Very quick q, recover weak field lim deriving schwar metric

1. Apr 14, 2015

binbagsss

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 $r\to\infty$ $g_{00} \to -1$

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.