A couple of questions on the ADM formalism in general relativity I've been reading up on the ADM formalism in general relativity and have been stuck on a couple of concepts. 


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*The first is to do with the foliation of spacetime into space-like hypersurfaces.  I understand that the hypersurfaces must be space-like as they are surfaces of constant time and hence any two points on a given surface must necessarily be space-like separated, but I'm slightly unsure as to why the normal to such a surface is necessarily time-like?! Does this follow because the normal is,  by definition, orthogonal to the surface, thus contains no space-like components, and so necessarily must be time-like? Can this be proven mathematically, or is it just a requirement (I understand it intuitively, as each hypersurface describes the 3-dimensional space at a given instant in time and thus consecutive hypersurfaces must be causally related by a time-like path). 

*The second question I have is what is the so-called lapse function, $N(t) $,  as in what is it physically describing? Is it just quantifying the change in description of time as one moves between two given hypersurfaces, or is there more to it than that?   
 A: *

*The spacelike hypersurface has three spacelike directions tangent to it.  Any vector that is normal to all three spacelike directions in the eneveloping space is necessarily timelike.  Equivalently, the spacelike surfaces can be thought to be labeled by a function $\tau$ which gives the "time coordinate"'s value on those surfaces.  the normal to the surface is therefore $\nabla \tau$ which must be a one-form pointing in a timelike dimension, since it is the gradient of a time coordinate.


*The shift caputres the fact that, as coordinate time evolves, it's not necessarily the case that "constant time" observers will stay stationary on the three-space.  There might be some drift involved.  This is not a physical effect, and is an effect of the coordinates, but one natural case where the physics of the situation and a real effect coincide is the case of frame-dragging near a spinning black hole.  More lazily, one can think of the shift vector as just being the $g^{ti}$ components of the metric tensor.


*The time lapse function captures the fact that coordinate time can pass more quickly in certain parts of the spacetime than in others.  For instance, in standard Schwarzschild spacetime, using the standard $t$ coordinate as the generator of the hypersurfaces, the lapse function becomes $\sqrt{1-\frac{2M}{r}}$, and captures the fact that proper time evolves more slowly near the black hole horizon.
A: I have tried to understand it too, for me.. I will describe it along with the normal and the shift vector.
For free particles free-falling in the spacetime (following the geodesic) their world-lines define the time flow. We call vector tangents to this flow at each point a $\textit{time-flow vector}\, t^\mu $. In their free-falling  frames at a particular point in the space-time 
the metric field to describe the particle motion is the flat Minkowksian metric adapted from metric for curved metric at that point. The observer in such frame may expect that a particle will follow the geodesic defined by the flat Minkowksian metric since she don't feel gravitation, but, indeed the shift in position occurs since a particle really follows the geodesic defined by general space-time metric. $\textit{Such shifts occur when the gravity is not uniform distribution}$, equivalently we can say that such free falling frame $\textit{ has inertia }$. Imagine that we are live inside the falling elevator in an uniform gravitational field and place the ball at some height above the floor of this elevator the deviation of path of such a ball will be not detected, but if so, the non-uniformity of gravitational field was detected because the shifts in position of this ball occurred. Such shifts are described by the shift vector $N^\mu (x)$.  At each infinitesimal region the shift vector is a projection of the time vector as shown in  figure.

Therefore we can associate the  vector perpendicular to the hypersurface  with the time-flow and the shift vectors 
\begin{equation}
 Nn^\mu := t^\mu - N^\mu \perp \Sigma_t , 
\end{equation}
where $n^\mu$ is a unit vector orthogonal to the hypersurface($\Sigma_t$)
\begin{eqnarray}
 & g_{\mu\nu} n^\mu n^\nu =-1 , & \\
 & g_{\mu\nu} n^\mu N^\nu =0 , &
\end{eqnarray}
and we call $N(x)$ the lapse function. It is dynamical and capture the deviation of speed of the clock from the local Minkowksian sense caused by non-uniformity of the gravitational field. 
The vector $n^\mu$ can be interpreted as a gradient of some scalar function that is constant at each hypersurface. Such time function, $T(x)%^\mu)
$, is usually defined via
\begin{equation}
 n_\alpha = -N\nabla_\alpha T. 
\end{equation}
A: I think the normal is always time-like because when you slice your space-time you do it in such a way such that the normal vector to this hyper-surface is time-like. Thus, time components of the original metric are absent in the induced metric. Which reference are you reading from?
