Four-wavevector ($K^{\mu }=(\omega /c,\mathbf{k})$)

The four-wavevector combines the angular frequency and spatial wavevector of a plane wave. Phase invariance gives $K^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{K}^{\nu }$. Under a boost $\mathbf{v}$, ${\omega }^{\prime }=\gamma (\omega -\mathbf{v}\cdot \mathbf{k})$, ${\mathbf{k}}_{\parallel }^{\prime }=\gamma ({\mathbf{k}}_{\parallel }-\omega \mathbf{v}/c^2)$ and ${\mathbf{k}}_{\perp }^{\prime }={\mathbf{k}}_{\perp }$. For metric signature $(+,-,-,-)$, $R_{\mu }{K}^{\mu }=\omega t-\mathbf{k}\cdot \mathbf{r}$ is a Lorentz scalar. A vacuum light wave has $K_{\mu }{K}^{\mu }=0$.

The wave phase labels a physical stage of an oscillation at a spacetime event. Treating it as a Lorentz scalar makes its gradient $K_{\mu }={\partial }_{\mu }\Phi$ a covector; raising the index gives the four-wavevector $K^{\mu }=(\omega /c,\mathbf{k})$. A boost along $x$ gives ${\omega }^{\prime }=\gamma (\omega -v{k}_x)$ and $k_x^{\prime }=\gamma (k_x-v\omega /c^2)$. Thus angular frequency and wavevector transform together rather than independently.