Born’s Reciprocal Relativity theory, curved phase space, Finsler geometry and the cosmological constant

Abstract

A very brief introduction of the history of Born’s Reciprocal Relativity Theory, Hopf algebraic deformations of the Poincare algebra, de Sitter algebra, and noncommutative spacetimes paves the road for the exploration of gravity in curved phase spaces within the context of the Finsler geometry of the cotangent bundle T∗M of spacetime. A scalar-gravity model is duly studied, and exact nontrivial analytical solutions for the metric and nonlinear connection are found that obey the generalized gravitational field equations, in addition to satisfying the zero torsion conditions for all of the torsion components. The curved base spacetime manifold and internal momentum space both turn out to be (Anti) de Sitter type. The most salient feature is that the solutions capture the very early inflationary and very-late-time de Sitter phases of the Universe. A regularization of the 8-dim phase space action leads naturally to an extremely small effective cosmological constant Λeff, and which in turn, furnishes an extremely small value for the underlying four-dim spacetime cosmological constant Λ, as a direct result of a correlation between Λeff and Λ resulting from the field equations. The rich structure of Finsler geometry deserves to be explored further since it can shine some light into Quantum Gravity, and lead to interesting cosmological phenomenology.