The gravitational energy‐momentum pseudo‐tensor of higher‐order theories of gravity

We derive the gravitational energy momentum tensor ταη for a general Lagrangian of any order L=L(gμν,gμν,i1,gμν,i1i2,gμν,i1i2i3,⋯,gμν,i1i2i3⋯in) and in particular for a Lagrangian such as Lg=(R¯+a0R2+∑k=1pakR□kR)−g. We prove that this tensor, in general, is not covariant but only affine, then it is a pseudo‐tensor. Furthermore, the pseudo‐tensor ταη is calculated in the weak field limit up to a first non‐vanishing term of order h2 where h is the metric perturbation. The average value of the pseudo‐tensor over a suitable spacetime domain is obtained. Finally we calculate the power per unit solid angle Ω carried by a gravitational wave in a direction x̂ for a fixed wave number k under a suitable gauge. These results are useful in view of searching for further modes of gravitational radiation beyond the standard two modes of General Relativity and to deal with nonlocal theories of gravity where terms involving □R are present. The general aim of the approach is to deal with theories of any order under the same standard of Landau pseudo‐tensor. The gravitational enenergy‐momentum tensor is related to the transport properties of gravitational waves. In this paper, it has been derived for metric theories of gravity of any derivative order pointing out further gravitational polarizations modes and non‐locality properties. These features could be extremely relevant in view of Quantum Gravity and gravitational waves detection.