376 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations
18.2.3 Differential Operators, Plane Waves
The 4D generalization of the nabla differential operator∇is
∂i=∂
∂xi:=
(
∂
∂r 1,
∂
∂r 2,
∂
∂r 3,
∂
∂ct)
. (18.29)
Clearly,∂ixi=4 is a scalar, thus∂iis a Lorentz-vector.
The second derivative
∂i∂i=− +∂^2
c^2 ∂t^2=− (18.30)
is a Lorentz scalar. Here =∇μ∇μis the 3D Laplace operator,is the d’Alembert
operator, cf. (7.62).
A plane wave proportional to
exp[−i(−kνrν+ωt)],is a solution of the wave equation...=0, cf. (7.64), provided that the wave
vectorkνand the circular frequencyωobey thedispersion relation kνkν=ω^2 /c^2 ,
or equivalently,ω=kc,cf.(7.65). Here,kis the magnitude of the wave vector.
The 4-wave vectorKiis defined by
Ki:=(
k 1 ,k 2 ,k 3 ,ω
c)
, Ki:=(
−k 1 ,−k 2 ,−k 3 ,ω
c)
. (18.31)
Thus the phase factor occurring in the expression for the plane wave is equal to the
Lorentz scalar
Kixi=Kixi=−kνrν+ωt.Let the functionΨ=Ψ(r,t)be the plane wave
Ψ∼exp[−iKnxn].Here, the summation index “i” is not used in order to avoid any confusion with the
imaginary uniti. Then one has
∂nΨ=−iKnΨ, ∂nΨ=−iKnΨ.Consequently, the wave equation
∂n∂nΨ=−KnKnΨ=−Ψ= 0