# Intuition behind the "Lapse Function"

I came across the following definite of the Lapse Function:

$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$

where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this review paper of the 3+1 formalism http://arxiv.org/pdf/gr-qc/0703035v1.pdf , where the lapse function is defined on page 41. However, both of these definitions seem purely technical to me.

What is the physical meaning behind the lapse function? Does it have anything to do with the asymptotical flatness of a spacetime?

The Einstein equations allow for a certain arbitrariness when one chooses to foliate space time into a three-dimensional spatial hypersurface with an orthogonal time direction. Two functions, the lapse and the shift, tell how to relate coordinates between two slices: the lapse $\alpha$ (or $N$) measures the proper time, while the shift $\beta$ measures changes in the spatial coordinates.
The proper time increment $\delta\tau$ is related to the time step $\delta t$ between two slices times the lapse function, according to $\delta\tau=\alpha(t,x)\delta t$.
A non-uniform $\alpha$ means that time advances with different rates at different positions. This non-uniformity is needed to avoid singularities in the numerical solution of the Einstein equations, such as appear on the black hole horizon.