Flat metric induced from Schwarzschild

Question

The task is to find a function $f(r)$ such that the induced metric from the Schwarzschild metric

$$ds^2 = -\left(1-\frac{2m}{r}\right) dt^2 + \frac{1}{1-\frac{2m}{r}} dr^2 + r^2 d\Omega^2 $$

on the level set $\{t=f(r)\}$ is flat. My first attempt was to guess

$$ dt=0 = f'(r)dr$$

but it led me nowhere. My other idea was to introduce advanced and retarded coordinates $v=t-r$ and $u=t+r$ but there also I'm stuck. Maybe someone could give a guidline, a hint or a direction.

Answer 1

You have $dt=f'dr$ so $ds^2=(\varphi^{-1}-f'^2\varphi)dr^2+r^2d\Omega^2$ with $\varphi:=1-\frac{2m}{r}$. We want the $dr^2$ coefficient to be $1$, so $f'^2=\frac{1-\varphi}{\varphi^2}$. You can take the rest from there.