Euclidean derivation of the black hole temperature; conical singularities
I am studying the derivation of the black hole temperature by means of the Euclidean approach, i.e. by <PERSON> rotating, compactifying the Euclidean time and identifying the period with the inverse temperature.
Consider the Schwarzschild case as an example. The Euclidean Schwarzschild metric is of course
$ds^2=\left(1-\frac{2M}{r}\right)d\tau^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2$
where $\tau=it$ the Euclidean time. Here $\tau\in[0,\beta]$ with $\beta=T^{-1}$ the inverse temperature, where the points $\tau=0$ and $\tau=\beta$ are equivalent. 394
(I ignore the 2-sphere part of the metric.)
Outside but close to the event horizon $r=2M$, we can (after simple some steps) write this as $ds^2=\frac{\sigma^2}{16M^2}d\tau^2+d\sigma^2$.
Here $\sigma^2 \equiv 8M(r-2M) $, though this is not really relevant for my question.
Now the next step in all the literature is to require that $\tau/4M$ is periodic with period $2\pi$ to prevent conical singularities. I have trouble understanding this.
* A conical singularity basically means that the point at $\sigma=0$ looks like the tip of a cone, right? So we have a singularity at $\sigma=0$. But don't we still have a singularity there for polar coordinates $(r,\theta)$, since the coordinate chart $\theta$ is not continuous there?
* If so, removing the point $\sigma=0$ would erase the conical singularity, right? Then why do we want to get rid of the conical singularity, why is it any worse than the polar coordinate singularity? Of course polar coordinates just describe flat space without the origin, is this not the case for the conical coordinates?
Clearly I have not understood the concept of a conical singularity...
Last question: suppose I did understand it, and continued the derivation to get the temperature $T=1/8\pi M$. Apparently this is the temperature as measured by an observer at infinity, how can I see this? I know the temperature gets redshifted, like frequency, but I don't see where the derivation identifies $T$ with the one measured at infinity. 272