# Coordinate transformation to flat space

1. Aug 27, 2007

### stampita

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

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,
$$u=T+ln(X)$$
$$v=T-ln(X)$$

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

Is there a systematic way of doing this?