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.

22 pages

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• Four-Dimensional Gravity on a Covariant Noncommutative Space (II)
• Noncommutative Gauge Theory and Gravity in Three Dimensions
• Noncommutative gravity and the relevance of the $\theta$-constant deformation
• Noncommutative Particles in Curved Spaces
• Poisson structure on a space with linear SU(2) fuzziness
• Gravity as a Gauge Theory on Three-Dimensional Noncommutative spaces
• Intertwining noncommutativity with gravity and particle physics
• Dirac field and gravity in NC $SO(2,3{)}_{\star }$ model
• Phase space quantization, noncommutativity and the gravitational field
• Central force problem in space with SU(2) Poisson structure
• Noncommutative $SO(2,3{)}_{\star }$ Gauge Theory of Gravity
• Noncommutative Gauge Theories and Gravity
• Eigenvalue problem for radial potentials in space with SU(2) fuzziness
• Gauge Theories on Fuzzy Spaces and Gravity
• Dimensional Reduction of Supersymmetric Gauge Theories