Set $A=1-2M/R=(R-2M)/R$ and
$$
F(R)=\frac{RM}{(R-M)(R-2M)},
\qquad dT=dt-F\,dR.
$$
Substitution into the <Schwarzschild metric> gives
$$
ds^2=-A\,dt^2+2AF\,dt\,dR
+\left(A^{-1}-AF^2\right)dR^2+R^2d\Omega^2.
$$
The cross coefficient and radial coefficient simplify to
$$
AF=\frac{M}{R-M},
\qquad
A^{-1}-AF^2=\frac{R^2}{(R-M)^2}.
$$
Therefore
$$
\boxed{
ds^2=-\left(1-\frac{2M}{R}\right)dt^2
+\frac{2M}{R-M}\,dt\,dR
+\frac{R^2}{(R-M)^2}\,dR^2+R^2d\Omega^2.
}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-357/2/a/solution