Before discussing black holes in detail we should acknowledge that such objects are still hypothetical. There undoubtedly exist bodies in the universe whose densities and gravitational intensities are extremely great, but the precise characteristics of the limiting cases are still largely conjectural, based on indirect evidence, so we shouldn't be too dogmatic about the actual physical limits of gravitation. What we describe in this section are the predictions of general relativity, the details of which have not (as yet) been corroborated by direct observations.
Furthermore, we should acknowledge that, even within the context of general relativity, the formal definition of a black hole may be impossible to satisfy. This is because, as discussed previously, a black hole is strictly defined as a region of spacetime that is not in the causal past of any point in the infinite future. Notice that this refers to the infinite future, because anything short of that could theoretically be circumvented by regions that are clearly not black holes. However, in some fairly plausible cosmological models the universe has no infinite future, because it re-collapses to a singularity in finite coordinate time. In such a universe (which, for all we know, could be our own), the boundary of any gravitationally collapsed region of spacetime would be contiguous with the boundary of the ultimate collapse, so it wouldn't really be a separate black hole in the strict sense. As Wald says, "there appears to be no natural notion of a black hole in a closed Robertson-Walker universe which recollapses to a final singularity", and further, "there seems to be no way to define a black hole in a closed universe, because it requires going to infinity, but there is no infinity in a closed universe."
It's interesting that this is essentially the same objection that is often raised by people when they first hear about black holes, i.e., they reason that if it takes infinite coordinate time for any object to cross an event horizon, and if the universe is going to collapse in a finite coordinate time, then it's clear that nothing can possess the properties of a true black hole in such a universe. Thus, in some fairly plausible cosmological models it's not strictly possible for a true black hole to exist. On the other hand, it is possible to have an approximate notion of a black hole in some isolated region of a closed universe, but of course many of the interesting transfinite issues raised by true (perhaps a better name would be "ideal") back holes are not strictly applicable to an "approximate" black hole.
Having said this, there is nothing to prevent us from considering an infinite open universe containing full-fledged black holes in all their transfinite glory. I use the word "transfinite" because ideal black holes involve singular boundaries at which the usual Schwarzschild coordinates for the external field of a gravitating body go to infinity - and back - as discussed in the previous section. There are actually two distinct kinds of "spacetime singularities" involved in an ideal black hole, one of which occurs at the center, r = 0, where the spacetime manifold actually does become unequivocally singular and the field equations are simply inapplicable (as if trying to divide a number by 0). It's unclear (to say the least) what this singularity actually means from a physical standpoint, but oddly enough the "other" kind of singularity involved in a black hole seems to shield us from having to face the breakdown of the field equations. This is because it seems (although it has not been proved) to be a characteristic of all realistic spacetime singularities in general relativity that they are invariably enclosed within an event horizon, which is a peculiar kind of singularity that constitutes a one-way boundary between the interior and exterior of a black hole. This is certainly the case with the standard black hole geometries based on the Schwarzschild and Kerr solutions. The idea that it may be true for all singularities is sometimes called the Cosmic Censorship Conjecture (Penrose). Whether or not this conjecture is true, it's a remarkable fact that at least some (if not all) of the singular solutions of Einstein's field equations automatically enclose the singularity within an amazing natural contrivance, an event horizon, that effectively shields the universe from direct two-way exposure to any regions in which the metric of spacetime breaks down.
Since we don't really know what to make of the true singularity at r = 0, we tend to focus our attention on the behavior of physics near the event horizon, which, for a non-rotating black hole, resides at the radial location r = 2m, where the Schwarzschild coordinates become singular. Of course, a singularity in a coordinate system doesn't necessarily represent a pathology of the manifold. (Consider travelling due East at the North Pole). Nevertheless, the fact that no true black hole can exist in a finite universe shows that the coordinate singularity at r = 2m is not entirely inconsequential, because it does (or at least can) represent a unique boundary between fundamentally separate regions of spacetime, depending on the cosmology. To understand the nature of this boundary, it's useful to consider hovering near the event horizon of a black hole. The components of the curvature tensor at r = 2m are on the order of 1/m2, so the spacetime can theoretically be made arbitrarily "flat" (Lorentzian) at that radius by making m large enough. Thus, for an observer "hovering" at a value of r that exceeds 2m by some small fixed amount Dr, the downward acceleration required to resist the inward pull can be made arbitrarily small. However, in order for the observer to be hovering close to 2m his frame must be tremendously "boosted" in the radial direction relative to an infalling particle. This is best seen in terms of a spacetime diagram such as the one below, which show the future light cones of two events located on either side of a black hole's event horizon.
