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.