= Averaged stress-energy of transverse gravitational waves
{title2=$t^{\rm GW}_{ab}=\frac1{32\pi G}\langle\partial_aH_{ij}\partial_bH_{ij}\rangle$}
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 $(16\pi G)^{-1}\langle\dot H_+^2+\dot H_\times^2\rangle$. This effective averaged stress is not an arbitrary pointwise gravitational energy <tensor>.
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