Deriving Schwarz Metric Weak Limit: Carroll's Lecture Notes 1997

[binbagsss]

I'm looking at Carroll's lecture notes 1997, intro to GR.

Equation 7.27 which is that he's 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 don't understand the limit given by [1], surely as $r\to\infty$ $g_{00} \to -1$

Thanks in advance.

 

[MarcusAgrippa]

You are looking for a power series in 1/r. The coefficient of $g_{00}$ needs no expansion. It is already a power series in 1/r. So correct to first order, this factor does not change. However, $g_{11}$ 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 $g_{00}$, you have a zeroth order approximation, which is too severe.