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
β = v/c, β = v/c, γ = (1 - β2)-½.
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/∂xdδ)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.
δαβ = (∂xα/∂xβ) is the Kroneker delta extended to 4 indices.
is the metric tensor.
The dot product between two contravariant 4-vectors is defined as A∙B = gαβAαBβ = AβBβ = AαBα. It is invariant under a Lorentz transformation, it is a Lorentz scalar.
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.
∂μ∂μ = ∂μ∂μ = (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.
(∂μ∂μ)Aμ = jμ/(ε0c2)
is the inhomogeneous wave equation for the potentials. It holds in every inertial frame.
All contravariant 4-vectors Aμ transform as A'0 = γ(A0
- β∙A), A'||= γ(A|| - βA0),
A'⊥ = A⊥.
The antisymmetric field strength tensor is defined through Fαβ = ∂αAβ - ∂βAα. It is a second rank contravariant 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/c2)×E)⊥.
Here || and ⊥ refer to the direction of the relative velocity.
E2 - c2B2 and (E∙B)2 are invariant under a Lorentz transformation.