# The Cartan Algorithm in Five Dimensions

## TL;DR

This paper presents a generalized Cartan algorithm for classifying five-dimensional spacetimes, extending previous methods from four dimensions, and demonstrates its application on notable black hole solutions.

## Contribution

It introduces a new algorithm for five-dimensional spacetime classification using the alignment classification, expanding the tools available for higher-dimensional general relativity.

## Key findings

- Successfully classifies known five-dimensional solutions
- Demonstrates the algorithm on Myers-Perry, Kerr-(A)dS, and black ring solutions
- Provides a framework for future applications in higher-dimensional gravity

## Abstract

In this paper we introduce an algorithm to determine the equivalence of five dimensional spacetimes, which generalizes the Karlhede algorithm for four dimensional general relativity. As an alternative to the Petrov type classification, we employ the alignment classification to algebraically classify the Weyl tensor. To illustrate the algorithm we discuss three examples: the singly rotating Myers-Perry solution, the Kerr (anti) de Sitter solution, and the rotating black ring solution. We briefly discuss some applications of the Cartan algorithm in five dimensions.