7.2 Background Space-Time/External Fields
161
With the electromagnetic field above as source, Einstein’s field equations for the
background space-time are
R (a)(b) = 2 E (a)(b) ,
(7.61)
where R (a)(b) are the components of the Ricci tensor calculated on the half null
tetrad defined via the 1-forms (7.27)–(7.30) and E (a)(b) are the tetrad components
of the electromagnetic energy-momentum tensor, given in terms of the tetrad
components of the Maxwell tensor F (a)(b) by
E (a)(b) = F (c)(a) F
(c)
(b) −
1
4
g (a)(b) F (c)(d) F
(c)(d) ,
(7.62)
with g (a)(b) given by (7.26). When the expansions (7.33)–(7.38) and (7.41)–(7.43)
are used to calculate the Weyl conformal curvature tensor on r = 0 we arrive at the
coefficients of the leading terms in (7.34)–(7.37) (the analogues of (7.51)):
α 2 =
1
6
P
2
0 C ij kl (u) k
i ∂k j
∂x
k
k ∂k l
∂x
,
(7.63)
β 2 =
1
6
P
2
0 C ij kl (u) k
i ∂k j
∂x
k
k ∂k l
∂y
,
(7.64)
a 1 =
2
3
P
2
0
C ij kl (u) k
i v
j k
k ∂k l
∂x
+ F
p
i (u) F pj (u) k
i ∂k j
∂x
,
(7.65)
b 1 =
2
3
P
2
0
C ij kl (u) k
i v
j k
k ∂k l
∂y
+ F
p
i (u) F pj (u) k
i ∂k j
∂y
,
(7.66)
and
c 2 = −C ij kl (u) k
i v
j k
k v
l
− 2
F
p
i (u) F pj (u) −
1
4
η ij F
pq (u) F pq (u)
k
i v
j
+F
p
i (u) F pj (u) k
i k
j .
(7.67)
Here, as in (7.51), F pj (u) = −F jp (u) (with F p i (u) = η pq F qi (u)) are the
components of the Maxwell tensor in coordinates X i calculated on r = 0 while
C ij kl (u) are the components of the Weyl conformal curvature tensor in coordinates
X i calculated on r = 0. Hence C ij kl satisfy C ij kl = −C jikl = −C ij lk = C klij ,
C ij kl + C ilj k + C iklj = 0 and η il C ij kl = 0.
The field equation R (3)(3) = 2 E (3)(3) yields
q 2 =
2
3
P
2
0 (L
2
2 + M
2
2 ) = −
1
6
F
p
i (u) F pj (u) k
i k
j .
(7.68)