Slowly rotating weak-field metric

Far from an isolated body of mass $M$ and angular momentum $\mathbf{J}$, a weak-field, leading-spin metric can be written in approximately Cartesian spatial coordinates asHere $r=|\mathbf{x}|$, the metric signature is $(-,+,+,+)$, and the retained terms are linear in $GM/(c^{2}r)$ and $GJ/(c^3{r}^2)$. Higher multipole moments and higher-order products are omitted.

For $\mathbf{J}=J\hat{\mathbf{z}}$, the mixed term is $4GJ(ydx-xdy)dt/(c^2{r}^3)$. Equivalently, it is $-4GJ{\sin }^2\theta dtd\phi /(c^{2}r)$. This agrees with the large-radius, leading-spin term of Kerr spacetime. The scalar part also depends on the chosen radial coordinate: the Cartesian form above uses the weak-field isotropic convention.^[ref]

The time-space metric components encode frame dragging, including Lense-Thirring precession. They represent an effect of moving matter that is absent from a description using only a scalar Newtonian gravitational potential.