the form of the natural extension of the special principle of
relativity. According to the special theory of relativity, the
equations which express the general laws of nature pass over into
equations of the same form when, by making use of the Lorentz
transformation, we replace the space-time variables _x, y, z, t_, of a
(Galileian) reference-body _K_ by the space-time variables _x , y , z ,
t _, of a new reference-body _K _. According to the general theory of
relativity, on the other hand, by application of _arbitrary
substitutions_ of the Gauss variables _x_1, _x_2, _x_3, _x_4, the
equations must pass over into equations of the same form; for every
transformation (not only the Lorentz transformation) corresponds to the
transition of one Gauss co-ordinate system into another.
If we desire to adhere to our old-time three-dimensional view of
things, then we can characterise the development which is being
undergone by the fundamental idea of the general theory of relativity
as follows: The special theory of relativity has reference to Galileian
domains, _i.e._ to those in which no gravitational field exists. In
this connection a Galileian reference-body serves as body of reference,
_i.e._ a rigid body the state of motion of which is so chosen that the
Galileian law of the uniform rectilinear motion of isolated material
points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian
domains to _non-Galileian_ reference-bodies also. A gravitational field
of a special kind is then present with respect to these bodies (cf.
Sections XX and XXIII).
In gravitational fields there are no such things as rigid bodies with
Euclidean properties; thus the fictitious rigid body of reference is of
no avail in the general theory of relativity. The motion of clocks is
also influenced by gravitational fields, and in such a way that a
physical definition of time which is made directly with the aid of
clocks has by no means the same degree of plausibility as in the
special theory of relativity.
For this reason non-rigid reference-bodies are used, which are as a
whole not only moving in any way whatsoever, but which also suffer
alterations in form _ad lib._ during their motion. Clocks, for which
the law of motion is of any kind, however irregular, serve for the
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