W. Rindler said:
In a beautiful application of SR, de Broglie proposed the following relation between the particle's 4-momentum ##\mathbf P## and the wave 4-vector of the associated wave ... :
$$ \mathbf P= h \mathbf L, \ \ \text{that is,} \ \ E(\frac{\mathbf u}{c^2},\frac{1}{c})=h\nu(\frac{\mathbf n}{w},\frac{1}{c}). \ \ \ \ \ \ \ \text{(51)}$$
In fact, if Planck's relation (50) is to be maintained for a material particle and its associated wave, then (51) is inevitable. For then the 4th components of the 4-vectors on either side of (51) are equal; by our earlier "zero-component lemma", the entire 4-vectors must therefore be equal! From (51) it then follows that the wave travels in the direction of the particle (##\mathbf n## ∝ ##\mathbf u##), but with a larger velocity ##w##, given by de Broglie's relation
$$uw=c^2, \ \ \ \ \ \ \ \text{(52)}$$
as can be seen by comparing the magnitudes of the leading 3-vectors. (However, the group velocity of the wave, which carries the energy, can be shown to be still ##u##.) The wave must necessarily travel at a speed other than the particle unless that speed is ##c##, for waves and particles aberrate differently, and a particle comoving with its wave would slide across it sideways in another frame.