I'm creating this thread to discuss some issues raised by kev in the Understanding maximally extended Schwarzschild solution thread, to avoid diverting that thread from its original question. As any fule kno, the problem with Schwarzschild coordinates is their coordinate singularity at the event horizon. In fact, really there are two sets of Schwarzschild coordinates, those strictly inside and strictly outside the event horizon, which happen to share the same metric equation, but which are otherwise independent. For example, in the outside coordinates, t represents warped time (approaching infinity for any particle approaching the horizon), whereas in the inside coordinates, t represents a warped distance. The two ts are different entities and there's no real connection between them. (For the avoidance of doubt, when I say the "inside coords", of course I refer to the vacuum solution inside the event horizon, not the "interior Schwarzschild solution" below the surface of a spherical mass.) There are other coordinates, e.g. Kruskal-Szekeres or Eddington-Finkelstein, which remove the coordinate singularity. They do this by transforming from the each of the two Schwarzschild coordinate systems inside and outside, then "tweaking the initial conditions" to get the two solutions to match approaching the horizon from either side. The solution can then be extended to the horizon itself as the unambiguous, finite limit, and it turns out that a single metric equation describes the whole solution. I have no problem with any of this, but the impression I get is that kev does have a problem. It is somewhat unsatisfactory in that we have to stitch two solutions together and add extra events (the event horizon) to bridge the gap. (The gap was there only because of a defect in the original coordinates (like the defect of the undefined longitude of the North Pole). There's no physical (i.e. spacetime-geometrical) gap. That last sentence seems to be a bone of contention with kev.) All the technical expositions of (non-rotating, symmetric, uncharged, asymptotically flat) black hole metrics I have ever seen begin with a derivation of Schwarzschild coords, and then derive other coords (Kruskal-Szekeres, Eddington-Finkelstein etc) from them. I wonder if there are any published works that do it the other way round, by deriving e.g. Kruskal-Szekeres coords "from first principles" without reference to Schwarzschild coords or event-horizon singularities?