USS Clueless - Non-Euclidean space

Stardate 20031031.1517

(Captain's log): I'm trying to explain something to Mark that I myself don't totally understand. (Which is always fun.) I'll do the best I can, but you get what you pay for. His original letter:

I have often seen Einstein's notion of curved space illustrated as a sheet where the sun lies in a big dent and the earth rolls around in that dent or in a trough.

But this is false. Space is not a smoothe sheet, but more like all ocean. How is it possible to curve the vacuum/space when it is filling up all "spaces", so to speak?

Also, I often see black holes illustrated as having funnels sucking in everything, but they could not be so since every point of a black hole would radiate attractive force and suck in directly, not tangentially.

My first response:

Have you ever heard of a book called "Flatland"? It's considered a milestone.

It was an attempt by a mathematician to explain how there could be four spacial dimensions. He does it by analogy; he creates a 2-dimensional world and puts Mr. A. Square in that world, who until a certain point encounters noone except other 2D people. Then one day A. Square meets a sphere, a 3-dimensional being.

All those pictures you're talking about were analogies, not literal descriptions. The curvature happens in a fourth dimension.

What the funnel picture you're seeing shows is how such curvature would have manifested in Flatland.

By which I meant that it was how Einsteinian space would look if the space was perceived as 2D by its residents. Mark responded:

Yes, I recall Flatland, but my question has to do with wondering if it is really possible to mathematically describe "curved" space when space can't really curve if it is actually oceanic.

The Earth doesn't roll along in a trough but would simply be tunneling along. How can curvature apply in a fourth dimension? I realize Time is called the fourth dimension, but it doesn't have a "real" dimension in space. Has not any spatial quality.

Mark is trying to apply Euclidean concepts to space. Which doesn't work, because in General Relativity space is non-Euclidean.

It's impossible to describe curved space in Euclidean geometry, but it is definitely possible in some non-Euclidean geometries.

Both theories of Relativity ended up with extremely surprising conclusions which weren't related to the original problem they tried to solve. The Special Theory was intended to explain the Michelson-Morley experiment, by showing how the speed of light could be the same in all inertial frames of reference, but it also told us that energy and mass are the same thing and that either could be converted into the other.

Likewise, as I understand it, General Relativity began as a way of trying to expand Special Relativity to deal with non-inertial frames of reference, and in particular to try to deal with certain esoteric issues about gravity and non-gravitational acceleration which I don't really want to try to go into, and ended up being a comprehensive theory of gravity, the only one we have to this day. Part of why I don't want to go into details is that I don't really understand General Theory. But I do know that one of its consequences was that space is non-Euclidean, and that gravity involves distortion of space-time. In terms of the General Theory, gravity isn't really a "force" at all, in the sense that the Strong Force, Weak Force and Electric Force are. I believe that gravity was described as being a side effect of the way that mass causes distortion of the fabric of spacetime.

Saying that time is "the fourth dimension" is not really helpful. It's not really clear that time is a dimension (though I believe that General Relativity implies that it is) but even if it were would be a fourth dimension, not the fourth dimension, and it would not prove that there are no others besides the three spatial dimensions and possibly time as a fourth. There's nothing that says there could not also be additional spatial dimensions.

Some new theories which are trying to explain just how mass and energy can be the same thing, and what mass truly is, and how mass-energy conversion takes place, require that there be many more spatial dimensions. I believe that one of them requires 11 dimensions, with 7 new spatial dimensions in addition to the four we already know about.

Euclidean geometry is famously based on five axioms, but there's an unspoken axiom of uniformity, which assumes that the universe is geometrically the same at every location and at all scales.

That's not true for some non-Euclidean geometries. In Euclidean geometry, lines will either intersect once or not at all. But in some non-Euclidean geometries, they can intersect two, or three, or an arbitrarily large number of times.

In Euclidean geometry, the fifth axiom was: if there is a line on a plane, and a point on that plane which is not on that line, then there is exactly one line on that plane passing through that point which is parallel to the other line.

For a long time, it seemed to many as if that didn't need to be an axiom, and much effort went into trying to prove it using the other four axioms, all of which failed.

In the 19th century, some mathematicians decided to try a different approach. One can prove a statement is false by presuming it is true and showing that leads to a contradiction. (Or vice versa.) So what they hoped was that they could try to show that the fifth axiom didn't need to be an axiom by showing that every alternative statement of it led to a contradiction. If successful, that would mean it was tautological and thus didn't need to be axiomatic.

But much to their surprise, when they changed it they discovered that the resulting geometries were consistent -- and useful, and fruitful, and different from Euclidean geometry. Two of the more famous results were spherical geometry and hyperbolic geometry, but in fact I believe there are potentially an infinite number of non-Euclidean geometries.

Spherical geometry is done on the surface of a sphere. In all of these kinds of geometries, some standard terms have to be redefined a bit, and in spherical geometry a "straight line" is a circumference of the sphere. Which means it is finite but unbounded; it has no endpoints, but it is not infinitely large.

In spherical geometry, the proper statement of the fifth axiom would be: ...there are no lines passing through the point which are parallel to the other line. (Though I don't think it's actually an axiom.) In spherical geometry, any two "straight lines" (i.e. circumferences of the sphere) which are not congruent always intersect exactly twice.

I had thought that hyperbolic geometry in 2 dimensions was performed not on a flat plane, but rather on a hyperbolic surface which is shaped something like a saddle (decribed as z=x*y). But I just looked it up in Encarta, and their description of it (which they also referred to as "Bolyai-Lobachevsky geometry") didn't sound like that, so I don't really know what it is. Nonetheless, I believe it's the case that the geometry is two-dimensional, but the plane itself requires 3 dimensions to exist because in Euclidean terms the hyperbolic plane is curved. Anyway, it's likely that anything or everything I say about hyperbolic geometry here is probably totally wrong.

