Wave vector
Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), and its direction is perpendicular to the wavefront.
It is common in several fields of physics to refer to the angular wave vector simply as the wave vector, in contrast to, for example, crystallography.
For light waves in vacuum, this is also the direction of the Poynting vector.
On the other hand, the wave vector points in the direction of phase velocity.
In other words, the wave vector points in the normal direction to the surfaces of constant phase, also called wavefronts.
In a lossless isotropic medium such as air, any gas, any liquid, amorphous solids (such as glass), and cubic crystals, the direction of the wavevector is the same as the direction of wave propagation.
The wave vector is always perpendicular to surfaces of constant phase.
[6] A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface.
A wavetrain (denoted by some variable X) can be regarded as a one-parameter family of such hypersurfaces in spacetime.
The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.
[7] The four-wavevector is a wave four-vector that is defined, in Minkowski coordinates, as: where the angular frequency
Alternately, the wavenumber k can be written as the angular frequency ω divided by the phase-velocity vp, or in terms of inverse period T and inverse wavelength λ.
When written out explicitly its contravariant and covariant forms are: In general, the Lorentz scalar magnitude of the wave four-vector is: The four-wavevector is null for massless (photonic) particles, where the rest mass
An example of a null four-wavevector would be a beam of coherent, monochromatic light, which has phase-velocity
The Lorentz matrix is defined as In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the Lorentz transformation as follows.
Applying the Lorentz transformation to the wave vector and choosing just to look at the
So As an example, to apply this to a situation where the source is moving directly away from the observer (
), this becomes: To apply this to a situation where the source is moving straight towards the observer (θ = 0), this becomes: To apply this to a situation where the source is moving transversely with respect to the observer (θ = π/2), this becomes:
