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Additional Note on Point Sets and Black Holes
(A response to an inquiry from a reader in Denmark)
There exist two fundamental entities which underlie all of mathematics: the complex number system and the generalized point set.
From the realm of the complex numbers and their properties, we may obtain, by selecting as special cases (or as subsets), such instances as the imaginary numbers, the real numbers, the rational numbers, the integers, and so on. We can also use the complex numbers to construct higher order systems such as quaternions, etc.
Once we have these various instances, we become able to construct quantitative theories of all sorts, both in mathematics and in physics. Despite the frightening-looking formalisms that appear in technical books and conference papers, everything boils down to numbers or number arrays and their defined properties.
Definition is important in mathematics. The numbers have only those properties with which we endow them. From the first simple, defined acts of counting, we are able to formulate the basic rules of arithmetic. From arithmetic, we are able to extend the rules to more general cases, and so, eventually, into the entire realm of higher mathematics.
Numbers are the basis of modern science and technology. Everything, from simple toys for children to such complex machines as the Space Shuttle, is described by numbers. Numbers are at the root of their respective designs; numbers are part and parcel of their moment-to-moment operation; numbers are intimately involved in their post-op evaluations (wherever appropriate).
In arithmetic, numbers combine in various ways to produce other numbers. But there exists an instance where numbers combine to produce a very curious entity that lives entirely outside the universe of numbers. This instance is division by zero. We say arithmetic is open for division by zero because it leads to a non-numerical entity called infinity.
Books have been written on infinity. I shall not deal with infinity further here except to note (because I am a physicist) that physicists take great pains to avoid it. Whenever and wherever infinity appears in a calculation, or in the development of a theory, it must be eliminated. In quantum mechanics, this elimination is done via a formidable technique called, “renormalization.”
It is clear why physicists are so “antsy” about infinity: the equations of physics are built entirely out of numbers and their defined properties; infinity is simply NOT a number; it is an abstract and sometimes contradictory-seeming concept whose exact nature and definition continues to challenge mathematicians and philosophers even to this day.
Ok, moving on: We have glanced at the numbers and glimpsed somewhat of we obtain from them. Now let’s take a glance at the other formalism we mentioned, the generalized point set.
From the generalized point set, we obtain, again as special cases, all sorts of geometrical figures such as the point, the line, the circle, the surface, the volume, the hypersurface, and so on. These specialized figures are all infinite aggregations of points with pre-defined properties. The various geometries consider, in detail, these properties. Even the generalized point set, itself, has properties. These properties are the proper subject matter of Point Set Topology.
Point Set Topology is, in many ways, very counterintuitive. I have already said that I am a physicist, not a mathematician; so you can imagine that I have a lot of fun trying to make sense out of the pronouncements of point set topologists (and other pure mathematicians). I have dozens of books on advanced mathematics, and I immerse myself in them from time to time, whenever the mood strikes. I like Point Set Topology, especially, because it is so general in nature, even if it IS very difficult to grasp!
Now, we have two apparently distinct sets of fundamental entities with two major branches of mathematics growing out of them. What happens when we combine them? As Descartes was among the first to point out, we acquire a greatly expanded mathematical facility. A tremendous amount of modern mathematics stems from the marriage of point sets and numbers.
Examples of such combinations include the number line, the Cartesian plane, The Euclidean n-space, the Riemannian n-space, and so on. As a further example, one may associate any denominate numerical quantity, along with its pre-defined properties (such as continuity), with the points in a pre-selected space and acquire a scalar field, a vector field, a tensor field, a coordinate space, or what have you.
The temperature across the hood of your car is an example of a scalar field. Gravitational, electric, and magnetic fields are examples of vector and/or tensor fields. The list is virtually endless…
Since we are going to be talking about black holes, it makes sense, at this juncture, to speak of the General Theory of Relativity (G.R.).
In G.R., Einstein considered a special group of generalized coordinate spaces with mass/energy distribution-dependent non-Euclidean properties. These generalized spaces became the extensions of the flat (Euclidean) spacetime introduced in Special Relativity. Their general properties had been mathematically formulated a century or so earlier by Lobachevsky, Bolyai, Riemann, and Gauss.
He successfully applied these generalized spaces to a description of the gravitational field, and created a revolution in modern physics whose ramifications are still being actively worked out at the time of this writing. In essence, he equated gravitation with geometry in a generalized four-dimensional world of variable curvature.
He further considered that physical space should be assigned the same characteristics as the mathematical space that best described it. And since he did so, people have been mystified by such terms as "curved space," (“warped space,” Star Trek) and "curved time." But this usage follows quite naturally from the use of curved geometry to describe physical space.
Black holes are one of the more pathological developments of G.R. What I am about to present is a description of a black hole in a two dimensional world, applying concepts from Point Set Topology. Perhaps I shall find a mathematical reader who can critique this idea, and/or extend it into a more complete development.
