On the Notions of Energy Tensors in Tetrad-Affine Gravity

Abstract

We are concerned with the precise modalities by which mathematical constructions related to energy-tensors can be adapted to a tetrad-affine setting. We show that, for fairly general gauge field theories formulated in that setting, two notions of energy tensor-the canonical tensor and the stress-energy tensor-exactly coincide with no need for tweaking. Moreover we show how both notions of energy-tensor can be naturally extended to include the gravitational field itself, represented by a couple constituted by the tetrad and a spinor connection. Then we examine the on-shell divergences of these tensors in relation to the issue of local energy-conservation in the presence of torsion.1 Indeed, a straightforward physical interpretation of the Ricci tensor in terms of energy is problematic as, for example, Schwarzschild spacetime has non-zero gravitational energy while the Ricci tensor vanishes.2 The notion of covariant differential of vector-valued forms, which has been variously present in the literature for several years, is strictly related to the Frölicher-Nijenhuis bracket [38,5,39,40,41]. In this paper we will just write down the needed coordinate expressions.

Gauge field theory without groups

“…Moreover, all fermionic sectors can be described by a unique overall antisymmetrized tensor algebra. 22 A similar observation holds true for the bosonic sectors. Furthermore, the mutual ordering of fermionic and bosonic sectors is regarded as inessential.…”

“…Let now the grades of a[ζ] and a * [z] be ⌊ζ⌉ and ⌊z⌉, respectively, and the grade of any composition the sum (mod 2) of the grades of all factors. The super-bracket (or supercommutator ) of X, Y ∈ op(V ⊗ ) is then defined by 22 If X and Y are any two vector spaces, then their antisymmetric tensor algebras fulfil the isomorphisms…”

A first-order Lagrangian theory of fields with arbitrary spin

“…Remark. The notion of soldering form considered here is essentially equivalent to the notion variously referred to in the literature [18,19,35,13] as a 'frame field', or a 'tetrad' or a 'vierbein' ('vielbein' for arbitrary dimension). A global soldering form, with the target bundle H constructed from the spinor bundle itself, yields a 'spin structure' on M (in this paper we are not concerned with the possible topological obstructions to the existence of such structures).…”