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,

φ 2 n+1=

ρ

2 n− 1

A 2 n

A 2 n− 1

,

ψ 2 n+1=

ωρ

2 n− 1

A

(2n+1)

1 , 2 n+1

A 2 n− 1

(ω

2

=−1). (6.10.46)

Hence the functionsζ+andζ−introduced in Section 6.2.8 can be expressed

as follows:

ζ+=φ 2 n+1+ωψ 2 n+1

=

ρ

2 n− 1

(A 2 n−A

(2n+1)

1 , 2 n+1

)

A 2 n− 1

, (6.10.47)

ζ−=φ 2 n+1−ωψ 2 n+1

=

ρ

2 n− 1

(A 2 n+A

(2n+1)

1 , 2 n+1

)

A 2 n− 1

. (6.10.48)

It is shown in Section 4.5.2 on symmetric Toeplitz determinants that if

An=|t|i−j||n, then

A 2 n− 1 =2Pn− 1 Qn,

A 2 n=PnQn+Pn− 1 Qn+1,

A

(2n+1)

1 , 2 n+1

=PnQn−Pn− 1 Qn+1, (6.10.49)

where

Pn=

1

2

∣

∣t

|i−j|−ti+j

∣

∣

n

Qn=

1

2

∣

∣t

|i−j|+ti+j− 2

∣

∣

n

. (6.10.50)

Hence,

ζ+=

ρ

2 n− 1

Qn+1

Qn

,

ζ−=

ρ

2 n− 1

Pn

Pn− 1

. (6.10.51)

In the present problem,tr=ω
r
ur(ω
2
=−1), whereuris a solution of the

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

ur=(−1)

r

2 n

∑

j=1

ejfr(xj)

√

1+x

2

j

,xj=

z+cj

ρ

, (6.10.52)

and then to choose

ej=(−1)

j− 1

Mj(c)e

ωθj

. (6.10.53)