Slowly rotating weak-field metric

Far from a slowly rotating isolated body of mass $m$ and angular momentum $\mathbf{J}$, the scalar part of the metric is the linearized point-mass field while $g_{0i}$ is the dipole field of $\mathbf{J}$. For rotation about the z-axis, the cross term is $4J{r}^{-3}(ydx-xdy)dt$ in a common convention.