Noncommutative gravity and the relevance of the $\theta$-constant deformation

TL;DR

This paper constructs a noncommutative gravity model on a constant noncommutative space-time, showing how noncommutativity induces curvature in Minkowski space while preserving torsion-free conditions, and emphasizes the importance of Fermi normal coordinates.

Contribution

It develops a noncommutative gravity framework using gauge theory and the Seiberg-Witten map, revealing noncommutativity as a source of curvature and identifying Fermi normal coordinates as natural for analysis.

Findings

Noncommutativity induces curvature in Minkowski space.

Minkowski space remains torsion-free despite noncommutativity.

Fermi normal coordinates are natural for studying noncommutative gravity.

Abstract

In this Letter we construct the noncommutative (NC) gravity model on the $\theta$-constant NC space-time. We start from the NC $SO(2,3)_\star$ gauge theory and use the enveloping algebra approach and the Seiberg-Witten map to construct the effective NC gravity action. The action and the equations of motion are expanded up to second order in the deformation parameter. The equations of motion show that the noncommutativity plays a role of a source for the curvature and/or torsion. Finally, we calculate the NC corrections to the Minkowski space-time and we show that in the presence of noncommutativity Minkowski space-time becomes curved, but remains torsion-free. The breaking of diffeomorphism invariance is understood in terms of the preferred coordinate system. We show that the coordinates we are using are the Fermi normal coordinates. This suggests that the natural coordinate system in which one should study NC gravity is given by the Fermi normal coordinates.