Einstein equations and inertial forces in axially symmetric stationary spacetimes

We investigate the relations between the inertial forces and the Einstein equations in axially symmetric stationary spacetimes. For the vacuum stationary axially symmetric spacetimes, we use the Geroch formalism to express the Einstein equations in terms of inertial forces. For the spacetimes with the perfect fluid sources, we use the formalism developed by Hansen and Winicour for establishing the relations. As expected intuitively, the gradients of inertial forces represent the field equations.