Smooth Gowdy-symmetric generalised Taub-NUT solutions with polynomial initial data

TL;DR

This paper extends the class of smooth Gowdy-symmetric generalised Taub-NUT solutions by allowing polynomial initial data of arbitrary degree, providing an algebraic construction method for the Ernst potential using soliton theory.

Contribution

It introduces a general algebraic algorithm for constructing solutions with polynomial initial data of any degree in this class of cosmological models.

Findings

Derived explicit determinant formula for Ernst potential

Developed a simple algebraic construction algorithm

Illustrated method with two explicit examples

Abstract

We consider smooth Gowdy-symmetric generalised Taub-NUT solutions, a class of inhomogeneous cosmological models with spatial three-sphere topology. They are characterised by existence of a smooth past Cauchy horizon and, with the exception of certain singular cases, they also develop a regular future Cauchy horizon. Several examples of exact solutions were previously constructed, where the initial data (in form of the initial Ernst potentials) are polynomials of low degree. Here, we generalise to polynomial initial data of arbitrary degree. Utilising methods from soliton theory, we obtain a simple algorithm that allows us to construct the resulting Ernst potential with purely algebraic calculations. We also derive an explicit formula in terms of determinants, and we illustrate the method with two examples.