A note on the gravitoelectromagnetic analogy

We discuss the linear gravitoelectromagnetic approach used to solve Einstein equations in the weak-field and slow-motion approximation, which is a powerful tool to explain, by analogy with electromagnetism, several gravitational effects in the Solar System, where the approximation holds true. In particular, we discuss the analogy according to which Einstein equations can be written as Maxwell-like equations and focus on the definition of the gravitoelectromagnetic fields in non stationary conditions. Furthermore, we examine to what extent, starting from a given solution of Einstein equations, gravitoelectromagnetic fields can be used to describe the motion of test particles using a Lorentz-like force equation