A COMPLEX FINSLER APPROACH OF GRAVITY

Abstract

In this paper our aim is mainly to obtain a two-dimensional complex Finsler model of the real gravitation space-time. We prove that, at least in the special case of the weakly gravitational field, this is possible and it leads to some interesting geometrical and physical aspects, such as the study of curvature invariants with respect to complex Berwald frame, intensively studied recently by us for a two-dimensional complex Finsler space. A generalization of the Klein–Gordon equation is proposed and we find solutions which are in concordance to the classical plane wave solution of momentum-energy relation. The last part of the paper is devoted to some applications in which the complex gravitational potential leads to the Bergman metric and to a more general case which leads to a non-purely Hermitian complex Finsler metric, with negative curvature invariant.