2.3 Bivectors and Gravitational Fields
31
Riemann curvature tensor, the components R ab of the Ricci tensor and the Ricci
scalar R by the formula:
C abcd = R abcd +
1
2
(g ad R bc + g bc R ad − g ac R bd − g bd R ac ) +
1
6
R(g ac g bd − g ad g bc ) .
(2.79)
The mathematical model of a vacuum gravitational field is a space-time for which
Einstein’s vacuum field equations, R ab = 0 (⇒ R = 0), hold and in this case
C abcd = R abcd . Thus the Riemann tensor describes a vacuum gravitational field
while a non-vacuum gravitational field is described by the Weyl tensor. Following
from the algebraic symmetries of the Riemann tensor, the Weyl tensor has the
algebraic symmetries
C abcd = C cdab , C abcd = −C bacd , C abcd = −C abdc ,
(2.80)
and
C abcd + C adbc + C acdb = 0 .
(2.81)
In addition the Weyl tensor has the symmetry
g
bd C abcd = 0 ,
(2.82)
which the reader can easily verify from (2.79). Since the Riemann tensor components R abcd are skew-symmetric in a, b and in c, d we can define a left dual
∗ R abcd =
1
2
η abpq R
pq
cd ,
(2.83)
and a right dual
R
∗
abcd =
1
2
η cdpq R ab
pq ,
(2.84)
and similarly for the Weyl tensor. Using the properties of the permutation tensor
η abcd the reader can demonstrate that
∗ R
∗
abcd =
1
4
η abpq η cdrs R
pqrs
= −R abcd − g bc R ad − g ad R bc + g bd R ac + g ac R bd
+
1
2
R(g ad g bc − g ac g bd ) .
(2.85)