What is nice about frequency (temporal and spatial) is that we can use the de Broglie hypothesis to get to energy and momentum as functions of space-time.

Take an idealized planar wave and rotate the coordinates. The component wavelengths do not make a vector.

It is not true that is a vector describing wavelength.

But a contravariant frequency vector does describe a wave.



This is simply derived from a displacement vector, X.



We apply the reciprocal function to each component of the vector as long as we take the bases frequency bases to be coparallel with the displacement bases.




What is the general principle involved?

Can we take any displacement vector, X, and subject it to any function and get a valid vector, as such

?