# I Schwarzschild metric asymptotically flat

1. May 11, 2017

### binbagsss

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. May 11, 2017

### stevendaryl

Staff Emeritus
What's important is not that $r \rightarrow \infty$, but that $\frac{r}{GM} \rightarrow \infty$. That is, we're assuming that $r \gg GM$, while still being finite.

3. May 12, 2017

### martinbn

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