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

Jörg Hennig

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.

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

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.

Paper Structure

This paper contains 13 sections, 84 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Metric and field equations
  3. Initial value problem with polynomial Ernst potential
  4. Sibgatullin's integral method
  5. Initial data
  6. Algebraic equations
  7. Calculation of the Ernst potential $\mathcal{E}$
  8. Calculation of the metric potentials $u$ and $Q$
  9. Example 1: Initial data of fourth degree with a double root
  10. Example 2: Initial data of sixth degree
  11. Discussion
  12. Calculation of three integrals
  13. Pseudocode

Figures (4)

  • Figure 1: The Ernst potential eq:EP1 for the solution with initial data of degree four (Example 1).
  • Figure 2: The metric potentials $\mathrm{e}^u$ and $Q$ for Example 1.
  • Figure 3: Ernst potential for the solution with initial data eq:ID2 of degree six. The Ernst potential clearly diverges at $x=y=-1$.
  • Figure 4: The metric potentials $\mathrm{e}^v\equiv (1-y^2)\mathrm{e}^u$ and $Q$ for the solution with initial data of degree six.