Energy–momentum current for coframe gravity

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. This must not 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 paper a general free parametric class of teleparallel models is considered. It includes a 1-parameter subclass of viable models with the Schwarzschild coframe solution.A new form of the coframe field equation is derived from the general teleparallel Lagrangian by introducing the notion of a 3-parameter conjugate field strength ℱa. The field equation turns out to have a form completely similar to the Maxwell field equation d * ℱa = \U0001d4afa. By applying the Noether procedure, the source 3-form \U0001d4afa is shown to be connected with the diffeomorphism invariance of the Lagrangian. Thus the source \U0001d4afa of the coframe field is interpreted as the total conserved energy–momentum current. The energy–momentum tensor for the coframe is defined. The total energy–momentum current of a system of a coframe and a material field is conserved. Thus a redistribution of the energy–momentum current between material and coframe (gravity) fields is possible in principle, unlike in the standard GR.For special values of parameters, when the GR is reinstated, the energy–momentum tensor gives up the invariant sense, i.e. becomes a pseudo-tensor. Thus even a small-parametric change of GR turns it into a well-defined Lagrangian theory.