four-dimensional points.
Thus, if we choose as time-variable the imaginary variable
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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.
XXVII.
THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A
EUCLIDEAN 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 XXVI, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the velocity of light. But
according to Section XXI the general theory of relativity cannot retain
this law. On the contrary, we arrived at the result that according to
this latter theory the velocity of light must always depend on the
co-ordinates when a gravitational field is present. In connection with
a specific illustration in Section XXIII, we found that the presence of
a gravitational field invalidates the definition of the coordinates and
the time, which led us to our objective in the special theory of
relativity.
In view of the resuIts of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
local variations of temperature, and with which we made acquaintance as
an example of a two-dimensional continuum. Just as it was there
impossible to construct a Cartesian co-ordinate system from equal rods,
so here it is impossible to build up a system (reference-body) from
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