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A Boundary conditions on the Euclidean Schwarzschild black hole

  1. Jun 15, 2017 #1
    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

    $$\phi \sim \phi + 8\pi M?$$
     
    Last edited: Jun 15, 2017
  2. jcsd
  3. Jun 15, 2017 #2

    PeterDonis

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    Staff: Mentor

    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##.
     
  4. Jun 16, 2017 #3
    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?
     
  5. Jun 16, 2017 #4

    PeterDonis

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    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.
     
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