
Higher dimensional gravity and Farkas property in oriented matroid theoryAbstract: We assume gravity in a ddimensional manifold M and consider a splitting of the form M = Mp x Mq, with d = p + q. The most general twoblock metric associated with Mp and Mq is used to derive the corresponding EinsteinHilbert action S. We focus on the special case of two distinct conformal factors and ( for the metric in Mp and for the metric in Mq), and we write the action S in the form S = Sp+Sq, where Sp and Sq are actions associated with Mp and Mq, respectively. We show that a simplified action is obtained precisely when = 1. In this case, we find that under the duality transformation 1, the action Sp for the Mpspace or the action Sq for the Mqspace 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.
