The goal of this paper is to study the geometry of the connected unit component of the real general linear Lie group four 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 the natural bi-invariant flat affine structure $\nabla$ (see \cite{AuMe}), we examine these structures and the relationships between them. The curvatures, tidal force, and Jacobi vector fields of $(G_0, k^+)$ are determined in Section 1. Section 2 discusses the causal structure of $(G_0,k^+)$, while Section 3 focuses on the developed map relative to $\nabla$ in the sense of C. Ehresmann.