Relationship between the inverse scattering techniques of Belinskii–Zakharov and Hauser–Ernst in general relativity

Abstract

We make a quantitative comparison between the pure-nonsoliton part of the inverse scattering method of Belinskii and Zakharov (BZ) and the homogeneous Hilbert problem of Hauser and Ernst (HE), these being two independent representations of an infinite-dimensional subgroup 𝒦 of the Geroch group K of invariance transformations for spacetimes with two commuting Killing vectors. An explicit formula for the BZ representing matrix function G0(λ) in terms of the HE representing matrix function u(t) is derived. It is shown how certain solution generating techniques (e.g., Harrison’s Bäcklund transformation, HKX transformation, generation of Weyl solution from flat space, generation of n-Kerr–NUT solution from n-Schwarzschild) can be derived directly from the BZ formalism, including the soliton part in some cases, thereby bringing our understanding of the BZ formalism up to the level of the more fully developed HE formalism. A technical point which needed to be resolved along the way was how to analytically continue the complex matrix potential F(t) across a quadratic branch cut and onto the second Riemann sheet. Finally, we consider how the subgroup 𝒦⊆K represented by the BZ and HE formalisms can be enlarged either by simple limiting transitions or by relaxing boundary conditions.