Gravito-electromagnetic approach for the space-time of a plane gravitational wave

Abstract

We build the Fermi frame associated to the world-line of a reference observer, arbitrary moving in a given space-time, and we show that local measurements can be described in terms of a gravitoelectromagnetic analogy, where the gravito-electric and gravito-magnetic fields are related to the non inertial features of the observer's motion and to the curvature of space-time. We apply this formalism to the space-time of a plane gravitational wave and show that the interaction of the wave with antennas can be explained in terms of gravito-electromagnetic forces acting on test masses. Moreover, we show that, besides the known gravito-electric effects, on which present gravitational waves antennas are based, gravito-magnetic effects could in principle lead to other kinds of detectors.

Gravitomagnetic Stern–Gerlach Force

“…The purpose of this section is to discuss spin-gravity coupling for linearized gravitational waves; in particular, we are interested in the corresponding Stern–Gerlach force. For related studies, see [ 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 ] and the references cited therein.…”

A Note on the Gravitoelectromagnetic Analogy

“…It is interesting to point out that the effects that are measured by current intereferometers are on the YZ plane, which is orthogonal to the wave propagation direction, since they are provoked by the gravitoelectric part of the wave field. On the other hand, this is effect (like other ones considered in Ruggiero and Ortolan [32,34]) is purely gravitomagnetic. As we have seen, it is simply described using this gravitoelectromagnetic approach, but it would be more complicated to understand in the framework of the TT gauge coordinates that are usually employed to describe gravitational waves.…”

“…As we discussed in our previous papers, Ruggiero and Ortolan [32] and Ruggiero [31], this approach can be applied to the study of the spacetime around a world-line of an observer in the field of a plane gravitational wave. Fermi coordinates were first applied, by Bini et al [33], to the study of a plane gravitational wave.…”

“…In particular, using Fermi coordinates (cT, X, Y, Z), for geodesic observers. The spacetime metric can be written (see, e.g., Ruggiero and Ortolan [32] and references therein) in the form given by Equation ( 24), with:…”

From Kerr to Heisenberg

“…It is always true that we may locally diagonalize the metric (this happens because, apart from singularities, there always is a local tangent flat space-time); the diagonalization is however not contextually possible everywhere. The nickname “gravitomagnetism” (GM) is, as known, due to the fact that, in weak field approximation, the Einstein equations in vacuo assume a form practically coinciding (but for a factor of 2) with the classical Maxwell equations of electromagnetism (EM) and may, in the same approximation, be treated as the components of a three-dimensional vector potential in analogy with EM [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ].…”