Conserved Current for General Teleparallel Models

Abstract

The obstruction for the existence of an energy momentum tensor for the gravitational field is connected with differential-geometric features of the Riemannian manifold. It has not to be valid for alternative geometrical structures. A teleparallel manifold is defined as a parallelizable differentiable 4D-manifold endowed with a class of smooth coframe fields related by global Lorentz, i.e., SO(1, 3) transformations. In this article a general 3-parameter class of teleparallel models is considered. It includes a 1-parameter subclass of models with the Schwarzschild coframe solution (generalized teleparallel equivalent of gravity). A new form of the coframe field equation is derived here from the general teleparallel Lagrangian by introducing the notion of a 3-parameter conjugate field strength F a . The field equation turns out to have a form completely similar to the Maxwell field equation d * F a = T a . By applying the Noether procedure, the source 3-form T a is shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source of the coframe field is interpreted as the total conserved energymomentum current of the system. A reduction of the conserved current to the Noether current and the Noether charge for the coframe field is provided. The energy-momentum tensor is defined as a map of the module of current 3-forms into the module of vector fields. Thus an energy-momentum tensor for the coframe field is defined in a diffeomorphism invariant and a translational covariant way. The total energy-momentum current of a system is conserved. Thus a redistribution of the energy-momentum current between material and coframe (gravity) field is possible in principle, unlike as in GR. The energy-momentum tensor is calculated for various teleparallel models: the pure Yang-Mills type model, the anti-Yang-Mills type model and the generalized teleparallel equivalent of GR. The latter case can serve as a very close alternative to the GR description of gravity.

The teleparallel equivalent of general relativity

“…They analysed the problem of consistently defining the gravitational energy-momentum and, in particular, the problem of regularization of the expression of the gravitational energy-momentum (see also Ref. [30]). This issue will also be addressed later on in the present geometrical framework.…”

ADM-like Hamiltonian formulation of gravity in the teleparallel geometry

“…defining the dynamics of a theory called Yang-Mills-type Teleparallel Model (YMTM) [42] canonical framework of which was studied in [38]. The phase space of that theory coincides with the phase space of TEGR described in this paper.…”

“…Let us finally note that by virtue of (3.40), (3.41) and (2.8) 42) which means that these variables are not completely independent-in fact, (θ A ⊥ w ) contains only one degree of freedom (per point) independent of (θ A w ). Indeed, if ( X i ), i = 1, 2, 3, is a (local) frame on Σ t then the functions (ξ A ) on Σ t can be alternatively defined by the first equation of (2.8) and X i θ A w ξ A = 0 (see (3.41)).…”

Kinematic quantum states for the Teleparallel Equivalent of General Relativity

“…In this paper we construct a space of kinematic quantum states for TEGR treated as a theory of cotetrad fields on a four-dimensional manifold. More precisely, the construction is valid for any theory of cotetrad fields the phase space of which coincides with that of TEGR-an example of such a theory is the Yang-Mills-type Teleparallel Model (YMTM) considered in [8,9].…”