Torsion-induced gravitational $\theta$ term and gravitoelectromagnetism

Motivated by the analogy between a weak field expansion of general relativity and Maxwell's laws of electrodynamics, we explore physical consequences of a parity violating $\theta$ term in gravitoelectromagnetism. This is distinct from the common gravitational $\theta$ term formed as a square of the Riemann tensor. Instead it appears as a product of the gravitoelectric and gravitomagnetic fields in the Lagrangian, similar to the Maxwellian $\theta$ term. We show that this sector can arise from a quadratic torsion term in nonlinear gravity. In analogy to the physics of topological insulators, the torsion-induced $\theta$ parameter can lead to excess mass density at the interface of regions where $\theta$ varies and consequently it generates a correction to Newton's law of gravity. We discuss also an analogue of the Witten effect for gravitational dyons.