The completeness problem on 3-dimensional non-unimodular Lie groups

We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix with nonzero trace. The case where $A$ is not diagonalizable over $\mathbb{C}$ was addressed in previous work by the authors, and the limiting case where $A$ is a scalar multiple of the identity is also known from the literature. In this paper, we determine all geodesically (in)complete left-invariant Lorentzian metrics for all other cases where $A$ is diagonalizable over $\mathbb{R}$. Additionally, we show that, when $A$ is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$, there exists at least one incomplete metric. As a consequence of prior work and our results, we obtain that every 3-dimensional non-unimodular Lie group admits an incomplete left-invariant Lorentzian metric.