Say that you place an asymptotically Euclidean metric on $\mathbb{R}^3,$ e.g. $\mathbb{R}^3$ is endowed Riemannian metric $g$ such that $\text{supp}(g^{ij}-\delta^{ij})\subseteq\{|x|\leq R\}$ for some large $R>0.$ Does it follow that $(\mathbb{R}^3,g)$ is geodesically complete? This seems to be applied in PDE literature quite often when working with the flow generated by the principle symbol of the Laplace-Beltrami operator, but I am unaware of a reference. It seems intuitively true: $\mathbb{R}^3=\{|x|\leq R\}\cup \{|x|>R\}.$ The first set in the union is compact, and the other is where our metric is just the standard metric.

If this is a simple question, please let me know in the comments and I will delete.