S S symmetry Article An Invariant Characterization of the Levi-Civita Spacetimes Cooper K. Watson 1,2,* , William Julius 1,2 , Matthew Gorban 1,2 , David D. McNutt 3 , Eric W. Davis 1 and Gerald B. Cleaver 1,2 1 Early Universe Cosmology and Strings (EUCOS) Group, Center for Astrophysics, Space Physics and Engineering Research (CASPER), Baylor University, Waco, TX 76798, USA;
[email protected] (W.J.);
[email protected] (M.G.);
[email protected] (E.W.D.);
[email protected] (G.B.C.) 2 Department of Physics, Baylor University, Waco, TX 76798, USA 3 Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway;
[email protected] * Correspondence:
[email protected] 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 researchers. 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. Keywords: Levi-Civita metric; general relativity; curvature invariant Citation: Watson, C.K.; Julius, W.; Gorban, M.; McNutt, D.D.; Davis, 1. Introduction E.W.; Cleaver, G.B. An Invariant In the years 1917–1919 Tullio Levi-Civita (LC) published nearly a dozen papers intro- Characterization of the Levi-Civita ducing and analyzing a variety of new solutions to Einstein’s field equations (collected Spacetimes.