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

## TL;DR

This paper explores how tetrad fields as reference frames relate to gravitational energy in free-fall conditions, showing that gravitational energy vanishes in free-fall frames within the teleparallel gravity framework, aligning with the equivalence principle.

## Contribution

It constructs tetrad fields adapted to free-fall observers in Schwarzschild spacetime and demonstrates that gravitational energy-momentum vanishes in these frames, supporting a viable energy definition.

## Key findings

- Gravitational energy-momentum vanishes in free-fall frames.
- Tetrad fields can be adapted to freely falling observers.
- Results align with the principle of equivalence.

## 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.