Tensors for Physics

384 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations

The inhomogeneous Maxwell equations (7.56), viz.

∇μDμ=ρ, εμνλ∇νHλ=jμ+

∂

∂t

Dμ,

are equivalent to
∂kHik=Ji. (18.62)

To complete the set of Maxwell equations, constitutive relations are needed which
link the field tensorsHikandFik. In vacuum, the simple linear relation

Hik=

1

μ 0

Fik (18.63)

applies. Hereμ 0 = 4 π 10 −^7 As/Vm is the magnetic induction constant of the vacuum,
As/Vm stands for the SI-units Ampere seconds/Volt meter.

18.4.6 Inhomogeneous Wave Equation

For currents and fields in vacuum, where (18.63) applies, the inhomogeneous
Maxwell equations, with (18.55) lead to

Ji=∂kHik=

1

μ 0

∂kFik=∂k(∂kΦi−∂iΦk).

Due to∂kΦk=0, cf. (18.54) and with the d’Alembert operator,cf.(7.62) and
(18.30), the inhomogeneous wave equation reads

Φi=−μ 0 Ji. (18.64)

18.4.7 Transformation Behavior of the Electromagnetic Fields

The field tensorFikis a Lorentz tensor which transforms according to (18.19).
For the special case where the primed coordinate system moves with the constant
velocityv=v 1 =βc, with respect to the original coordinate system, the resulting
transformed tensor is

(F′)ik:=γ

⎛

⎜

⎜

⎝

0 F^12 +βF^24 F^13 +βF^34 γ−^1 F^14

−(F^12 +βF^24 ) 0 γ−^1 F^23 F^24 +βF^12

−(F^13 +βF^34 ) −γ−^1 F^230 F^34 +βF^13

−γ−^1 F^14 −(F^24 +βF^12 )−(F^34 +βF^13 ) 0

⎞

⎟

⎟

⎠.

(18.65)