Relativistic E&M

Electrodynamics in relativistic notation

A contravariant 4-vector is a set of 4 quantities which transform under a Lorentz transformation like (ct,r) = (x0,x1,x2,x3). 
(A0,A1,A2,A3) is a contravariant 4-vector if A'α = (∂x'α/∂xβ)Aβ.  The repeated index β is summed over.

A covariant 4-vector is a set of 4 quantities which transform under a Lorentz transformation like (ct,-r) = (x0,x1,x2,x3). 
(A0,A1,A2,A3) is a covariant 4-vector if A'α = (∂xβ/∂x'α)Aβ.
If the primed coordinate system moves with velocity v = βci with respect to the unprimed one, then

contravariant vector transformation

β = v/c,   β = v/c,  γ = (1 - β2).

A contravariant tensor of second rank is a set of 16 quantities which transform under a Lorentz transformation according to F'αβ = (∂x'α/∂xγ)(∂x'β/∂x)Fγδ.
A covariant tensor of second rank transforms under a Lorentz transformation according to G'αβ = (∂xγ/∂x'α)(∂xδ/∂x'β)Gγδ,
and a mixed tensor transforms according to H'αβ = (∂x'α/∂xγ)(∂xδ/∂x'β)Hγδ.

Special tensors:
δαβ = (∂xα/∂xβ) is the Kroneker delta extended to 4 indices.

metric tensor 
is the metric tensor.
The dot product between two contravariant 4-vectors is defined as AB = gαβAαBβ = AβBβ = AαBα.  It is invariant under a Lorentz transformation, it is a Lorentz scalar.


Important 4-vectors:

xμ = (ct,r), 
uμ = (γc,γv) = 4-vector velocity,
pμ = (γmc,γmv) = (E/c,p) = (p0,p) = 4-vector momentum,
(∂/∂xμ) = ∂μ = (∂/∂x0,-) = 4-dimensional gradient,
jμ = (cρ,j) = 4-vector current,
Aμ = (Φ/c,A) = 4-vector potential.

The divergence of a 4-vector ∂μAμ = ∂A0/∂x0 + A  is a Lorentz scalar.

Examples
:
μμ = ∂μμ = (1/c2)∂2/∂t2  - 2  is the D'Alambertian.  It is invariant under a Lorentz transformation.

μjμ  = ∂μjμ = ∂ρ/∂t - ∇∙j  is the statement of charge conservation.
If charge is conserved in one inertial frame, it is conserved in every inertial frame.

μAμ = ∂μAμ = (1/c2)∂Φ/∂t - A = 0
is the Lorentz condition.  If the Lorentz condition holds in one inertial frame, it holds on every inertial frame.

(∂μμ)Aμ = jμ/(ε0c2)
is the inhomogeneous wave equation for the potentials.  It holds in every inertial frame.


Transformation of the fields

All contravariant 4-vectors Aμ transform as A'0 = γ(A0 - βA), A'||= γ(A|| - βA0), A' = A.
The antisymmetric field strength tensor is defined through Fαβ = ∂αAβ - ∂βAα.  It is a second rank contravariant tensor.

fitld strength tensor
The transformation F'αβ = (∂x'α/∂xγ)(∂x'β/∂xδ)Fγδ yields
E'|| = E||B'|| = B||,
E' = γ(E + v×B),
B' = γ(B - (v/c2E).
Here || and ⊥ refer to the direction of the relative velocity.


Invariants

E2 - c2B2 and (EB)2 are invariant under a Lorentz transformation.