Classical Strings - constructing explicit solutions

[Illuminatum]

Hi all,

Can anyone advise me on the following...I'm trying to get a more intuitive feel to classical strings and have always found following exercises in textbooks and online lecture notes to be useful. However, I'd like to have a sanity check on some of my solutions and ask for some help with a system I'm having trouble describing...

I'm notionally working with the Nambu-Goto action,
$\int d\sigma d\tau ~\sqrt{-h}$
for $h$ the determinant of the pullback of the space-time metric onto the worldsheet. This has eom (at least for a fixed metric, such as Minkowski +---)
$\partial_{\tau} p^{\tau}_{\mu} + \partial_{\sigma} p^{\sigma}_{\mu} = 0$
for the worldsheet momenta $p^{\tau}_{\mu},~ p^{\sigma}_{\mu}$.

Now I'm trying to construct explicit solutions for the embedding $X^{\mu}\left(\tau, \sigma\right)$. I then check my solutions fit the equations of motion, either by calculating the worldsheet momenta explicitly and checking the equation above, or by checking that whatever dynamical fields I write in my solution minimise the action (maybe this will be clearer below).

The first situation I considered was the case of a simple closed string in the $X^{1}-X^{2}$ plane. I choose throughout the static gauge $X^{0} = \tau$. Actually this problem is easiest in cylindrical polars (then $g_{\theta \theta}$ is non-constant so the eom above are changed a little)...I chose
$X^{0} = \tau;~X^{1} = r(\tau);~X^{2} = \sigma; X^{3} = 0.$
where I allowed the radius of the string to depend on time because I expect the string to try to minimise its worldsheet area, so will symmetrically collapse (though I then expect a kinetic energy cost associated with changing the shape of the worldsheet, so this function may be oscillatory...). Anyway, when I did this I found an effective action for the $r$ field
$S' = \int d\tau d\sigma ~r\sqrt{1 - \dot{r}^{2}}$
which is minimised for $r = \sin{\tau}$ which fits in with the discussion above.

Actually in cartesian three space this corresponds to
$X^{0} = \tau;~X^{1} = \sin{\tau}\cos{\sigma};~X^{2} = \sin{\tau}\sin{\sigma};~X^{3} = 0$
which gives the spatial components as a sum of left-movers and right movers, $X^{1} \propto \sin{(\sigma + \tau)} + \sin{(-\sigma + \tau)}$ which is as expected, and will be useful for the next problem.

First question - does this solution make sense...is it correct?

I then moved on to trying to find a solution for *spinning* closed string. Actually my instinct was that it might be possible to find a solution at fixed radius, since there can be an interplay between reducing the wordsheet area and the kinetic energy associated with spinning, thought this wasn't what I found...(?) I used
$X^{0} = \tau;~X^{1} = r(\tau);~X^{2} = \sigma + \tau; X^{3} = 0.$
where $X^{2}$ is to be taken mod $2\pi$. This ansatz has a solution for $r$ which minimises its effective action: again $r = \sin{\tau}$. In Cartesians:
$X^{0} = \tau;~X^{1} = \sin{\tau}\cos{(\sigma + \tau)};~X^{2} = \sin{\tau}\sin{(\sigma + \tau)};~X^{3} = 0$
which is a right-mover alone.

Second question - is this correct? I can't find a solution with fixed radius...is this to be expected?

Third question - I consider what happens if we perturb these solutions slightly. I haven't made calculations yet because at first glance the maths is really nasty, but I am wondering whether my reasoning is correct. Suppose we take a slightly elliptical string - one direction is pulled by a tiny amount. My instinct is that the string will move to counter this perturbation and end up circular...? Or suppose we have a radius with a slight $\sigma$ dependence - again my instinct is that such a perturbation should decay to leave the circular string...?

Now I'm moving onto a problem for which I haven't been able to construct a solution, thought it may be because I don't understand the question... this is the problem I am trying to solve:
"Take an open string - give it Neumann boundary conditions at either end and localise to a two plane in Minkowski spacetime."
To me, localising to a 2-plane means that, if we want it to "exist" for some time period, we had better localise to the $X^{0}-X^{1}$ plane (just choose the spatial direction, Cartesian axes). So that in space we have a straight line - we can set the other spatial embeddings to be a constant, 0.

So I need to parameterise a straight line...easy...but I need the Neumann boundary conditions $\partial_{\sigma} X^{\mu} = 0$ at $\sigma = 0, \pi$. So I can't just take some linear function of $\sigma$. So I try an obviously period function
$X^{1} = -A(\tau)cos(\sigma)$
which is at least always increasing and satisfies the boundary conditions, and I allow the string's length to change. The problem? The effective action for the length, A, is
$S' = \int d\tau d\sigma~A\sin{\sigma}$
which means that A is not dynamical, and so must be constant...my instinct is that it should be oscillatory again, to minimise the worldsheet area (and trade off the kinetic energy of collapsing the length of the string)...? This does seem to satisfy the eom, but seems a bit odd to me, that the string can have fixed length. Should I have allowed my embedding to have some time dependence? I also would like to explicitly see in my solution that the endpoints move at the speed of light...

Any comments on any of the above would be greatly appreciated, and I'm also grateful to anyone who has taken the time to read all of this and get to this point :-)

Thanks,
J

 

[AndyW]

ennyHi Jenny,

Thank you for sharing your progress and questions with us. It seems like you have a good understanding of the Nambu-Goto action and are applying it well to solve problems related to classical strings. I will try my best to answer your questions and provide some feedback.

First of all, your solution for a simple closed string in the X^{1}-X^{2} plane looks correct to me. Your reasoning for choosing the specific form for the radius function is also sound. As for your second question about a spinning closed string, it is indeed possible to find a solution with a fixed radius. However, it may not be as simple as the one you have proposed. You may need to consider more complex functions for the embedding in order to minimize the action. I would suggest exploring different forms for the radius function and see if you can find a solution that satisfies the equations of motion.

Your reasoning for perturbing the solutions slightly also seems correct. The string should move to counter the perturbation and eventually reach a circular shape. However, it would be helpful to make some calculations to confirm this intuition.

Moving on to your problem with an open string and Neumann boundary conditions, your approach seems reasonable. However, I think your choice of the linear function for X^{1} might be causing the issue with the effective action for the length. Have you tried using a non-linear function for X^{1}? This may help in finding a more suitable solution.

Overall, I think you are on the right track and have a good understanding of the concepts. I would suggest continuing to explore different forms for the embedding functions and making calculations to confirm your solutions. It may also be helpful to consult with other experts in the field or refer to other resources for guidance.

Best of luck with your research!