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Spatially homogeneous solutions of vacuum Einstein equations in general dimensions

Yuichiro Sato, Takanao Tsuyuki

TL;DR

This work addresses time-dependent compactification under the vacuum Einstein equations in $(n+1)$ dimensions by focusing on spatially homogeneous spacetimes generated by almost abelian Lie groups. It develops an explicit higher-dimensional generalization of Bianchi type II for $G=H_{3} \times \mathbb{R}^{n-3}$, derives the associated Ricci-flat metric, and shows diagonalization is always possible in these zero-moduli cases, linking to known 4D Taub II and 5D Christodoulakis solutions. A key finding is that, at late times, not all spatial dimensions can expand or contract simultaneously; instead, selective dimensional growth (e.g., three expanding dimensions) is possible, illustrating dynamical compactification scenarios. The results extend the classical Bianchi II solutions to higher dimensions and provide new generalized Bianchi II-type metrics with potential implications for higher-dimensional cosmology and string/M-theory compactifications.

Abstract

We study time-dependent compactification of extra dimensions. We assume that the spacetime is spatially homogeneous, and solve the vacuum Einstein equations without cosmological constant in more than three dimensions. We consider globally hyperbolic spacetimes in which almost Abelian Lie groups act on the spaces isometrically and simply transitively. We give left-invariant metrics on the spaces and solve Ricci-flat conditions of the spacetimes. In the four-dimensional case, our solutions correspond to the Bianchi type II solution. By our results and previous studies, all spatially homogeneous solutions whose spaces have zero-dimensional moduli spaces of left-invariant metrics are found. For the simplest solution, we show that each of the spatial dimensions cannot expand or contract simultaneously in the late-time it.

Spatially homogeneous solutions of vacuum Einstein equations in general dimensions

TL;DR

This work addresses time-dependent compactification under the vacuum Einstein equations in dimensions by focusing on spatially homogeneous spacetimes generated by almost abelian Lie groups. It develops an explicit higher-dimensional generalization of Bianchi type II for , derives the associated Ricci-flat metric, and shows diagonalization is always possible in these zero-moduli cases, linking to known 4D Taub II and 5D Christodoulakis solutions. A key finding is that, at late times, not all spatial dimensions can expand or contract simultaneously; instead, selective dimensional growth (e.g., three expanding dimensions) is possible, illustrating dynamical compactification scenarios. The results extend the classical Bianchi II solutions to higher dimensions and provide new generalized Bianchi II-type metrics with potential implications for higher-dimensional cosmology and string/M-theory compactifications.

Abstract

We study time-dependent compactification of extra dimensions. We assume that the spacetime is spatially homogeneous, and solve the vacuum Einstein equations without cosmological constant in more than three dimensions. We consider globally hyperbolic spacetimes in which almost Abelian Lie groups act on the spaces isometrically and simply transitively. We give left-invariant metrics on the spaces and solve Ricci-flat conditions of the spacetimes. In the four-dimensional case, our solutions correspond to the Bianchi type II solution. By our results and previous studies, all spatially homogeneous solutions whose spaces have zero-dimensional moduli spaces of left-invariant metrics are found. For the simplest solution, we show that each of the spatial dimensions cannot expand or contract simultaneously in the late-time it.
Paper Structure (12 sections, 10 theorems, 74 equations, 2 figures, 1 table)

This paper contains 12 sections, 10 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spatially homogeneous spacetimes
  4. Almost abelian Lie groups
  5. Higher-dimensional vacuum solutions
  6. $G=\mathbb{R}^n,\ \mathbb{R}\mathrm{H}^n$ cases
  7. $G=H_{3} \times \mathbb{R}^{n-3}$ case
  8. Diagonalizability of metrics
  9. Expansion and contraction of space
  10. New solutions for other generalizations of Bianchi type II
  11. Block-diagonal case
  12. One off-diagonal and diagonal components

Key Result

Theorem 1.1

Except for the Minkowski spacetimes, all general solutions of vacuum Einstein equations for the spatially homogeneous spacetimes $I\times G$, where $G$ has zero-dimensional moduli space of left-invariant Riemannian metrics, are eq:ds2, eq:kas, eq:solV and eq:mil.

Figures (2)

  • Figure 1: Time evolution of the scale factors $a_i(t)$ of the solution (\ref{['eq:ds2']}) with the spatial dimension number $n=9$. The initial conditions are chosen as $a_i(0) = 1$ and (\ref{['eq:h9']}).
  • Figure 2: Time evolution of scale factors $a_i(t)$ of the solution (\ref{['eq:ds2']}) with the spatial dimension number $n=10$. The initial conditions are chosen as $a_i(0) = 1$ and (\ref{['eq:h11']}).

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Theorem 2.5: MR1958155MR2783396
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • proof
  • ...and 4 more