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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.