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

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

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