Angular Dimensions

[Enola Straight]

I was under the impression that dimensions wer linear; straight line measurements, representing length, width, depth, and time, at right angles to each other.

When Kaluza Klein theory arose, adding a fifth dimension to 4D General Relativity, it produced Gravity AND Electromagnetism.

Adding more dimensions gave rise to Supergravity/Supersymmetry/Superstrings/M Theory, encompasing all forces.

Dimensions 5 and up are considered "compactified"...curled up on the order of the Planck Scale, about 10^-32 meter.

Are these curled up dimensions not linear, but angular...something like orbits and spin?

Is the lack of curled up dimensions one of the reasons quantum gravity is not observed?

[Surlethe]

I'm not particularly sure, but I've tended to think of any unobservable dimensions as cyclic, so that after a certain distance traveled in that dimension you come back to your starting point. Does anybody know if that's at all accurate?

[Winston Blake]

I'm not particularly sure, but I've tended to think of any unobservable dimensions as cyclic, so that after a certain distance traveled in that dimension you come back to your starting point. Does anybody know if that's at all accurate?

I don't understand any of it, really, but that was my understanding too.

A heuristic I came across is to imagine an ant walking along a thin cylinder. From a long distance, the ant appears to be walking in a straight line (1D). Up close, however, it can actually walk in two dimensions - along the cylinder or around it. The round direction in this case is compact at long distances.

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One point I've wondered about is whether you can have velocity in these other dimensions. For example, in the game Portal, if you shoot a portal onto the ceiling and another onto the floor, you could fall through at any velocity. If the two portals were extremely close to each other, then 'you' would stay in the same place, but you could still have any velocity.

So if you have a electron, could it be moving at near the speed of light in a fourth spatial dimension, which is so tiny that the particle appears to be stationary in 3-space? If it accelerates in this dimension, does it still radiate energy? Is it even correct to think of these extra dimensions as spatial?

[Kuroneko]

In terms of manifold (as the context you're talking about explicitly treats spacetime as a manifold), the number of dimensions is the number of independent vectors needed to span a tangent space of that manifold. Without delving too deeply into all of that, let's try a more pedestrian example.

Think of a circle. If you look very closelely at it, by which I mean your scale is small compared to it circumference, it will look straight, just as the Earth may look flat if your observations are confined just to your immediate neighborhood. Directions on the circle at a point may thus be described as vectors on a tangent line at that point (e.g., think of a circular orbit--velocity vector is tangent to the trajectory). Thus, at any given point on a circle, you need only one vector to span the "tangent space" (here, a line), from which we can conclude that the circle is one-dimensional. Repeating this for, say, a sphere, gives the "tangent spaces" as planes tangent to the sphere. The planes are two-dimensional, so the sphere is two-dimensional. [1]

In general, a "manifold" is a space that locally "looks like" Euclidean space [2]. You can think of it simply as this: at any point, there is a "tangent space" of all possible directions from that point that we can represent as ordinary flat space.

Dimensions are "linear" in the sense that each tangent space is linear--they are vector spaces. But there is nothing problematic for the manifold to be of finite size, or finite in one direction only. Also notice that manifold are required to "look flat" only locally; there is also no problem for them to twist and turn in various ways. This has nothing to do with spin per se [3].

As for non-observed dimensions for our spacetime, they better be small--otherwise, we'd have already noticed them.


[1] This is sometimes a source of confusion: depending on context, a standard sphere may be called either a "2-sphere" (because it is a two-dimensional manifold) or a "3-sphere" (because it 'lives' in Euclidean 3-space). Personally, I find the former convention preferable, because it is independent of embedding in any higher space.
[2] I really should say "pseudo-Euclidean", but the distinction isn't important here. For the topologically inclined, "looks like" should be read as "homeomorphic to".
[3] Indeed, Kaluza-Klein theory theory cannot geometrically represent spin at all, nor can GTR or any other metric theory of gravity. This doesn't mean that we cannot have a theory of gravity in which gravity couples to spin, however, and indeed is a generalization of GTR called Einstein-Cartan theory which this occurs.

[Kuroneko]

So if you have a electron, could it be moving at near the speed of light in a fourth spatial dimension, which is so tiny that the particle appears to be stationary in 3-space?

Sure; there's no reason for this not to be the case, if you have a particle in more than three dimensions. Note that in Kaluza-Klein theory, no particle actually "lives" in four spatial dimensions in the ordinary sense--instead, ordinary (3+1-dimensional) matter and charge is represented as 4+1-dimensional vacuum curvature.

If it accelerates in this dimension, does it still radiate energy?

Well, that depends on the theory. In Kaluza-Klein theory, it radiates charge.

Is it even correct to think of these extra dimensions as spatial?

Geometrically, yes. Physically, with some caveats as illustrated above.

[Enola Straight]

In general, a "manifold" is a space that locally "looks like" Euclidean space [2]. You can think of it simply as this: at any point, there is a "tangent space" of all possible directions from that point that we can represent as ordinary flat space.
.

If higher dimensions are curled up to Planck size...the smallest physically possible size...there is no Euclidean-looking local flat space.

[Surlethe]

In general, a "manifold" is a space that locally "looks like" Euclidean space [2]. You can think of it simply as this: at any point, there is a "tangent space" of all possible directions from that point that we can represent as ordinary flat space.
.

If higher dimensions are curled up to Planck size...the smallest physically possible size...there is no Euclidean-looking local flat space.

Sure there is. Planck size is finite, so you can still 'zoom in' further mathematically. The local "Euclideanidity" is a property of manifolds, which are used to model space.

[Enola Straight]

Mathematically, yes...physically, not really.

Around the Planck scale, such basic concepts as left/right, back/forth, up/down, even before/after get mixed up in the quantum foam at that level. Imagine how mixed up it gets with an 11d manifold.

[Kuroneko]

Let's not get too far ahead of ourselves. We're talking about interpretations of physical theories, most of which happen to employ manifolds. Both GTR and Kaluza-Klein are classical theories, in which quantum fluctuations don't even exist. And even in quantum theories, space and time themselves do not have to be quantized (e.g., a free particle can have any of a continuum of positions and energies, although they become quantized under a potential). This is also true of super[symmetric]-string theories and so forth mentioned in the OP. Although there are (for example) QFTs in which space is not continuous, the point is that this is theory-dependent.