An Invariant Characterization of the Levi-Civita Spacetimes

TL;DR

This paper reviews Levi-Civita's solutions to Einstein's equations, presenting them accessibly and invariantly characterized using Cartan-Karlhede and Carminati-Mclenaghan invariants, serving as a practical reference.

Contribution

It provides a modern, accessible review of Levi-Civita's solutions and demonstrates their invariant characterization using two distinct methods.

Findings

Invariant characterization of solutions achieved

Comparison of Cartan-Karlhede and Carminati-Mclenaghan invariants

Practical application of the Cartan-Karlhede algorithm

Abstract

In the years 1917-1919 Tullio Levi-Civita published a number of papers presenting new solutions to Einstein's equations. This work, while partially translated, remains largely inaccessible to English speaking authors. In this paper we review these solutions, and present them in a modern, readable manner. We will also compute both Cartan-Karlhede and Carminati-Mclenaghan invariants such that these solutions are invariantly characterized by two distinct methods. These methods will allow for these solutions to be totally, and invariantly characterized. Because of the variety of solutions considered here, this paper will also be a useful reference for those seeking to learn to apply the Cartan-Karlhede algorithm in practice.