The Conserved Effective Stress Tensor of Gravitational Wave

We present a detailed study of the effective stress tensor of gravitational wave (GW) as the source for the background Einstein equation and examine three candidates in literature. The second order perturbed Einstein tensor $G^{(2)}_{\mu\nu}$, up to a coefficient, proposed by Brill, Hartle, and Isaacson, has long been known to be covariantly nonconserved with respect to the background spacetime. We observe that $G^{(2)}_{\mu\nu}$ is not a true tensor on the background spacetime. More importantly, we find that, by expressing $G^{(2)}_{\mu\nu}$ in terms of the perturbed Hilbert-Einstein actions, the nonconserved part of $G^{(2)}_{\mu\nu}$ is actually canceled out by the perturbed fluid stress tensors in the back-reaction equation, or is vanishing in absence of fluid. The remaining part of $G^{(2)}_{\mu\nu}$ is just the conserved effective stress tensor $\tau_{\mu\nu}$ proposed by Ford and Parker. As the main result, we derive $\tau_{\mu\nu}$ for a general curved spacetime by varying the GW action and show its conservation using the equation of GW. The stress tensor $T_{\text{MT}}^{\mu\nu}$ proposed by MacCallum and Taub was based on an action $J_2$. We derive $T_{\text{MT}}^{\mu\nu}$ and find that it is nonconserved, and that $J_2$ does not give the correct GW equation in presence of matter. The difficulty with $J_2$ is due to a background Ricci tensor term, which should be also canceled out by the fluid term or vanishing in absence of fluid. We also demonstrate these three candidates in a flat Robertson-Walker spacetime. The conserved $\tau_{\mu\nu}$ has a positive energy density spectrum, and is adequate for the back-reaction in a perturbation scheme, while the two nonconserved stress tensors have a negative spectrum at long wavelengths and are unphysical.