In A Relativist's toolkit by Poisson the defining of the ADM mass starts by introducing an asymptotic Lorenztian frame and in the asymptotic portion of the hypersurface the flow vector becomes:
$t^{\alpha}\rightarrow N\Big(\frac{\partial{x^{\alpha}}}{\partial{\bar{t}}}\Big)_{y^{a}}+N^a\Big(\frac{\partial{x^{\alpha}}}{\partial{y^{a}}}\Big)_{\bar{t}}$
where $\bar{t}$ is the proper time in Lorentzian frame.
In defining ADM mass we make the choice of lapse funcion $N=1$ and shift vector $N^a=0$ (it is a free choice because they are non-dynamical variables), so that $t^{\alpha}\rightarrow\frac{\partial{x^{\alpha}}}{\partial{\bar{t}}}$, so the flow generates an asymptotic time translation. I guess that this means that we are chosing a static observer: is it right?
Now, my doubt follows from a statement of my teacher: "the $N=1$ and $N^a=0$ choice imply that the $(t,y^a)$ coordinates of spacetime become the $(\bar{t},x^i)$ coordinates of asymptotic Minkowski spacetime". Since he also says that "asymptotic frame has infinite possible choices of coordinates", it is not clear to me what does it means that "the $(t,y^a)$ coordinates of spacetime become the $(\bar{t},x^i)$ coordinates of asymptotic Minkowski spacetime".
By the previous relation it is clear the corrispondence between the choice of the flow vector and the choice of $N$ and $N^a$. Then, Poisson states that we can "select a foliation of spacetime by specifying the lapse $N$ and the shift $N^a$ as function of $x^{\alpha}=(t,y^a)$ and the choice of foliation is completely arbitrary": I guess that by this statement we can see the corrispondence between the choice of $N$, $N^a$ and the coordinates, but is not clear to me what does it means that foliation is determined by this choice.
In fact, what I understood was that the foliation is an embedding of hypersurfaces that is independent by the congruence of curves we define on it, and that those curves are geodesics if the tangent vector is parallel transported along them, but we always consider them geodesics? Are they determined by the extrinsic curvature of the hypersurfaces (or viceversa they determine the embedding, so "select a foliation")? Or, as I understood, given a generic spacetime we can freely choose a congruence of curves, and this determines a particular coordinate frame? So what does the choice $N=1$, $N^a=0$ mean in a curved spacetime, since we are saying that the lapse function is constant along the hypersurfaces?
