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}