The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.
Covariant objects
Preliminary four-vectors
Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:
four-displacement: x α = ( c t , x ) = ( c t , x , y , z ) . {\displaystyle x^{\alpha }=(ct,\mathbf {x} )=(ct,x,y,z)\,.}
Four-velocity: u α = γ ( c , u ) , {\displaystyle u^{\alpha }=\gamma (c,\mathbf {u} ),} where γ(u) is the Lorentz factor at the 3-velocity u. Four-momentum: p α = ( E / c , p ) = m 0 u α {\displaystyle p^{\alpha }=(E/c,\mathbf {p} )=m_{0}u^{\alpha }} where p {\displaystyle \mathbf {p} } is 3-momentum, E {\displaystyle E} is the total energy, and m 0 {\displaystyle m_{0}} is rest mass. Four-gradient: ∂ ν = ∂ ∂ x ν = ( 1 c ∂ ∂ t , − ∇ ) , {\displaystyle \partial ^{\nu }={\frac {\partial }{\partial x_{\nu }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-\mathbf {\nabla } \right)\,,}
The d'Alembertian operator is denoted ∂ 2 {\displaystyle {\partial }^{2}} , ∂ 2 = ∂ ν ∂ ν = 1 c 2 ∂ 2 ∂ t 2 − ∇ 2 . {\displaystyle \partial ^{2}=\partial ^{\nu }\partial _{\nu }={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-\nabla ^{2}.}
The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is (+ − − −), corresponding to the Minkowski metric tensor:
η μ ν = ( 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ) {\displaystyle \eta ^{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}
Electromagnetic tensor
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.
F α β = ( 0 E x / c E y / c E z / c − E x / c 0 − B z B y − E y / c B z 0 − B x − E z / c − B y B x 0 ) {\displaystyle F_{\alpha \beta }={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}
and the result of raising its indices is
F μ ν = d e f η μ α F α β η β ν = ( 0 − E x / c − E y / c − E z / c E x / c 0 − B z B y E y / c B z 0 − B x E z / c − B y B x 0 ) , {\displaystyle F^{\mu \nu }\mathrel {\stackrel {\mathrm {def} }{=}} \eta ^{\mu \alpha }\,F_{\alpha \beta }\,\eta ^{\beta \nu }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}\,,}
where E is the electric field, B the magnetic field, and c the speed of light.
Four-current
The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:
J α = ( c ρ , j ) . {\displaystyle J^{\alpha }=(c\rho ,\mathbf {j} )\,.}
Four-potential
The electromagnetic four-potential is a covariant four-vector containing the electric potential (also called the scalar potential) ϕ and magnetic vector potential (or vector potential) A, as follows:
A α = ( ϕ / c , A ) . {\displaystyle A^{\alpha }=\left(\phi /c,\mathbf {A} \right)\,.}
The differential of the electromagnetic potential is
F α β = ∂ α A β − ∂ β A α . {\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,.}
In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form A = A α d x α {\displaystyle A=A_{\alpha }dx^{\alpha }} and a 2-form F = d A = 1 2 F α β d x α ∧ d x β {\textstyle F=dA={\frac {1}{2}}F_{\alpha \beta }dx^{\alpha }\wedge dx^{\beta }} respectively. Here, d {\displaystyle d} is the exterior derivative and ∧ {\displaystyle \wedge } the wedge product.
Electromagnetic stress–energy tensor
The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum four-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall stress–energy tensor:
T α β = ( ε 0 E 2 / 2 + B 2 / 2 μ 0 S x / c S y / c S z / c S x / c − σ x x − σ x y − σ x z S y / c − σ y x − σ y y − σ y z S z / c − σ z x − σ z y − σ z z ) , {\displaystyle T^{\alpha \beta }={\begin{pmatrix}\varepsilon _{0}E^{2}/2+B^{2}/2\mu _{0}&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{pmatrix}}\,,}
where ε 0 {\displaystyle \varepsilon _{0}} is the electric permittivity of vacuum, μ0 is the magnetic permeability of vacuum, the Poynting vector is
S = 1 μ 0 E × B {\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} }
and the Maxwell stress tensor is given by
σ i j = ε 0 E i E j + 1 μ 0 B i B j − ( 1 2 ε 0 E 2 + 1 2 μ 0 B 2 ) δ i j . {\displaystyle \sigma _{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-\left({\frac {1}{2}}\varepsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right)\delta _{ij}\,.}
The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:
T α β = 1 μ 0 ( η α ν F ν γ F β γ − 1 4 η α β F γ ν F γ ν ) {\displaystyle T^{\alpha \beta }={\frac {1}{\mu _{0}}}\left(\eta ^{\alpha \nu }F_{\nu \gamma }F^{\beta \gamma }-{\frac {1}{4}}\eta ^{\alpha \beta }F_{\gamma \nu }F^{\gamma \nu }\right)}
where η is the Minkowski metric tensor (with signature (+ − − −)). Notice that we use the fact that
ε 0 μ 0 c 2 = 1 , {\displaystyle \varepsilon _{0}\mu _{0}c^{2}=1\,,}
which is predicted by Maxwell's equations. Another way to covariant expression for the electromagnetic stress-energy tensor which may be simpler since it does not involve covariant and contravariant indices is this one:
T = − 1 μ 0 ( F ∗ η ∗ F ′ − 1 4 tr ( F ∗ η ∗ F ′ ∗ η ) ) {\displaystyle T=-{\frac {1}{\mu _{0}}}(F*\eta *F'-{\frac {1}{4}}\operatorname {tr} (F*\eta *F'*\eta ))}
Where F' is the transposed electromagnetic tensor or equivalently -F and the asterisk denotes matrix multiplication.
