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 7
(4)
is a Weyl scalar of weight Therefore,
(5)
is a Weyl invariant of weight . In combination with (1) and (2), this result pro-
vides a generalization of the geodesic line according to the method outlined by
Wirtinger. Of course, in order to judge the significance of this and the following re-
sults, it is a question of great importance whether or not J is the only Weyl invariant
of weight that does not contain higher than second derivatives of the .
Based upon what has been developed so far, it is now easy to assign a Weyl ten-
sor to every Riemann tensor and with this to establish laws of nature in the form of
differential equations that depend only upon the ratios of the . If we put
,
then
is an invariant that depends only upon the ratios of the . All Riemann tensors
formed as fundamental invariants from in the customary manner are—when
seen as functions of the and their derivatives—Weyl tensors of weight 0. This
can be symbolically expressed as follows. If is a Riemann tensor which de-
pends not only upon the and their derivatives but also upon other quantities,
e.g., the components of an electromagnetic field, then —seen as a func-
tion of the and their derivatives—is also a Weyl tensor of weight 0. Therefore,
to every law of nature of the general theory of relativity, there corre-
sponds a law which contains only the ratios of the .
This result becomes even more distinct by the following consideration. Since
there is an arbitrary factor in the , it will be possible to select this factor such
that everywhere
(6)
where is a constant. The are then equal to the up to a constant factor;
and the laws of nature in the new theory again take the form
H
HiklmHiklm
=
–2.
J H =
–1
–1
[8]
gμν
[p. 264]
gμν
g′μν Jgμν =
dσ2
gμν
′
dxμdxν =
gμν
dσ
gμν
T g) (
gμν
φμν T g′) (
gμν
T g) ( 0 =
T g′) ( 0, = gμν
gμν
J J
°
, =
J
°
gμν ′ gμν