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I Schwarzschild metric asymptotically flat

  1. May 11, 2017 #1
    This is probably a stupid question but so as ##r \to \infty ## it is clear that
    ##-(1-GM/r)dt^2+(1-GM/r)^{-1}dr^2 \to -dt^2 +dr^2 ##

    However how do you consider ## \lim r \to \infty (r^2d\Omega^2 )##..?

    Schwarschild metric: ##-(1-GM/r)dt^2+(1-GM/r)^{-1}dr^2+r^2 d\Omega^2##
    flat metric : ##-dt^2+dr^2+r^2 d\Omega^2##

    i.e without doing this limit the result is clear, but what happens to this limit?
     
  2. jcsd
  3. May 11, 2017 #2

    stevendaryl

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    What's important is not that [itex]r \rightarrow \infty[/itex], but that [itex]\frac{r}{GM} \rightarrow \infty[/itex]. That is, we're assuming that [itex]r \gg GM[/itex], while still being finite.
     
  4. May 12, 2017 #3

    martinbn

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    The difference of the two metrics approaches zero as ##r## goes to infinity.
     
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