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.