In this drawing r is the radial Schwarzschild coordinate and t' is an Eddington-Finkelstein mapping of the Schwarzschild time coordinate, i.e.,
The right-hand ray of the cone for the event located just inside the event horizon is tilted just slightly to the left of vertical, whereas the cone for the event just outside 2m is tilted just slightly to the right of vertical. The rate at which this "tilt" changes with r is what determines the curvature and acceleration, and for a sufficiently large black hole this rate can be made negligibly small. However, by making this rate small, we also make the outward ray more nearly "vertical" at a given Dr above 2m, which implies that the hovering observer's frame needs to be even more "boosted". (Another way of looking at this is to think of the gravitational potential, which need not be changing very steeply at r = 2m, but overall it has changed by a huge amount relative to infinity. We must be very deep in a potential hole in order for the light cones to be tilted that far, even though the rate at which the tilt has been increasing can be arbitrarily slow. This just means that for a supermassive black hole they started tilting a long ways out.)
As can be seen from the diagram, relative to the frame of a particle falling in from infinity, a hovering observer must be moving outward at near light velocity. Consequently his axial distances are tremendously contracted, to the extent that, if the value of Dr is normalized to his frame of reference, he is actually a great distance (perhaps even light-years) from the r = 2m boundary, even though he is just 1 inch above 2m in terms of Schwarzschild coordinate r. Also, the closer he tries to hover, the more radial boost he needs to hold that value of r, and the more contracted his radial distances become. Thus he is living in a thinner and thinner shell of Dr, but from his own perspective there's a world of room. Assuming he brought enough rocket fuel to accelerate himself up to this "hovering frame" at that radius 2m + Dr (or actually to slow himself down to a hovering frame), he would thereafter just need to resist the very slight downward tug to maintain that frame of reference.
Quantitatively, for an observer hovering at a small Schwarzschild distance Dr above the horizon of a black hole, the radial distance Dr' to the event horizon with respect to the observer's local coordinates would be
which approaches 
as Dr goes to zero. This shows that as the observer hovers closer to the horizon in terms of Schwarzschild coordinates, his "proper distance" remains relatively large until he is nearly at the horizon. Also, the derivative of Dr' with respect to Dr in this range is 
, which goes to infinity as Dr goes to zero. (Bear in mind that these relations pertain to a truly static observer, so they don't apply when the observer is moving from one radial position to another, unless he moves sufficiently slowly.)
Incidentally, it's amusing to note that if a hovering observer's radial distance contraction factor at r was 1-2m/r instead of the square root of that quantity, his scaled distance to the event horizon at a Schwarzschild distance of Dr would be Dr' = 2m + Dr. Thus when he is precisely at the event horizon his scaled distance from it would be 2m, and he wouldn't achieve zero scaled distance from the event horizon until arriving at the origin r = 0 of the Schwarzschild coordinates. This may seem rather silly, but it's actually quite similar to one of Einstein's proposals for avoiding what he regarded as the unpleasant features of the Schwarzschild solution at r = 2m. He suggested replacing the radial coordinate r with r = 
, and noted that the Schwarzschild solution expressed in terms of this coordinate behaves regularly for all values of r. Whether or not there is any merit in this approach, it clearly shows how easily we can "eliminate" poles and singularities simply by applying coordinates that have canceling zeros (much as one does in the design of control systems) or otherwise restricting the domain of the variables. However, we shouldn't assume that every arbitrary system of coordinates has physical significance.
