# Coordinate transformation to flat space

by stampita
Tags: coordinate, flat, space, transformation
 P: 1 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?