Gravitational wave

Remarks

Every such $h$ can be changed by a pure gauge term to a field in transverse-traceless (TT) gauge: $h_{0\mu} = 0$, $\partial^j h_{jk} = 0$ and $h_{jj} = 0$ (sum over $j$). In TT gauge condition 1 becomes the wave equation $\square h_{jk} = 0$ with $\square = -\partial_t^2 + \Delta$. So the waves travel at the speed of light and have two independent polarisations, called plus and cross.

Example: let $A \ne 0$ and $\omega > 0$, and put $h_{11} = -h_{22} = A \cos(\omega(t - x^3))$, all other components $0$. Each component is a function of $t - x^3$, so $\square h_{jk} = 0$. The trace $h_{11} + h_{22}$ is $0$, and $h$ has no $t$ or $x^3$ components and does not depend on $x^1, x^2$, so $\partial^j h_{jk} = 0$. This is a plus-polarised plane wave moving in the $x^3$ direction; it is not pure gauge, because its linearised Riemann tensor, with components such as $R_{1010} = -\frac{1}{2}\partial_t^2 h_{11}$, is not zero.

Non-example: $h_{\mu\nu} = \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$ satisfies clause 1 for every $\xi$, since it only describes a change of coordinates, but it fails clause 2. Its linearised curvature vanishes, and it carries no energy.