Solve String Theory Problem 6.5: Why Do Endpoints Move at Light Speed?

[ehrenfest]

Homework Statement

http://books.google.com/books?id=Xm...5Dw&sig=6cUrZKqmPMoe0QBRTSYNnipNRw4#PPA114,M1
For problem 6.5 I did a lot of manipulation and I get that the following holds at the endpoints of an open string:

$$c = |\frac{\partial{\vec{X}}}{\partial{t}}|$$

This probably seems like silly intuitive queston, but I cannot explain why this implies that the string endpoints move at the speed of light. Since X is a four-vector not a three-vector, why can we draw that conclusion?

Homework Equations

The Attempt at a Solution

 

[Jhero]

Thank you for bringing up this interesting problem. You are correct that the equation c = |\frac{\partial{\vec{X}}}{\partial{t}}| holds at the endpoints of an open string. This is known as the open string endpoint condition and it is a fundamental property of string theory.

To understand why this implies that the string endpoints move at the speed of light, we need to first understand the concept of four-vectors in string theory. In string theory, the position of a string is described by a four-dimensional vector, known as the worldsheet, which is represented by the equation X = (X^0, X^1, X^2, X^3). The first component X^0 represents time and the remaining three components X^1, X^2, and X^3 represent the three spatial dimensions.

Now, the equation c = |\frac{\partial{\vec{X}}}{\partial{t}}| can be rewritten as c = |\frac{\partial{X^0}}{\partial{t}}|, since the derivative with respect to time only affects the first component of the worldsheet. This means that the string endpoint is moving at a speed equal to the derivative of the time component of the worldsheet.

In string theory, the speed of light is represented by the parameter c, which is a constant. This means that the endpoint of the string is moving at a constant speed, which is equal to the speed of light. This is why we can draw the conclusion that the string endpoints move at the speed of light.

I hope this explanation helps to clarify your understanding of the open string endpoint condition and its implications. Please let me know if you have any further questions.