Localized energetics of linear gravity: Theoretical development

Abstract

We recently developed a local description of the energy, momentum and angular momentum carried by the linearized gravitational field, wherein the gravitational energy-momentum tensor displays positive energy density and causal energy flux, and the gravitational spin tensor describes purely spatial spin [1,2]. We now investigate the role these tensors play in a broader theoretical context, demonstrating for the first time that (a) they do indeed constitute Noether currents associated with the symmetry of the linearized gravitational field under translation and rotation and (b) they are themselves a source of gravity, analogous to the energy momentum and spin of matter. To prove (a) we construct a Lagrangian for linearized gravity (a covariantized Fierz-Pauli Lagrangian for a massless spin-2 field) and show that our tensors can be obtained from this Lagrangian using a standard variational technique for calculating Noether currents. This approach generates formulae that uniquely generalize our gravitational energy-momentum tensor and spin tensor beyond harmonic gauge: we show that no other generalization can be obtained from a covariantized Fierz-Pauli Lagrangian without introducing second derivatives in the energy-momentum tensor. We then construct the Belinfante energy-momentum tensor associated with our framework (combining spin and energy momentum into a single object) and as our first demonstration of (b) we establish that this Belinfante tensor appears as the second-order contribution to a perturbative expansion of the Einstein field equations, generating the gravitational field in a manner equivalent to the (Belinfante) energy-momentum tensor of matter. By considering a perturbative expansion of the Einstein-Cartan field equations, we then demonstrate that (b) can be realized without forming the Belinfante tensor: our energy-momentum tensor and spin tensor appear as the quadratic terms in separate field equations, generating gravity as distinct entities. Finally, we examine the role of field redefinitions within these perturbative expansions; in contrast to our tensors, the Landau-Lifshitz tensor is found to require a nonlocal field redefinition in order to be cast as a source of the gravitational field. In an appendix, we also give a brief treatment of the global quantities that our framework defines and verify their equivalence (within the quadratic approximation) to the energy momentum and angular momentum of Arnowitt, Deser, and Misner.