Stationary multiple euclidon solutions to the vacuum Einstein equations

TL;DR

This work develops the Euclidon method to generate exact stationary, axisymmetric vacuum solutions of the Einstein equations by nonlinear superposition with a seed metric. By recasting the problem in Weyl canonical and prolate coordinates and exploiting Ernst potentials, the authors obtain compact, tractable formulas for multi-Euclidon configurations, including a two-stationary solution that reproduces Kerr–NUT and maps to Boyer–Lindquist form; they further extend to higher-order multi-source solutions and establish an algebraic structure for composing these solutions. The physical interpretation assigns the Euclidon parameters to mass, angular momentum, NUT charges, and relative accelerations, while Bonnor's theorem and the Kerr–Sen extension illustrate the broader applicability to electrovacuum and string-theory-inspired models. The paper thus provides a unified framework for constructing and analyzing complex rotating spacetimes, including regularity, ergoregions, and observable multipole signatures, with potential applications to gravitational lensing and frame-dragging studies.

Abstract

The non-linear superposition of the stationary euclidon solution with an arbitrary axially symmetric stationary gravitational field on the basis of the method of variation of parameters was constructed. Stationary soliton solution of the Einstein equations was generalized to the case of a stationary seed metric. The formulae obtained have a simple and compact form, permitting an effective non-linear "addition" of the solutions. These euclidon solutions serve as building block of the theory, which allows for the construction of almost all known solutions to the vacuum static axially-symmetric Einstein equations, including such important ones as the Kerr-NUT solution. The stationary euclidon solution has a clear physical interpretation as a relativistic accelerated non-inertial reference frame, which provides a different perspective on the physical interpretation of well-known solutions, such as the Kerr solution.