= Lorentz invariance of plane-wave phase
{c}
{title2=$\Phi=K_\mu x^\mu$}
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 $\omega'=\gamma(\omega-vk_x)$ and $k_x'=\gamma(k_x-v\omega/c^2)$. Thus <angular frequency> and <wavevector> transform together rather than independently.
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