The energy-momentum of a Poisson structure

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. © 2008 Springer-Verlag / Società Italiana di Fisica.