On conserved quantities for a moving black hole in TEGR

In the framework of the Teleparallel Equivalent of General Relativity (TEGR), where the field variables are tetrad components, mass and momentum for a moving (uniformly with respect to distant observers) Schwarzschild black hole (SBH) are constructed. A formalism developed by the authors earlier for constructing conserved quantities in TEGR, where currents and superpotentials are covariant with respect both to coordinate transformations and to local Lorentz rotations of tetrads is applied. This advantage has been reached by introducing inertial spin connection (ISC) and using the Noether theorem with preservation of a displacement vector in final expressions. A set of pairs (tetrad and related ISC) connected by smooth transformations we call as a “gauge”, it is the equivalence class. The quantity ISC is an external one, therefore we define it with making the use of the introduced by us generalized “turning off gravity” principle. But, even this a reasonable principle leads to different values of ISCs for the same tetrad that leads to different results. Here, on the example of the moving SBH we 1) demonstrate advantages of our fully covariant formalism, 2) study the ambiguity in definition of ISC as well. In calculations, we the use analogies with a moving mater ball in Minkowski space only in the “static gauge”. Expected mass and momentum have been obtained. Next we compare “static gauge” and “moving gauge”. It was found that they coincide. In the result, in the case of a moving SBH aforementioned ambiguity is absent because in both the cases the same mass and momentum are obtained.