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The field equations of noncommutative gravity can be obtained by replacingall exterior products by twist-deformed exterior products in the actionfunctional of general relativity, and are here studied by requiring that thetorsion 2-form should vanish, and that the Lorentz-Lie-algebra- valued part ofthe full connection 1-form should be self-dual. Other two conditions,expressing self-duality of a pair 2-forms occurring in the full curvature2-form, are also imposed. This leads to a systematic solution strategy, heredisplayed for the first time, where all parts of the connection 1-form arefirst evaluated, hence the full curvature 2-form, and eventually all parts ofthe tetrad 1-form, when expanded on the basis of {\gamma}-matrices. By assumingasymptotic expansions which hold up to first order in the noncommutativitymatrix in the neighbourhood of the vanishing value for noncommutativity, wefind a family of self-dual solutions of the field equations. This is generatedby solving first a inhomogeneous wave equation on 1-forms in a classical curvedspacetime (which is itself self-dual and solves the vacuum Einstein equations),subject to the Lorenz gauge condition. In particular, when the classicalundeformed geometry is Kasner spacetime, the above scheme is fully computableout of solutions of the scalar wave equation in such a Kasner model.

Self-dual road to noncommutative gravity with twist: a new analysis