A linearized gravitational wave with + polarization can be written in light-cone coordinates as $$d s^{2}=2 d u d v+(1+h(u)) d x^{2}+(1-h(u)) d y^{2}$$

where $|{h(u)}|<<1$. To linearized order, I want to show that this metric is equivalent to that of a pp-wave in Brinkmann coordinates

$$d s^{2}=2 d u d V+\frac{1}{2}\left(X^{2}-Y^{2}\right) \ddot{h}(u) d u^{2}+d X^{2}+d Y^{2}$$

(Dot denotes derivative with respect to u). I want to find the linearized transformation between (v,x,y) and (V,X,Y). I know that the transformation should reduce to the trivial case (v=V, x=X, y=Y) when $h(u) \rightarrow 0$. I know I should start by finding the transformation for constant $h$, but I'm not sure how to solve this problem.


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