Retarded quadrupole field (${\bar{h}}_{ij}(t,\mathbf{x})\simeq 2{\ddot{I}}_{ij}(t-r)/r$)

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.

Dropping the position-dependent source delay in the leading far-zone retarded field requires source size $d$ to be small compared with its variation time $\tau$, in units $c=1$. The retarded quadrupole field also requires $d/r\ll 1$. For a typical nonrelativistic bound source, $\tau \sim d/v$ implies this long-wavelength condition; an arbitrarily rapidly oscillating small displacement is not controlled merely by its instantaneous speed.