Einstein Gravity as a Nonholonomic Almost Kahler Geometry, Lagrange-Finsler Variables, and Deformation Quantization

TL;DR

This paper develops a geometric framework that transforms (pseudo) Riemannian metrics into almost Kahler structures using Lagrange-Finsler variables, enabling a novel approach to quantum gravity via deformation quantization on nonholonomic spacetimes.

Contribution

It introduces a method to represent Einstein gravity within Lagrange-Finsler and almost Kahler geometries on nonholonomic spacetimes, facilitating deformation quantization of gravity.

Findings

Reformulation of Einstein equations in Lagrange-Finsler variables

Construction of almost Kahler structures on nonholonomic spacetimes

Application of deformation quantization to quantum gravity models

Abstract

A geometric procedure is elaborated for transforming (pseudo) Riemanian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in their turn, can be equivalently represented as almost Kahler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2 splitting with associate nonlinear connection structure. We also show how the Einstein equations can be redefined in terms of Lagrange-Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.