body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the observation
of the encounter of the hands of clocks with particular points on the
dials. It is just the same in the case of space-measurements by means
of measuring-rods, as a little consideration will show.
The following statements hold generally: Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events _A_ and _B_. In terms of
Gaussian co-ordinates, every such statement is expressed by the
agreement of their four co-ordinates _x_1, _x_2, _x_3, _x_4. Thus in
reality, the description of the time-space continuum by means of Gauss
co-ordinates completely replaces the description with the aid of a body
of reference, without suffering from the defects of the latter mode of
description; it is not tied down to the Euclidean character of the
continuum which has to be represented.
XXVIII.
EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the provisional formulation of the
general principle of relativity given in Section XVIII by an exact
formulation. The form there used, All bodies of reference _K, K _,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion, cannot be maintained, because the use of rigid
reference-bodies, in the sense of the method followed in the special
theory of relativity, is in general not possible in space-time
description. The Gauss co-ordinate system has to take the place of the
body of reference. The following statement corresponds to the
fundamental idea of the general principle of relativity: _All Gaussian
co-ordinate systems are essentially equivalent for the formulation of
the general laws of nature._
We can state this general principle of relativity in still another
form, which renders it yet more clearly intelligible than it is when in
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