Page:De Raum Zeit Minkowski 018.jpg

 always earlier, every point on the aft side of O, necessarily later than O. The limit $$c = \infty$$ corresponds to a complete folding up of the wedge-shaped cross-section between the cones in the plane manifoldness $$t = 0$$. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards x, y, z, t into its four components x, y, z, t. If the directions of the two vectors are respectively the directions of the radius vector OR of O at one of the surfaces $$\pm F = 1$$, and additionally a tangent RS at the point R of the relevant surface, then the vectors shall be called normal to each other. Accordingly

$c^{2}tt_{1}-xx_{1}-yy_{1}-zz_{1}=0,\,$

which is the condition that the vectors with the components x, y, z, t and $$\left(x_{1}\ y_{1}\ z_{1}\ t_{1}\right)$$ are normal to each other.

For the sums of vectors in different directions, the unit measuring rods are to be fixed in the following manner; — a space-like vector from to $$-F = 1$$ is always to have the sum 1, and a time-like vector from O to $$+F = 1$$, $$t>0$$ is always to have the sum $$1/c$$.

Let us now fix our attention upon the world-line of a substantial point running through the world-point P(x, y, z, t); then as we follow the progress of the line, the quantity

$d\tau=\frac{1}{c}\sqrt{c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}},$|undefined

corresponds to the time-like vector-element dx, dy, dz, dt.

The integral $$\tau=\int d\tau$$ of this sum, taken over the world-line from any fixed initial point $$P_{0}$$ to any variable endpoint P, may be called the "proper-time" of the substantial point in P. Upon the world-line, we may regard x, y, z, t, i.e., the components of the vector OP, as functions of the "proper-time" $$\tau$$; let $$\left(\dot{x},\ \dot{y},\ \dot{z},\ \dot{t}\right)$$ denote their first differential-quotients with respect to $$\tau$$, and $$\left(\ddot{x},\ \ddot{y},\ \ddot{z},\ \ddot{t}\right)$$ their second differential quotients with respect to $$\tau$$, and denote the corresponding vectors, i.e. the derivation of the vector OP with respect to $$\tau$$ the motion-vector in P, and the derivation of this motion-vector with respect to $$\tau$$ the acceleration-vector in P. There we have

{{Center|$$\left.\begin{array}{l} c^{2}\dot{t}^{2}-\dot{x}^{2}-\dot{y}^{2}-\dot{z}^{2}=c^{2}\\ c^{2}\dot{t}\ddot{t}-\dot{x}\ddot{x}-\dot{y}\ddot{y}-\dot{z}\ddot{z}=0\end{array}\right\} $$}}

i.e., the motion-vector is the time-like vector in the direction of the world-line at P of sum 1, the acceleration-vector at P is normal to the motion-vector at P, and is in any case a space-like vector.