On Edge

The twisted-universe model I wrote about last time clearly must involve curved space. The idea that space can be curved is pretty familiar by now, mostly because of Einstein’s idea that gravity results from a bending or curving of space by a mass of matter. That seems to be the consensus these days about how gravity works; the most popular alternative, that gravity is somehow transmitted from one mass to another, suffers somewhat from the failure so far to detect any of the “gravity waves” this theory would require.

I always had a hard time figuring out how gravity could have an effect on space, which essentially has no properties of its own to be affected. (Under relativity and quantum mechanics, empty space seems to be occupied by a “quantum vacuum” that does have some properties, but that’s not the same thing as space.) However, when I started thinking about the properties of my twisted space, I realized that any space does have one property: dimension. Which means that the only kind of change you can make to a space is a change of dimension.

As I explained last time, giving a spherical universe a half-twist through the fourth dimension raises the dimension of the universe as a whole to 4. The dimension at any point within the universe would appear to a normal physical observer to be 3, but in fact it would be slightly more than 3; we’ll say it’s D=3+(n<1). And if you add up that n<1 for all the points (at some arbitrarily selected but uniform scale, such as a light year or a parsec) in an orbit of the universe, the sum will be 1.

Now, even though this cosmos is finite, it’s still very large. So the n<1 – which I’m going to declare a fundamental universal constant, and call Ü, mostly because I like umlauts – is going to be very small, almost ininitesmal. The 3+Ü that exists at any point would then be the natural dimension of space in this bent universe of mine.

This whole idea of a non-integer dimension is exactly what Benoit Mandelbrot means by the term “fractal” that he coined to describe objects with a “fractional dimension.” And he has demonstrated that a very wide variety of objects are fractals, which means that many (perhaps most) of the objects we think of as, say, three-dimensional are in fact three-plus dimensional. The more complex the object, the higher the fractional excess; so a big, many-branched tree would be “more fractal,” if you will, than, say, a bowling ball.

But it seems likely that the fractional dimension of even a fairly smooth 3D+ object, like a planet, would have to be greater than the near-infinitesmal value of Ü. And as a result, the planet would “stretch” the dimension of the space around it, causing the kind of contour that Einstein’s conjecture associates with gravity. This would be true even with very small objects, which would account for the kind of “clumping” that scientists believe took place in the early universe, leading eventually to the formation of galaxies and so on.

There’s something else that the structure of the twisted-universe model might help account for that has puzzled me for a long time. I think some visual aids may help here.

We often hear airplane pilots talking about flying “straight and level.” But they’re actually doing anything but. What they’re really doing is flying at a constant altitude above the Earth’s surface. Because the Earth’s surface is curved, however, the plane’s actual path is also curved, as shown above. What would happen if an airplane really flew “straight and level” looks like this:

Now, the thing I’ve wondered about for years is this: We all know that the speed of light is a sort of universal speed limit, that nothing can go faster without violating all sorts of natural laws. But I’ve always wondered why it’s precisely the speed it is, 299,792.5 kilometers per second, or about 186,000 miles per second.

We’re used to the idea, again thanks to Einstein, of light traveling a curved path around massive, high-gravity objects. But since I’m supposing here that all light must travel a curved path in a curved, twisted universe, the “normal” path of light would look something like this:

Obviously, the curvature of this “universe” is highly exaggerated, but it illustrates how the rays of light in a sense “flow” along the contour of the space. What I’m going to suggest is that the speed at which light (and of course other forms of radiation) travels is actually determined by that contour or curvature, because if it travelled at a higher speed, this would happen:

What this would actually mean is hard to say. It might mean that the energy disappears into the fourth dimension, or it could even mean that it exits the universe, whatever that might entail.

Mention of the fourth dimension brings up one final point I want to make before closing this largely pointless expostulation. You’ll remember the Möbius strip from last time:

In looking at this illustration, I want you to imagine that the strip is actually transparent, because what we’re talking about here is empty space, not paper. So there’s really nothing separating point A from point B, or C from D, except space; or rather, except the twisted structure of this space. But the separation is nevertheless complete and inviolable: The only way to get from point A to point B is to go around the strip; you can’t go through it.

Why not? Well, if you travel around the strip from A to B, you’re in effect adding up Ü units, or in a sense travelling uphill dimensionally. By the time you reach point C, you’re in a dimension that’s 0.5 higher in relation to A, and when you reach B, the space you’re in is a full 1 dimension away from A. It’s still 2+ÜD from a local point of view (in the illustration; in the twisted universe, it would be 3+ÜD), but A is 3D from the perspective of B (4D in the real universe), and vice versa. So naturally, there’s no way to perceive one space from the other, much less to go there directly.

