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-

itation.

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

dσ

dσ2

g11dx1

2

2g12dx1dx2 . . . g44dx4

2

+ + + =

{4}

g11 . . . g44 , , x1, x2, x3, x4

K

dσ2 dx1

2

dx2

2

dx3

2 dx42

+ + + =

K′

x′1, x′2, x′3, x′4) ( K

dσ K′

{4}

gμν

K′

g11 . . . g44 , ,

{5}

[p. 34]