String theory reparameterisation/ transformation law metric

[binbagsss]

Homework Statement

Attached

Homework Equations

The Attempt at a Solution

[/B]
where $\tau$ and $\sigma$ are world-sheet parameters.

where $h_{ab}$ is the world-sheet metric.

To be honest, I am trying to do analogous to general relativity transformations, since this is new to me, so in that case an tensor with two lower indicies transforms as:

$h'_{ab}=h_{cd}\Lambda^{c}_{a}\Lambda^{d}_{b}$

So, to write it all out to make sure I understand clearly...

$h'_{11}=h_{cd}\Lambda^{c}_{1}\Lambda^{d}_{1}=h_{11}\Lambda^{1}_{1}\Lambda^{1}_{1} +h_{22}\Lambda^{2}_{1}\Lambda^{2}_{1}$

and

$h'_{22}=h_{cd}\Lambda^{c}_{2}\Lambda^{d}_{2}=h_{11}\Lambda^{1}_{2}\Lambda^{1}_{2} + h_{22}\Lambda^{2}_{2}\Lambda^{2}_{2}$

And, since $h_{ab}$ is the metric on the world-sheet is a funciton of $\tau$ and $\sigma$, let $X^{u}=(\tau,\sigma)$ , then were $\Lambda^{u}_{v}=\frac{\partial x^{u}}{\partial x'_v}$

(Apologies the convention I believe is to denote the above with $\Lambda^{-1 u}_v$)

So here $X'^{u}= (\tau',\sigma')=(\tau^2/2,1)$

So then $\Lambda^{1}_{1}=\frac{1}{\tau}$

$\Lambda^{1}_{2}=0=\Lambda^{2}_{1}$
So

$h'_{11}=h_{cd}\Lambda^{c}_{1}\Lambda^{d}_{1}=h_{11}\Lambda^{1}_{1}\Lambda^{1}_{1} =-\tau^2 / \tau^2=-1$

Now I am a bit confused with $h'_{22}$, so to recover the Minkowski metric I need this to be $1$.

I have

$h'_{22}=h_{cd}\Lambda^{c}_{2}\Lambda^{d}_{2}=h_{22}\Lambda^{2}_{2}\Lambda^{2}_{2} = \Lambda^{2}_{2}\Lambda^{2}_{2}$

Now I have both $X^2=X'^2=1$ , so what is $\Lambda^2_2=\frac{\partial x^{2}}{\partial x'_2}=\frac{0}{0}$, i.e how do we get $1$ for this, $\Lambda^2_2=1$? how should this be formally done?

Many thanks in advance.

 

[mark726]

Hi there,

It looks like you are on the right track with your calculations. Just to clarify, the transformation you are considering is a conformal transformation, which preserves angles but not distances. In this case, the metric components will transform as you have written, but the actual values of the metric components will change. So for example, for the component h'11, you are correct that it will be -1, but this does not mean that the Minkowski metric is being recovered. Rather, this is just showing that the angle between the two axes is preserved under this transformation.

For h'22, you are correct that the value of the metric component will be 1, but this does not mean that the Minkowski metric is being recovered either. In fact, it is not possible to recover the Minkowski metric using a conformal transformation. This is because the Minkowski metric is not conformally invariant. So even though the values of the metric components may match those of the Minkowski metric, the actual metric itself will not be the same.

As for your question about how to formally get a value of 1 for $\Lambda^{2}_{2}$, since $\Lambda^{2}_{2} = 0/0$, this is an indeterminate form and cannot be determined using just the information given. However, we can use the fact that the transformation is conformal to impose additional constraints on the transformation functions. For example, we could require that the determinant of the transformation matrix be equal to 1, which would give us an additional equation to solve for $\Lambda^{2}_{2}$. This is just one possible approach, and there may be other ways to determine the value of $\Lambda^{2}_{2}$, depending on the specific problem at hand.

I hope this helps to clarify things for you. Let me know if you have any further questions or need any additional clarification. Good luck with your work!