Page 306 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 291
2
E p1

= − ∂a pq E E .
nq
A ∂z
p q
Hence, referring to Lemma 6.19,
∂E n1 A
2 ∂A n1
+ ω = − ∂e pq + ω ∂a pq E E nq
pq
∂ρ E ∂z ∂ρ ∂z
p q
1 p1
= (p − q)e pq E E nq
ρ
p q

1 p1 p1
= pE e pq E nq − qE nq e pq E
ρ
p q q p

1 p1
= pE δ pn − qE δ q1
nq
ρ
p q
1 n1 n1
= (nE − E ),
ρ
which is equivalent to (a).
∂A n1 ∂A 1n
= = − ∂a pq A A 1q
pn
∂ρ ∂ρ ∂ρ
p q
∂E n1 p1
= − ∂e pq E E nq
∂z ∂z
p q
2

A
= − ∂e pq A A .
pn
1q
E ∂z
p q
Hence,
∂A n1 E
2 ∂E n1
+ ω
∂ρ A ∂z

∂a pq ∂e pq
= − + ω A A 1q
pn
∂ρ ∂z
p q
1
= − (p − q +1)a pq A A 1q
pn
ρ
p q

1
= qA 1q a pq A pn − (p +1)A pn a pq A 1q
ρ
q p p q

1
= qA δ qn − (p +1)A δ p1
1q
pn
ρ
q p
1 n1
= (nA 1n − 2A )(A 1n = A ),
1n
ρ