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String theory reparameterisation/ transformation law metric

  1. Jun 16, 2017 #1
    1. The problem statement, all variables and given/known data

    Attached question.jpg

    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?

    Many thanks in advance.
     
  2. jcsd
  3. Jun 21, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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