (2), and the directions of the tangents to (2) at A' are to be called normal to each other.

Let us now follow a definite position. of matter in its course through all time t. The totality of the space-time points x, y, z, t, which correspond to the positions at different times t, shall be called a space-time line.

The task of determining the motion of matter is comprised in the following problem: — It is required to establish for every space-time point the direction of the space-time line passing through it.

Transforming a space-time point P(x, y, z, t) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference x', y', z', t', in which the t' axis has the direction OA', OA' indicating the direction of the space-time line passing through P. The space t' = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P. To the increment dt of the time of P corresponds the increment

(3)	$d\tau ={\sqrt {dt^{2}-dx^{2}-dy^{2}-dz^{2}}}=dt{\sqrt {1-{\mathfrak {w}}^{2}}}={\frac {dx_{4}}{w_{4}}}$[1]

of the newly introduced time parameter t'. The value of the integral

$\int d\tau =\int {\sqrt {-(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2})}}$,

when calculated upon the space-time line from a fixed initial point P° to the variable point P, (both being on the space-time line), shall be called the Proper-time of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)

If we take a body $R^{0}$ which has got extension in space at time $t^{0}$, then the region comprising all the space-time line passing through $R^{0},t^{0}$ shall be called a space-time filament.

If we have an analytical expression $\Theta (x,\ y,\ z,\ t)$ so that $\Theta (x,\ y,\ z,\ t)=0$ is intersected by every space time line of the filament at one pointy — whereby

1. The notation with indices and the symbols ${\mathfrak {w}},w$ we again use in the form as it was defined before. (s. § 8 and § 4).