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.
Highlights
We present a new form of this metric which is not related to previous solutions by any known transform, real or complex: ds2 = −e2z dt2 + a2 dz2 + a2 dΩ2
With the form of line element given by (21), it is possible to choose coordinates, such that ρ0 may be eliminated from the metric, this will result in φ not being parameterized from (0, 2π )
We presented two different invariant characterizations of these solutions, using both the CK algorithm and CM invariants
Summary
In the years 1917–1919 Tullio Levi-Civita (LC) published nearly a dozen papers introducing and analyzing a variety of new solutions to Einstein’s field equations (collected works in Italian available in Volume IV at [1]). To generate an invariant coframe and the corresponding scalar quantities which uniquely characterize these spacetimes. The variety of solutions considered in this work will result in the CK algorithm running in several markedly different ways. In several of the cases considered the spacetime is sufficiently special, such that only a subset of the CM invariants are needed [13]. We note that all spacetimes considered here are I non-degenerate, as the only case considered with constant scalar invariants is homogeneous [16]. We note that the CK algorithm will always generate a complete classification of the spacetime, there is no possibility this may fail for the specific cases considered here. We will consider several generalizations of these solutions in cases where our methods of characterization extend directly, and in an instructive manner.
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