I'm going to use the notation in this Wiki article, and refer to the diagram therein. I assume the information here is essentially correct, aside from the fact the axes on the diagram should be T and R. Firstly, I wonder what sort of uniqueness properties this space-time is supposed to have. Suppose I consider the general extension problem Let M be a space-time. Do there exist other space-times N such that M is a submanifold of N?If M is region I -- i.e. the exterior Schwarzschild solution, then the maximally extended Schwarzschild solution (call it E) is obviously a solution to the extension problem. And E is maximal in the sense that the extension problem for E has no solutions. However, there is another extension of region I, constructed as follows: (1) Start with E (2) Remove the origin (3) Identify region I with region III. (Specifically, by identifying (T, -R, theta, phi) in region I with (T, R, pi - theta, phi + pi) in region III) Call this extension F. (Question: do I need to remove the origin? I can't convince myself either way) F appears to be another maximal extension in the sense that the extension problem for F has no solutions. However, F is clearly not a "best" extension, because it is constructed as a quotient space of (a subset of) E. This prompts a question: are all extensions of region I actually quotient spaces of subsets of E? If not, then what if we invoke other constraints (such as being vacuum solutions)? Okay, the hard mathematical question is out of the way. I have some more lowbrow questions. My first observation from the diagram is that in these coordinates, region I bears a very strong resemblance to a diagram one might draw for a family of uniformly accelerating special relativistic observers. In this space-time: (1) Curves of constant Schwarzschild time are lines through the origin (2) Curves of constant Schwarzschild radii are hyperbolae asymptotic to the event horizon (3) Along a line of constant t, the spatial separation between two hyperbolae of constant r is fixed Uniformly accelerating SR observers: (1) Lines of simultaneity are lines through the origin (2) Observers' worldlines are hyperbolae asymptotic to the light cone from the origin (3) Along a line of constant t, the spatial separation between two hyperbolae of constant r is fixed Of course, the spatial distance between hyperbola differ between the two settings. I had previously intuited the perspective of an observer hovering at a fixed distance above an event horizon as looking very much like that of a uniformly accelerating observer in SR, and the above seems to cement that. So my question is is my intuition here actually correct? I want to make sure I'm not misleading myself! Physically, this space-time looks like it's describing a white-hole that has been collapsing since the beginning of time... and when it finally collapses, it immediately forms a black hole that remains for the rest of eternity. It seems obvious that a 'real' black holes aren't required to have matching white holes. (right?) I would expect a 'real' black hole to look more or less like the solution F I constructed above, but instead of having region IV in the past, you instead have region I (= region III) continuing backwards. (right?) But this does prompt a hypothetical: can there be a space-time with a black hole with the following properties? (1) There isn't a region IV (2) Regions I and III can communicate before the formation of the black hole (3) Regions I and III are forever separated once the black hole forms Hrm, I thought I had more questions, but I can't think of them. So I'll settle for just asking these ones.