Question about extra dimensions

breadmonkey

I've just read 'The Universe In A Nutshell' and Hawking keeps saying something along the lines of "but these extra dimensions would be curled up so small that we wouldn't notice them".

What does this mean?

Son Goku

breadmonkey wrote:

I've just read 'The Universe In A Nutshell' and Hawking keeps saying something along the lines of "but these extra dimensions would be curled up so small that we wouldn't notice them".

What does this mean?

A basic explanation is that our universe is very thin in those directions, which allows for very little movement through them for large objects. Although I admit this doesn't explain why he says curled up, instead of just thin.

I can give a much better explanation, a correct one actually. Just one question though, have you studied special relativity and complex analysis(or even just complex numbers, particularly their depiction as a plane, i.e. Argand diagram)?
(Sorry for the intrusive questions, but if you have done these things, then the explanation is fairly simple.)

breadmonkey

Unfortunately I haven't studied General Relativity or complex analysis but I have studied complex numbers so I do know what an Argand diagram is etc.

Son Goku

breadmonkey wrote:

Unfortunately I haven't studied General Relativity or complex analysis but I have studied complex numbers so I do know what an Argand diagram is etc.

I wouldn't worry about General Relativity, you only need Special Relativity to understand the explanation. However after thinking about it, you don't even really need Special Relativity that much.

I'll post the explanation later today.

Professor_Fink

I'm sure Son Goku will provide a better explanation, as he always seems to, but here is a quick answer.

Particles have internal degrees of freedom, this is called spin for massive particles and helicity for massless particles. In bosonic string theory the first excited state of a string has the spin properties of a massless particle (more technically it's internal states form a representation of the spatial rotation group SO(D-2) rather than SO(D-1) where D is the number of dimensions).

The mass for particles is given by m^2 = 1/a' (N + (2-D)/24). Here a' is a constant known as the Regge slope, and N is the excitation number.

Since it is the first excitated state we are concerned with, N=1. Thus m^2 = 1/a' (26-D)/24.

Clearly the only way to make this zero is to set D=26. This is where all those extra dimensions come from. We can do the same treatment using super-strings (strings incorporating super-symmetric) and we get D=10. In any case we are stuck with D>4, which is what we observe.

So how do you make a large number of dimensions look like smaller number of dimensions? The answer is to roll up the extra dimionsions and have them on a different lenght scale to the dimensions we experience. One way of doing this is with Calabi-Yau compactification.

I know this isn't a particularly instructive answer, but to work through the maths is pretty nasty. Hope it's of some use.

breadmonkey

Thanks for that. Why do you set D=0?

Son Goku

Alright here is an attempt at an explanation:

Think of a sphere and an infinite plane. Both a sphere and a plane are 2-D, but a sphere is a finite(compact) 2-D space, where as a infinite plane isn't.

Now let's call the sphere the finite companion of a plane.
Every plane has a companion like this, even if the "planes" are infinite 3-D space or infinite spaces of even greater dimension.
It even works if the "plane" is the complex plane. Or course then it'll be a sphere with i^2=-1 built into it.

First we start of with spacetime from Einstein's special relativity. Spacetime is a 4-D space where your position is given by 4 coordinates (t,x,y,z). t for your position in time and x,y,z for your position in space.

Now, in the following pictures I'm not going to draw the z axis, because I can't draw 4-D, but just imagine it's there.



So basically you take three complex planes and attach them to every point throughout spacetime. One complex plane has 2 dimensions, so three complex planes have 6 dimensions. So every point in spacetime has 6 dimensions attached to it.

Now take this six-dimensional object, now to be found at every point, and turn it into its finite companion. Now punch three holes in it and you've got a Calabi-Yau space. Do this to the complex planes at every point and you'll eventually end up with spacetime having a 6-D Calabi-Yau space attached at every point.

So now you have the 4-D from spacetime and the 6-D from the Calabi-Yaus, giving you 10-D.

Now what's with the smallness?
Well basically, as I said the Calabi-Yaus are the finite cousins of the three complex planes. So we already know the extra dimensions are finite, making them much smaller than the other 4. Punching the three holes into the Calabi-Yau makes it even smaller, because it requires it to curve up on itself a bit.
Then we shrink the Calabi-Yaus until the point the physics needs them to be shrunk to give sensible answers. At this point the Calabi-Yaus are micro-microscopic, so only things like Strings are able to vibrate through them or "feel" them to put it another way.
(The thing to remember is that the Calabi-Yaus still have i^2=-1 built into them because the were built out of complex planes, not regular planes.)

I hope that made a lick of sense, but that's modern theory for you.

Professor_Fink

breadmonkey wrote:

Thanks for that. Why do you set D=0?

I didn't set D=0, because this would give a non-zero mass for the N=1 case, which we know must have zero mass, because the internal degrees of freedom form an SO(D-2) group.

Tar.Aldarion

I saw some little flash thing that did a fair job of explaining dimensions, see if I can find it tomorrow.

Professor_Fink

Tar.Aldarion wrote:

I saw some little flash thing that did a fair job of explaining dimensions, see if I can find it tomorrow.

I think I know the one you mean. It's called imagining the tenth dimension. It's horribly innacurate, mixing up space-like and time-like dimensions. Also they come up for a justification as to why there are ten based on "now we can't imagine any more" which is complete rubbish.

Tar.Aldarion

Oh no, not that, I think I posted in that thread or some thread on baords(maybe in The A/A forum) pointing out innacuracies. I don't know what site this one is on though, hmm.