Lapse function and spacetime splicing

Yes, note more precisely that [itex]n^ah_{ab}=0[/itex] (and itex]v^av^bh_{ab}=v^av^bg_{ab}[/itex]). So this is really what you see as the induced metric: it gives the correct inner product for vectors tangent to the hypersurfaces (and it's not a metric on the 4D space, since there it is degenerate).

On the lapse and shift thing: note that there is no reason for [itex]\nabla_\mu t[/itex] to be parallel to the normal (or orthogonal to the tangent space to the hypersurfaces). So just call the component tangent to the hypersurface the shift vector, and the component of the normal the shift vector
[tex]\nabla_\mu t = N n_\mu + N_\mu.[/tex]

While you can find 3 independent vectors satisfying this, they simply do not span the space tangent to your hypersurface.

I can't figure out what George is getting at, sorry. The induced metric on should only be 3x3.

Generically what you're doing is you have some global time function [itex]t[/itex] with gradient [itex]dt[/itex]. You want to find the induced metric on the level sets of [itex]dt[/itex]. The level sets of [itex]dt[/itex] are generated by a triplet of linearly-independent vector fields X, Y, Z such that

[tex]dt(X) = dt(Y) = dt(Z) = 0.[/tex]
In order that each level set be a surface, this set of vector fields needs to be integrable; that is, the set should be closed under the Lie bracket. This should hold automatically, given that [itex]t[/itex] is a global time function, and X, Y, Z are everywhere perpendicular to [itex]dt[/itex].

Then the induced metric [itex]h[/itex] can be given by (where X and Y are some vectors within the level surface)

[tex]\begin{align}h(X,Y) &= g(X,Y) = g_{tt} dt(X) dt(Y) + g_{ti} \Big( dt(X) dx^i(Y) + dt(Y) dx^i(X) \Big) + g_{ij} dx^i(X) dx^j(Y) \\ &= 0 + 0 + g_{ij} X^i Y^j. \end{align}[/tex]
So perhaps this is what George means by "think of [itex]h[/itex] as 4x4". Note that I'm assuming the vectors X and Y are already tangent to the level surfaces of [itex]dt[/itex]. One can imagine instead a 4x4 metric on general vectors that includes some extra terms to project those vectors onto the level surfaces of [itex]dt[/itex]. I think that is what George wrote down. But I wouldn't call that the "induced metric", since it acts on a vector space of the wrong dimension.

I also see that I haven't used the covariant derivative [itex]\nabla_\mu t[/itex] anywhere; I'm not sure exactly why this is needed. Probably the method I am outlining above is a different route to the same result, rather than the method Wald (?) is using.

I also see that I haven't used the covariant derivative [itex]\nabla_\mu t[/itex] anywhere; I'm not sure exactly why this is needed. Probably the method I am outlining above is a different route to the same result, rather than the method Wald (?) is using.

The set of coordinates on the initial hypersurface has some origin somewhere.

Is this origin carried to the subsequent hypersurface by this vector t or by the normal vector to the hypersurface? Meaning, if I started at the origin and move along this vector t to the next hypersurface, do I stay at the origin, or do I move away from the origin of coordinates? Or is this arbitrary based on how I define my coordinates?

I'm confused here because intuitively to me it seems like I would prefer to have my coordinates move along with the normal

but MTW makes a statement that the shift vectors tell you where these "normal struts" end up on the above hypersurface via the equation:

[tex]x_{new} ^i=x^i-N^idt[/tex]

Which seems to suggest that these "normal struts" (normal vectors) do not start and end at the same coordinates.