Introduction to Lorentzian and Flat Affine Geometry of $\mathsf{G}\mathsf{L}(2,\mathbb{R})$

The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group $4$ dimensional $G_0$ as a Lorentzian and flat affine manifold. As the group $G_0$ is naturally equipped with a bi-invariant Hessian metric $k^{+}$, relative to a bi-invariant flat affine structure $\mathrm{\nabla }$, we examine both structures and the relationships between them. Both structures are defined using the Lie algebra $\mathfrak{g}$, the first one through the trace $k(u,v):=\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(u\circ v)$ and the second by the composition ${\mathrm{\nabla }}_{u^{+}}{v}^{+}:=(u\circ v{)}^{+}$, where $u,v\in \mathfrak{g}$. The curvatures, tidal force, and Jacobi vector fields of $(G_0,k^{+})$ are determined in Section 1. Section 2 discusses the causal structure of $k^{+}$, while Section 3 focuses on the developed map relative to $\mathrm{\nabla }$ in the sense of C. Ehresmann.