Determinants and Their Applications in Mathematical Physics

244 6. Applications of Determinants in Mathematical Physics

difference–differential equations. These solutions are reproduced with mi-

nor modifications in Section 6.10.2. In 1986, Kyriakopoulos approached the

same problem from another direction and obtained the same determinant

in a different form.

The Nakamura–Vein solutions are of great interest mathematically but

are not physically significant since, as can be seen from (6.10.21) and

(6.10.22),φnandψncan be complex functions when the elements ofBnare

complex. Even when the elements are real,ψnandψ

′

nare purely imaginary

whennis odd. The Nakamura–Vein solutions are referred to as intermediate

solutions.

The Neugebauer family of solutions published in 1980 contains as a par-

ticular case the Kerr–Tomimatsu–Sato class of solutions which represent

the gravitational field generated by a spinning mass. The Ernst complex

potentialξin this case is given by the formula

ξ=F/G (6.2.17)

whereF andGare determinants of order 2nwhose column vectors are

defined as follows:

InF,

Cj=

[

τj cjτj c

2

jτj···c

n− 2

j τj^1 cj c

2

j...c

n

j

]T

2 n

, (6.2.18)

and inG,

Cj=

[

τj cjτj c

2

j

τj···c

n− 1

j

τj 1 cj c

2

j

...c

n− 1

j

]T

2 n

, (6.2.19)

where

τj=e

ωθj

[

ρ

2

+(z+cj)

2

] 1

2

(ω

2

=−1) (6.2.20)

and 1≤j≤ 2 n. Thecjandθjare arbitrary real constants which can be

specialized to give particular solutions such as the Yamazaki–Hori solutions

and the Kerr–Tomimatsu–Sato solutions.

In 1993, Sasa and Satsuma used the Nakamura–Vein solutions as a start-

ing point to obtain physically significant solutions. Their analysis included

a study of Vein’s quasicomplex symmetric Toeplitz determinantAnand a

related determinantEn. They showed thatAnandEnsatisfy two equa-

tions containing Hirota operators. They then applied these equations to

obtain a solution of the Einstein equations and verified with the aid of

a computer that their solution is identical with the Neugebauer solution

for small values ofn. The equations satisfied byAnandEnare given as

exercises at the end of Section 6.10.2 on the intermediate solutions.

A wholly analytic method of obtaining the Neugebauer solutions is

given in Sections 6.10.4 and 6.10.5. It applies determinantal identities and

other relations which appear in this chapter and elsewhere to modify the

Nakamura–Vein solutions by means of algebraic B ̈acklund transformations.