Exact relation between canonical and metric energy-momentum tensors for higher derivative tensor field theories

Abstract

We discuss the relation between canonical and metric energy-momentum tensors for field theories with actions that can depend on the higher derivatives of tensor fields in a flat spacetime. In order to obtain it we use a modification of the Noether's procedure for curved space-time. For considered case the difference between these two tensors turns out to have more general form than for theories with no more than first order derivatives. Despite this fact we prove that the difference between corresponding integrals of motion still has the form of integral over 2-dimensional surface that is infinitely remote in the spacelike directions. *

Energy–Momentum Pseudotensor and Superpotential for Generally Covariant Theories of Gravity of General Form

“…Despite the strong connection with the Noether procedure described in formulae (5)- (22), the choice of J ρ (ξ) precisely in the form (9) for construction of the recurrent chain (20)-(22) is not unique. Indeed, for theories discussed in the Section 4 with the change in the independent variables (57) in action one may start from the identities (8) for the original theory instead of the new ones and rewrite them in form:…”

“…The analogous recurrent chain arises in [22] in the context of generalized Belinfante relation. One may refine the recurrent chain (20)-(22) by following the procedure described in [22] on-shell in order to obtain the general form of superpotential for pEMT of theory (1). The resulting superpotential is not nescessarily antisymmetric.…”

“…In particular, it means that the total action S is invariant under certain global transformations, which, due to the Noether theorem, lead to the on-shell local conservation laws (continuity equations). In order to study these quantities, we follow the route presented in [18,22,23]. We consider an infinitesimal diffeomorphism x µ → x µ + ξ µ (x) and by using the arbitrariness of the vector ξ µ we will obtain a recurrent chain of identities.…”

“…One of the requirements imposed on the action (1) was the separate general covariance of both gravitational and matter parts. It allows one to obtain the recurrent chains similar to the chain (18), (21) and (22) for each of these action contributions. In this section, we show that such chains allow one to lend certain physical sense to the gravitational and matter contributions into the superpotential (23).…”

“…is the metric (Hilbert) EMT of the matter. The identity (40) arising from the second Noether theorem may be considered independently from the chain Equations (20)- (22). As one shall see below, it can be used to simplify them further.…”