On conservation laws in teleparallel gravity

Abstract

Applying directly the Noether theorem in the framework of the Teleparallel Equivalent of General Relativity (TEGR), we construct conserved quantities, currents and superpotentials. They are covariant both under coordinate transformations and under local Lorentz rotations, unlike earlier approaches. This advantage is achieved by a presence in expressions of conservation laws of a displacement vector that can be interpreted as a Killing vector, as a proper vector of an observer, etc. We introduce, as well, a principle for a definition of an inertial spin connection that is an undetermined quantity in TEGR in the original formulation. The new expressions for conserved quantities and the introduced principle are applied to calculate mass for the Schwarzschild black hole and energy density for an observer freely falling in spatially flat Friedmann world.

On conserved quantities for the Schwarzschild black hole in teleparallel gravity

“…We will demonstrate that both of these points can be understood by considering the second definition based on the diffeomorphism invariance and the Noether theorem. This is the definition first introduced by Obukhov et al [18][19][20] in the language of differential forms and derived in a more common tensorial language in the recent papers [24,25]. Here we will only refer to the main result and briefly outline the general idea, while all the details can be found in [24,25].…”

“…• A a bρ is not determined by the field equations but, as we have discussed in Introduction, it can be chosen by the principle of "switching-off" gravity [3,17,22,23]. According to [24,25], this means that for the tetrad h a μ we calculate the Levi-Civita connection (2.6) and the Riemann tensor…”

“…It was then advocated that the underlying principle can be altered to choosing the spin connection in a such way that the teleparallel Lagrangian vanishes asymptotically [3,22,23]. In [24,25], the idea of "switching-off" gravity was further formalized in order to be less dependent on the asymptotic behaviour. Nevertheless, at least for the asymptotically Minkowski spacetimes, all these various approaches reduce to the original principle of "switching-off" gravity, and the inertial spin connection is determined from the Levi-Civita spin connection by taking a parameter that controls the strength of gravity to zero or considering the limit r → ∞.…”

On the kinematic interpretation of cosmological redshifts

“…A complex study of more distant astronomical objects raises several disturbing questions of the physical interpretation of cosmological dynamics, see e.g. (Bolós & Klein, 2012, Bunn & Hogg, 2009, Chodorowski, 2011, Davis & Lineweaver, 2004, Emtsova & Toporensky, 2019, Grøn & Elgarøy, 2007, Harrison, 1993, 1995, 2000, Kaya, 2011, Klein & Collas, 2010, Klein & Randles, 2011, Liebscher, 2007, Murdoch, 1977, Narlikar, 1994, Peacock, 1999, Peebles, 1993, Peebles et al, 1991, Pössel, 2019, Silverman, 1986, Stuckey, 1992, Whiting, 2004.…”

Unique definition of relative speed along the line of sight of a luminous object in a Riemannian space-time

“…This provides a new perspective to solve startling difficulties of superluminal`proper´recession velocities, which the conventional scenario of expanding universe of standard cosmological model presents (see e.g. Bolós & Klein, 2012, Bunn & Hogg, 2009, Chodor-owski, 2011, Davis & Lineweaver, 2004, Emtsova & Toporensky, 2019, Grøn & Elgarøy, 2007, Harrison, 1993, 1995, Kaya, 2011, Klein & Collas, 2010, Klein & Randles, 2011, Murdoch, 1977, Narlikar, 1994, Peacock, 1999, 2008, Peebles, 1993, Peebles et al, 1991, Silverman, 1986, Whiting, 2004. In some instances (in earlier epochs), the distant astronomical objects are observed to exhibit redshifts in excess of unity, and only a consistent theory could tackle the key problems of a dynamics of such objects.…”