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350 CONTRIBUTIONS TO SCIENCE
true of the new generalization, which is an attempt to comprise
the properties of the total field. In the generalized theory the
procedure of deriving from the premises of the theory conclusions
that Can be confronted with empirical data is so difficult
that so far no such result has been obtained. In favor of this
theory are, at this point, its logical simplicity and its "rigidity."
Rigidity means here that the theory is either true or false, but
not modifiable.
The greatest inner difficulty impeding the development of
the theory of relativity is the dual nature of the problem, indicated
by the two questions we have asked. This duality is the
reason why the development of the theory has taken place in two
steps so widely separated in time. The first of these steps, the
theory of gravitation, is based on the principle of equivalence
discussed above and rests on the following consideration: According
to the theory of special relativity, light has a constant
velocity of propagation. If a light ray in a vacuum starts from
a point, desigoated by the coordinates Xl, X, and Xs in a three
dimensional coordinate system, at the time X., it spreads as a
spherical wave and reaches a neighboring point (Xl + dx].,
x, + dx" Xs + dXa) at the time x. + dx,. Introducing the
velocity of light, c, we write the expression:
VdXl' + dx,'+ dxs' = cdx.
This can also be written in the form:
dXl' + dx,' + dx,' - c' dx.' = 0
This expression represents an objective relation between
neighboring space· time points in four dimensions, and it holds
for all inertial systems, provided the coordinate transformations
are restricted to those of special relativity. The relation loses
this form, however, if arbitrary continuous transformations of
the coordinates are admitted in accordance with the principle
of general relativity. The relation then assumes the more general
form:
:E g,. dx, dXk = 0
'k
The g'k are certain functions of the coordinates which transform
in a definite way if a continuous coordinate transformation is
applied. According to the principle of equivalence, these g ..