Abstract
The current paper is devoted to the investigation of the general form of the energy–momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpotential is also given. We discuss the structure of the obtained expressions and the conditions for the derived pEMT conservation laws to be satisfied independently (fully or partially) by the equations of motion. Deformations of the superpotential form for theories with a change in the independent variables in action are also considered. We apply these results to some interesting particular cases: general relativity and its modifications, particularly mimetic gravity and Regge–Teitelboim embedding gravity.
Highlights
One of the traditional issues of General Relativity (GR) is the correct definition of the conserved quantities related to the space–time symmetries, especially the energy
We study the generalized Noether procedure to obtain explicit formulae for pEMT and the corresponding superpotential for the maximally general theories of gravity with diffeomorphism-invariant action
Despite the lack of the antisymmetricity for the superpotential, we show that the conserving energy–momentum vector can always be expressed through the integral over an infinitely remote surface
Summary
One of the traditional issues of General Relativity (GR) is the correct definition of the conserved quantities related to the space–time symmetries, especially the energy. The currents obtained from standard Noether theorem [14] are suitable for such purposes This fact opens new possibilities for investigations of these quantum identities in the wide class of theories with help of well-known classical results. We study the generalized Noether procedure to obtain explicit formulae for pEMT and the corresponding superpotential for the maximally general theories of gravity with diffeomorphism-invariant action. It is worth noting, that the analysis made in [18] is limited to the most common case of action with no more than first derivatives of the fields with simple transformation law with respect to the gauge symmetry.
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