Page:The principle of relativity (1920).djvu/173

 co-ordinates; X_{4} is the corresponding time-co-ordinate measured by some suitable measuring clock. These co-*ordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure). The expression

(1) ds^2 = - dX_{1}^2 - dX_{2}^2 - dX_{3}^2 + dX_{4}^2

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system. Let us take ds as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If ds^2 belonging to the element (dX_{1} dX_{2}, dX_{3}, dX_{4}) be positive we call it with Minkowski, time-like, and in the contrary case space-like.

To the line-element considered, i.e., to both the infinitely near point-events belong also definite differentials dx_{1}, dx_{2}, dx_{3}, dx_{4}, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a local system of the above kind given for the case under consideration, dX's would then be represented by definite linear homogeneous expressions of the form

(2) dX_{v} = Σ_{σ}a_{νσ}dx_{σ}

If we substitute the expression in (1) we get

(3) ds^2 = Σ_{στ}g_{στ}dx_{σ}dx_{τ}

where g_{στ} will be functions of x_{σ}, but will no longer depend upon the orientation and motion of the 'local' co-ordinates; for ds^2 is a definite magnitude belonging to two point-*events infinitely near in space and time and can be got by