Pp-wave spacetime

Example

Let $H \colon \R^3 \to \R$ be a smooth function of $(u, x, y)$. On $\R^4$ with coordinates $(u, v, x, y)$, the pp-wave metric in Brinkmann form is $$g = H(u, x, y)\, du^2 + 2\, du\, dv + dx^2 + dy^2.$$ Check that $g$ is Lorentzian: the $(u, v)$ block has determinant $-1$, and $dx^2 + dy^2$ is positive, so the signature is $(-, +, +, +)$.

The vector field $k = \partial_v$ is null, since $g_{vv} = 0$, and it is covariantly constant, $\nabla k = 0$. The only nonzero component of the Ricci curvature is $$R_{uu} = -\frac{1}{2}\left(\frac{\partial^2 H}{\partial x^2} + \frac{\partial^2 H}{\partial y^2}\right).$$ So $g$ is a vacuum solution exactly when $H$ is harmonic in $(x, y)$ for each $u$. For example $H = f(u)\,(x^2 - y^2)$, with any smooth $f$, gives a vacuum plane gravitational wave of the $+$ polarization and profile $f$. If $H$ is not harmonic but $\Delta H \le 0$, then $R_{ab}$ is a nonnegative multiple of $k_a k_b$. This is the stress–energy of null dust or of a null electromagnetic wave.