Geometry of non-symmetric metrics and its application to theoretical physics

‎General ‎relativity ‎is ‎model ‎of ‎nature, ‎especially, ‎of ‎gravity.‎‎ ‎Its ‎cent‎ral ‎assumption ‎is‎ that space, time, and gravity are all aspects of a single entity, ‎cal‎led ‎space-‎time, which is modeled by a 4-dimensional Lorentzian manifold. It analyzes ‎space-‎time, electromagnetism, matter, and their mutual ‎infl‎uences. ‎But ‎the ‎effects ‎of ‎matter ‎and ‎electromagnetism ‎are ‎added ‎to ‎the ‎model ‎in a‎ ‎way‎ ‎which ‎is ‎not ‎directly ‎related ‎to ‎geometry ‎of ‎space-time ‎manifold. ‎In ‎fact, ‎in‎fluences ‎of ‎matter ‎and ‎electromagnetism ‎fields ‎are ‎added ‎to ‎theory ‎under ‎notion ‎of ‎stress-energy ‎tensor. ‎Hence, considering non-symmetric metrics extends this geometry and make a good apparatus to describe other physical quantities.

In this paper, we consider the geometry of a non-symmetric semi-Riemannian metric on a manifold M. A special class of such metrics contains a semi-Riemannian metric and a symplectic structure on M, simultaneously. Similar to the Levi-Civita connections in semi-Riemannain manifold, we define a new connection which is torsion free and compatible with our symplectic structure. With the help of semi-Riemannian metric, we define and compute the Rcci and scalar curvature of this new connection.

Using a natural Lagrangian (which is a generalization of Hilbert-Einstein action) and calculus of variations we derive some new field equations. The equations show that the symmetric part of semi-Riemannain metrics is directly related to gravity and the symplectic part is capable of describing quantities related to matter. 

In this work, we present a completely geometric theory of gravity. The Riemannian geometry, which is usually used to formulate gravitational theories adds the notion of matter to space time manifold as the way which is not directly related to geometry of the theory. In this framework, we retrieve Einstein’s field equation and we will show that the distribution of matter in space-time is directly related to symplectic part of our geometry