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 the 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 relatrivity, 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.

The teleparallel equivalent of general relativity

“…A set of tetrad fields is defined by four orthonormal, linearly independent vector fields in space-time, {e (0) μ , e (1) μ , e (2) μ , e (3) μ }, which establish the local reference frame of an observer that moves along a trajectory C, represented by the worldline x μ (τ ) [32][33][34] The tetrad fields e a μ allow the projection of vectors and tensors in space-time in the local frame of an observer. In order to measure field quantities with magnitude and direction, an observer must project these quantities on the frame carried by the observer.…”

“…(32) and (33). We are led to interpret t λμ as the gravitational energy-momentum tensor, and the quantity on the left-hand side of Eq.…”

Erratum: The energy-momentum of plane-fronted gravitational waves in the teleparallel equivalent of GR [Phys. Rev. D78, 047502 (2008)]

“…The tetrad fields are interpreted as reference frames adapted to preferred fields of observers in spacetime. This interpretation is possible by identifying the e (0) µ components of the frame with the four-velocities u µ of the observers, e (0) µ = u µ [8]. Therefore two different sets tetrad fields that yield the same metric tensor, and which are related by a local Lorentz transformation, represent different frames in the same spacetime.…”

“…(10) under the local SO(3,1) group reflects the frame dependence of the definition. We have argued [8] that this dependence is a natural feature of P a , since in the TEGR each set of tetrad fields is interpreted as a reference frame in spacetime.…”

Gravitational energy and radiation of a charged black hole

“…[2]). This gauge dependence, typical of any definition of gravitational energy is physically meaningful in GR || when we interpret the tetrad field as a reference system [3].…”