I'm a bit puzzled about the concepty of observer in 3+1 or ADM decomposition in GR.
The decomposition is typically described as starting with a scalar field $t$, whose spacelike level surface $\Sigma_t$ form a foliation. Then take a vector field $\vec{t}$, with $\vec{t}(t)=1$ or $dt(\vec{t})=1$. Denote the unit future timelike normal of $\Sigma_t$ by $\vec{n}$. There is then decomposition $$\vec{t}=N\vec{n}+\vec{\beta}$$ where $N$ and $\vec{\beta}$ are lapse and shift vector, respectively.
I see people take $\vec{n}$ as the four-velocity of "Eulerian" observer. But since spatial coordinates $(x_1,x_2,_3)$ is constant along integral curve of $\vec{t}$, I would think $\frac{\vec{t}}{\sqrt{-|\vec{t}|^2}}$ as the four velocity of "the" observer in this setting. But then it comes with an issue that $\vec{t}$ is non-timelike when $N^2\leq |\vec{\beta}|^2$.
Any clarification or suggestion in thinking about observors here? Thanks in advance!
