## 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**) = (x^{0},x^{1},x^{2},x^{3}).

(A^{0},A^{1},A^{2},A^{3}) is a contravariant
4-vector if A'^{α} = (∂x'^{α}/∂x^{β})A^{β}. 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**) = (x_{0},x_{1},x_{2},x_{3}).

(A_{0},A_{1},A_{2},A_{3}) is a covariant
4-vector if A'_{α} = (∂x^{β}/∂x'^{α})A_{β}.

If the primed coordinate system moves with velocity **v** = βc**i**
with respect to the unprimed one, then

β = v/c, **β** = **v**/c, γ = (1 - β^{2})^{-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}/∂x^{γ})(∂x'^{β}/∂x^{dδ})F^{γδ}.

A **covariant tensor of second rank** transforms under a Lorentz
transformation according to G'_{αβ} = (∂x^{γ}/∂x'^{α})(∂x^{δ}/∂x'^{β})G_{γδ},

and a **mixed tensor** transforms according to H'^{α}_{β} =
(∂x'^{α}/∂x^{γ})(∂x^{δ}/∂x'^{β})H^{γ}_{δ}.

Special tensors:

δ^{α}_{β} = (∂x^{α}/∂x^{β}) 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**
= g_{αβ}A^{α}B^{β} = A_{β}B^{β} = A^{α}B_{α}.
It is invariant under a Lorentz transformation, it is a **Lorentz scalar**.

### Important 4-vectors:

x^{μ} = (ct,**r**),

u^{μ} = (γc,γ**v**) = 4-vector velocity,

p^{μ} = (γmc,γm**v**) = (E/c,**p**) = (p_{0},**p**)
= 4-vector momentum,

(∂/∂x_{μ}) = ∂^{μ} = (∂/∂x_{0},-**∇**) =
4-dimensional gradient,

j^{μ} = (cρ,**j**) = 4-vector current,

A^{μ} = (Φ/c,**A**) = 4-vector potential.

The **divergence** of a 4-vector ∂_{μ}A^{μ} = ∂A^{0}/∂x^{0}
+ **∇**∙**A** is a Lorentz scalar.

Examples:

∂^{μ}∂_{μ} = ∂_{μ}∂^{μ} = (1/c^{2})∂^{2}/∂t^{2}
- **∇**^{2}
is the **D'Alambertian**. It is invariant under a Lorentz transformation.

∂^{μ}j_{μ} = ∂_{μ}j^{μ}
= ∂ρ/∂t
- **∇∙j** is
the statement of **charge conservation**.

If charge is conserved in one inertial frame, it is conserved in every
inertial frame.

∂^{μ}A_{μ} = ∂_{μ}A^{μ}
= (1/c^{2})∂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})A^{μ} = j^{μ}/(ε_{0}c^{2})

is the **inhomogeneous wave equation for the potentials**. It holds in
every inertial frame.

### Transformation of the fields

All contravariant 4-vectors A^{μ} transform as A'^{0} = γ(A^{0}
- **β**∙**A**), **A**'_{||}= γ(**A**_{||} - βA^{0}),
**A**'_{⊥} = **A**_{⊥}.

The **antisymmetric field strength tensor** is defined through F^{αβ}
= ∂^{α}A^{β} - ∂^{β}A^{α}. It is a second rank
contravariant tensor.

The transformation F'^{αβ} = (∂x'^{α}/∂x^{γ})(∂x'^{β}/∂x^{δ})F^{γδ}
yields

**E**'_{||} = **E**_{||}, **B**'_{||} = **
B**_{||},

**E**'_{⊥} = γ(**E** + **v**×**B**)_{⊥},

**B**'_{⊥} = γ(**B** - (**v**/c^{2})×**E**)_{⊥}.

Here || and ⊥ refer to the direction of the relative velocity.

### Invariants

**E**^{2 }- c^{2}**B**^{2} and (**E**∙**B**)^{2}
are invariant under a Lorentz transformation.