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Very quick q, recover weak field lim deriving schwar metric

  1. Apr 14, 2015 #1
    I'm looking at Carroll's lecture notes 1997, intro to GR.

    Equation 7.27 which is that hes argued the S metric up to the form ##ds^{2}=-(1+\frac{\mu}{r})dt^{2}+(1+\frac{\mu}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##

    And argues that we expect to recover the weak limit as ##r \to \infty##.
    So he then has ##g_{00}(r\to\infty)=-(1+\frac{\mu}{r}) ## [1]
    where ##g_{00}=-(1+2\phi)## and equates these.

    The reasoning is fine to me, but I dont understand the limit given by [1], surely as ##r\to\infty## ##g_{00} \to -1##

    Thanks in advance.
     
  2. jcsd
  3. Apr 14, 2015 #2

    MarcusAgrippa

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    Gold Member

    You are looking for a power series in 1/r. The coefficient of [itex] g_{00} [/itex] needs no expansion. It is already a power series in 1/r. So correct to first order, this factor does not change. However, [itex] g_{11} [/itex] can be expanded in terms of 1/r by the binomial expansion. He then truncates this so that it is accurate to first order.

    If you neglect 1/r in [itex] g_{00} [/itex], you have a zeroth order approximation, which is too severe.
     
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