On some models of geometric noncommutative general relativity

TL;DR

This paper develops models of geometric noncommutative vacuum gravity using Fedosov deformation quantization, deriving covariant field equations and explicit metric corrections, and explores their relation to Seiberg-Witten maps and spacetime noncommutativity.

Contribution

It introduces new models of noncommutative gravity based on Fedosov theory, extending classical vacuum solutions with explicit deformation corrections.

Findings

Derived coordinate covariant field equations up to second order in deformation

Solved for some models, providing explicit Ricci-flat metric corrections

Explored the connection between Fedosov *-product and spacetime noncommutativity

Abstract

Using Fedosov theory of deformation quantization of endomorphism bundle we construct several models of pure geometric, deformed vacuum gravity, corresponding to arbitrary symplectic noncommutativity tensor. Deformations of Einstein-Hilbert and Palatini actions are investigated. Coordinate covariant field equations are derived up to the second order of the deformation parameter. For some models they are solved and explicit corrections to an arbitrary Ricci-flat metric are pointed out. The relation to the theory of Seiberg-Witten map is also studied and the correspondence to the spacetime noncommutativity described by Fedosov *-product of functions is explained.