What is a maximal analytic extension? Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm not sure how analytic continuation is used, since I can't really see how the process of extending the domain of a complex function has anything to do with extending a manifold through a coordinate change.
 A: Dear dbrane, some basic terminology refinement could be useful. The words "maximal" and "analytic" are two more or less independent adjectives of the "extension". The word "maximal" means that "it cannot be extended further". On the other hand, the word "analytic" refers to the standard "analytic functions".
Analytic functions are infinitely differentiable functions such that if you write the Taylor expansion around any point, it converges to the exact original function.
Real vs complex variables in analyticity
Despite Tim's warning, you are very right that the word "analytic" in the context of general relativity is linked to "analytic" in the context of functions of complex variable. The only difference is that in general relativity, we typically substitute real values for the spacetime coordinates only.
However, the definition of an analytic function of a complex variable and an analytic function of a real variable is totally analogous. For complex functions of complex variables, it's still true that analytic functions are infinitely differentiable so that the Taylor expansion converges to the full function at each point of the "domain".
Nevertheless, analytic functions of the complex variable are much more constrained than "analytic functions of two real variables", namely the real and imaginary part: that's because the analytic functions of complex variables must be holomorphic - independent of the complex conjugate variable. In fact, "analytic" and "holomorphic" are exactly equivalent adjectives when it comes to functions of complex variables. So the right analogy is between holomorphic functions and real analytic functions of one (rather than two) real variable.
Extending solutions in general relativity
But let's return to general relativity. In that case, the spacetime coordinates are real. A solution we start with - e.g. the Schwarzschild solution - is usually not maximally extended to start with: it has coordinate singularities and one can't get beyond them by reading the solution, even though geodesics continue through those points in the real space.
An extension of this solution can be obtained if we cleverly redefine the spacetime coordinates so that the space around the coordinate singularity - in this case, the Schwarzschild event horizon - becomes regular. By this step, we get rid of the coordinate singularity and the metric tensor becomes nondegenerate even in the locus of the previous coordinate singularity (event horizon).
Once we do so, it becomes clear that the locus of the horizon appears at a finite locus in spacetime, and because the metric is smooth on one side, the metric tensor may be continued as a collection of analytic functions of the new spacetime coordinates. This continuation is totally analogous to the continuation in the complex case: we may write the Taylor expansion around a given point near the previous boundary and simply extrapolate it as far as we can.
We may continue it as far as we can and if the Taylor expansion diverges somewhere, we may try to continue from another point to get even further. Again, even in the new coordinates, we may encounter coordinate singularities as we are extending the spacetime. In order to extend the solution further, we may choose even better coordinates, and so on.
This process ultimately stops because the spacetime is surrounded either by asymptotic infinity - infinite volume where trajectories may be extended to an infinite proper length - or by genuine (curvature) singularities that cannot be extended by any coordinates. Geodesics physically terminate at those real singularities.
My description above is pretty much a mechanical recipe how to proceed. However, in practice, one always needs to be clever at each point, to know which coordinates should be chosen to get as far as you can, and so on. He may also find out that there are several maximal extensions although I am not sure and I cannot mention any well-known examples now.
Example
A useful example are the kruskal-Szekeres coordinates for the Schwarzschild solution

http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates

which are the maximal analytic continuation of the neutral black hole geometry.
