2 2 6 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
By “Riemann tensor” or “Riemann invariant” we understand a tensor or invari-
ant (relative to an arbitrary point transformation), resp., such that their invariant
character is assured under the postulated invariance of . Further-
more, we understand as “Weyl tensor” or “Weyl invariant” (of weight n), resp., a
Riemann tensor or a Riemann invariant, resp., with the following additional prop-
erty: the value of a tensor component or invariant, resp., is multiplied by if
is replaced by , where is an arbitrary function of the coordinates. This con-
dition can be expressed symbolically by the equation
.
Now, if J is a Weyl invariant of weight depending only upon the and their
derivatives, the
(1)
is an invariant of weight 0, i.e., an invariant that depends only upon the ratios of the
. The desired generalization of the geodesic line is then given by the equation
. (2)
This solution, of course, presupposes the existence of a Weyl invariant of the kind
defined above. Weyl’s investigations show the way to one such invariant. He has
shown that the tensor
(3)
is a Weyl tensor of weight 1. is here the Riemann curvature tensor, and
the tensor of rank 2 that results from the previous one by means of
one contraction; R is then the scalar resulting from one further contraction, and d
is the number of dimensions. From this, one immediately gets that
[p. 263]
ds2
gμνdxμdxν =
λn
gμν
λgμν λ
T( λg)
λnT(g)
=
–1, gμν
dσ2
Jgμν dxμdxν =
gμν
δ
∫dσ
⎩ ⎭
⎨ ⎬
⎧ ⎫
0 =
[7]
Hiklm Riklm
1
d 2 –
-----------gilRkm -
gkmRil gimRkl gklRim – – + –
1
d 1)( – ( d 2) –
------------------------------ -
gilgkm gimgkl)R – ( +
=
Riklm
Rkm
gilRiklm
=