Let $v=\sqrt{2m/r}$. The proposed coframe gives
$$
-(e^0)^2+(e^1)^2+(e^2)^2+(e^3)^2
=-dt^2+(dr+vdt)^2+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2,
$$
which expands to the stated metric. Hence it is the <orthonormal coframe in Painlevé–Gullstrand coordinates> with signature $(-,+,+,+)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 4 / a / Solution

An orthonormal coframe is
$$
e^0=dt,
\qquad e^1=dr+\sqrt{\frac{2M}{r}}\,dt,
\qquad e^2=r\,d\theta,
\qquad e^3=r\sin\theta\,d\phi.
$$


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