Page 316 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 301

Hence,
2n−1

2 G
= V 2n (c) ,
P n
P n−1 ρ F
2n−1

Q n+1 2 F ∗
= − V 2n (c) (6.10.61)
ρ G ∗
Q n

F ∗
2n−1
ζ + = −2 V 2n (c) ,
G ∗

G
2n−1
ζ − =2 V 2n (c) . (6.10.62)
F
Finally, applying the B¨acklund transformation ε in Appendix A.12 with
b =2 2n−1 V 2n (c),
ζ − − 2 2n−1 V 2n (c)

ζ =
+
ζ − +2 2n−1 V 2n (c)
1 − (F/G)
= .
1+(F/G)
Similarly,
1 − (F /G )
∗
∗

ζ = . (6.10.63)
− ∗ ∗
1+(F /G )
Discarding the primes, ζ − = ζ . Hence, referring to (6.2.13),
∗
+
1
1
φ = (ζ + + ζ − )= (ζ + + ζ ),
∗
2 2 +
1 1 2
ψ = (ζ + − ζ − )= (ζ + − ζ )(ω = −1), (6.10.64)
∗
+
2ω 2ω
which are both real. It follows that these solutions are physically significant.
Exercise. Prove the following identities:
A 2n = α n (GG − FF ),
∗
∗
∗
A 2n+1 = β n F G,
A 2n−1 = β n−1 FG ,
∗
(2n+1) ∗ ∗
A 1,2n+1 = α n (GG + FF ),
where
n 2n(n−1)
(−1) 2 V 2(n−1)
α n = 2n ,
ρ 2n(n−1) 2n ε i
i=1
2 2n−1
(−1) n−1 2n V
2
β n = 2n .
ρ 2n 2 −1 2n ε ∗
i
i=1