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

Abstract

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, andWeinberg. 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 ?k gravity governed by the gravitational Lagrangian L1 = (R + a0R2 + Pp k=1 akR?kR) ??g and generally for n-order gravity described by the gravitational Lagrangian L = L (g??, g??,i1, 1??,i1i2, g??,i1i2i3 ,..., g??,i1i2i3...in). 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 ? emitted by a massive system and carried by a gravitational wave in the direction ? x 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.