The energy-momentum tensor of gravitational waves, Wyman spacetime and freely falling observers

A good definition for the energy momentum tensor of gravity (EMTG) in General Relativity (GR) is a hard, if not impossible, task. On the other hand, in its teleparallel version, known as The Teleparallel Equivalent of General Relativity (TEGR), one can define the EMTG in a very satisfactory way. In this paper, it is proved that the EMTG of TEGR for linearized gravitational waves (GWs) is the same as the version of GR that is usually given in the literature. In addition, the exact version of the EMTG for a $pp-$wave with a $+$ polarization is obtained in a freely falling frame (FFF). Unlike the previous case, the energy density can be either positive or negative, depending on the details of the wave. The gravitational energy density for the Wyman spacetimes is obtained both in a static frame and in a FFF. It turns out that observers in free fall can measure the effects of gravity.