- #1
da_willem
- 594
- 1
I am reading Zwiebach's 'a first course in string theory' and I have a question. As an example of a solution to the equations of motion he uses a rigidly rotating string with length l in the x-y plane with it's center at the origin. I would like to do some calculations using as parameters
\tau=t and \sigma =s
with s the distance from the origin, and t just time. s runs from -l/2 to l/2. This should be possible right, as I'm free to choose which way to paramterize the string.
Now the solution is (given that the string lies along the x-axis at t=0) ofcourse:
\vec{X} (t,s) = s(cos(\omega t), sin(\omega t))
As the endpoints are not tied down to anything I have to use free enpoint conditions which means the endpoints move at the speed of light. This yields for the enpoints c= \omega l/2. Futhermore you can calculate that l is related to the energy of the string E by:
\frac{2}{\pi} \frac{E}{T_0}
With T0 the tension of the string. Now the solution is complete as both l and omega are found.
Now I would like to calculate the angular momentum, so I can find \alpha '[/tex] the constant of proprtianality between the angular momentum and the energy squared (in terms of hbar). I tried:<br /> <br /> J= \int _{-l/2} ^{l/2} X_1 P_2 - X_2 P_1 ds should be \frac{E^2}{2 \pi T_0 x}<br /> <br /> but now I don't know what expression to use for P the momentum density. I tried:<br /> <br /> \vec{P} =\frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}<br /> <br /> but this yields a result a factor 4/3pi off. So I guess there is something wrong with my expression for P. T_0/c^2 is the rest mass density of a relativistic string. But my string is moving, so I tried<br /> <br /> \vec{P}=\gamma (v) \frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}<br /> <br /> with v= \omega s, but this diverges bacause gamma explodes at the string endpoints. What's wrong? How to calculate the angular momentum using \sigma =s?
\tau=t and \sigma =s
with s the distance from the origin, and t just time. s runs from -l/2 to l/2. This should be possible right, as I'm free to choose which way to paramterize the string.
Now the solution is (given that the string lies along the x-axis at t=0) ofcourse:
\vec{X} (t,s) = s(cos(\omega t), sin(\omega t))
As the endpoints are not tied down to anything I have to use free enpoint conditions which means the endpoints move at the speed of light. This yields for the enpoints c= \omega l/2. Futhermore you can calculate that l is related to the energy of the string E by:
\frac{2}{\pi} \frac{E}{T_0}
With T0 the tension of the string. Now the solution is complete as both l and omega are found.
Now I would like to calculate the angular momentum, so I can find \alpha '[/tex] the constant of proprtianality between the angular momentum and the energy squared (in terms of hbar). I tried:<br /> <br /> J= \int _{-l/2} ^{l/2} X_1 P_2 - X_2 P_1 ds should be \frac{E^2}{2 \pi T_0 x}<br /> <br /> but now I don't know what expression to use for P the momentum density. I tried:<br /> <br /> \vec{P} =\frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}<br /> <br /> but this yields a result a factor 4/3pi off. So I guess there is something wrong with my expression for P. T_0/c^2 is the rest mass density of a relativistic string. But my string is moving, so I tried<br /> <br /> \vec{P}=\gamma (v) \frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}<br /> <br /> with v= \omega s, but this diverges bacause gamma explodes at the string endpoints. What's wrong? How to calculate the angular momentum using \sigma =s?