What "acceleration of gravity" would a hovering observer feel locally near the event horizon of a black hole? In terms of the Schwarzschild coordinate r and the proper time t of the particle, the path of a radially free-falling particle can be expressed parametrically in terms of the parameter q by the equations
where R is the apogee of the path (i.e., the highest point, where the outward radial velocity is zero). These equations describe a cycloid, with q = 0 at the top, and they are valid for any radius r down to 0. We can evaluate the second derivative of r with respect to t as follows
At q = 0 the path is tangent to the hovering worldline at radius R, and so the local gravitational acceleration in the neighborhood of a stationary observer at that radius equals - m/R2, which implies that if R is approximately 2m the acceleration of gravity is about - 1/(4m). Thus the acceleration of gravity in terms of the coordinates r and t is finite at the event horizon, and can be made arbitrarily small by increasing m.
However, this acceleration is expressed in terms of the Schwarzschild radial parameter r, whereas the hovering observer's radial distance r' must be scaled by the "gravitational boost" factor, i.e., we have dr' = 
. Substituting this expression for dr into the above formula gives the proper local acceleration of a stationary observer
This value of acceleration corresponds to the amount of rocket thrust an observer would need to hold position, and we see that it goes to infinity as r goes to 2m. However, the form suggests interpreting the effect at 2m not as "infinite local gravity" but as finite local gravity combined with infinite global boost (i.e., the light cones tipping over). We will see below that indeed the gravitational field strength at the event horizon is finite.
Also, it remains true that for any fixed Dr above the horizon we can make the proper acceleration arbitrarily small by increasing m. To see this, note that if r = 2m + Dr for a sufficiently small increment Dr we have m/r ~ 1/2, and we can bring the other factor of r into the square root to give
Still, these formulas contain a slight "mixing of metaphors", because they refer to two different radial parameters (r' and r) with different scale factors. To remedy this, we can define the locally scaled radial increment Dr' = 
as the hovering observer's "proper" distance from the event horizon. Then, since Dr = r - 2m, we have Dr' = 
and so r = 
. Substituting this into the formula for the proper local acceleration gives the proper acceleration of a stationary observer at a "proper distance" Dr' above the event horizon of a (non-rotating) object of mass m is given by
This is nice because as (Dr'/M) gets small the acceleration approaches -1/(2Dr'), which is the asymptotic proper acceleration at a small "proper distance" Dr' from the event horizon of a large black hole. Thus, for a given proper distance Dr' the proper acceleration can't be made arbitrarily small by increasing m. Conversely, for a given proper acceleration g our hovering observer can't be closer than 1/2g of proper distance, even as m goes to infinity. For example, the closest an observer can get to the event horizon of a supermassive black hole while experiencing no more than 1g proper acceleration is about half a light-year of proper distance. At the other extreme, if (Dr'/m) is very large, as it is in normal circumstances between gravitating bodies, then this acceleration approaches m/(Dr')2, which is just Newton's inverse-square law of gravity in geometrical units.
We've seen that the amount of local acceleration that must be overcome to hover at a radial distance r increases to infinity at r = 2m, but this doesn't imply that the gravitational curvature of spacetime at that location becomes infinite. Unfortunately, the components of the curvature tensor depend to some extent on the choice of coordinate systems, so we can't simply examine the components of Rabgd to ascertain whether the intrinsic curvature is actually singular at the event horizon. For example, with respect to the Schwarzschild coordinates the non-zero components of the covariant curvature tensor are
along with the components related to these by symmetry. The two components relating the radial coordinate to the spherical surface coordinates are singular at r = 2m, but this is again related to the fact that the Schwarzschild coordinates are not well-behaved on this manifold near the event horizon. A more suitable system of coordinates in this region (as noted by Misner, et al) is constructed from the basis vectors
where g = 
. With respect to this "hovering" orthonormal system of coordinates the non-zero components of the curvature tensor (up to symmetry) are
Interestingly, if we transform to the orthonormal coordinates of a free-falling particle, the curvature components remain unchanged. Plugging in r = 2m, we see that these components are all proportional to 1/m2 at the event horizon, so the intrinsic spacetime curvature at r = 2m is finite. Indeed, for a sufficiently large mass m the curvature can be made arbitrarily mild at the event horizon. As a result, if we imagine the light cone at a point extremely close to the horizon, with its outermost ray pointing just slightly in the positive r direction, we could theoretically boost ourselves at that point so as to maintain a constant radial distance r, and thereafter maintain that position with very little additional acceleration. Thus when we say "the gravitational acceleration goes to infinity as our radial position approaches 2m" we really mean that the amount of acceleration required to boost us into a hovering frame goes to infinity, but we only need to apply that that massive boost once, after which we're just fighting the local geodesic deviation, which can be arbitrarily small for a sufficiently massive black hole. Of course, as m increases the depth of the potential "well" also goes to infinity.
