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The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

D. Catalano Ferraioli, M. Marvan

Abstract

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the $2$-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants $C_ρ$ and $\ell_{\mathcal C}$ are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all $Λ$-vacuum metrics that are generic in the above sense, except that either $C_ρ$ or $\ell_{\mathcal C}$ is zero. In this way we extend the Kundu class to $Λ$-vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

Abstract

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the -dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants and are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterise the Van den Bergh metric. To understand the non-generic cases, we also find all -vacuum metrics that are generic in the above sense, except that either or is zero. In this way we extend the Kundu class to -vacuum metrics. The results of the paper can be exploited for invariant characterisation of classes of metrics and for extension of the set of known solutions of the Einstein equations.

Paper Structure

This paper contains 21 sections, 27 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. The pseudogroup and the metric
  3. Pseudogroup prolongation and differential invariants
  4. Orbit metric
  5. First-order invariants
  6. Scalar invariants $C_{\rho}$, $C_{\chi}$, $Q_{\chi}$, $Q_{\gamma}$ and the semi-invariant orthogonal frame $\{ \mathcal{X}, \mathcal{X}^\perp \}$ on $\mathcal{S}$
  7. The semi-invariant vector field $\mathcal{C}$ and scalar invariant $\ell_{\mathcal{C}}$
  8. O'Neill tensors $\mathbf A$ and $\mathbf T$. The invariant and semi-invariant vector fields $\mathcal{H}$ and $\mathcal{H}^\perp$ in the case when $C_\rho \neq 0$
  9. The semi-invariant orthogonal frame $\{ \mathcal{H}, \mathcal{H}^\perp, \mathcal{C}, \mathcal{C}^\perp \}$ in the case when $C_\rho\,\ell_\mathcal{C}\neq 0$
  10. Further first-order scalar invariants
  11. Additional second-order invariants
  12. $\Lambda$-vacuum Einstein equations for $G_2$ metrics, and their solutions in the special cases $C_\rho=0$ and $\ell_\mathcal{C}=0$
  13. The case when $\tilde{\mathbf{g}}$ is Lorentzian and explicit solutions with $C_\rho=0$
  14. The case when $\tilde{\mathbf{g}}$ is Riemannian and explicit solutions with $C_\rho=0$
  15. Exact solutions in the case when $\ell_{\mathcal{C}}=0$
  16. ...and 6 more sections

Key Result

Proposition 2.1

The Lie pseudogroup $\mathfrak{G}$ is formed by transformations $P:\mathcal{M}\rightarrow\mathcal{M}$ which in adapted coordinates have the form where $\phi^{i}(t^{1},t^{2})$ and $\psi^{i}(t^{1},t^{2})$ are arbitrary differentiable functions satisfying and $\alpha_{j}^{i}\in\mathbb{R}$, with $(\alpha_{j}^{i})\in \mathrm{GL}(2, \mathbb{R})$. Infinitesimal generators of $\mathfrak{G}$ have the for

Theorems & Definitions (57)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Proposition 4.1: Geroch Ger1
  • proof
  • ...and 47 more