Higher dimensional gravity and Farkas property in oriented matroid theory

J.A. Nieto and E.A. León

Facultad de Ciencias Físico–Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, México, e–mails: nieto@uas.uasnet.mx; ealeon@uas.edu.mx

Recibido el 17 de octubre de 2011.
Aceptado el 16 de febrero de 2012.

Abstract

We assume gravity in a d–dimensional manifold M and consider a splitting of the form M = M_p × 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 S. We focus on the special case of two distinct conformal factors ψ and φ (ψ for the metric in M_p and φ for the metric in M_q), and we write the action S in the form S = S_p+S_q, where S_p and S_q are actions associated with M_p and M_q, respectively. We show that a simplified action is obtained precisely when ip = p^–1. In this case, we find that under the duality transformation φ ↔ φ^–1, the action S_p for the M_p–space or the action S_q 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.

Keywords: Higher dimensional gravity; gravitational duality; 2t physics.

PACS: 04.60.–m; 04.65.+e; 11.15.–q; 11.30.Ly

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