= Solution
Use geometrized units $G=c=1$ and signature $(-,+,+,+)$. The original PDF starts with the <d'Alembert operator> $\Box\bar h_{\mu\nu}=-16\pi T_{\mu\nu}$; the local TeX incorrectly transcribes its derivative indices. The harmonic condition is the <Lorenz gauge in linearized gravity>.
Choose the retarded solution, excluding an incoming homogeneous wave. For a spatially localized source, the <Linearized Einstein equations> give
$$
\bar h_{\mu\nu}(t,\mathbf x)
=4\int\frac{T_{\mu\nu}(t-|\mathbf x-\mathbf y|,\mathbf y)}{|\mathbf x-\mathbf y|}\,d^3y.
$$
Let the source size be $D$ and the characteristic angular frequency be $\omega$. Assume the <weak-field approximation>, nonrelativistic source velocities, and $D\ll\omega^{-1}\ll r$ for a radiative far-zone measurement. Then $|\mathbf x-\mathbf y|=r-\mathbf n\cdot\mathbf y+O(D^2/r)$. At leading order in $D/r$ and $\omega D$, one may replace the denominator by $r$ and the retarded time throughout the source by $t-r$, obtaining
$$
\bar h_{ij}(t,\mathbf x)\simeq\frac4r\int T_{ij}(t-r,\mathbf y)\,d^3y.
$$
To identify this integral, use <stress-energy conservation> $\partial_\mu T^{\mu\nu}=0$ for a symmetric source tensor. Compact support or sufficient decay permits <integration by parts> with no boundary flux. Because $T^{00}=T_{00}$ in this signature, the <second mass moment tensor> satisfies
$$
\begin{aligned}
\dot I_{ij}
&=-\int\partial_kT^{k0}\,y_i y_j\,d^3y
=\int(T^{i0}y_j+T^{j0}y_i)\,d^3y,\\
\ddot I_{ij}
&=-\int\bigl(\partial_kT^{ki}\,y_j+\partial_kT^{kj}\,y_i\bigr)\,d^3y
=2\int T^{ij}\,d^3y.
\end{aligned}
$$
Spatial indices are raised with $\delta_{ij}$, so $T^{ij}=T_{ij}$. Substitution gives the <retarded quadrupole field>:
$$
\boxed{\bar h_{ij}(t,\mathbf x)\simeq\frac2r\ddot I_{ij}(t-r).}
$$
The assumptions are an isolated conserved source, retarded boundary conditions, weak gravity, a source small compared with the wavelength, and observation far from it. The physical radiative field follows by the <transverse-traceless projector> applied to this <trace-reversed metric perturbation>.
Back to article page
