A plane-wave spacetime has line element

$$d s^{2}=H d u^{2}+2 d u d v+d x^{2}+d y^{2}$$

where $H=x^{2}-y^{2}$. Show that the line element is unchanged by the coordinate transformation

$$u=\bar{u}, \quad v=\bar{v}+\bar{x} e^{\bar{u}}-\frac{1}{2} e^{2 \bar{u}}, \quad x=\bar{x}-e^{\bar{u}}, \quad y=\bar{y}$$

Show more generally that the line element is unchanged by coordinate transformations of the form

$$u=\bar{u}+a, \quad v=\bar{v}+b \bar{x}+c, \quad x=\bar{x}+p, \quad y=\bar{y}$$

where $a, b, c$ and $p$ are functions of $\bar{u}$, which you should determine and which depend in total on four parameters (arbitrary constants of integration).

Deduce (without further calculation) that the line element is unchanged by a 6parameter family of coordinate transformations, of which a 5 -parameter family leave invariant the surfaces $u=$ constant.

For a general coordinate transformation $x^{a}=x^{a}\left(\bar{x}^{b}\right)$, give an expression for the transformed Ricci tensor $\bar{R}_{c d}$ in terms of the Ricci tensor $R_{a b}$ and the transformation matrices $\frac{\partial x^{a}}{\partial \bar{x}^{c}}$. Calculate $\bar{R}_{\bar{x} \bar{x}}$ when the transformation is given by $(*)$ and deduce that $R_{v v}=R_{v x}$