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

TL;DR

This paper provides an invariant-based, algorithmic characterization of type N vacuum solutions in general relativity, determining the dimension of their isometry groups using the IDEAL approach, with practical implementation examples.

Contribution

It introduces an IDEAL classification for type N vacuum solutions, linking invariant concomitants to isometry group dimensions and enabling computational implementation.

Findings

Invariant conditions for isometry group dimensions are established.

The IDEAL approach facilitates explicit classification of solutions.

Practical algorithms are demonstrated using Mathematica's xAct package.

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.