Solution
= Solution
Along the radial curve at fixed $t,\theta,\varphi$, the spatial line element is $dS=\sqrt{1+2M/r}\,dr$. To first order,
$$
S=\int_{r_1}^{r_2}\left(1+\frac Mr\right)dr,
$$
so
$$
\boxed{S=(r_2-r_1)+M\log\frac{r_2}{r_1}}.
$$
= Solution
Along the radial curve at fixed $t,\theta,\varphi$, the spatial line element is $dS=\sqrt{1+2M/r}\,dr$. To first order,
$$
S=\int_{r_1}^{r_2}\left(1+\frac Mr\right)dr,
$$
so
$$
\boxed{S=(r_2-r_1)+M\log\frac{r_2}{r_1}}.
$$