Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 243

which yields only two independent scalar equations, namely

φ

(

φρρ+

1

ρ

φρ+φzz

)

−φ

2

ρ

−φ

2

z

+ψ

2

ρ

+ψ

2

z

=0, (6.2.8)

φ

(

ψρρ+

1

ρ

ψρ+ψzz

)

−2(φρψρ+φzψz)=0. (6.2.9)

The second equation can be rearranged into the form

∂

∂ρ

(

ρψρ

φ

2

)

+

∂

∂z

(

ρψz

φ

2

)

=0.

Historically, the scalar equations (6.2.8) and (6.2.9) were formulated before

the matrix equation (6.2.1), but the modern approach to relativity is to

formulate the matrix equation first and to derive the scalar equations from

them.

Equations (6.2.8) and (6.2.9) can be contracted into the form

φ∇

2

φ−(∇φ)

2

+(∇ψ)

2

=0, (6.2.10)

φ∇

2

ψ− 2 ∇φ·∇ψ=0, (6.2.11)

which can be contracted further into the equations

1

2

(ζ++ζ−)∇

2

ζ±=(∇ζ±)

2

, (6.2.12)

where

ζ+=φ+ωψ,

ζ−=φ−ωψ (ω

2

=−1). (6.2.13)

The notation

ζ=φ+ωψ,

ζ

∗

=φ−ωψ, (6.2.14)

whereζ
∗
is the complex conjugate ofζ, can be used only whenφandψare

real. In that case, the two equations (6.2.12) reduce to the single equation

1

2

(ζ+ζ

∗

)∇

2

ζ=(∇ζ)

2

. (6.2.15)

In 1983, Y. Nakamura conjectured the existence two related infinite sets of

solutions of (6.2.8) and (6.2.9). He denoted them by

φ

′

n,ψ

′

n,n≥^1 ,

φn,ψn,n≥ 2 , (6.2.16)

and deduced the first few members of each set with the aid of the pair of

coupled difference–differential equations given in Appendix A.11 and the

B ̈acklund transformationsβandγgiven in Appendix A.12. The general

Nakamura solutions were given by Vein in 1985 in terms of cofactors as-

sociated with a determinant of arbitrary order whose elements satisfy the