Coordinate transformation to flat space

  1. 1. The problem statement, all variables and given/known data
    find the transformation that turns this metric:
    [tex]ds^2=-X^2dT^2+dX^2[/tex]

    into the minkowski metric: diag(-1,1).

    3. The attempt at a solution
    attempt 1:
    I transformed the above metric into the coordinates that use the lines that define the light cone as the axes. Namely,
    [tex]u=T+ln(X)[/tex]
    [tex]v=T-ln(X)[/tex]

    attempt 2:
    If I can transform the above, into the coordinates u and v such that i have
    [tex] ds^2=-dudv [/tex]
    then I'm done because with another transformation:
    [tex]dt'=\frac{du+dv}{2}[/tex]
    [tex]dx'=\frac{du-dv}{2}[/tex]
    I have the relation
    [tex]-(dt')^2+dx^2=-(\frac{du+dv}{2})^2+(\frac{du-dv}{2})^2=-dudv.[/tex]

    Is there a systematic way of doing this?
     
  2. jcsd
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