define the 4-wavevector A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation The 4-wavevector is the 4-gradient of the negative phase (or the negative 4-gradient of the phase) of a wave in Minkowski Space: This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave): where 4-position , is the temporal angular frequency, is the spatial 3-space wavevector, and is the Lorentz scalar invariant phase. with the assumption that the plane wave and are not explicit functions of or . The explicit form of an SR plane wave can be written as: where is a (possibly complex) amplitude. A general wave would be the superposition of multiple plane waves: Again using the 4-gradient, or which is the 4-gradient version of complex-valued plane waves As the d'Alembertian operator In special relativity, electromagnetism and wave theory, the d'Alembert operator, also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert. The square of is the 4-Laplacian, which is called the d'Alembert operator: As it is the dot product of two 4-vectors, the d'Alembertian is a Lorentz invariant scalar. Occasionally, in analogy with the 3-dimensional notation, the symbols and are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol is reserved for the d'Alembertian. Some examples of the 4-gradient as used in the d'Alembertian follow: In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): In the wave equation for the electromagnetic field (using Lorenz gauge ): In vacuum: With a 4-current source, not including the effects of spin: With quantum electrodynamics source, including effects of spin: where: Electromagnetic 4-potential is an electromagnetic vector potential 4-current density is an electromagnetic current density Dirac Gamma matrices provide the effects of spin In the wave equation of a gravitational wave (using a similar Lorenz gauge ) where is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source). Further conditions on are: Purely spatial: Traceless: Transverse: In the 4-dimensional version of Green's function: where