1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.