Conformal Gravity on Noncommutative Spacetime

TL;DR

This paper develops a formulation of conformal gravity on noncommutative spacetime using Moyal star products and Seiberg-Witten maps, deriving the modified field equations to explore quantum geometric effects.

Contribution

It introduces a novel approach to noncommutative conformal gravity by combining Moyal star products with Seiberg-Witten maps to preserve local conformal invariance.

Findings

Generalized conformal gravity action formulated with noncommutative geometry

Field equations derived incorporating noncommutative effects

Maintains local conformal invariance in a noncommutative setting

Abstract

Conformal gravity on noncommutative spacetime is considered in this paper. The presupposed gravity action consists of the Brans-Dicke gravity action with a special prefactor of the term, where the Ricci scalar couples to the scalar field, to maintain local conformal invariance and the Weyl gravity action. The commutation relations between the coordinates defining the noncommutative geometry are assumed to be of canonical shape. Based on the moyal star product, products of fields depending on the noncommutative coordinates are replaced by generalized expressions containing the usual fields and depending on the noncommutativity parameter. To maintain invariance under local conformal transformations with the gauge parameter depending on noncommutative coordinates, the fields have to be mapped to generalized fields by using Seiberg-Witten maps. According to the moyal star product and the thus induced Seiberg-Witten maps the generalized conformal gravity action is formulated and the corresponding field equations are derived.