D O C . 3 1 I D E A S A N D M E T H O D S 1 4 7
general Gaussian coordinates are—in mathematical terminology—covariant under
arbitrary transformations of the coordinates.
21. Physical and Mathematical Content of the Principle of General Relativity
After these more formal intermediary considerations, we pick up the thread of our
main objective. We have seen how the extension of the principle of relativity upon
non-acceleration-free relative movements of coordinate systems helped us to un-
derstand the essential equality of an inertial and gravitational mass. On the other
hand, however, it turned out to be impossible to introduce into finite domains
space-time coordinates such that the spatial coordinates could be measured with
identically constructed measuring rods, and the timelike coordinates could be di-
rectly measured with clocks. There are no physical objects whatsoever to represent
the straight line, whereupon, consequently, it becomes impossible to distinguish in
a physically meaningful manner between rectilinear-orthogonal (Cartesian) and
curvilinear coordinate systems. Therefore, we have here for the four-dimensional
space-time continuum of physics a case that is precisely analogous to the geomet-
rical problem of Gauss and Riemann.
But the analogy goes even further. Just as the metric behavior of infinitesimal
pieces of a surface (or Riemannian space) is amenable to Euclidean geometry, such
that there exist in the infinitesimal locally Cartesian coordinate systems in which
the distance , measured with a measuring rod, can be expressed in (locally) di-
rectly measurable coordinates according to the Pythagorean formula
;
exactly in the same way there exist everywhere in the space-time world of the the-
ory of general relativity local coordinate systems where the simple metric relations
of the theory of special relativity obtain. Just as in the theory of special relativity,
the space-time coordinates are directly connected to results of measurements that
can be obtained with measurement rods and clocks; and, also, the Minkowski in-
variant given by the Pythagorean
formula[50]
(15)
ds
X1, X2) (
ds2
dX1
2
dX2
2
+ =
[p. 33]
dσ,
dσ2 dX1
2
dX2
2
dX3
2
dX4
2
+ + + =
{4}