Averaged stress-energy of transverse gravitational waves ($t_{ab}^{\mathrm{GW}}=\frac{1}{32\pi G}⟨{\partial }_a{H}_{ij}{\partial }_b{H}_{ij}⟩$)

In the weak-field, short-wavelength averaging regime, a transverse-traceless tensor strain $H_{ij}$ carries the displayed effective stress-energy tensor in units $c=1$. The average spans several wave periods while remaining small compared with the background curvature scale. The stress is second order in amplitude, so it is absent from a first-order vacuum Linearized Einstein equations calculation. For a plane wave, summing the two equal transverse diagonal contributions gives flux $(16\pi G{)}^{-1}⟨{\dot{H}}_{+}^2+{\dot{H}}_{\times }^2⟩$. This effective averaged stress is not an arbitrary pointwise gravitational energy tensor.