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In Wikipedia's Gravitational plane wave article a metric for a gravitational plane wave is given by \begin{equation} ds^{2}=\left[a\left(u\right)\left(x^{2}-y^{2}\right)+2b\left(u\right)xy\right]du^{2}+2dudv+dx^{2}+dy^{2}.\ \ \ \ \ \ \ \ \ \ (1) \end{equation} I don't understand how this can be a gravitational plane wave because usually it is said that a gravitational plane wave distorts spacetime (changes lengths) only in the directions perpendicular to the propagation of the wave, but since $u=\frac{1}{\sqrt{2}}(z-t)$ this metric seems to distort spacetime in $z,t$-plane and not at all in $x,y$-plane since there is nothing front of $dx^{2}$ and $dy^{2}$ terms.

In 'Schutz, A First Course in General Relativity' an ansatz for gravitational plane wave is given by \begin{equation} ds^{2}=2dudv+f(u)^{2}dx^{2}+g(u)^{2}dy^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ (2) \end{equation} which makes sense because this metric clearly changes lengths only in $x,y$-plane. So how on earth is (1) also a metric for a gravitational plane wave?

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    $\begingroup$ In short, because they are related by a coordinate transformation $\endgroup$ – mmeent Sep 20 '19 at 13:04
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you can transformed equation (1) to get equation (2) how?

the metric of equation (1) $\quad( ds^2=(..)dU^2+2\,dU\,dV$) is :

$$G_1=\left[ \begin {array}{cc} a \left( u \right) \left( {x}^{2}-{y}^{2} \right) +2\,b \left( u \right) xy&-1\\ -1&0 \end {array} \right] $$

the metric of equation (2) $\quad( ds^2=2\,du\,dv$) is :

$$G_2=\left[ \begin {array}{cc} 0&1\\ 1&0\end {array} \right] $$

Step I:

we transformed $G_2\mapsto T^T\,G_2\,T$ to get $G_1$

the transformation matrix $T$ is arbitrary $2\times 2$ matrix

$T= \left[ \begin {array}{cc} T_{{1,1}}&T_{{1,2}}\\ T_{ {2,1}}&T_{{2,2}}\end {array} \right] $

so we have to solve the matrix equation $\quad T^T\,G_2\,T=G_1$, we have four equations for the four unknowns $T_{i,j}$

$\Rightarrow $

$$\begin{bmatrix} du\\ dv\\ \end{bmatrix}=T\,\begin{bmatrix} dU\\ dV\\ \end{bmatrix}\tag 1$$

Step II:

from equation (1) we can calculate the coordinate transformation $ \quad G_1\mapsto G_2$

$$\begin{bmatrix} dU\\ dV\\ \end{bmatrix}=T^{-1}\,\begin{bmatrix} du\\ dv\\ \end{bmatrix}\tag 2$$

result

$$T^{-1}=\left[ \begin {array}{cc} 0&-1\\ 1& \left( -1/2\,{x }^{2}+1/2\,{y}^{2} \right) a \left( u \right) -b \left( u \right) xy \end {array} \right] $$

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