Maxwell's equations in vacuum
In vacuum (or for the microscopic equations, not including macroscopic material descriptions), Maxwell's equations can be written as two tensor equations. The two inhomogeneous Maxwell's equations, Gauss's law and Ampère's law (with Maxwell's correction) combine into (with (+ − − −) metric):
The homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form ∂ σ F μ ν + ∂ μ F ν σ + ∂ ν F σ μ = 0 {\displaystyle \partial ^{\sigma }F^{\mu \nu }+\partial ^{\mu }F^{\nu \sigma }+\partial ^{\nu }F^{\sigma \mu }=0} , which may be written using Levi-Civita duality as:
where Fαβ is the electromagnetic tensor, Jα is the four-current, εαβγδ is the Levi-Civita symbol, and the indices behave according to the Einstein summation convention. Each of these tensor equations corresponds to four scalar equations, one for each value of β. Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation can also be written more compactly as:
F [ α β , γ ] = 0. {\displaystyle F_{[\alpha \beta ,\gamma ]}=0.}
In the absence of sources, Maxwell's equations reduce to:
∂ ν ∂ ν F α β = def ∂ 2 F α β = def 1 c 2 ∂ 2 F α β ∂ t 2 − ∇ 2 F α β = 0 , {\displaystyle \partial ^{\nu }\partial _{\nu }F^{\alpha \beta }\mathrel {\stackrel {\text{def}}{=}} \partial ^{2}F^{\alpha \beta }\mathrel {\stackrel {\text{def}}{=}} {1 \over c^{2}}{\partial ^{2}F^{\alpha \beta } \over {\partial t}^{2}}-\nabla ^{2}F^{\alpha \beta }=0\,,}
which is an electromagnetic wave equation in the field strength tensor.
Maxwell's equations in the Lorenz gauge
The Lorenz gauge condition is a Lorentz-invariant gauge condition. (This can be contrasted with other gauge conditions such as the Coulomb gauge, which if it holds in one inertial frame will generally not hold in any other.) It is expressed in terms of the four-potential as follows:
∂ α A α = ∂ α A α = 0 . {\displaystyle \partial _{\alpha }A^{\alpha }=\partial ^{\alpha }A_{\alpha }=0\,.}
In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
∂ 2 A σ = μ 0 J σ . {\displaystyle {\partial }^{2}A^{\sigma }=\mu _{0}\,J^{\sigma }\,.}
Lorentz force
Charged particle
Electromagnetic (EM) fields affect the motion of electrically charged matter: due to the Lorentz force. In this way, EM fields can be detected (with applications in particle physics, and natural occurrences such as in aurorae). In relativistic form, the Lorentz force uses the field strength tensor as follows. Expressed in terms of coordinate time t, it is:
d p α d t = q F α β d x β d t , {\displaystyle {dp_{\alpha } \over {dt}}=q\,F_{\alpha \beta }\,{\frac {dx^{\beta }}{dt}},}
where pα is the four-momentum, q is the charge, and xβ is the position. Expressed in frame-independent form, we have the four-force
d p α d τ = q F α β u β , {\displaystyle {\frac {dp_{\alpha }}{d\tau }}\,=q\,F_{\alpha \beta }\,u^{\beta },}
where uβ is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by dt = γdτ.
Charge continuum
The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by
f α = F α β J β . {\displaystyle f_{\alpha }=F_{\alpha \beta }J^{\beta }.}
and is related to the electromagnetic stress–energy tensor by
f α = − T α β , β ≡ − ∂ T α β ∂ x β . {\displaystyle f^{\alpha }=-{T^{\alpha \beta }}_{,\beta }\equiv -{\frac {\partial T^{\alpha \beta }}{\partial x^{\beta }}}.}
Conservation laws
Electric charge The continuity equation:
J β , β = def ∂ β J β = ∂ β ∂ α F α β / μ 0 = 0. {\displaystyle {J^{\beta }}_{,\beta }\mathrel {\overset {\text{def}}{\mathop {=} }} \partial _{\beta }J^{\beta }=\partial _{\beta }\partial _{\alpha }F^{\alpha \beta }/\mu _{0}=0.}
expresses charge conservation.
Electromagnetic energy–momentum Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector
T α β , β + F α β J β = 0 {\displaystyle {T^{\alpha \beta }}_{,\beta }+F^{\alpha \beta }J_{\beta }=0}
or
η α ν T ν β , β + F α β J β = 0 , {\displaystyle \eta _{\alpha \nu }{T^{\nu \beta }}_{,\beta }+F_{\alpha \beta }J^{\beta }=0,}
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
Covariant objects in matter
Free and bound four-currents In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, Jν. Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations;
J ν = J ν free + J ν bound , {\displaystyle J^{\nu }={J^{\nu }}_{\text{free}}+{J^{\nu }}_{\text{bound}}\,,}
where
J ν free = ( c ρ free , J free ) = ( c ∇ ⋅ D , − ∂ D ∂ t + ∇ × H ) ,