This is precisely what constitutes the boundary or “edge” of the universe, this dimensional barrier. And what that means is that every point in the universe is on the edge of the universe.

Somehow, that reminds me of the famous Hermetic saying quoted by Giordano Bruno and Pascal, among others: “God is an intelligible sphere whose center is everywhere and whose circumference is nowhere.” In the twisted universe, the circumference is everywhere, but I’m not sure whether there’s a center anywhere.

The Vast Unknown

As we know, there are known knowns: There are things we know we know. We also know there are known unknowns; that is to say, we know there are some things we do not know. But there are also unknown unknowns, the ones we don't know we don't know.

- Former Secretary of Defense Donald Rumsfeld, news briefing, Feb. 12, 2002

Some people in the media ridiculed the above statement when "Rummy" said it back in 2002. And as a response to questions about the Iraq War, it certainly had some inadequacies. But from a purely epistemological point of view, it's not entirely lacking in merit. Certainly, "we know there are some things we do not know," and I would add that there are some things we cannot know.

I mean this in a strictly rational, scientific sense (at least for now): There are some things that are inherently unknowable. Heisenberg's principle of uncertainty (or indeterminacy) provides a famous example: In quantum physics, it's impossible to know simultaneously the position of a particle and its velocity; the more precisely you measure one property, the less you know about the other. And Werner Heisenberg, the physicist who first formulated this principle, called attention to the fact that this uncertainty isn't just a result of insufficiently precise measuring tools or processes, it's a fundamental property of quantum systems; that is, of matter as such.

Equally fundamental, and possibly even more so, are Kurt Gödel's incompleteness theorems. Without going into great detail (partly because trying to do so would probably give me a severe headache), these theorems prove that many formal logical systems cannot prove all of their own possible axioms and/or cannot prove their own consistency. One reading of the implications is that if the result of a formal logical proof is something we can label as "knowledge," Gödel's theorems show that there must remain some things that are "unknowable" in this sense.

These examples may seem somewhat nitpicky or irrelevant: Surely it doesn't matter much in the larger scheme of things if we can't precisely locate every particle of matter/energy in the universe or if we can't absolutely prove or disprove every possible statement.

But what if one of the inherently unknowable things is the largest scheme of things itself - the universe? In other words, what if there's an inherent unknowability at the smallest scale, the microcosmic, and also at the largest, the cosmic, and an unprovability about any guesses we might make as a substitute for direct knowing?

Within my limited and decidedly math-impaired understanding, this does appear to be precisely the case.

When we look up at the sky on a clear night, we can see millions of glowing objects in the sky - and not one of them is actually located where it appears to be. The reason, of course, is that it takes time for light to travel through space, and during the time the light is traveling from its source (a star, galaxy or planet, for instance) to our eyes, we and that source are moving. Even our nearest neighbors in space are far away enough for there to be a time lag, and thus a displacement, between their emission of light and our reception of it: It takes light about 9 minutes to travel from the sun to the Earth. And obviously, the farther away the emitter is, the longer the time lag and the larger the spatial displacement become.

What this means is that the picture our perceptions (even as we extend them through technology such as telescopes) give us of the universe is inevitably geocentric. We can adjust the picture to some extent, in effect creating a mental or conceptual map of the real current locations of celestial bodies, and this procedure obviously works well enough for us to send space probes to the Moon, Mars and so on. But as the distance involved increases, so must the uncertainty of our conceptual map.

In other words, any model we propose for the structure of the universe in its entirety will always have a major theoretical component. It's likely that we are safe in supposing that the natural physical laws that operate within our zone of certainty will also operate outside that zone, so it's fairly safe to hypothesize that the most distant regions of the universe will be like ours in a general, qualitative sense. But we cannot know the precise structure or appearance of those regions, or of the universe as a whole, in "real time," that is, as they are at any one moment.

One implication of this unknowability is that we (and by "we" I mean all intelligent beings who happen not to be blessed with supernatural omniscience) may ultimately have to accept multiple, and possibly mutually contradictory, models of the universe. As long as a conceptual model doesn't conflict with natural universal laws, insofar as we can understand those, and does account as much as possible for the phenomena that we can observe directly, we probably must accept that it's "true," even if there exist one or more alternative models with equal claims to be "true."

In some sense, we already do live with multiple universes - that is, with multiple explanations of the structure and origin of the cosmos - though there's considerable argument about which, if any, are true. And maybe a realistic understanding of the limits of certainty should prompt us to be a bit less fiercely argumentative about our varying understandings.

Perhaps to add fuel to the fire, or perhaps to help diffuse it, I'll be writing next time about an alternative model that as far as I know - and obviously, that can't be very far - hasn't previously been proposed and is very unlikely to be provable.