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

2

+ψ

2

,

ψ

′

=−

ψ

φ

2

+ψ

2

, (6.10.15)

thenP

′

(φ

′

,ψ

′

) is also a solution. Transformationβstates that ifP(φ, ψ)

is a solution and

φ

′

=

ρ

φ

,

∂ψ

′

∂ρ

=−

ωρ

φ

2

∂ψ

∂z

,

∂ψ

′

∂z

=

ωρ

φ

2

∂ψ

∂ρ

(ω

2

=−1), (6.10.16)

thenP
′
(φ
′
,ψ
′
) is also a solution. The theorem can therefore be proved by

showing that the application of transformationγtoPngivesP
′
n
and that

the application of TransformationβtoP
′
n
givesPn+1.

Applying the Jacobi identity (Section 3.6) to the cofactors of the corner

elements ofA,

A

2

n+1

−A

2

1 n

=AnAn− 2. (6.10.17)

Hence, referring to (6.10.15),

φ

2

n+ψ

2

n=

(

ρ

n− 2

An− 2

) 2

(

A

2

n− 1 −E

2

n− 1

)

=

(

ρ

n− 2

An− 2

) 2

(

A

2

n− 1 −A

2

1 n

)

=

ρ

2 n− 4

An

An− 2

, (6.10.18)

φn

φ

2

n

+ψ

2

n

=

An− 1

ρ

2 n− 2

An

(An− 1 =A 11 )

=

A

11

ρ

2 n− 2

=φ

′

n,

ψn

φ

2

n+ψ

2

n

=

ωEn− 1

ρ

2 n− 2

An

=

(−1)

n− 1

ωA

1 n

ρ

n− 2

=−ψ

′

n

.