Consider a Euclidean plane. In the plane, draw a circle. By a slight variation on the Jordan Curve Theorem, the plane is now subdivided into three parts: the interior of the circle, the exterior of the circle, and the boundary of the circle. (I believe that Jordan considered the boundary as belonging simultaneously to both the interior and the exterior.)
The boundary of the circle has a very special, negative property: For any point P in the boundary, it is not possible to find a finite open neighborhood of P that does not contain both interior and exterior points.
Einstein once said, in an essay on G.R., that much information can be derived from negative statements. Specifically, he considered the statement, "There is no perpetual motion," and argued that it was foundational to thermodynamics. He then went on to derive G.R., from the negative premise, "There is no rigid matter." It is the most brilliant single exposition on G.R. that I have ever seen!!! (and the first one that I ever really understood!)
Going back to the circle in the plane, the statement, "For any point in the boundary, it is not possible to find an open neighborhood that does not contain both interior and exterior points," holds only for boundary points; it does not hold for interior or exterior points. For any interior and exterior point P, it is always possible to find a finite open neighborhood of P which contains interior and/or exterior points only 1. Thus, we have a ready means for differentiating between boundary and non-boundary points.
OK. If I now remove the boundary from the plane by deleting all points which have the aforementioned negative property, I am left with two disjoint sets: one consisting only of interior points, and the other consisting only of exterior points. These two sets have the further property that both are open sets. (The boundary, which has a different dimension than either of the other two plane sets, is still a closed set.)
Now, in the next step, I shall further remove all the interior points from the plane and retain only the exterior points. I call the open set consisting of exterior points only, "the Universe." In this little model, the boundary-less hole that remains in the plane is the topological equivalent of a 2-dimensional black hole.
(An analogous operation may be carried on in a three or four dimensional space where the circle is replaced by a sphere or a hypersphere – a “ball,” in topology – with interior, exterior, and boundary.)
The deleted interior of the circle represents the interior of the black hole. Notice that is contains no points whatsoever belonging to the universe. It represents a genuine "elsewhere." To an observer living in the universe, this interior region is not reachable without going outside of the universe. The deleted boundary itself represents the event horizon.
From G.R., we have the definition that the event horizon and the interior of the black hole are “outside” of our universe; they also represent a genuine elsewhere. The event horizon may be classically defined as that surface upon which the escape velocity is everywhere equal to the speed of light in free space. Thus, no information can ever be obtained from the event horizon by any observer in the universe.
Now suppose that such an observer (situated at a safe distance from the hole) has dropped a test probe into the gravitational field of the black hole. He/she watches it fall toward the hole. Probe-time (i.e., time told by a clock on the probe, riding the probe into oblivion) is seen to slow as the probe recedes toward the event horizon. The observer sees the probe becoming redder and slower, always approaching the event horizon but never reaching it.
This situation is entirely appropriate since, for our observer, nothing can exist outside of the universe proper; moreover, nothing can ever enter or leave the universe. By definition, the universe is all there is. If there were something more (say an exterior), it would still have to be part of the universe by simple virtue of the fact that it exists at all. The point here is a philosophical one, but extremely important!
By another, more physical, argument, strict conservation laws do not permit matter/energy to vanish from the universe, either into a black hole or anywhere else. The hole has mass, as attested by its gravitational field. As matter falls into the hole, it adds to the hole’s mass, and the field around the hole increases by an appropriate amount.
From the perspective of the probe, the situation is very different than it was for the observer. The probe experiences itself approaching the hole, falling through the event horizon, and entering the interior of the hole all within a finite (and relatively short) time. But, once having reached the event horizon, it looses all contact with the universe.
Observer-time (i.e., time told by a clock stationary with the observer) would be seen, by the probe, to go faster and faster. And, just prior to entering the event horizon, the entire history of the universe will flash before the probe. Then the probe leaves the universe forever.
Complicated? Yes. Counter-intuitive? Yes. But this is G.R.
I believe that the only element missing from the above description is any reference to the so-called embedding diagram. I chose to describe everything on a plane. Curvature was taken care of by observing the behavior of local clocks on the probe and with the observer, each as seen by the other. But there IS another picture…
Someone has proven that the plane of Special Relativity (the Minkowski plane) may be replaced by a curved surface that takes the form of a rubber sheet deformed by a weight placed on it and viewed from a higher-dimensional Euclidean space. If you do not invoke this higher space, of course, my description is perfectly adequate. In the embedding diagram view, the plane is simply replaced by the rubber sheet, and the curvature becomes “visible.”
I prefer the my description in that it does NOT invoke a higher dimensional space, which is an unnecessary artifact and can lead to confusion if not handled carefully. But, different people have different preferences. In the end, either description seems to work.
1 Another way to look at this situation is to argue that each and every interior point is COMPLETELY SURROUNDED by other interior points; every exterior point is COMPLETELY SURROUNDED by other exterior points; all regardless of their proximity to the boundary! Not easily intuited, I'll grant you; but provable within the context of point set topology all the same!!!
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