Lorentzian geometry in four extended spatial dimensions

A vector space defined as inertial 4 space (I$^4$) is described as an extension of Minkowski four dimensional spacetime (M$^4$). I$^4$ shares metric signature (- + + +) with M$^4$ and is also shown as a subspace of a non-temporal symmetrical vector space defined as primary 4-space (P$^4$) where the momentum of mass is manifested as a wave. The collective 4-space geometry where $\exists P^4:P^4\to I^4\to M^4$ is shown to be compatible with special relativity. In the 4-space system, the three spatial dimensions in an M$^4$ subspace can be considered a modified 3-brane embedded in a 4 dimensional bulk. The 4$^{th}$ special dimensions is occupied by the wave property of mass resulting in the creation of a time dimension and the suppression of a space dimension.