1 4 8 D O C . 3 1 I D E A S A N D M E T H O D S
does exist. This formula is uniquely determined for all locations once a unit rod of
measurement has been chosen.
Hence, it is clear that justice is done to the metric conditions that rule the four-
dimensional world when we introduce general Gaussian coordinates
into the four-dimensional world by perceiving the world as a four-dimensional
Riemannian space in which the metric invariant —which belongs to two neigh-
boring space-time points—is physically uniquely defined by the generalized
Pythagorean invariant
. (16)
The coefficients depending on not only determine the
metric behavior of the world, i.e., the behavior of measuring rods and clocks, but
also the phenomena of inertia and gravitation, as can be concluded from the hy-
pothesis of equivalence.
For let us next look at a finite domain where the theory of special relativity is
valid with sufficient approximation. With a suitable choice of the coordinate sys-
tem (inertial system) and measurement of time, the formula
is valid in such a subdomain. If we now introduce a new coordinate system
that is accelerated relative to , then this is mathematically
equivalent to the introduction of new space-time variables that are connected to the
original ones by a nonlinear transformation. One finds by direct calculation that the
metric invariant in the new system is represented by a formula like (16),
whereby not all coefficients are constant. On the other hand, we know from
the hypothesis of equivalence that there is a gravitational field relative to .
We thus arrive at the result: the physical behavior of the space-time continuum
is governed by 10 quantities that determine the metric qualities (be-
havior of measuring rods and clocks) as well as the phenomena of inertia and grav-
The question of the mathematical formulation of the principle of general relativ-
ity is thereby decided. While the special theory of relativity demanded the
covariance of the equations that express laws of nature under certain linear trans-
formations of coordinates (Lorentz transformations), the general theory of
relativity demands covariance under arbitrary transformations. In this theory,
coordinates merely have the role of arithmetic parameters, devoid of any direct
x1, x2, x3, x4

2g12dx1dx2 . . . g44dx4
+ + + =
g11 . . . g44 , , x1, x2, x3, x4
dσ2 dx1
2 dx42
+ + + =
x′1, x′2, x′3, x′4) ( K
g11 . . . g44 , ,
[p. 34]
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