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Kruzkal coordinates for inside the horizon

  1. Jun 10, 2013 #1
    We start by defining two coordinates ##u=t+r^*## and ##v=t-r^*##. Then we define another two coordinates ##u'=e^{u/4GM}## and ##v'=-e^{-v/4GM}##. But from what I have understood this is true for ##r>2GM##. How do we define ##u'## and ##v'## for ##r<2GM##?

    I think it's ##u'=e^{u/4GM}## and ##v'=e^{-v/4GM}## for ##r<2GM## so that the line element is the same for ##r<2GM## and ##r>2GM##.
     
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  3. Jun 10, 2013 #2

    Bill_K

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    Why don't you just look up Kruskal coordinates on Wikipedia.
     
  4. Jun 11, 2013 #3
    Yes, I looked at it. But I asked something specific and the wiki article doesn't answer it.
    I'm reading Carroll's notes and there he defines u' and v' in the similar way, but only for r>2GM.
    Same in Wald's book.
     
    Last edited: Jun 11, 2013
  5. Jun 11, 2013 #4

    WannabeNewton

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    The extended Kruskal representation of the Schwarzschild metric ##ds^{2} = \frac{32M^{3}e^{\frac{-r}{2M}}}{r}(-dT^{2} + dX^{2}) + r^{2}d\Omega^{2}## is valid for all ##r > 0## which from the definition ##(\frac{r}{2M} - 1)e^{\frac{r}{2M}} = X^{2} - T^{2}## yields ##-1 < X^{2} - T^{2}##. This is made clear in both the Wiki article and in Wald (page 154). Compare this with the Rindler space-time.
     
  6. Jun 11, 2013 #5
    In Carroll's notes on page 186 he defines u' and v' only for r>2GM. We need also to make transformation for r<2GM and than write the line element for both regions.

    Please check the page 186. I found there another mistake.
     
  7. Jun 11, 2013 #6

    WannabeNewton

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    Ah ok. Yes Carroll isn't being complete. The way he wrote it down, the chart has to be defined piecewise for ##r > r_g## and ##r < r_g## (it will agree continuously at ##r = r_g## of course).
     
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