On reference frames in spacetime and gravitational energy in freely falling frames

Abstract

We consider the interpretation of tetrad fields as reference frames in spacetime. Reference frames may be characterized by an antisymmetric acceleration tensor, whose components are identified as the inertial accelerations of the frame (the translational acceleration and the frequency of rotation of the frame). This tensor is closely related to gravitoelectromagnetic field quantities. We construct a set of tetrad fields adapted to observers that are in free fall in the Schwarzschild spacetime, and show that the gravitational energy–momentum constructed out of this set of tetrad fields, in the framework of the teleparallel equivalent of general relativity, vanishes. This result is in agreement with the principle of equivalence, and may be taken as a condition for a viable definition of gravitational energy.