Kaluza-Klein theory

[PWrong]

What would forces be like in a simple Kaluza-Klein sort of universe? For starters, lets look at gravity with n extended dimensions, and one curled up dimension. In other words, the universe is wrapped around a cylinder, cubinder, or some similar rotatope. On a large scale, we see nD behaviour, but on smaller scales, we get (n+1)D behaviour. For convenience, let the circumference of the extra dimension be exactly 1.

I'm making a lot of assumptions here, but the way I see it, there are many ways a graviton can get from A to B: a direct route, one clockwise loop, two clockwise loops, e.t.c.
The more loops you make, the further you travel and the weaker the force. We have to sum the force for k loops over all integers k (negative k implies anticlockwise loops). So the force between two particles should be something like this:
Sum over k from -inf to inf:
r/(r^2 + k^2)^[(n+1)/2]
For n=3, this sum turns out to be an elementary function of r, and it also has a nice taylor series with zeta functions.

Does anyone know about this stuff? I hope I'm on the right track, but if not I might follow it anyway.

[jinydu]

Whoa, you've studied Kaluza-Klein theory, PWrong? I admit I haven't.

[PWrong]

Nope, I just read about in Brian Greene's "The Elegent Universe". I've never seen the maths behind it, I'm just making it up . My example is a lot simpler than the real Kaluza-Klein theory; a completely flat, cylindrical universe, no quantum stuff, no calabi-yau shapes. It should be simple enough for me to deal with using only 1st-year maths and mathematica.

I think it's a bit like being in a corridor with mirrors on each side. You can see infinitely many copies of any object, getting progressively smaller. Yet the total amount of light absorbed is finite, because the sum converges. Gravity in a cylindrical/cubindrical universe would work the same way. The total force is just the sum over all the "copies". Each copy is further away than the last, so the sum converges.

If you've studied infinite series in maths, it's pretty simple to find an expression for the total force, and prove it converges with the comparison test. Of course, it's much harder to actually sum the series. I have no idea how mathematica does it. Unfortunately, it can only find an elementary function when n is odd.

The next step is to find out how planets behave under this force. It's apparently possible to do this for any given force, although it involves integrating a lot of stuff, and I don't know if even mathematica will be able to cope.

[thigle]

quote1: "...a lot simpler than the real Kaluza-Klein theory; a completely flat, cylindrical universe..."

quote2: "...I think it's a bit like being in a corridor with mirrors on each side. You can see infinitely many copies of any object, getting progressively smaller. Yet the total amount of light absorbed is finite, because the sum converges. Gravity in a cylindrical/cubindrical universe would work the same way..."

ad1: why is cylindrical universe considered flat ? because just 1 dimension is curved while others are not?
ad2: how do you get from 2 flat facing mirrors to cylinder ? is cylinder inwardly mirriring ?

ad1: well, topologically, you get a klein bottle (universe?) by taking your cylindrical universe and joining its 2 circular-edges with a twist. a torus with a twist. or you deform the circle into lemniscate by twisting it around any of its diameter axis, through the centre of the circle by pi, and then join the two 8-like edges with a half twist.

a major question (for my imagination faculty) is then: what kind of projection is happening in/out a klein-bottle with its surface being a mirror ? is there a mapping that inverts klein-bottle into itself (around its singularity - a circle of ambiguity for certain embeddings into 3space)? some self-reflexive structuring? i mean, what happens to the total amount of light absorbed ? is it still finite ?

ad2: your 2 facing mirrors can be interpreted as the complex numbers. ( and, after t smith & onar aam, Reals as 1 mirror, Quaternions as 3 mirrors, and Octonions as an inwardly mirroring tetrahedron. (whatsmore, sedenions are a funky mirrorhouse !) does it help anyhow ?
well, i hope this ain't too off topic.

[PWrong]

ad1: why is cylindrical universe considered flat ? because just 1 dimension is curved while others are not?

To be honest, I'm not sure why. But if you draw a triangle on a cylinder, the angles add up to 180 degrees. This isn't true on a sphere.

ad2: how do you get from 2 flat facing mirrors to cylinder ? is cylinder inwardly mirriring ?

My analogy wasn't completely accurate. In the mirror corridor, light may bounce several times off the mirrors before it reaches your eyes. That makes it look like there are infinitely many copies of any object. Similarly, on the cylinder, light can loop around several times before reaching your eyes, while still travelling in a straight line. Again, you can see infinitely many copies of an object. If you don't see what I mean, try this experiment. Roll up some paper around a cylinder, and draw two points. Now see how many straight lines you can draw between the two points.

ad1: well, topologically, you get a klein bottle (universe?) by taking your cylindrical universe and joining its 2 circular-edges with a twist. a torus with a twist. or you deform the circle into lemniscate by twisting it around any of its diameter axis, through the centre of the circle by pi, and then join the two 8-like edges with a half twist.

That's a bit too complicated mathematically. Remember, we're trying to look at gravity in this universe.

i mean, what happens to the total amount of light absorbed ? is it still finite ?

Well, physically speaking, the amount of light absorbed can't possibly be more than the amount emitted in the first place. But when dealing with infinite series, it's a good idea to be careful.

ad2: your 2 facing mirrors can be interpreted as the complex numbers. ( and, after t smith & onar aam, Reals as 1 mirror, Quaternions as 3 mirrors, and Octonions as an inwardly mirroring tetrahedron. (whatsmore, sedenions are a funky mirrorhouse !) does it help anyhow ?

The mirrors are only an analogy to explain why there are infinitely many copies of an object. Complex numbers don't really apply here. Vectors are much more useful.

Mathematica solved the equation (numerically of course), and it turned out exactly the way I hoped . At large distances, planets move in nD orbits. At smaller distances, they move in an (n+1)D orbit. I'm surprised at how easy this is.

I might try Schrodinger's equation next.

[bo198214]

Of course, it's much harder to actually sum the series. I have no idea how mathematica does it. Unfortunately, it can only find an elementary function when n is odd.

Due to some research its now possible to (alogrithmicly) decide whether a hypergeometric sum has a closed form or not and if it has one to give the one, and if it hasnt then give a proof for it.
Hypergeometric means that the quotient of two consecutive summands is a rational function in the summing index.
Thatswhy you get a solution from mathematica for odd n (because (n+1)/2 is then integer, and the quotient {(r^2 + k^2)/(r^2+(k+1)^2)}^[(n+1)/2] a rational function in k).

For even n I would guess there isnt any closed form.