= Solution
Use $G=c=1$, select the retarded source field with no added homogeneous radiation, and work to leading order in the <weak-field approximation> and <long-wavelength source approximation>. The <retarded fundamental solution> of the <Linearized Einstein equations> in <Lorenz gauge in linearized gravity> is
$$
\bar h_{ij}(t,\mathbf x)=4\int\frac{T_{ij}(t-|\mathbf x-\mathbf x'|,\mathbf x')}{|\mathbf x-\mathbf x'|}\,d^3x'.
$$
For $r\gg d$, the denominator is $r$ to leading order. The delay is $t-r+\mathbf n\cdot\mathbf x'+O(d^2/r)$, where $\mathbf n=\mathbf x/r$. In the usual slowly evolving source regime, its characteristic time $\tau$ obeys $d/\tau\ll1$, so the source-size part of the delay is negligible:
$$
\bar h_{ij}(t,\mathbf x)\simeq\frac4r\int T_{ij}(t-r,\mathbf x')\,d^3x'.
$$
The <stress-energy conservation> equation $\partial_\mu T^{\mu\nu}=0$ is required at the approximation order used; it follows from the divergence of the gauge-fixed field equation. <Compact support> removes the <integration by parts> surface terms. Since $T^{00}=T_{00}$ and $T^{ij}=T_{ij}$ in this signature,
$$
\dot I_{ij}=\int(T^{0i}x^j+T^{0j}x^i)\,d^3x,
\qquad
\ddot I_{ij}=\int(T^{ji}+T^{ij})\,d^3x=2\int T_{ij}\,d^3x.
$$
Combining these identities gives the <retarded quadrupole field>
$$
\boxed{\bar h_{ij}(t,\mathbf x)\simeq\frac2r\ddot I_{ij}(t-r).}
$$
For an ordinary nonrelativistic bound source, $\tau\sim d/v$ makes $d/\tau\sim v\ll1$. More generally the size relative to the variation timescale must also be small; speed alone does not exclude a rapidly varying small-amplitude motion.
The <radiation boundary condition for linearized gravity> is essential for a statement about the full field. Without it, the displayed implication is false: take $T_{\mu\nu}=0$ and add $\bar h_{11}=A\cos(k(t-z))$, $\bar h_{22}=-\bar h_{11}$, all other components zero. This weak plane <gravitational wave> satisfies both the homogeneous wave equation and <Lorenz gauge in linearized gravity>, while every $I_{ij}$ is zero. The proved formula is for the retarded source contribution, with the standard slow-source qualification. Its transverse–traceless projection gives the physical radiative strain.
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