\subsection{The Space-Time Continuum of the Special Theory of Relativity Considered as a
Euclidean Continuum}
From Relativity: The Special and General Theory by Albert Einstein
We are now in a position to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in section 17. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these ``Galileian
co-ordinate systems." For these systems, the four co-ordinates </math>x, y,
z, t by the space co-ordinate
differences and the time-difference . With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are . Then these
magnitudes always fulfill the condition \footnotemark.
The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude
\noindent which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace , , by </math>x_1,
x_2, x_3, x_4
\noindent is independent of the choice of the body of reference. We call the
magnitude ds the "distance" apart of the two events or
four-dimensional points.
Thus, if we choose as time-variable the imaginary variable
instead of the real quantity , we can regard the space-time
continuum---accordance with the special theory of relativity---as a
"Euclidean" four-dimensional continuum, a result which follows from
the considerations of the preceding section.
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. Einstein Reference Archive which is under the FDL copyright.
\footnotetext{Cf. Appendixes I and 2. The relations which are derived
there for the co-ordinates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).}