Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)? If not, what assumptions do we need on $\Omega$ for the metric to be asymptotically flat?

Any references are really appreciated.