  In General Relativity, the issue of defining the gravitational energy
contained in a given spatial region is still unresolved, except for particular
cases of localized objects where the asymptotic flatness holds for a given
spacetime. In principle, a theory of gravity is not self-consistent, if the
whole energy content is not uniquely defined in a specific volume. Here we
generalize the Einstein gravitational energy-momentum pseudotensor to non-local
theories of gravity where analytic functions of the non-local integral operator
$\Box^{-1}$ are taken into account. We apply the Noether theorem to a
gravitational Lagrangian, supposed invariant under the one-parameter group of
diffeomorphisms, that is, the infinitesimal rigid translations. The invariance
of non-local gravitational action under global translations leads to a locally
conserved Noether current, and thus, to the definition of a gravitational
energy-momentum pseudotensor, which is an affine object transforming like a
tensor under affine transformations. Furthermore, the energy-momentum complex
remains locally conserved, thanks to the non-local contracted Bianchi
identities. The continuity equations for the gravitational pseudotensor and the
energy-momentum complex, taking into account both gravitational and matter
components, can be derived. Finally, the weak field limit of pseudotensor is
performed to lowest order in metric perturbation in view of astrophysical
applications.
