Time integral of worldsheet current

In summary: The "t" that is integrated over is not the physical time of the ambient space, it is just a parameter used to label points along the string. it is under reparametrization of that variable that the integral must be made invariant. With respect to that variable, the P is a scalar.What is still confusing is that ##g## is the determinant of the 2-dimensional metric (or is it?), while the integral is only 1-dimensional. Shouldn't one have ##\int dt \sqrt{-g_{00}}## instead? When the metric is diagonal, where does the additional factor ## \sqrt{g_{rr}}## comes from? Is there, perhaps
  • #1
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I'm reading this paper and have no problem with its calculations. But there is one little thing I'm not sure I understand. At some point, the author tries to find the amount of momentum carried across the spatial direction of the worldsheet in a time ## \Delta t ## by calculating the following integral:

## \Delta P_1=\int dt \sqrt{-g} P^r_{\ \ x^1} ##

where g is the induced metric on the worldsheet and r and t are the coordinates on the worldsheet which are also two of the background spacetime coordinates along with ## x^1 ##.

My problem is, I'm not sure I understand the reason for the presence of ## \sqrt{-g} ## in the integrand instead of integrating only ## P^r_{\ \ x^1} ##. Could anyone explain?

Thanks
 
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  • #2
It's the Jacobian, I guess they change variables in the integration. (I didn't read the paper).
 
  • #3
MathematicalPhysicist said:
It's the Jacobian, I guess they change variables in the integration. (I didn't read the paper).
Yeah, Its the Jacobian but there is no change of variable involved. The worldsheet coordinates are r and t which are also two of the coordinates of the background spacetime so no change of variable is needed.
 
  • #4
Shyan said:
Yeah, Its the Jacobian but there is no change of variable involved. The worldsheet coordinates are r and t which are also two of the coordinates of the background spacetime so no change of variable is needed.
It is necessary to make the expression reparametrization invariant. Look up for example the Lagrangian of general relativity, it is basically ##\int \sqrt{-g} R ##.
 
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  • #5
nrqed said:
It is necessary to make the expression reparametrization invariant. Look up for example the Lagrangian of general relativity, it is basically ##\int \sqrt{-g} R ##.

Thanks, that helps. But just to get it more clearly, I have one question.

In the case of the EH Lagrangian, We know that ## \int R d^4 x ## is not a scalar because R is a scalar but ##d^4 x ## is a weight ## \pm 1 ## scalar density, so we need to put ##\sqrt{-g} ##(which is a ##\mp 1 ## scalar density) there to make the whole thing a scalar.(I'm not sure about the weights of ## d^4 x ## and ##\sqrt{-g}##).

But here we're dealing with ## P^r_{\ x^1} ## which is one of the components of a tensor which mixes with other components under a transformation and so is not a scalar. How should we come to the conclusion that putting ## \sqrt{-g} ## besides it, makes the whole integral reparametrization invariant?
 
  • #6
Shyan said:
Thanks, that helps. But just to get it more clearly, I have one question.

In the case of the EH Lagrangian, We know that ## \int R d^4 x ## is not a scalar because R is a scalar but ##d^4 x ## is a weight ## \pm 1 ## scalar density, so we need to put ##\sqrt{-g} ##(which is a ##\mp 1 ## scalar density) there to make the whole thing a scalar.(I'm not sure about the weights of ## d^4 x ## and ##\sqrt{-g}##).

But here we're dealing with ## P^r_{\ x^1} ## which is one of the components of a tensor which mixes with other components under a transformation and so is not a scalar. How should we come to the conclusion that putting ## \sqrt{-g} ## besides it, makes the whole integral reparametrization invariant?
The "t" that is integrated over is not the physical time of the ambient space, it is just a parameter used to label points along the string. it is under reparametrization of that variable that the integral must be made invariant. With respect to that variable, the P is a scalar.
 
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  • #7
What is still confusing is that ##g## is the determinant of the 2-dimensional metric (or is it?), while the integral is only 1-dimensional. Shouldn't one have ##\int dt \sqrt{-g_{00}}## instead? When the metric is diagonal, where does the additional factor ## \sqrt{g_{rr}}## comes from? Is there, perhaps, a hidden integration over ##\int dr## involved? (For instance, perhaps ##P^r_1## should really be ##\int dr P^r_1##, or something like that.) In any case, the author seems to be rather sloppy in that equation.
 
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1. What is the definition of the time integral of worldsheet current?

The time integral of worldsheet current is a mathematical concept used in string theory to describe the flow of energy and momentum along a string's worldsheet. It is calculated by integrating the current over the entire time duration of the string's motion.

2. How is the time integral of worldsheet current related to conservation laws?

The time integral of worldsheet current is closely related to the conservation of energy and momentum in string theory. It is a way of quantifying the flow of these quantities along the string's worldsheet, and can be used to verify that energy and momentum are conserved in a particular string interaction.

3. Can the time integral of worldsheet current be calculated experimentally?

Yes, the time integral of worldsheet current can be measured experimentally by studying the interactions of strings in high-energy particle colliders. By analyzing the resulting particle trajectories, scientists can infer the flow of energy and momentum along the strings' worldsheets and calculate the time integral of the current.

4. How does the time integral of worldsheet current differ from the time-averaged worldsheet current?

The time integral of worldsheet current takes into account the full duration of the string's motion, while the time-averaged worldsheet current only considers an average over a specific time period. This means that the time integral provides a more comprehensive picture of the energy and momentum flow along the string's worldsheet.

5. What are some applications of the time integral of worldsheet current in string theory?

The time integral of worldsheet current is a fundamental concept in string theory and has many applications, including the calculation of scattering amplitudes, understanding the behavior of strings in different backgrounds, and studying the dynamics of black holes. It is also used in the development of new theories and models in string theory.

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