6.10 The Einstein and Ernst Equations 291=−
(
E
A
) 2
∑
p∑
q∂apq∂zE
p 1
Enq
.Hence, referring to Lemma 6.19,
∂E
n 1∂ρ+ω(
A
E
) 2
∂A
n 1∂z=−
∑
p∑
q(
∂epq∂ρ+ω∂apq∂z)
E
pq
Enq=
1
ρ∑
p∑
q(p−q)epqEp 1
Enq=
1
ρ[
∑
ppEp 1∑
qepqEnq
−∑
qqEnq∑
pepqEp 1]
=
1
ρ[
∑
ppEp 1
δpn−∑
qqEnq
δq 1]
=
1
ρ(nEn 1
−En 1
),which is equivalent to (a).
∂A
n 1∂ρ=
∂A
1 n∂ρ=−
∑
p∑
q∂apq∂ρA
pn
A1 q∂E
n 1∂z=−
∑
p∑
q∂epq∂zE
p 1
Enq=−
(
A
E
) 2
∑
p∑
q∂epq∂zA
pn
A1 q
.Hence,
∂A
n 1∂ρ+ω(
E
A
) 2
∂E
n 1∂z=−
∑
p∑
q(
∂apq∂ρ+ω∂epq∂z)
A
pn
A1 q=−
1
ρ∑
p∑
q(p−q+1)apqApn
A1 q=
1
ρ[
∑
qqA1 q∑
papqApn
−∑
p(p+1)Apn∑
qapqA1 q]
=
1
ρ[
∑
qqA1 q
δqn−∑
p(p+1)Apn
δp 1]
=
1
ρ(nA1 n
− 2 A1 n
)(A1 n
=An 1
),