= Far-zone quadrupole strain
{title2=$H^{\mathrm{TT}}_{ij}=\frac{2G}{r}\Lambda_{ij,kl}(\mathbf n)\ddot Q^{kl}(t-r)$}
In units $c=1$, a slowly moving isolated source has trace-free <mass quadrupole moment> $Q^{ij}=\int T^{00}(x^ix^j-\delta^{ij}|\mathbf x|^2/3)\,d^3x$. The retarded <linearized gravity> solution and the <stress-energy quadrupole identity> give the displayed physical spatial strain, with $\mathbf n=\mathbf x/r$ and the <transverse-traceless projector> $\Lambda$. If the spatial metric is written $g_{ij}=-\delta_{ij}-2E_{ij}$ for <metric signature> $(+---)$, its trace-free Lorenz-gauge coefficient before the transverse projection is $E_{ij}=G\ddot Q_{ij}(t-r)/r$. Thus $H^{\mathrm{TT}}=2E^{\mathrm{TT}}$. Far distance alone does not permit dropping source-dependent retardation: the quadrupole approximation also requires the source size to be small compared with the radiation wavelength.
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