Lorentz invariance of plane-wave phase ($\Phi =K_{\mu }{x}^{\mu }$)

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.