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Use geometrized units $G=c=1$ and signature $(-,+,+,+)$. The original PDF starts with the d'Alembert operator $\square {\bar{h}}_{\mu \nu }=-16\pi T_{\mu \nu }$; 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 giveLet 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

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 satisfiesSpatial indices are raised with ${\delta }_{ij}$, so $T^{ij}=T_{ij}$. Substitution gives the retarded quadrupole field: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.

For a slowly moving particle in the weak-field approximation, $dt/d\tau =1+O(\Phi ,v^2)$ and terms containing two spatial velocities may be neglected. The spatial geodesic equation reduces toFor $g_{00}=-(1+2\Phi )$, the Levi-Civita connection isHence the Newtonian limit of general relativity is

Move the positive cosmological constant to the source side and defineTo first order about Minkowski spacetime, $T^{(\Lambda )}=-\Lambda /(2\pi )$. Its contribution to the trace-reversed $00$ source isRepeating the Newtonian limit of general relativity calculation givesFor a point mass, ${\nabla }^2{r}^2=6$, soThe radial acceleration isThus positive $\Lambda$ supplies an outward acceleration and balances the point-mass attraction at the static radiusThe weak-field approximation requires both $M/r\ll 1$ and $\Lambda r^2\ll 1$, giving parametrically

For a compact slowly moving source in the weak-field approximation, with no incoming homogeneous wave and $G=c=1$, the retarded Linearized Einstein equations give this leading far-zone trace-reversed metric perturbation. The key consequence of stress-energy conservation is ${\ddot{I}}_{ij}=2\int T_{ij}{d}^{3}x$. The physical gravitational wave is obtained by the transverse-traceless projector.