In spherical geometry, there is no assumption of uniformity, quite. In Euclidean geometry, the sum of the angles of a triangle always add up to exactly 180 degrees, no matter where it is nor how large it is. But that sum is not a constant in spherical geometry. The sum varies from 180 to 360 540 degrees as a function of how large the triangle is compared to the underlying sphere. (Actually, it limits at 180 or 360 540 degrees but cannot exactly equal either.)

There's an interesting relationship between spherical geometry and Euclidean geometry: as the ratio between the diameter of the sphere to the objects being manipulated on the surface of the sphere rises (i.e. as the objects get relatively smaller compared to the sphere) then the differences between spherical geometry and Euclidean geometry get smaller and smaller. As the sphere limits at being infinitely large, they become identical.

The particular curved plane which I thought formed the basis for hyperbolic geometry requires three dimensions to describe. I thought that the formula for it is z=x*y, and though the geometry is 2D, operating on a flat surface, it requires three dimensions to exist. But I may have been misinformed.

Spherical geometry definitely does require an extra spatial dimension. It describes 2D objects on the surface of a sphere, but the sphere cannot exist unless there's a third dimension to contain it.

But all of these geometries can be extrapolated to an arbitrary number of dimensions. Euclidean geometry was originally described in terms of 2D objects on a plane, but it makes sense describing 3D objects in space as well. It also makes sense describing 4D objects in a hyperspace.

A circle is described as all the points on a plane which are equidistant from a single point. A sphere is all the points in space which are equidistant from a single point. A hypersphere is all the points in a 4-dimensional hyperspace which are equidistant from a single point. The circumference of a circle is a curved line; the surface of a sphere is a curved plane. The surface of a hypersphere is a curved 3D space.

Spherical geometry can be extrapolated to describe manipulation of 3D objects on the "surface" of a hypersphere. It takes 4 dimensions to hold the hypersphere, but the objects being manipulated only have 3 dimensions, and the "surface" of the hypersphere is a 3-dimensional space.

Abbot's "Flatland" described Mr. A. Square living on an infinite Euclidean plane. But A. Square could have been living on the surface of a sphere. In that case, he'd think he was flat, and he'd see everything around him as flat, but as he explored further he'd start noticing things that didn't make sense. For instance, he might discover that if he traveled in a straight line, eventually he'd return to his starting point without turning around. Nonetheless, his direct local examination of his environment would seem to be 2D, and A. Square himself would still be 2D even though the sphere on which he lived was 3D. And if he was small and the sphere was huge, he might live his entire life and never notice that it wasn't actually a Euclidean plane.

By the same token, his cousin A. Cube could live in an infinite Euclidean flat space, but could also be living on the "surface" of a hypersphere. A. Cube would be 3 dimensional as would be all the things around him and the space in which he lived, but he might discover that if he traveled in a "straight line" in any direction far enough he'd return to his starting position without having turned around. That's the nature of the spatial "surface" of a hypersphere, just as it is for the flat surface of a regular sphere. But if the hypersphere were huge, and if A. Cube didn't wander far, he might spend his entire life thinking he was in an infinite flat Euclidean space, instead of on the finite unbounded "surface" of a hypersphere.

The actual geometry of space is far more complicated and messy. It is a "space", in the sense that geometry uses the term; it's 3 dimensional and contains 3-dimensional objects. But it is curved and warped in ways which require 4 spatial dimensions to contain. The 4th dimension exists but isn't directly apparent to the 3D objects inside that space (i.e. Mr. A. Cube and you and me) and it may take observation of unusual conditions to notice that all is not as Euclid would have predicted.

One of the ways we evaluate esoteric theories is to use them to make surprising predictions, and to see whether the predictions are correct. Newton's theory of Universal Gravitation was a triumph, but it wasn't perfect. Predictions of Mercury's orbit based on it didn't come out right. General Relativity fixed that, by explaining that Mercury was close enough to the sun so that the spatial distortion caused by the sun's mass altered the calculation. (Or so I've read.) General Relativity also predicted that the spatial distortion caused by mass should bend light. Light follows a straight line, but the spatial distortion caused by mass could make it seem to curve when viewed in Euclidean terms.

That prediction was spectacularly vindicated by observations taken during a solar eclipse. There were direct observations of certain stars whose apparent position was very near the eclipsed sun, and their apparent positions were shifted by exactly the amount the theory predicted, because the light was being "bent" by the mass of the sun. Since then, more powerful telescopes have found several cases of what's now called "gravitational lensing" where we can see several different images of the same object, usually quite heavily distorted, in different apparent locations because the light was bent by an intervening mass.

It's really hard to get to the point where one can conceptually understand four or more spatial dimensions. I've been working on it for several decades and I think I can do it. Abbot's "Flatland" was intended to help people with that exact problem, because in the end A. Square excitedly starts asking the visiting sphere whether there might not be even more than three dimensions, something the sphere hadn't actually thought of. Abbot is trying to use the analogy of 2D-to-3D as a way of getting 3D objects, like the sphere and Abbot's readers, to conceptualize a fourth spatial dimensions.

If you can't do it, then the entire idea of "warped space" will seem inexplicable.

But warping of space is an established observable fact. Anyone who tries to claim that it isn't happening has a lot of explaining to do, because there's now a lot of good astronomical observations which currently can only be explained in terms of the kind of spatial distortion that General Relativity predicts.

It's also entirely possible to describe it in mathematical terms, as long as you're willing to abandon Euclid.

Update: Dave correctly points out that in spherical geometry, the sum of the angles in a triangle limits at 540 degrees, not 360 degrees.

Update 20031102: A reader provides some better information about General Relativity.

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