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.