= 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 as
$$
ds^2=-\left(1-\frac{2GM}{c^2r}\right)c^2dt^2+\left(1+\frac{2GM}{c^2r}\right)d\mathbf x^2
-\frac{4G}{c^2r^3}(\mathbf J\times\mathbf x)\cdot d\mathbf x\,dt.
$$
Here $r=|\mathbf x|$, the <metric signature> is $(-,+,+,+)$, and the retained terms are linear in $GM/(c^2r)$ and $GJ/(c^3r^2)$. Higher <multipole moments> and higher-order products are omitted.
For $\mathbf J=J\widehat{\mathbf z}$, the mixed term is $4GJ(y\,dx-x\,dy)dt/(c^2r^3)$. Equivalently, it is $-4GJ\sin^2\theta\,dt\,d\varphi/(c^2r)$. 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.<General Relativity by David Tong>[Section 6.3.1, equation 6.282: Kerr mixed term and its large-radius expansion]{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>.
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