OK, thanks.
I think I'm getting it - attached is my color coded version of the Schwarzschild geometry based on Marolf's paper.
http://arxiv.org/PS_cache/gr-qc/pdf/9806/9806123v3.pdf
It is an embedding of the r-t radial plane of a Schwarzschild black hole in a 3 dimensional Mikowski geometry, as described by the paper. The only advantage of this diagram over the ones in the link is that it's color coded. Drawing it was very useful to me in understanding the paper, however.
It has four regions, as it is the fully extended Schwarzschild spacetime, which is a non-traversable wormhole connecting two different asymptotically flat space-times.
The two asymptotically flat spacetimes are colored green and blue, which represent the exterior region of the black hole outside the event horizon. I think of the green region as "our" space-time (for no particularly good reason).
Note that these are the same four regions that are shown on a penrose diagram of a black hole. For readers unfamiliar with Penrose diagrams see for instance:
http://en.wikipedia.org/wiki/Image:PENROSE2.PNG
the diagram for the Schwarzschild geometry is the one labelled "static wormhole".
There are also two interior regions, colored red and pink. The pink region represents the interior of a white hole, the red region is the interior of a black hole. As T increases, any object in the pink region must eventually leave it and enter the blue or green regions.
The singularity itself is located at R=0, which corresponds to Y = -infinity (which also implies T = +infinity or T=-infinity, as per the Penrose diagram). The attachment is only drawn for Y greater than -5, however.
The event horizon, at r=1, is a lightlike (null) surface where differing colors intersect.
The coordinate labeled T is the time coordinate of the Minkowski geometry, X and Y are space coordinates. T increasing determines the direction of increasing time for any (timelike) worldline.
Lines of constant r are planes of constant Y on the diagram. Y>0 corresponds to r>2M, i.e. one of the two exterior regions. Y<0 corresponds to r<2M, i.e. one of the two interior regions. Y=0 is the event horizon.
The equations used to construct it are interesting, and unfortunately are a bit obfuscated in the paper (in my opinion).
We wish to create a map from (r,t) to (X,Y,T), where (r,t) are the Schwarzschild coordinates, and (X,Y,T) are the coordinates of our embedding. The Schwarzschild radius is assumed to be unity (i.e the mass of the black hole is 1/2 in geometric units).
The coordinate Y can be expressed as an integral, and depends only on r
\int_1^r \left( 1+1/R+1/R^2+1/R^3 \right) dR
By construction, when r=1, Y=0, i.e. the event horizon is located at Y=0.
The X and T coordinates are functions of both r and t. For the exterior region, the formula is:
X = \pm 2 \sqrt{1-1/r} \cosh t/2
T = 2 \sqrt{1-1/r} \sinh t/2
The plus and minus sign gives two separate regions, the green and blue, representing the two different asymptotically flat space-times in the exterior region.
In the interior region, the formula is slightly different
T = \pm 2 \sqrt{1-1/r} \cosh t/2
X = 2 \sqrt{1-1/r} \sinh t/2
Again, the plus and minus signs represent different regions.
I won't go through the algebra in detail, but one can confirm that
-dT^2 + dX^2 + dY^2
yields the Schwarzschild metric when re-expresed in terms of dr and dt, i.e. one substitutes
dT = (dT/dt)*dt + (dT/dr)*dr
dX = (dX/dt)*dt + (dX/dr)*dr
dY = (dY/dr)*dr
and gets the Schwarzschild metric
(-1+1/r) dt^2 + 1/(1-1/r) dr^2
One may note that r=1 from the above transformation equations corresponds to a point, T=0 and X=0, but the event horizon is actually a line. This happens because Schwarzschild coordinates are ill-behaved. The event horizon really has the topology of a null surface and because we are modeling only 1+1 dimensions said "null surface" is a pair of lines on the diagram.
