wave equation thing

[`hýsøŕ]

i was looking at the official string theory site, over here http://www.superstringtheory.com/basics/basic4a.html, and in the page i linked there it says that the idea of a string uses the usual wave equation, namely d^2 y/dt^2 = v^2 d^2 y/dx^2 (d's are meant to be partials)

 

i remember when i was learning about the wave equation, its derivation involves use of the approximation that there are small vibrations, as it simplifies the differential equation into the one above, instead of a much more complicated non linear one. now if this equation is used to help define the physics of a string, shouldn't this approximation be done away with and the more complicated equation, which takes into account large amplitude vibrations, be used instead?

[studiot]

The derivatives will only be partials in more than one (spatial) dimension.

 

The equation, as written, is linear and also conservative.

 

I confess I am not a fan of string theory but the small vibrations requirement is to ensure the linearity and therefore the conservation.

So small means small enough for the restoring force to be linear.

 

If a non linear equation is used it will be dissipative at best or dispersive at worst, neither of which model observed reality that the strings (or whatever) endure and do not fade away.

 

What other equation would you suggest?

[ajb]

This may be a bit "chicken and egg", but if you pick the simplest action, thinking of string theory as a sigma-model then you get the wave equation as the equation of motion (also you need to take care of the boundary conditions).

 

If you were to start which another action then you could get something else as the equations of motion.

[`hýsøŕ]

@ajb this sounds kinda beyond me tbh lol, i'll take your word for it at least for now

 

@studiot well i tried working out what the wave equation would be, if instead of the approximation of small vibrations, the opposite (namely large vibrations) was used, but i get an unsolvable different equation :S

the equation might not be completely unsolvable but i sure as heck can't get it lol, not even convinced its the right equation, but here it is just for the record:

 

(dy/dx)^3 d^2 y/dt^2 = (T/ρ) d^2y/dx^2 .. T and ρ are constants also with the partial derivatives, aren't they partials because y=y(x, t)? so you'd need a partial to differentiate with respect to x, or with respect to t, seperately

[studiot]

 

aren't they partials because y=y(x, t)? so you'd need a partial to differentiate with respect to x, or with respect to t, seperately

 

 

Yes you are quite right. Silly me.

 

How did you come to a product of differential coefficients?

[ajb]

There are higher order wave equations like the KdV, this also has a Lagrangian formuation. I don't know if you could try to construct sigma models following this. You would have KdV like string theory!

[`hýsøŕ]

@ajb interesting equation o.o haven't heard of this before, btw what do you mean by sigma models? :S

 

@studiot well, here's roughly what i did:

 

here's a derivation of the usual wave equation (for reference sake): http://www.math.ubc.ca/~feldman/m256/wave.pdf

 

about the start of the second page, it's explaining how to get those cos's and sin's in terms of the derivatives of u(x,t). after this it goes on to make the small vibrations approximation where they use the small angle approximation with theta.

So i tried to start off back with all thsoe cos's and sin's and use a different approximation, by using a series expansion of 1/sqrt(1-1/x^2). when i put them back into the soon-to-be differential wave equation, there were terms involving adding two fractions with the derivatives of y on the bottoms, so i tried combining them (giving the product of the differential coefficients) and doing some long .. handwavy algebra I finally got to that equation i listed above namely

(dy/dx)^3 d^2 y/dt^2 = (T/ρ) d^2y/dx^2

 

(oh also the y here is the same as their u, i just prefer y lol)

Edited August 20, 2014 by `hýsøŕ

[ajb]

@ajb interesting equation o.o haven't heard of this before, btw what do you mean by sigma models? :S

For example, bosonic string theory is a sigma model. It is a theory where the classical fields are maps between a two dimensional source manfold and a higher dimensional target manifold. These maps give you the world sheet, i.e. how the two dimensional suerface is embedding into the target manifold. You need some extra structure to define the models, classically you need at least a metric on the source and target. Writting down just about the simplest action you can have gives you the Polyakov action. In reality, you need a bit more than just the metrics to define everything properly (even classically) and have a well-defined (as well as they ever are) quantum theory.

 

Even classical mechanics is really a kind of sigma model. You are thinking of maps between the real line, a loop or an interval and your configuration manifold.

[`hýsøŕ]

umm, i sort of see .. don't quite get much of this though, like what is the configuration manifold? also why the mapping or the target/source thing .. i don't see where all this stuff comes from

[ajb]

... like what is the configuration manifold?

All possible positions your particle could take.

 

also why the mapping or the target/source thing .. i don't see where all this stuff comes from

Really it comes from thinking about physics in terms of geometry. If you look at the formulation of string theory you see that it is a theory of maps from a 2-d surface to space-time. A lot, in fact I would say all most all, of physics can be described geometrically.

 

In classical mechanics, what are trajectories? Well they describe a curve on the set of all possible positions your particle can take. We will take this set to be a smooth manifold and then you see that trajectories really are (smooth) maps from the real line (or an interval) to the configuration manifold: this is just the mathematical definition of a curve.

 

String theory is very similar, you just replace the line with a 2d object! I over simplify, but basically that is the idea.

[`hýsøŕ]

OHH i see, that's an interesting idea imo, thanks. i've heard string theory also predicts branes of higher dimensions, im guessing those would be target manifolds of higher dimensions? like the curve thing, only instead of a line it'd be like a sheet or a volume or.. uhh .. .. something (?)

[ajb]

i've heard string theory also predicts branes of higher dimensions, im guessing those would be target manifolds of higher dimensions?

D-branes are the "sheets" that open string can start and end on. They are really the boundary conditions of your open strings. At first D-branes were not thought of as being very important things, but in the modern setting D-branes are considered just as fundamental as the stings themselves.

 

 

like the curve thing, only instead of a line it'd be like a sheet or a volume or.. uhh .. .. something (?)

This sounds more like the world sheet or world volume.

[`hýsøŕ]

things like these that makes me wish we all had incredibly powerful brains, so I could understand the whole idea at the age of like 20 ish xD