The Energy-momentum of a Poisson structure

TL;DR

This paper explores how a Poisson structure derived from noncommutative geometry relates to Riemann curvature and applies this to analyze high-frequency gravitational waves, extending classical results to noncommutative settings.

Contribution

It introduces a natural map from Poisson structures to Riemann curvature in noncommutative geometry and extends classical gravitational wave results to this framework.

Findings

Established a map from Poisson structures to Riemann curvature.

Extended dispersion relation and conservation law to noncommutative gravity.

Linked high-frequency gravitational phenomena to cocycle conditions in Poisson structures.

Abstract

Consider the quasi-commutative approximation to a noncommutative geometry. It is shown that there is a natural map from the resulting Poisson structure to the Riemann curvature of a metric. This map is applied to the study of high-frequency gravitational radiation. In classical gravity in the WKB approximation there are two results of interest, a dispersion relation and a conservation law. Both of these results can be extended to the noncommutative case, with the difference that they result from a cocycle condition on the high-frequency contribution to the Poisson structure, not from the field equations.