Page:SahaSpaceTime.djvu/12

at the point R of the 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 measurement of vectors in different directions, the unit measuring rod is to be fixed in the following manner; — a space-like vector from to -F = 1 is always to have the measure unity, and a time-like vector from O to +F = 1 , t >0 is always to have the measure ${\tfrac {1}{c}}$.

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

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

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

The integral $\tau =\int d\tau$, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the "Proper-time" of the substantial point at P₀ 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" τ; let $\left({\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}\right)$ denote the first differential-quotients, and $\left({\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}\right)$ the second differential quotients of (x, y, z, t) with regard to τ, then these may respectively