Page 308 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 293

Proof. The proof is by induction and applies the B¨acklund transforma-
tion theorems which appear in Appendix A.12 where it is proved that if
P(φ, ψ) is a solution and
φ

φ = ,
φ + ψ 2
2
ψ

ψ = − 2 2 , (6.10.15)
φ + ψ
then P (φ ,ψ ) is also a solution. Transformation β states that if P(φ, ψ)

is a solution and
ρ
φ = ,

φ
∂ψ ωρ ∂ψ
= − ,
∂ρ φ 2 ∂z
∂ψ ωρ ∂ψ 2
= (ω = −1), (6.10.16)
∂z φ 2 ∂ρ
then P (φ ,ψ ) is also a solution. The theorem can therefore be proved by

showing that the application of transformation γ to P n gives P and that

n
the application of Transformation β to P gives P n+1 .

n
Applying the Jacobi identity (Section 3.6) to the cofactors of the corner
elements of A,
A 2 n+1 − A 2 1n = A n A n−2 . (6.10.17)
Hence, referring to (6.10.15),

2
ρ 2 2
n−2
2
2
φ + ψ = A n−1 − E n−1
n n
A n−2

2
n−2
ρ 2 2
= A − A
n−1
A n−2 1n
ρ 2n−4
= A n , (6.10.18)
A n−2
A n−1
= (A n−1 = A 11 )
φ n
2
φ + ψ 2 ρ 2n−2
n n A n
A 11
=
ρ 2n−2

= φ ,
n
ωE n−1
=
ψ n
2
φ + ψ 2 ρ 2n−2
n n A n
(−1) n−1 ωA 1n
=
ρ n−2

= −ψ .
n