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6.10 The Einstein and Ernst Equations 287

Hence, referring to (6.9.28) and applying the Jacobi identity,

Q n+1,n+1
(−1) F = Q n+1,n+2 − QQ n+1,n+2;n+1,n+2
n
Q n+2,n+1 Q n+2,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
1 2 2 2 2

− φ − φ + ψ + ψ =0, (6.10.1)
ρ ρ z ρ z
φ φ ρρ + φ ρ + φ zz
1

− 2(φ ρ ψ ρ + φ z ψ z )=0, (6.10.2)
ρ
φ ψ ρρ + ψ ρ + ψ zz
but before the theorems can be stated and proved, it is necessary to define
a function u r , three determinants A, B, and E, and to prove some lemmas.
2
The notation ω = −1 is used again as i and j are indispensable as row
and column parameters, respectively.
6.10.2 Preparatory Lemmas
Let the function u r (ρ, z) be defined as any real solution of the coupled
equations

∂u r+1 ∂u r ru r+1
+ = − , r =0, 1, 2,..., (6.10.3)
∂ρ ∂z ρ
∂u r−1 ∂u r ru r−1
− = , r =1, 2, 3 ..., (6.10.4)
∂ρ ∂z ρ
which are solved in Appendix A.11.
Define three determinants A n , B n , and E n as follows.

where A n = |a rs | n
2
a rs = ω |r−s| u |r−s| , (ω = −1). (6.10.5)
B n = |b rs | n ,
where
u r−s , r ≥ s

b rs =
(−1) s−r u s−r ,r ≤ s