Higher Dimensional Gravity and Farkas Property in Oriented Matroid Theory

We assume gravity in a $d$-dimensional manifold $M$ and consider a splitting of the form $M=M_{p}\times M_{q}$, with $d=p+q$. The most general two-block metric associated with $M_{p}$ and $M_{q}$ is used to derive the corresponding Einstein-Hilbert action $\mathcal{S}$. We focus on the special case of two distinct conformal factors $\psi$ and $\phi$ ($\psi$ for the metric in $M_{p}$ and $\phi$ for the metric in $M_{q}$), and we write the action $\mathcal{S}$ in the form $\mathcal{S=S}_{p}\mathcal{+S}$, where $\mathcal{S}_{p}$ and $\mathcal{S}_{q}$ are actions associated with $M_{p}$ and $M_{q}$, respectively. We show that a simplified action is obtained precisely when $\psi =\phi ^{-1}$. In this case, we find that under the duality transformation $\phi \leftrightarrow \phi ^{-1}$, the action $\mathcal{S}_{p}$ for the $M_{p}$-space or the action $\mathcal{S}$ for the $M_{q}$-space remain invariant, but not both. This result establishes an analogy between Farkas property in oriented matroid theory and duality in general relativity. Furthermore, we argue that our approach can be used in several physical scenarios such as 2t physics and cosmology.