Rigidly rotating relativistic string

In summary, the conversation is about using parameters \tau=t and \sigma=s to calculate the solution for a rigidly rotating string with length l in the x-y plane. The solution is given by \vec{X}(t,s)=s(cos(\omega t), sin(\omega t)) and the endpoints move at the speed of light due to free endpoint conditions. The energy of the string is related to l by \frac{2}{\pi} \frac{E}{T_0}. The conversation then moves on to calculating the angular momentum using \sigma=s as a parameter and the expression for P the momentum density. Various attempts are made, but there is uncertainty about the correct expression for P and how to calculate the constant of
  • #1
da_willem
599
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

[tex]\tau=t[/tex] and [tex]\sigma =s[/tex]

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:

[tex] \vec{X} (t,s) = s(cos(\omega t), sin(\omega t))[/tex]

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 [tex]c= \omega l/2[/tex]. Futhermore you can calculate that l is related to the energy of the string E by:

[tex]\frac{2}{\pi} \frac{E}{T_0}[/tex]

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 [itex]\alpha '[/tex] the constant of proprtianality between the angular momentum and the energy squared (in terms of hbar). I tried:

[tex]J= \int _{-l/2} ^{l/2} X_1 P_2 - X_2 P_1 ds [/tex] should be [tex]\frac{E^2}{2 \pi T_0 x}[/tex]

but now I don't know what expression to use for P the momentum density. I tried:

[tex]\vec{P} =\frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}[/tex]

but this yields a result a factor 4/3pi off. So I guess there is something wrong with my expression for P. [itex]T_0/c^2[/itex] is the rest mass density of a relativistic string. But my string is moving, so I tried

[tex]\vec{P}=\gamma (v) \frac{T_0}{c^2} \frac{\partial \vec{X}}{\partial t}[/tex]

with [itex]v= \omega s[/itex], but this diverges bacause gamma explodes at the string endpoints. What's wrong? How to calculate the angular momentum using [itex] \sigma =s[/itex]?
 
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  • #2
would be nice to have an answer to this question...I would be very interested to see how alpha prime is derived.
 

1. What is a rigidly rotating relativistic string?

A rigidly rotating relativistic string is a theoretical concept in physics that describes a one-dimensional object rotating at a constant speed close to the speed of light. It is often used as a simplified model for larger objects, such as cosmic strings, in order to study their behavior and properties.

2. What are the properties of a rigidly rotating relativistic string?

A rigidly rotating relativistic string has a fixed length and a constant angular velocity, meaning that it rotates at a consistent speed. It also exhibits tension, as the string must maintain its rigid shape while rotating. Additionally, it follows the principles of special relativity, including time dilation and length contraction.

3. What is the significance of studying rigidly rotating relativistic strings?

Studying rigidly rotating relativistic strings allows scientists to better understand the behavior of larger objects, such as cosmic strings, that are difficult to observe directly. It also helps to further our understanding of special relativity and its effects on objects moving at high speeds.

4. How are rigidly rotating relativistic strings relevant to real-world applications?

While rigidly rotating relativistic strings are primarily studied in a theoretical context, they have potential applications in the fields of astrophysics and cosmology. For example, they may play a role in the formation and evolution of galaxies and can be used to study the early universe.

5. What are some current research areas related to rigidly rotating relativistic strings?

Some current research areas related to rigidly rotating relativistic strings include investigating their behavior in different spacetime geometries, exploring their interactions with other objects in the universe, and using them to better understand the properties of cosmic strings. Additionally, researchers are studying the potential role of rigidly rotating relativistic strings in the formation of large-scale structures in the universe.

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