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 ds² belonging to the element (dX₁, dX₂, dX₃, dX₄) 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 dx₁, dx₂, dx₃, dx₄, 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) dXν = σσaνσdxσ
If we substitute the expression in (1) we get
(3) ds² = σστgστdxσdxτ
where gστ will be functions of xσ, but will no longer depend upon the orientation and motion of the ‘local’ co-ordinates; for ds² is a definite magnitude belonging to two point-events infinitely near in space and time and can be got by measurements with rods and clocks. The gτσ’s are here to be so chosen, that gτσ = gστ; the summation is to be extended over all values of σ and τ, so that the sum is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.
From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviour of gστ in a finite region it is possible to choose the system of co-ordinates in such a way that gστ assumes constant values—
{ -1, 0, 0, 0
(4) { 0 -1 0 0
{ 0 0 -1 0