On the Equivalence of Spacetimes, the Cartan-Karlhede Algorithm

TL;DR

This paper introduces the Cartan-Karlhede algorithm, a systematic method for determining whether two spacetimes in general relativity are equivalent, simplifying the complex process of checking metric equivalence.

Contribution

The paper explains and illustrates the Cartan-Karlhede algorithm, providing practical examples to help readers understand how to determine spacetime equivalence.

Findings

The algorithm provides a finite procedure for spacetime equivalence.

It simplifies the process of checking metric equivalence in general relativity.

Illustrative examples demonstrate the algorithm's application.

Abstract

It is well known that in general relativity theory two spacetimes whose metrics are related by a coordinate transformation are physically equivalent. However, given two line elements, it is virtually impossible to implement the most general coordinate transformation in order to check the equivalence of the spacetimes. In this paper we present the so-called Cartan-Karlhede algorithm, which provides a finite sequence of steps to decide whether or not two metrics are equivalent. The point of this note is to illustrate the method through several simple examples, so that the reader can learn the fundamentals and details of the algorithm in practice.