7. Cosmology 1 Is the Universe Closed?

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  of the particle, the path of a radially free-falling particle can be expressed parametrically in terms of the parameter  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  = 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  as follows

At  = 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  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' = dr/(12m/r)^1/2. 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 in order to hold position, and we see that it goes to infinity as r goes to 2m. Nevertheless, for any ratio r/(2m) greater than 1 we can still make this acceleration arbitrarily small by choosing a sufficiently large m. On the other hand, an enormous amount of effort would be required to accelerate the rocket into this hovering condition for values of r/(2m) very close to 1. This amount of “boost” effort cannot be made arbitrarily small, because it essentially amounts to accelerating (outwardly) the rocket to nearly the speed of light relative to the frame of a free-falling particle from infinity.

Interestingly, as the preceding figure suggests, an outward going photon can hover precisely at the event horizon, since at that location the outward edge of the light cone is vertical. This may seem surprising at first, considering that the proper acceleration of gravity at that location is infinite. However, the proper acceleration of a photon is indeed infinite, since the edge of a light cone can be regarded as hyperbolic motion with acceleration “a” in the limit as “a” goes to infinity, as illustrated in the figure below.

 

Also, it remains true that for any fixed r above the horizon we can make the proper acceleration arbitrarily small by increasing m. To see this, note that if  r = 2m + r  for a sufficiently small increment r we have m/r ~ 1/2, and we can bring the other factor of r into the square root to give

Notice that as (r'/M) becomes small the acceleration approaches -1/(2r'), which is the asymptotic proper acceleration at a small "proper distance" r' from the event horizon of a large black hole. Thus, for a given proper distance r' 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 super-massive black hole while experiencing no more than 1g proper acceleration is about half a light-year of proper distance. At the other extreme, if (r'/m) is very large, as it is in normal circumstances between gravitating bodies, then this acceleration approaches m/(r'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. 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 R_ 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  = . With respect to this "hovering" orthonormal system of coordinates the non-zero components of the curvature tensor (up to symmetry) are

Of course, it isn’t possible to hover precisely at (or inside) the event horizon, but remarkably, 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. If we imagine the light cone at a radial coordinate r extremely close to the horizon (i.e., such that r/(2m) is just slightly greater than 1), 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 (for sufficiently large m). But, as noted above, the work that must be expended to achieve this hovering condition from infinity cannot be made arbitrarily small, since it requires us to accelerate to nearly the speed of light.

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 in-falling 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 in-falling 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  t_j  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 traveling, those time slices are compressed infinitely close together, so the in-falling 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  [, , , s]  where  and  are just ordinary latitude and longitude angles to determine a direction in space,  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 t_j 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 in-falling 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.