Page 314 - Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 299

6.10.5 Physically Significant Solutions
From the theorem in Section 6.10.2 on the intermediate solution,
ρ 2n−1
φ 2n+1 = A 2n ,
A 2n−1
ωρ 2n−1 A (2n+1)
1,2n+1
2
ψ 2n+1 = (ω = −1). (6.10.46)
A 2n−1
Hence the functions ζ + and ζ − introduced in Section 6.2.8 can be expressed
as follows:
ζ + = φ 2n+1 + ωψ 2n+1
ρ 2n−1 (A 2n − A (2n+1) )
1,2n+1
= , (6.10.47)
A 2n−1
ζ − = φ 2n+1 − ωψ 2n+1
ρ 2n−1 (A 2n + A (2n+1) )
1,2n+1
= . (6.10.48)
A 2n−1
It is shown in Section 4.5.2 on symmetric Toeplitz determinants that if
A n = |t |i−j| | n , then
A 2n−1 =2P n−1 Q n ,
A 2n = P n Q n + P n−1 Q n+1 ,
(2n+1)
A = P n Q n − P n−1 Q n+1 , (6.10.49)
1,2n+1
where

P n = 1
2 t |i−j| − t i+j n

Q n = 1 t |i−j| + t i+j−2 . (6.10.50)

2
n
Hence,
ρ 2n−1 Q n+1
ζ + = ,
Q n
ρ 2n−1
ζ − = P n . (6.10.51)
P n−1
2
In the present problem, t r = ω u r (ω = −1), where u r is a solution of the
r
coupled equations (6.10.3) and (6.10.4). In order to obtain the Neugebauer
solutions, it is necessary first to choose the solution given by equations
(A.11.8) and (A.11.9) in Appendix A.11, namely
e j f r (x j )
2n

u r =(−1) r 3 , x j = z + c j , (6.10.52)
j=1 1+ x 2 ρ
j
and then to choose
e j =(−1) j−1 M j (c)e ωθ j . (6.10.53)