Having discussed the prospects for hovering near a black hole, let's review the process by which an object may actually fall through an event horizon. If we program a space probe to fall freely until reaching some randomly selected point outside the horizon and then accelerate back out along a symmetrical outward path, there is no finite limit on how far into the future the probe might return. This sometimes strikes people as paradoxical, because it implies that the infalling probe must, in some sense, pass through all of external time before crossing the horizon, and in fact it does, if by "time" we mean the extrapolated surfaces of simultaneity for an external observer. However, those surfaces are not well-behaved in the vicinity of a black hole. It's helpful to look at a drawing like this:
This illustrates schematically how the analytically continued surfaces of simultaneity for external observers are arranged outside the event horizon of a black hole, and how the infalling object's worldline crosses (intersects with) every timeslice of the outside world prior to entering a region beyond the last outside timeslice. The dotted timeslices can be modeled crudely as simple "right" hyperbolic branches of the form tj - T = 1/R. We just repeat this same -y = 1/x shape, shifted vertically, up to infinity. Notice that all of these infinitely many time slices curve down and approach the same asymptote on the left. To get to the "last timeslice" an object must go infinitely far in the vertical direction, but only finitely far in the horizontal (leftward) direction.
The key point is that if an object goes to the left, it crosses every single one of the analytically continued timeslice of the outside observers, all the way to their future infinity. Hence those distant observers can always regard the object as not quite reaching the event horizon (the vertical boundary on the left side of this schematic). At any one of those slices the object could, in principle, reverse course and climb back out to the outside observers, which it would reach some time between now and future infinity. However, this doesn't mean that the object can never cross the event horizon (assuming it doesn't bail out). It simply means that its worldline is present in every one of the outside timeslices. In the direction it is travelling, those time slices are compressed infinitely close together, so the infalling object can get through them all in finite proper time (i.e., its own local time along the worldline falling to the left in the above schematic).
Notice that the temporal interval between two definite events can range from zero to infinity, depending on whose time slices we are counting. One observer's time is another observer's space, and vice versa. It might seem as if this degenerates into chaos, with no absolute measure for things, but fortunately there is an absolute measure. It's the absolute invariant spacetime interval "ds" between any two neighboring events, and the absolute distance along any specified path in spacetime is just found by summing up all the "ds" increments along that path. For any given observer, a local absolute increment ds can be projected onto his proper time axis and local surface of simultaneity, and these projections can be called dt, dx, dy, and dz. For a sufficiently small region around the observer these components are related to the absolute increment ds by the Minkowski or some other flat metric, but in the presence of curvature we cannot unambiguously project the components of extended intervals. The only unambiguous way of characterizing extended intervals (paths) is by summing the incremental absolute intervals along a given path.
An observer obviously has a great deal of freedom in deciding how to classify the locations of putative events relative to himself. One way (the conventional way) is in terms of his own time-slices and spatial distances as measured on those time slices, which works fairly well in regions where spacetime is flat, although even in flat spacetime it's possible for two observers to disagree on the lengths of objects and the spatial and temporal distances between events, because their reference frames may be different. However, they will always agree on the ds between two events. The same is true of the integrated absolute interval along any path in curved spacetime. The dt,dx,dy,dz components can do all sorts of strange things, but observers will always agree on ds.
