Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 287

Hence, referring to (6.9.28) and applying the Jacobi identity,

(−1)

n

F=

∣

∣

∣

∣

Qn+1,n+1 Qn+1,n+2

Qn+2,n+1 Qn+2,n+2

∣

∣

∣

∣

−QQn+1,n+2;n+1,n+2

=0,

which completes the proof of the theorem.

6.10 The Einstein and Ernst Equations..............

6.10.1 Introduction

This section is devoted to the solution of the scalar Einstein equations,

namely

φ

(

φρρ+

1

ρ

φρ+φzz

)

−φ

2

ρ−φ

2

z+ψ

2

ρ+ψ

2

z=0, (6.10.1)

φ

(

ψρρ+

1

ρ

ψρ+ψzz

)

−2(φρψρ+φzψz)=0, (6.10.2)

but before the theorems can be stated and proved, it is necessary to define

a functionur, three determinantsA,B, andE, and to prove some lemmas.

The notationω
2
=−1 is used again asiandjare indispensable as row

and column parameters, respectively.

6.10.2 Preparatory Lemmas

Let the functionur(ρ, z) be defined as any real solution of the coupled

equations

∂ur+1

∂ρ

+

∂ur

∂z

=−

rur+1

ρ

,r=0, 1 , 2 ,..., (6.10.3)

∂ur− 1

∂ρ

−

∂ur

∂z

=

rur− 1

ρ

,r=1, 2 , 3 ..., (6.10.4)

which are solved in Appendix A.11.

Define three determinantsAn,Bn, andEnas follows.

An=|ars|n

where

ars=ω

|r−s|

u|r−s|, (ω

2

=−1). (6.10.5)

Bn=|brs|n,

where

brs=

{

ur−s,r≥s

(−1)

s−r

us−r,r≤s