12 - Gravitoelectric and gravitomagnetic fields and applications

Summary

We define gravitoelectric and gravitomagnetic fields, that is, splitting the gravitational field into “electric” and “magnetic” components. We first use an electromagnetic analogy for perturbative fields, which however only works in the static case. Next, we use a covariant formulation, first defining it by using Weyl tensor components. But the best definition, that is found to be in complete analogy to the electromagnetic case, is by using the Riemann tensor components, to define tidal tensor, for the tidal effect on neighboring geodesics. To define the analogy, we first define tidal tensors in electromagnetism and write the Maxwell’s equations in terms of them, then define the tidal tensors in gravity, and find that the Einstein’s equations are also written in terms of the gravitational tidal tensors. As a first application, we find the Lense–Thirring effect, for the precession of satellites in orbit due to “frame-dragging,” and as a second application, the clock effect, for the effect of the period of the clocks in orbit.