Black Holes and Point Set Topology

Consider a Euclidean plane. In this plane, draw a circle. Now mark the interior, exterior, and boundary of the circle. These three parts of the circle may be considered subsets of the plane.

The plane is a point set. We will call it the universe. The three subsets we have just created are also point sets. We know a great deal about the plane from our school geometry. Let's examine the subsets we have created from the circle and develop a language to describe how they are related to the plane.

First, the interior and the exterior of the circle have no points in common (disjoint). We say that their intersection is the null set.

Second, all the points in either of these sets are the same. Consider the interior. Any point in the interior of the circle is completely surrounded by other interior points. We say that, for each point of the interior, there exists a neighborhood (another subset containing the point in its interior) that is completely contained inside the circle. A similar argument may be made for points in the exterior.

The boundary may be attached either to the interior or to the exterior but not to both. The subset that has the boundary attached to it is called a closed subset. The other subset is called an open subset. The boundary is distinguished from the interior and the exterior in that each point of the boundary has no neighborhood that is completely contained in the interior or the exterior. The boundary is also of a different dimension than the interior and the exterior. It is a one dimensional point set, while the other two are two dimensional point sets (as is the plane as a whole).

Now consider a black hole. The black hole is the remnant of a collapsed star that has all of its mass now concentrated in a singularity at the center of the hole. The singularity is surrounded by a surface of finite radius called the event horizon. The event horizon is that surface on which the escape velocity is everywhere equal to the speed of light in free space: c = 3 ´ 108 m/sec.
Like the circle in the plane, the event horizon of a black hole separates the universe into two disjoint regions: the exterior of the hole and the interior of the hole. Call the exterior the known universe-the universe we all inhabit. The event horizon itself is the boundary between the two regions.

Question: To which of the regions does the boundary attach?

If it attaches to the exterior, then the known universe is topologically a closed set with respect to the black hole. If it attaches to the interior, then the known universe is topologically an open set.

From the perspective of an observer stationed outside of the hole (and safe from harm, we hope), General Relativity teaches that a clock dropped into the hole will appear to run slower and slower as it approaches the event horizon. The time between each "tick" (as measured by a clock local to the observer) will increase without limit, approaching infinity as the clock approaches the event horizon. The "last tick" before the clock drops into the hole will never be observed in finite time. The clock will always appear inside the known universe and outside the hole.

This behavior is very suggestive of an open set in which every point is surrounded by a neighborhood completely contained inside the set. For our observer, the clock never actually reaches the event horizon. It only approaches it in the limit of infinite time. Thus, the clock always has a 'way to go' before encountering the event horizon. It is always surrounded by a neighborhood completely contained within the known universe.

This experiment is certainly not unique, but it is typical of behavior near a black hole. No experiment performed by an observer exterior to the hole can ever reveal the event horizon or the interior of the hole that lies beyond. These regions are forever shrouded in cosmic secrecy.

If observers were to plunge into the hole, they would drop through the event horizon in finite time as measured by a clock carried along on the trip (and assuming an ability to survive the severe space-time distortions near the hole). But once through, the observers could never again hope to send a signal to a receiver stationed outside the hole in the known universe.

And the receiver would never be able to witness the dropping of the travelers through the event horizon in finite time as measured by a clock local to the travelers. As far as the receiver is concerned, the time between the travelers' signals would appear to increase without limit, just as in the previous experiment. (Relativity is really relative to the where-when-how of making an observation!!!)

It appears, therefore, that the event horizon attaches to the interior of the black hole, leaving the known universe topologically an open set with respect to the hole.

This brief argument is just one of many attempts, made by many people, to model a black hole outside the equations of General Relativity itself. Hopefully, its use of topology will shed some light of understanding for at least a few readers.