Event horizon in different coordinate systems

[Matterwave]

On approach to the singularity, there is no pileup or unusual feature of received signals. There is asymptotically a well defined last signal received from some outside source on approach to the singularity, and it is not very long after the signal received from said outside source on free faller's horizon crossing. The catastrophe for an infaller is tidal - infinite stretching in the direction of the extra spacelike killing vector (extra being in addition to the two angular killing vectors), and compression in the other spatial directions - a little ball approaches death as a line. That, and 'no future' - geodesic incompleteness. [This is, of course, for the ideal SC BH, which doesn't exist in nature; real BH interiors are much more complex and not known, in that the Kerr interior is unstable against perturbation.]

Ah ok, that makes sense. I could only recall that an observer reaches the singularity in finite proper time (as must be the case for geodesic incompleteness to occur), I could not recall the result of which signals he can still receive from the outside as he's falling in.

 

[PeterDonis]

when we define a metric in curved space-time, what have we defined in general?

pervect's response is a good one, but might be a bit "heavy", though it's already a condensed version of the paper he linked to, so I'll try to condense it a bit more. ;) A metric defines a geometry--a shape, composed of points and curves with particular relationships between them--distances and angles. So, for example, if you know the metric of the Earth's surface, you know the distance between any two points on it (say, the intersection of 5th Avenue and 59th Street in Manhattan, and Nelson's Column in Trafalgar Square in London), and you know the angle between any two curves where they intersect (say, the angle between the great circle--note that all distances are measured along great circles, i.e., geodesics--connecting the two points I just named, and the great circle connecting the North Pole with Rio de Janeiro, Brazil, at the point where they intersect). But you can use many different coordinate charts to describe the same geometry, so just knowing a geometry does not give you any coordinates. (Notice that I used no coordinates in describing the distances and angles above.)

The geometry of spacetime works the same way except that you have to add time to the specification of a "point", i.e., an event. So, for example, in spacetime there is a particular "distance" (actually a proper time) between, say, the event of a rocket in free fall flying past the Moon on its way to Mars, and the same rocket flying past Phobos just before it reaches Mars. Assuming the rocket has been in free fall the whole time, its path through spacetime will be a geodesic, and the proper time elapsed on its clock between the two events will give the spacetime "distance" between them. Also, the "angle" between a pair of timelike worldlines is the relative velocity between them when they intersect, so, for example, knowing the geometry of spacetime means knowing the angle--the relative velocity--between the above rocket and another rocket in free fall on its way from Venus to Jupiter that happens to pass the first rocket, when they pass. But again, just knowing the geometry--all the distances and angles--doesn't specify any coordinates (once again, I used no coordinates in my description of events above).