In my lecture notes, they claim that the family of Schwarzschild metrics form a smooth one-parameter family of smooth metrics, but the proof of this is left to the reader as an exercise and I am struggling with convincing myself of it.
Let $(t,x_1,x_2,x_3)$ denote the standard coordinatees on $\mathbb{R}^{3+1}$ and $r= \sqrt{x_1^2+x_2^2+x_3^2}$. Then consider for fix $R>0$ the manifold $\mathcal{M}_R=\{r>R\}\subset \mathbb{R}^{3+1}$. Now we can consider on $\mathcal{M}_R$ the family of metrics with parameter $M\in (-\infty,R/2]$ given by $$ds^2= -(1-2M/r)dt^2+(1-2M/r)^{-1}dr^2 + r^2(d\theta^2 + sin^2\theta d\phi^2).$$
I would like to show that this forms a smooth one-parameter family of smooth metrics (and are Ricci flat).
Any help would be greatly appreciated!