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On local equivalence problem of spacetimes with two orthogonally transitive commuting Killing fields

M. Marvan, O. Stolin

Abstract

Considered is the problem of local equivalence of generic four-dimensional metrics possessing two commuting and orthogonally transitive Killing vector fields. A sufficient set of eight differential invariants is explicitly constructed, among them four of first order and four of second order in terms of metric coefficients. In vacuum case the four first-order invariants suffice to distinguish generic metrics.

On local equivalence problem of spacetimes with two orthogonally transitive commuting Killing fields

Abstract

Considered is the problem of local equivalence of generic four-dimensional metrics possessing two commuting and orthogonally transitive Killing vector fields. A sufficient set of eight differential invariants is explicitly constructed, among them four of first order and four of second order in terms of metric coefficients. In vacuum case the four first-order invariants suffice to distinguish generic metrics.

Paper Structure

This paper contains 15 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The metric and the pseudogroup
  3. The metric
  4. The pseudogroup
  5. The equivalence problem
  6. First-order invariants
  7. Invariant frame
  8. Second-order scalar invariants
  9. Weyl--Lewis--Papapetrou parametrization
  10. A solution to the equivalence problem
  11. Relation to $\Psi_2$
  12. Genericity
  13. Vacuum case
  14. Example
  15. Conclusions

Key Result

Proposition 1

Let two metrics ${\mathbf g}$ and ${\mathbf g}'$ on $\mathcal{M}$ possess a unique two-dimensional commutative algebra $\mathfrak K$ of Killing vectors, which induces one and the same decomposition $T_a \mathcal{M} = \Xi_a \oplus \Xi_a^\bot$, $a \in \mathcal{M}$. Then ${\mathbf g}' = \Phi^*{\mathbf

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Example 1
  • ...and 1 more