Abstract
We examine various methods of constructing conserved quantities in the Teleparallel Equivalent of General Relativity (TEGR). We demonstrate that in the covariant formulation the preferred method are the Noether charges that are true invariant quantities. The Noether charges depend on the vector field xi and we consider two different options where xi is chosen as either a Killing vector or a four-velocity of the observer. We discuss the physical meaning of each choice on the example of the Schwarzschild solution in different frames: static, freely falling Lemaitre frame, and a newly obtained generalised freely falling frame with an arbitrary initial velocity. We also demonstrate how to determine an inertial spin connection for various tetrads used in our calculations, and find a certain ambiguity in the “switching-off” gravity method where different tetrads can share the same inertial spin connection.
Highlights
Our primary goal is to clarify the situation with conserved quantities in teleparallel gravity where two distinct approaches to their definition exist
This follows from the fact that in the non-covariant formulation, we can uniquely identify the vector field ξ characterizing the observer with the tetrad variable in the field equations
It is interesting to observe that applying the same logic to (3.4), we find that Pμ are the Noether charges for the set of vector fields ξ(μ) = δ(μ)ν∂ν, i.e. diffeomorphisms generated by the coordinates
Summary
Our primary goal is to clarify the situation with conserved quantities in teleparallel gravity where two distinct approaches to their definition exist. The Noether conserved charges P(ξ ), on the other hand, are true invariant quantities but depend on the additional vector field ξ characterizing the observer that needs to be specified. To our surprise, we find that Pa defines the energy–momentum uniquely only in the case of the non-covariant formulation of TEGR where the spin connection is assumed to be zero This follows from the fact that in the non-covariant formulation, we can uniquely identify the vector field ξ characterizing the observer with the tetrad variable in the field equations. In the covariant formulation, on the other hand, the freedom to use an arbitrary tetrad in the field equations forces us to treat the vector field ξ independently and use the Noether conserved charges in order to obtain meaningful results
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