Noncommutative general relativity

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be a noncommutative extension of the unimodular theory of gravitation. We compute the leading order noncommutative correction to the action and derive the noncommutative correction to the equations of motion of the weak gravitation field.