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If I am given a metric how do I decide which coordinate is periodic? Eg. can I look at metric in plane polar coordinates and tell that θ direction is periodic.

Also How do I calculate the period of the periodic coordinate?

what I wanted to ask is related to following question. Given the schwarzschild metric we can transform the coordinates as $ \tau $ = i*t and $ \rho^2 $ = $ r - r_h $ (where $ r_h $ = horizon radius of black hole ) and show that $\tau $ is periodic if there is no conical singularity at $ \rho = 0 $, and then find the period of $\tau$. (This period is related to temperature of black hole).

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  • $\begingroup$ Given a metric and nothing else, you don't, as Slereah's answer states. Concerning your edit, the absence of the conical singularity is an additional constraint, so this is what forces $\tau$ to be periodic in some specific period. $\endgroup$ Apr 29, 2016 at 13:32
  • $\begingroup$ In that case how do I find the period of $\tau $ ? $\endgroup$
    – AMIT SINGH
    Apr 29, 2016 at 17:13
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    $\begingroup$ I spelled it out here. $\endgroup$ May 2, 2016 at 9:56

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You don't. Two given spacetimes can have their metrics written in the same way but may have different coordinate ranges. A simple example is just a spacetime with spatial coordinate identified , such as the cylinder spacetime :

$$ds^2 = -dt^2 + d\theta$$

Identical to Minkowski space, which is its universal cover.

Of course, two things to watch out for :

  • Spacetimes that vary in coordinate ranges may not all be geodesically complete. If they possess a maximal extension that is geodesically complete, probably go for that one.
  • For coordinates to be periodic, the metric has to be able to join up with itself at the point of identification. Which means that at the very least, the value of the metric and its first derivatives should be the same.
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