Electromagnetic duality in general relativity

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to tho se in electromagnetic theory. It turns out that the duality transformation implies the Einstein vacuum equation without the cosmological term. The vac uum equation is invariant under interchange of active and passive electric parts, giving rise to the same vacuum solutions but with the opposite sign for the gravitational constant. Further, by modifying the equation it is po ssible to construct interesting dual solutions to vacuum as well as to flat spacetimes.