Page 353 - Determinants and Their Applications in Mathematical Physics

338 Appendix
c

∇ζ = − ∇ζ + ,
+ 2
ζ
+
c 2
2 2 2
∇ ζ = − ∇ ζ + − (∇ζ + ) .
+ 2
ζ + ζ +
Hence,
c 2
1 2 2 2 1 2 2
+
+
+
2 (ζ + ζ )∇ ζ − (∇ ζ ) = − 3 2 (ζ + + ζ − )∇ ζ + − (∇ζ + )
−
+
ζ ζ −
=0.

This identity remains valid when ζ and ζ are interchanged, which proves
+
−
the validity of transformation γ. It follows from the particular case in which
c = 1 and d = 0 that if the pair P(φ, ψ) is a solution of (A.12.4) and (A.12.5)
and
φ
φ = ,

2
φ + ψ 2
ψ

ψ = − ,
2
φ + ψ 2

then the pair P (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.
Transformation ε
Combining transformation γ and δ with a = d = 1 and c = −2b, it is found
that if ζ + and ζ − are solutions of (A.12.1) and
ζ − − b

ζ = ,
+
ζ − + b
b + ζ +
ζ = ,

−
b − ζ +
then ζ and ζ 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
∇V = , , (A.12.6)
∂ρ ∂z
1 ∂
∇· F = (ρF ρ )+ ∂F z , (A.12.7)
ρ ∂ρ ∂z
2
2
∂ V 1 ∂V ∂ V
2
∇ V = + + , (A.12.8)
∂ρ 2 ρ ∂ρ ∂z 2
∇· (V F)= V ∇· F + F · V, (A.12.9)