This suggests that rather than trying to map the universe with a "grid" composed of time slices and spatial distances on those slices, an observer might be better off using a sort of "polar" coordinate system, with himself at the center, and with outgoing geodesic rays in all directions and at all speeds. Then for each of those rays he measures the total ds between himself and whatever is "out there". This way of "locating" things could be parameterized in terms of the coordinate system [q, f, b, s] where q and f are just ordinary latitude and longitude angles to determine a direction in space, b is the velocity of the outgoing ray (divided by c), and s is the integrated ds distance along that ray as it emanates out from the origin to the specified point along a geodesic path. (Incidentally, these are essentially the coordinates Riemann used in his 1854 thesis on differential geometry.) For any event in spacetime the observer can now assign it a location based on this system of coordinates. If the universe is open, he will find that there are things which are only a finite absolute distance from him, and yet are not on any of his analytically continued time slices! This is because there are regions of spacetime where his time slices never go, specifically, inside the event horizon of a black hole. This just illustrates that an external observer's time slices aren't a very suitable set of surfaces with which to map events near a black hole, let alone inside a black hole.
For this reason it's best to measure things in terms of absolute invariant distances rather than time slices, because time slices can do all sorts of strange things and don't necessarily cover the entire universe, assuming an open universe. Why did I specify an open universe? The schematic above depicted an open universe, with infinitely many external time slices, but if the universe is closed and finite, there are only finitely many external time slices, and they eventually tip over and converge on a common singularity, as shown below
In this context the sequence of tj slices eventually does include the vertical slices. Thus, in a closed universe an external observer's time slices do cover the entire universe, which is why there really is no true event horizon in a closed universe. An observer could use his analytically continued time slices to map all events if he wished, although they would still make an extremely somewhat ill-conditioned system of coordinates near an approximate black hole.
One common question is whether a man falling (feet first) through an even horizon of a black hole would see his feet pass through the event horizon below him. As should be apparent from the schematics above, this kind of question is based on a misunderstanding. Everything that falls into a black hole falls in at the same local time, although spatially separated, just as everything in your city is going to enter tomorrow at the same time. We generally have no trouble seeing our feet as we pass through midnight tonight, although it is difficult one minute before midnight trying to look ahead and see your feet one minute after midnight. Of course, for a small black hole you will have to contend with tidal forces that may induce more spatial separation between your head and feet than you'd like, but for a sufficiently large black hole you should be able to maintain reasonable point-to-point co-moving distances between the various parts of your body as you cross the horizon.
On the other hand, we should be careful not to understate the physical significance of the event horizon, which some authors have a tendency to do, perhaps in reaction to earlier over-estimates of its significance. Section 6.4 includes a description of a sense in which spacetime actually is singular at r = 2m, even in terms of the proper time of an infalling particle, but it turns out to be what mathematicians call a "removable singularity", much like the point x = 0 on the function sin(x)/x. Strictly speaking this "curve" is undefined at that point, but by analytic continuation we can "put the point back in", essentially by just defining sin(x)/x to be 1 at x = 0. Whether nature necessarily adheres to analytic continuation in such cases is an open question.
Finally, we might ask what an observer would find if he followed a path that leads across an event horizon and into a black hole. In truth, no one really knows how seriously to take the theoretical solutions of Einstein's field equations for the interior of a black hole, even assuming an open infinite universe. For example, the "complete" Schwarzschild solution actually consists of two separate universes joined together at the black hole, but it isn't clear that this topology would spontaneously arise from the collapse of a star, or from any other known process, so many people doubt that this complete solution is actually realized. It's just one of many strange topologies that the field equations of general relativity would allow, but we aren't required to believe something exists just because it's a solution of the field equations, . On the other hand, from a purely logical point of view, we can't rule them out, because there aren't any outright logical contradictions, just some interesting transfinite topologies.
Unless the giddy heaven fall,
And earth some new convulsion tear,
And, us to join, the world should all
Be cramped into a planisphere.
As lines so loves oblique may well
Themselves in every angle greet;
But ours, so truly parallel,
Though infinite, can never meet.
Therefore the love which us doth bind,
But Fate so enviously debars,
Is the conjunction of the mind,
And opposition of the stars.
Andrew Marvell (1621-1678)