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There are five unimodular simply connected three dimensional unimodular nonabelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitarygroup $\mathrm{SU}(2)$, the universal covering group$\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, thesolvable Lie group $\mathrm{Sol}$ and the universal covering group$\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclideangroup. For each $G$ among these Lie groups, we give explicitly the list of allLorentzian left invariant metrics on $G$, up to un automorphism of $G$.Moreover, for any Lorentzian left invariant metric in this list we give itsRicci curvature, scalar curvature, the signature of the Ricci curvature and weexhibit some special features of these curvatures. Namely, we give all themetrics with constant curvature, semi-symmetric non locally symmetric metricsand the Ricci solitons.

Lorentzian left invariant metrics on three dimensional unimodular Lie groups and their curvatures