382 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations
18.4.3 Field Tensor Derived from the 4-Potential
In the 3D formulation of electrodynamics, theB-field and theE-field are related to
the vector and scalar potential functions by
B=∇×A, E=−∇φ−∂A
∂t.
The first components, e.g. of these equations are
B 1 =
∂A 3
∂r 2−
∂A 2
∂r 3=
∂Φ 2
∂x^3−
∂Φ 3
∂x^2, E 1 =−
∂φ
∂r 1−
∂A 1
∂t=c(
∂Φ 1
∂x^4−
∂Φ 4
∂x^1)
.
The equations for the other components can be inferred by analogy. All these equa-
tions are combined by introducing the second rank field tensorF:
Fik:=∂Φi
∂xk−
∂Φk
∂xi=∂kΦi−∂iΦk. (18.55)Its contra-variant version is
Fik:=∂kΦi−∂iΦk.The field tensor is antisymmetric:
Fik=−Fki. (18.56)In matrix notation, the field tensor is related to the components of the magnetic and
electric fields by
Fik:=⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0 B 3 −B (^21) cE 1
−B 3 0 B (^11) cE 2
B 2 −B (^101) cE 3
−^1 cE 1 −^1 cE 2 −^1 cE 3 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
. (18.57)
Notice, the top-left 3×3 part of this antisymmetric 4×4 matrix is just the mag-
netic field tensor introduced in Sect.7.5.5. Thus (18.57) can be regarded as the 4-
dimensional extension of (7.70) made such that the components ofEare also incor-
porated. This works because an antisymmetric tensor, in 4D, has 6 components, just
likeBandEtogether.
The matrix for the contra-variant tensorFikis given by an expression analogous
to (18.57) where the terms involving theE-field have the opposite sign.