Boundary conditions on the Euclidean Schwarzschild black hole

[Afonso Campos]

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf).

The Euclidean Schwarzschild black hole

$$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} + r^{2}d\Omega_{2}^{2}$$

is obtained from the Lorentzian Schwarzschild black hole via Wick rotation $t \to -i\tau$.

Why does the fact that the coordinates must be regular at the origin imply that the angular coordinate must be identified as

$$\phi \sim \phi + 8\pi M?$$

 

[PeterDonis]

Why does the fact that the coordinates must be regular at the origin imply that the angular coordinate must be identified

It isn't the angular coordinate $\phi$ that's identified, it's the "angular" coordinate $\tau$--the one that is derived via $t \rightarrow - i \tau$. The "origin" is $r = 2M$, and the solution has no interior, so the only range covered is $r \ge 2M$.

 

[Afonso Campos]

It isn't the angular coordinate $\phi$ that's identified, it's the "angular" coordinate $\tau$--the one that is derived via $t \rightarrow - i \tau$. The "origin" is $r = 2M$, and the solution has no interior, so the only range covered is $r \ge 2M$.

How can we see explicitly that the range $r < 2M$ in the Lorentzian Schwarzschild black hole is not covered by the $\tau$ coordinate in the Euclidean Schwarzschild black hole?

 

[PeterDonis]

How can we see explicitly that the range $r < 2M$ in the Lorentzian Schwarzschild black hole is not covered by the ττ\tau coordinate in the Euclidean Schwarzschild black hole?

The question doesn't make sense. To see whether the range $r < 2M$ is covered or not, you look at the behavior of the $r$ coordinate, not any other coordinate.