Noncommutative Einstein equations

We study a noncommutative deformation of general relativity (called matrix gravity) proposed recently by one of the authors, in which the gravitational field is described by a matrix-valued symmetric two-tensor field instead of a pseudo-Riemannian metric. In this theory all the constructions of Riemannian geometry, including connection and curvature, become matrix valued, and the action functional is constructed by using a matrix-valued generalization of scalar curvature and a matrix-valued measure. In the present paper, we compute the first variation of the action functional and derive the equations of motion of matrix gravity. Interestingly, the genuine noncommutative part of the dynamical equations is described only in terms of a particular tensor density that vanishes identically in the commutative limit. A noncommutative generalization of the energy-momentum tensor for the matter field is studied as well