Extended lagrangian theory of electromagnetism

Abstract

The present paper proposes a new generalized model for unifying the gravitational and electromagnetic fields. The model goes for beyond Finsler geometry and is based on the concept of the Lagrange space originated in [17]. The Lagrangian of a charged particle moving in gravitational and electromagnetic fields determines a special Lagrange space L n 0 studied in Chapter II. The h-covariant (horizontal-covariant) deflection tensor of L n 0 proves to coincide with the electromagnetic tensor, while the v-covariant (vertical-covariant) deflection tensor of L n 0 is exactly the gravitational metric tensor a ij ( x). It is natural, hence, to treat, the skew-symmetric part F of the h-covariant deflection tensor of a general Lagrange space L n to be an extended electromagnetic tensor associated with L n . The first pair of the generalized Maxwell equations satisfied by F are deduced only from the properties of the canonical metrical d-connection of L n . The attempts to unify the gravitational theory with the electromagnetism on the basis of new geometrical ideas are known to have a long history (see, e.g., the works [3, 5, 6, 24] and the references therein). The present paper is a logical continuation of these attempts. We shall not propose a suitable Lagrangian for deriving the second pair of Maxwell equations in terms of our approach, postponing this to future publications.