Definition of asymptotic flatness This question is about an idea in general relativity, but its underlying realm is basically geometric. Let me begin with a snapshot of Geometric Relativity by Dan A. Lee:

I'd like to ask two questions. The first one is, why does Lee talk about the Euclidean background metric? Are we endowing $M$ with another metric? I mean, in the beginning of the definition, we are given a manifold $M$ already equipped with a Riemannian metric $g$. Can we impose another metric, possibly different from $g$, on $M$? Do geometers  really equip one single manifold with, if necessary, a couple of (distinct) metrics?
Let me ask one more question, please. In the inequality circled in red, I have no idea about those absolute values. What exactly is meant by $|x|$? That $x$ seems to be an arbitrary point in $M$, but how can we take its absolute value? Are we working in some coordinate chart, or something?
Thank you for your patience.
 A: On each end $M_k$, one has a particular chart
$\Phi_k \colon M_k \to \Bbb R^n \setminus \bar B_1(0)$, which is by definition a diffeomorphism.
On the LHS, you have the Riemannian metric $g$, and on the RHS, you have the euclidean metric $g_{eucl}$. One can pullback the euclidean metric on the LHS thanks to the diffeomorphism and define
$b_k = (\Phi_k)^*g_{eucl}$.
This is another Riemannian metric on the LHS.
One can show that the definition of $M_k$ to be asymptotically flat is equivalent to
$$
|{b_k}-g|_{b_k} = \mathcal{O}(|x|_{b_k}^{-q}).
$$
In other words, when you compare $b_k$ and $g$, where the measure is taken with reference to $b_k$, the error $e_k=b_k-g$ is small. In that case, $|x|_{b_k}$ is a short notation for $d(\Phi_k(x),0)$ is $\Bbb R^n$. Note that this asymptotic behaviour says that the geometry at infinity of $M_k$ is really close to be that of the euclidean space.
The reason we are comparing things on $M_k$ with respect to $b_k$ and not with respect to $g$ is for convinience: since this metric is euclidean, gradients, divergence etc. are easy to compute, and they are useful tool to define asymptotic geometric invariants.
