# Homework Help: String theory reparameterisation/ transformation law metric

1. Jun 16, 2017

### binbagsss

1. The problem statement, all variables and given/known data

Attached

2. Relevant equations

3. The attempt at a solution

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?