Far-zone quadrupole strain ($H_{ij}^{\mathrm{TT}}=\frac{2G}{r}{\Lambda }_{ij,kl}(\mathbf{n}){\ddot{Q}}^{kl}(t-r)$)

In units $c=1$, a slowly moving isolated source has trace-free mass quadrupole moment $Q^{ij}=\int T^{00}(x^i{x}^j-{\delta }^{ij}|\mathbf{x}{|}^2/3)d^{3}x$. 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.

For a weakly self-gravitating bound source of size $R$, mass $M$ and nonspherical dynamical mass quadrupole moment $Q=O(M{R}^2)$, the virial theorem gives $v^2=O(GM/R)=O(ϵ)$. Its time scale is $R/v$, so $\ddot{Q}=O(M{v}^2)$. The far-zone quadrupole strain then has a spatial half-strain coefficient $E=O(GM{v}^2/r)=O({ϵ}^{2}R/r)$ in units $c=1$. The physical transverse-traceless strain is $2E^{\mathrm{TT}}$. Spherical or stationary sources can have zero radiation, so this is a characteristic scale for a changing nonspherical quadrupole, not a positive lower bound. The radiation zone also requires $r\gg R/v$.

Two equal masses $m$ orbiting their centre of mass at radius $R$ have separation $2R$. Newton's law of universal gravitation gives the displayed orbital frequency. Their mass quadrupole moment varies at twice that frequency, producing gravitational radiation with two gravitational wave polarizations. For inclination $\iota$, its leading plus and cross amplitudes are proportional to $1+{\cos }^2\iota$ and $2\cos \iota$, respectively. Face-on radiation is circularly polarized; edge-on radiation is linearly polarized.