Page 259 - 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), φ n and ψ n can be complex functions when the elements of B n are

complex. Even when the elements are real, ψ n and ψ are purely imaginary
n
when n is 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
potential ξ in this case is given by the formula

ξ = F/G (6.2.17)
where F and G are determinants of order 2n whose column vectors are
defined as follows:
In F,

2 n−2 2
C j = τ j c j τ j c τ j ··· c τ j 1 c j c ...c n T , (6.2.18)
j j
j j 2n
and in G,

2 n−1 2 n−1 T
C j = τ j c j τ j c τ j ··· c τ j 1 c j c ...c , (6.2.19)
j j j j
2n
where
1
2 2 2 2
τ j = e ωθ j ρ +(z + c j ) (ω = −1) (6.2.20)
and 1 ≤ j ≤ 2n. The c j and θ j are 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 determinant A n and a
related determinant E n . They showed that A n and E n satisfy 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 of n. The equations satisfied by A n and E n are 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.