18.4 Maxwell Equations in 4D-Formulation 381
18.4 Maxwell Equations in 4D-Formulation
18.4.1 Electric Flux Density and Continuity Equation
The4-flux density Jiis defined by
Ji:=(j 1 ,j 2 ,j 3 ,cρ), (18.51)wherejis the 3D electric flux density andρis the charge density. The continuity
equation, cf. (7.59),
∂ρ
∂t+∇μjμ= 0 ,is equivalent to
∂iJi= 0. (18.52)
The differential operator∂iis a Lorentz vector, thus the 4-flux density is also a Lorentz
vector.
18.4.2 Electric 4-Potential and Lorentz Scaling.
In terms of the 3D vector potentialAand the scalar potentialφ, the electrodynamic
4-potential is defined by
Φi:=(
A 1 ,A 2 ,A 3 ,
φ
c)
,Φi:=(
−A 1 ,−A 2 ,−A 3 ,
φ
c)
. (18.53)
The Lorentz scaling, cf. (7.67),
∂
∂tφ+∇λAλ= 0 ,corresponds to
∂iΦi= 0. (18.54)
Clearly, the 4-potential is a Lorentz vector.