Non-Euclidean geometry

Jeremy Kahn

The nature of light and space

In the mathematical abstraction, it is possible to speak of the ideal space where beams (some projectile phenomenon) are bent by very massive objects, but the space itself is Euclidean.(see Banchoff, p. 190)

But under every conception of modern light physics that I know of, a beam of light bends under the influence of a very massive object because those massive objects actually deform space--this is the insight of Einstein. The conception of space and the definition of light, under this paradigm are parallel (so to speak) in that the shortest distance between two points (in a vacuum) is by definition the path that light travels to get from one to another.

I think the new physicists can get away with this because they do not perceive a beam of light as a stream of particles; instead it behaves as a wave--a wave, rather counter-intuitively, of probability that the particle would appear. My understanding is a bit fuzzy, though, and I don't doubt that there are others in this class who can explain this better or at least more thoroughly than I can.

On this topic I recommend The Cosmic Code, by Heinz R. Pagels, and Relativity, by the man himself, Albert Einstein. Both are eminently readable. Not quite so readable is A Brief History of Time, by Stephen Hawking, which covers a lot of this material in its first four or five chapters but reaches out to incomprehensibility in its last chapters. Or at least that's how I remember it when I read it several years ago.

Thoughts on deformations of space

So if very massive objects deform space, could it be possible to induce a change from an orientable space to a non-orientable space?

It depends on whether the deformations made by massive objects follow the topologist's rules about not breaking surfaces or not. If the deformations indeed follow those rules, there should be no trouble with nasty things like changing regions of space into Klein bottles--as interesting as it might be. But if space can be deformed to the point of breaking off and reattaching to other parts, then we might very well discover that the insides of a particular space might be no different from the outside--we could find that space had changed from an orientable surface to a non-orientable one.

The other possibility, of course, is that we are already in a non-orientable (hyper-)surface. The implication of that is that a traveler going far enough in one direction would return reflected. What a bizarre thought.

Jeremy Kahn x6753

Prof. Banchoff's comments