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 = √(dt² - dx² - dy²) - dz² = dt√(1 - u²)
of the newly introduced time parameter t′. The value of the integral
∫ dτ = ∫ √(-(dx₁² + dx₂² + dx₃² + dx₄²))
when calculated upon the space-time line from a fixed initial point P₀ to the variable point P, (both being on the space-time line), is known as the ‘Proper-time’ of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)