If we replace x, y, z, sq. rt. -I . ct , by x[1],
x[2], x[3], x[4], we also obtaill the result that
ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
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 sq. rt. -I
. ct instead of the real quantity t, we can regard the space-time
contintium -- accordance with the special theory of relativity -- as a
", Euclidean " four-dimensional continuum, a result which follows from
the considerations of the preceding section.
Notes
*) Cf. Appendixes I and 2. The relations which are derived
there for the co-ordlnates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).
THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A
ECULIDEAN CONTINUUM
In the first part of this book we were able to make use of space-time
co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to Section 26, can be regarded
as four-dimensional Cartesian co-ordinates.
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