We present a detailed study of the effective stress tensor of a gravitational wave (GW) as the source for the background Einstein equation and examine three candidates in literature. The second order perturbed Einstein tensor $G_{\mu \nu }^{(2)}$, 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 $G_{\mu \nu }^{(2)}$ is not a true tensor on the background spacetime. More importantly, we find that, by expressing $G_{\mu \nu }^{(2)}$ in terms of the perturbed Hilbert-Einstein actions, the nonconserved part of $G_{\mu \nu }^{(2)}$ is actually canceled out by the perturbed fluid stress tensors in the backreaction equation, or is vanishing in the absence of fluid. The remaining part of $G_{\mu \nu }^{(2)}$ 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 $T_{\mathrm{MT}}^{\mu \nu }$ proposed by MacCallum and Taub was based on an action $J_2$. We derive $T_{\mathrm{MT}}^{\mu \nu }$ and find that it is nonconserved, and that $J_2$ does not give the correct GW equation in the 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 the 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 backreaction in a perturbation scheme, while the two nonconserved stress tensors have a negative spectrum at long wavelengths and are unphysical.

Close

PrevNext

View figure in article

• General relativity
• Gravitational waves