Determinants and Their Applications in Mathematical Physics

338 Appendix

∇ζ

′

+

=−

c

ζ

2

+

∇ζ+,

∇

2

ζ

′

+=−

c

ζ

2

+

[

∇

2

ζ+−

2

ζ+

(∇ζ+)

2

]

.

Hence,

1

2

(ζ

′

+

+ζ

′

−

)∇

2

ζ

′

+

−(∇

2

ζ

′

+

)

2

=−

c

2

ζ

3

+

ζ−

[

1

2

(ζ++ζ−)∇

2

ζ+−(∇ζ+)

2

]

=0.

This identity remains valid whenζ

′

+andζ

′

−are interchanged, which proves

the validity of transformationγ. It follows from the particular case in which

c= 1 andd= 0 that if the pairP(φ, ψ) is a solution of (A.12.4) and (A.12.5)

and

φ

′

=

φ

φ

2

+ψ

2

,

ψ

′

=−

ψ

φ

2

+ψ

2

,

then the pairP
′
(φ
′
,ψ
′
) is also a solution of (A.12.4) and (A.12.5). This

relation is applied in Section 6.10.2 on the intermediate solution of the

Einstein equations.

Transformationε

Combining transformationγandδwitha=d= 1 andc=− 2 b, it is found

that ifζ+andζ−are solutions of (A.12.1) and

ζ

′

+=

ζ−−b

ζ−+b

,

ζ

′

−=

b+ζ+

b−ζ+

,

thenζ
′

• andζ
  ′
  −
  are also solutions of (A.12.1). This transformation is ap-

plied in Section 6.10.4 on physically significant solutions of the Einstein

equations.

The following formulas are well known and will be applied later. (ρ, z)

are cylindrical polar coordinates:

∇V=

(

∂V

∂ρ

,

∂V

∂z

)

, (A.12.6)

∇·F=

1

ρ

∂

∂ρ

(ρFρ)+

∂Fz

∂z

, (A.12.7)

∇

2

V=

∂

2

V

∂ρ

2

+

1

ρ

∂V

∂ρ

+

∂

2

V

∂z

2

, (A.12.8)

∇·(VF)=V∇·F+F·∇V, (A.12.9)