Page:The principle of relativity (1920).djvu/129

 A space time null point 0 (x, y, z, t = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation.

The figure -x^2 - y^2 - z^2 + t^2 = 1, t > 0  (2)

which represents a hyper boloidal shell, contains the space-time points A (x, y, z, t = 0, 0, 0, 1), and all points A´ which after a Lorentz-transformation enter into the newly introduced system of reference as (x´, y´, z´, t´ = 0, 0, 0, 1).

The direction of a radius vector 0A´ drawn from 0 to the point A´ of (2), and the directions of the tangents to (2) at A´ are to be called normal to each other.

Let us now follow a definite position of matter in its course through all time t. The totality of the space-time points (x, y, z, t) which correspond to the positions at different times t, shall be called a space-time line.

The task of determining the motion of matter is comprised in the following problem:—It is required to establish for every space-time point the direction of the space-time line passing through it.

To transform a space-time point P (x, y, z, t) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference (x´, y´, z´, t´), in which the t´ axis has the direction 0A´, 0A´ indicating the direction of the space-time line passing through P. The space t´ = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.

To the increment dt of the time of P corresponds the increment

dτ = [sqrt](dt^2 - dx^n - dy^2) - dz^2 = dt[sqrt](1 - u^2)

of the newly introduced time parameter t´. The value of the integral

[integral]dτ = [integral][sqrt](-(dx_{1}^2 + dx-{2}^2 + dx_{3}^2 + dx_{4}^2))