The gravitational energy-momentum pseudo-tensor in higher-order theories of gravity

The problem of non-localizability and the non-uniqueness of gravitational energy in general relativity has been considered by many authors. Several gravitational pseudo-tensor prescriptions have been proposed by physicists, such as Einstein, Tolman, Landau, Lifshitz, Papapetrou, Moller, and Weinberg. We examine here the energy-momentum complex in higher-order theories of gravity applying the Noether theorem for the invariance of gravitational action under rigid translations. This, in general, is not a tensor quantity because it is not a covariant object but only an affine tensor, that is, a pseudo-tensor. Therefore we propose a possible prescription of gravitational energy and momentum density for square(k) gravity governed by the gravitational Lagrangian L-g = ((R) over bar + a(0)R(2) + Sigma(p)(k=1) a(k)R square R-k) root-g and generally for n-order gravity described by the gravitational Lagrangian L = L (g(u,v),g(uv,)i(1,)g(uv,)i(1,)i(2,)i(3,) ..., g(uv),i(1)i(2)i(3) ... i(n) ) . The extended pseudo-tensor reduces to the one introduced by Einstein in the limit of general relativity where corrections vanish. Then, we explicitly show a useful calculation, i.e., the power per unit solid angle Omega emitted by a massive system and carried by a gravitational wave in the direction (x) over cap for a fixed wave number k. We fix a suitable gauge, by means of the average value of the pseudo-tensor over a spacetime domain and we verify that the local pseudo-tensor conservation holds. The gravitational energy-momentum pseudo-tensor may be a useful tool to search for possible further gravitational modes beyond the two standard ones of general relativity. Their finding could be a possible observable signatures for alternative theories of gravity.