D O C . 5 4 A D D I T I O N T O G E N E R A L R E L A T I V I T Y 2 2 5
(measuring rods). Furthermore, Riemannian geometry requires that the state
(length) of a measuring rod in one place is independent of the path used to get to
this place, i.e., it contains two assumptions:
I. The existence of transferable measuring rods
II. The independence of their length from the path of transfer.
Weyl’s generalization of Riemann’s metric retains (I) but drops (II). He allows the
measured length of a measuring rod to depend upon its path of transfer by means
of an integral extended over this path of transfer; in general, the integral
depends on this path where the are space functions which, consequently, co-
determine the metric. In the physical interpretation of the theory, these are then
identified with the electromagnetic potentials.
Notwithstanding the admirable consistency and beauty of Weyl’s framework of
ideas, it does not—in my opinion—measure up to physical reality. We do not know
things in nature that can be utilized in measuring and whose relative extension de-
pends upon their past history. It also does not appear that the straightest line,
introduced by Weyl, and the electric potentials explicitly occurring in its equation
and in the other equations of Weyl’s theory, have direct physical meaning.
On the other hand, the idea elaborated by Weyl under (a) seems to me to be a
lucky and natural one, even though one cannot a priori know whether or not it can
lead to a useful physical theory. Under these circumstances, one can ask if a distinct
theory can be obtained by dropping from the beginning not only Weyl’s assumption
(II), but also assumption (I) about the existence of transferable measuring rods (and
clocks, resp.). In what follows, it shall be shown that one arrives freely and easily
at a theory by starting out only from the invariant meaning of the equation:
without using the concept of distance ds, or—to put it in terms of physics—without
using the concepts of measuring rods or measuring clocks.
In my effort to formulate such a theory, my colleague Wirtinger in Vienna gave
me efficient support. I asked him if there is a generalization of the equation of a
geodesic line such that only the ratios of the play a role. He answered me as
gμνdxμdxν 0, = =