Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 301

Hence,

Pn

Pn− 1

=

{

2

ρ

} 2 n− 1

V 2 n(c)

{

G

F

}

,

Qn+1

Qn

=−

{

2

ρ

} 2 n− 1

V 2 n(c)

{

F

∗

G

∗

}

(6.10.61)

ζ+=− 2

2 n− 1

V 2 n(c)

{

F

∗

G

∗

}

,

ζ−=2

2 n− 1

V 2 n(c)

{

G

F

}

. (6.10.62)

Finally, applying the B ̈acklund transformationεin Appendix A.12 with

b=2

2 n− 1

V 2 n(c),

ζ

′

+

=

ζ−− 2

2 n− 1

V 2 n(c)

ζ−+2

2 n− 1

V 2 n(c)

=

1 −(F/G)

1+(F/G)

.

Similarly,

ζ

′

−

=

1 −(F

∗

/G

∗

)

1+(F

∗

/G

∗

)

. (6.10.63)

Discarding the primes,ζ−=ζ

∗

+. Hence, referring to (6.2.13),

φ=

1

2

(ζ++ζ−)=

1

2

(ζ++ζ

∗

+),

ψ=

1

2 ω

(ζ+−ζ−)=

1

2 ω

(ζ+−ζ

∗

+)(ω

2

=−1), (6.10.64)

which are both real. It follows that these solutions are physically significant.

Exercise.Prove the following identities:

A 2 n=αn(GG

∗

−FF

∗

),

A 2 n+1=βnF

∗

G,

A 2 n− 1 =βn− 1 FG

∗

,

A

(2n+1)

1 , 2 n+1

=αn(GG

∗

+FF

∗

),

where

αn=

(−1)

n

2

2 n(n−1)

V

2(n−1)

2 n

ρ

2 n(n−1)

(^2) ∏n
i=1
εi

,

βn=

(−1)

n− 1

2

2 n

2

V

2 n− 1

2 n

ρ^2 n

(^2) − 1
(^2) ∏n
i=1
ε
∗
i

.