Coordinate singularity in Schwarzschild solution

[Confused Physicist]

Hi! I have the following problem I don't really know how to approach. Could someone give me a hand?

The line element of a black hole is given by: $$ds^2=\Bigg(1-\frac{2m}{r}\Bigg)d\tau ^2+\Bigg(1-\frac{2m}{r}\Bigg)^{-1} dr^2+r^2\Big(d\theta ^2+\sin^2(\theta)d\phi ^2\Big)$$

It has an apparent singularity at $r=0$. By making $\tau$ an angular coordinate, show that this singularity is a coordinate singularity (not physical) and find the period of $\tau$ that makes it possible. (consider expanding the metric functions about $r=2m$).

Thanks for the help!

 

[PeterDonis]

The line element of a black hole

Note that this is not the usual Schwarzschild metric; there is no minus sign in front of the first term on the RHS. (If there were, the question would not make sense because $\tau$ would not be able to be treatd as an angular coordinate.)

 

[PeterDonis]

@Confused Physicist I have moved this thread to the homework forum. You will need to at least show some attempt at a solution. You could start by taking the hint in the parenthetical in the OP.

 

[PeterDonis]

It has an apparent singularity at $r=0$. By making $\tau$ an angular coordinate, show that this singularity is a coordinate singularity

Are you sure the problem statement says $r = 0$ and not $r = 2m$?

 

[Confused Physicist]

Are you sure the problem statement says $r = 0$ and not $r = 2m$?

Thanks PeterDonis, I will post my future questions in the homework forum. I've been trying to squeeze my head around it, but I haven't posted my attempt because I literally don't have a decent one.

Yes, the problem says $r=0$, but you're right. I believe it's a mistake and it should say $r=2m$.

 

[Confused Physicist]

Note that this is not the usual Schwarzschild metric; there is no minus sign in front of the first term on the RHS. (If there were, the question would not make sense because $\tau$ would not be able to be treatd as an angular coordinate.)

What does it mean to treat $\tau$ as an angular coordinate? Is it a specific change of variable?

 

[PeterDonis]

What does it mean to treat $\tau$ as an angular coordinate?

It means it only covers the range $0$ to $2 \pi$ instead of $- \infty$ to $+ \infty$.