# Counting the number of Killing vectors in a 3D spacetime

## TL;DR

This paper presents an algorithm to determine the number of symmetries in 3D Lorentzian spacetimes using Ricci tensor invariants, enabling classification of highly symmetric solutions like Lifshitz and pp-waves.

## Contribution

It introduces a novel algorithm based on Ricci tensor invariants to count and classify Killing vectors in 3D spacetimes, including a complete classification for four symmetries.

## Key findings

- Classified Lifshitz and pp-wave spacetimes by symmetry level
- Developed an algorithm for counting Killing vectors using Ricci invariants
- Provided a complete classification of spacetimes with four Killing vectors

## Abstract

We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the values of differential invariants arising from the compatibility conditions of the Killing equation. As illustrating examples, we classify the Lifshitz and pp-wave spacetimes into a hierarchy based on their level of symmetry. A complete classification of spacetimes admitting 4 Killing vectors is also presented.