6.10 The Einstein and Ernst Equations 287
Hence, referring to (6.9.28) and applying the Jacobi identity,
(−1)
n
F=
∣
∣
∣
∣
Qn+1,n+1 Qn+1,n+2
Qn+2,n+1 Qn+2,n+2
∣
∣
∣
∣
−QQn+1,n+2;n+1,n+2
=0,
which completes the proof of the theorem.
6.10 The Einstein and Ernst Equations..............
6.10.1 Introduction
This section is devoted to the solution of the scalar Einstein equations,
namely
φ
(
φρρ+
1
ρ
φρ+φzz
)
−φ
2
ρ−φ
2
z+ψ
2
ρ+ψ
2
z=0, (6.10.1)
φ
(
ψρρ+
1
ρ
ψρ+ψzz
)
−2(φρψρ+φzψz)=0, (6.10.2)
but before the theorems can be stated and proved, it is necessary to define
a functionur, three determinantsA,B, andE, and to prove some lemmas.
The notationω
2
=−1 is used again asiandjare indispensable as row
and column parameters, respectively.
6.10.2 Preparatory Lemmas
Let the functionur(ρ, z) be defined as any real solution of the coupled
equations
∂ur+1
∂ρ
+
∂ur
∂z
=−
rur+1
ρ
,r=0, 1 , 2 ,..., (6.10.3)
∂ur− 1
∂ρ
−
∂ur
∂z
=
rur− 1
ρ
,r=1, 2 , 3 ..., (6.10.4)
which are solved in Appendix A.11.
Define three determinantsAn,Bn, andEnas follows.
An=|ars|n
where
ars=ω
|r−s|
u|r−s|, (ω
2
=−1). (6.10.5)
Bn=|brs|n,
where
brs=
{
ur−s,r≥s
(−1)
s−r
us−r,r≤s