Reality in Noncommutative Gravity

Abstract

We study the problem of reality in the geometric formalism of the 4D noncommutative gravity using the known deformation of the diffeomorphism group induced by the twist operator with the constant deformation parameters $\vt^{mn}$. It is shown that real covariant derivatives can be constructed via $\star$-anticommutators of the real connection with the corresponding fields. The minimal noncommutative generalization of the real Riemann tensor contains only $\vt^{mn}$-corrections of the even degrees in comparison with the undeformed tensor. The gauge field $h_{mn}$ describes a gravitational field on the flat background. All geometric objects are constructed as the perturbation series using $\star$-polynomial decomposition in terms of $h_{mn}$. We consider the nonminimal tensor and scalar functions of $h_{mn}$ of the odd degrees in $\vt^{mn}$ and remark that these pure noncommutative objects can be used in the noncommutative gravity.

Latex file, 14 pages, corrected version to be publised in CQG

References in corpus (3)

• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• A Gravity Theory on Noncommutative Spaces
• Noncommutative Geometry and Gravity

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• Connes spectral distance and nonlocality of generalized noncommutative phase spaces
• Deformation quantization for systems with second-class constraints in deformed fermionic phase space