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Dimension of the isometry group in type N vacuum solutions: an IDEAL approach

Juan Antonio Sáez, Salvador Mengual, Joan Josep Ferrando

TL;DR

The paper presents an IDEAL characterization of type N vacuum spacetimes with a cosmological constant by exploiting Weyl-invariant concomitants and the existence of a Riemann frame to algorithmically determine the isometry-group dimension. It partitions these spacetimes into seven regular Weyl-concomitant classes $C_n$ plus a degenerate family $\uhat{C7}$, deriving explicit invariant conditions and constructing $R$-frames when possible; for regular classes the isometry dimension is obtained via an established flow diagram, while the $amily$ is handled with a separate invariant-driven analysis. The results yield a detailed, implementable classification that includes pp-waves, Robinson-Trautman, Kundt and Siklos families, with concrete relations among Weyl invariants, Ricci constraints, and the resulting symmetry groups. This IDEAL framework enables automated verification and generation of symmetry information from curvature data, and the authors discuss ongoing work to implement these algorithms in the xAct/xIdeal suite for broader applicability and new geometries.

Abstract

The necessary and sufficient conditions for a type N vacuum solution (with cosmological constant) to admit a group of isometries of dimension $r$ are given in terms of the invariant concomitants of the Weyl tensor. This study requires defining several invariant classes, and for each class, the conditions that determine the dimension are given. Thus, an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterisation of these spacetimes follows. Some examples show that our algorithmic results can easily be implemented on the \textit{xAct Mathematica} suite of packages. The relation between our classes and already known families of solutions of Einstein equations is outlined.

Dimension of the isometry group in type N vacuum solutions: an IDEAL approach

TL;DR

The paper presents an IDEAL characterization of type N vacuum spacetimes with a cosmological constant by exploiting Weyl-invariant concomitants and the existence of a Riemann frame to algorithmically determine the isometry-group dimension. It partitions these spacetimes into seven regular Weyl-concomitant classes plus a degenerate family , deriving explicit invariant conditions and constructing -frames when possible; for regular classes the isometry dimension is obtained via an established flow diagram, while the is handled with a separate invariant-driven analysis. The results yield a detailed, implementable classification that includes pp-waves, Robinson-Trautman, Kundt and Siklos families, with concrete relations among Weyl invariants, Ricci constraints, and the resulting symmetry groups. This IDEAL framework enables automated verification and generation of symmetry information from curvature data, and the authors discuss ongoing work to implement these algorithms in the xAct/xIdeal suite for broader applicability and new geometries.

Abstract

The necessary and sufficient conditions for a type N vacuum solution (with cosmological constant) to admit a group of isometries of dimension are given in terms of the invariant concomitants of the Weyl tensor. This study requires defining several invariant classes, and for each class, the conditions that determine the dimension are given. Thus, an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterisation of these spacetimes follows. Some examples show that our algorithmic results can easily be implemented on the \textit{xAct Mathematica} suite of packages. The relation between our classes and already known families of solutions of Einstein equations is outlined.

Paper Structure

This paper contains 23 sections, 12 theorems, 70 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Isometries in spacetimes with a Riemann frame
  3. Notation and invariants in type N spacetimes
  4. Invariant algebraic elements associated with a type N Weyl tensor
  5. Riemann frames in type N spacetimes
  6. Some regular classes of type N spacetimes
  7. Summary: classification and characterisation theorem
  8. Some suitable remarks
  9. Constraints on the Ricci tensor
  10. Isometries in type N vacuum metrics with cosmological constant
  11. Classes ${\rm C}n$, $n \leqslant 7$, compatible with $\, 6\,Q= \Lambda g$
  12. Invariant scalars and constraints in the family ${\rm \widehat{C5}}$ with $6\,Q= \Lambda g$
  13. Vacuum solutions ($\Lambda=0$) in the family ${\rm \widehat{C5}}$: pp-waves
  14. Analyses of the family $\widehat{\rm C7}$ with $\Lambda \not=0$
  15. Summary: classification and characterisation theorem for the family $\widehat{\rm C7}$
  16. ...and 8 more sections

Key Result

Proposition 1

The fundamental bivector ${\cal L}$ and the fundamental vector $\ell$ of a type N Weyl tensor (${\cal W}={\cal L} \otimes {\cal L}$, $\ell \wedge {\cal L} =0$) can be obtained as: where ${\cal X}$ is an arbitrary bivector such that ${\cal W}({\cal X}) \not= 0$ and $x$ an arbitrary time-like vector.

Figures (2)

  • Figure 1: This flow diagram distinguishes the regular classes C$n$, $n \leqslant 7$ and the complementary family $\widehat{\rm C7}$.
  • Figure 2: This flow diagram distinguishes the different classes in the family $\widehat{\rm C7}$, and gives de dimension of the group and orbits for the singular ones.

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 1
  • Theorem 1
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 4 more