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New Axisymmetric Stationary Solutions of Five-dimensional Vacuum Einstein Equations with Asymptotic Flatness

Takashi Mishima, Hideo Iguchi

TL;DR

This work extends the construction of higher-dimensional black objects by applying four-dimensional solitonic solution-generation techniques to the five-dimensional vacuum Einstein equations. By reducing to a 4D Ernst problem with an accompanying harmonic field, the authors derive a four-parameter family of asymptotically flat, axisymmetric solutions from a Minkowski seed, uncovering a new branch of one-rotation black rings with horizon topology $S^2\times S^1$ for $\lambda>1$ and a distinct rotation plane from Emparan and Reall. They analyze asymptotics, mass and angular momentum parameters, and horizon/ergoregion structures, and identify regularity conditions that avoid Dirac–Misner pathologies and potential CTCs. Limit cases connect to Myers–Perry and Kerr/NUT-like regimes, illustrating how the method expands the landscape of higher-dimensional black objects and provides a systematic approach to generating new solutions, with avenues for extended rotations via inverse scattering.

Abstract

New axisymmetric stationary solutions of the vacuum Einstein equations in five-dimensional asymptotically flat spacetimes are obtained by using solitonic solution-generating techniques. The new solutions are shown to be equivalent to the four-dimensional multi-solitonic solutions derived from particular class of four-dimensional Weyl solutions and to include different black rings from those obtained by Emparan and Reall.

New Axisymmetric Stationary Solutions of Five-dimensional Vacuum Einstein Equations with Asymptotic Flatness

TL;DR

This work extends the construction of higher-dimensional black objects by applying four-dimensional solitonic solution-generation techniques to the five-dimensional vacuum Einstein equations. By reducing to a 4D Ernst problem with an accompanying harmonic field, the authors derive a four-parameter family of asymptotically flat, axisymmetric solutions from a Minkowski seed, uncovering a new branch of one-rotation black rings with horizon topology for and a distinct rotation plane from Emparan and Reall. They analyze asymptotics, mass and angular momentum parameters, and horizon/ergoregion structures, and identify regularity conditions that avoid Dirac–Misner pathologies and potential CTCs. Limit cases connect to Myers–Perry and Kerr/NUT-like regimes, illustrating how the method expands the landscape of higher-dimensional black objects and provides a systematic approach to generating new solutions, with avenues for extended rotations via inverse scattering.

Abstract

New axisymmetric stationary solutions of the vacuum Einstein equations in five-dimensional asymptotically flat spacetimes are obtained by using solitonic solution-generating techniques. The new solutions are shown to be equivalent to the four-dimensional multi-solitonic solutions derived from particular class of four-dimensional Weyl solutions and to include different black rings from those obtained by Emparan and Reall.

Paper Structure

This paper contains 5 sections, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic equations
  3. Generation of new solutions
  4. Results and Discussion
  5. summary

Figures (3)

  • Figure 1: Schematic diagram of a local ringlike object which resides in the spacetime. Generally some singular behavior appears near the horizon.
  • Figure 2: The behavior of $0$-$0$ component of the metric in the case of $(\alpha,\, \beta,\, \lambda)=(1/2,\, -0.195752,\,2)$. The region where the component function is above the level zero corresponds to the ergo-region.
  • Figure 3: The behavior of $\phi$-$\phi$ component of the metric in the case of $(\alpha,\, \beta,\, \lambda)=(1/2,\, -0.195752,\,2)$ which satisfies the Eq.(\ref{['eq:noCTC']}). The corresponding component always has non-negative values, while for general case the component becomes negative near the horizon, which means the existence of CTC-regions.