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.
This article uses the classical treatment
of tensors and Einstein summation convention
throughout and 
the Minkowski metric has the form diag (+1,
−1, −1, −1). Where the equations are
specified as holding in a vacuum, one could
instead regard them as the formulation of
Maxwell's equations in terms of total charge
and current.
For a more general overview of the relationships
between classical electromagnetism and special
relativity, including various conceptual implications
of this picture, see Classical electromagnetism
and special relativity.
== Covariant objects ==
=== Preliminary 4-vectors ===
Lorentz tensors of the following kinds may
be used in this article to describe bodies
or particles:
4-Displacement:
x
α
=
(
c
t
,
x
)
=
(
c
t
,
x
,
y
,
z
)
.
{\displaystyle x^{\alpha }=(ct,\mathbf {x}
)=(ct,x,y,z)\,.}
4-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.4-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.4-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}}
.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 }=\left({\begin{matrix}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{matrix}}\right)\,}
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 }\,{\stackrel {\mathrm
{def} }{=}}\,\eta ^{\mu \alpha }\,F_{\alpha
\beta }\,\eta ^{\beta \nu }=\left({\begin{matrix}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{matrix}}\right)\,.}
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
}\,.}
=== Electromagnetic stress–energy tensor
===
The electromagnetic stress–energy tensor
can be interpreted as the flux density of
the momentum 4-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}\epsilon
_{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 ε0 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}=\epsilon _{0}E_{i}E_{j}+{\frac
{1}{\mu _{0}}}B_{i}B_{j}-\left({\frac {1}{2}}\epsilon
_{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
γ
ν
F
α
γ
F
ν
β
+
1
4
η
α
β
F
γ
ν
F
γ
ν
)
{\displaystyle T^{\alpha \beta }={\frac {1}{\mu
_{0}}}\left(-\eta _{\gamma \nu }F^{\alpha
\gamma }F^{\nu \beta }+{\frac {1}{4}}\eta
^{\alpha \beta }F_{\gamma \nu }F^{\gamma \nu
}\right)}
where η is the Minkowski metric tensor. Notice
that we use the fact that
ϵ
0
μ
0
c
2
=
1
,
{\displaystyle \epsilon _{0}\mu _{0}c^{2}=1\,,}
which is predicted by Maxwell's equations.
== 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):
while the homogeneous equations – Faraday's
law of induction and Gauss's law for magnetism
combine to form:
where Fαβ is the electromagnetic tensor,
Jα is the 4-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
=
d
e
f
∂
2
F
α
β
=
d
e
f
1
c
2
∂
2
F
α
β
∂
t
2
−
∇
2
F
α
β
=
0
,
{\displaystyle \partial ^{\nu }\partial _{\nu
}F^{\alpha \beta }\,\ {\stackrel {\mathrm
{def} }{=}}\ \,{\partial }^{2}F^{\alpha \beta
}\,\ {\stackrel {\mathrm {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
it 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.
In the co-moving reference frame, this yields
the 4-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
α
,
α
=
d
e
f
∂
α
J
α
=
0
.
{\displaystyle {J^{\alpha }}_{,\alpha }\,{\stackrel
{\mathrm {def} }{=}}\,\partial _{\alpha }J^{\alpha
}\,=\,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 4-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
)
,
{\displaystyle {J^{\nu }}_{\text{free}}=(c\rho
_{\text{free}},\mathbf {J} _{\text{free}})=\left(c\nabla
\cdot \mathbf {D} ,-\ {\frac {\partial \mathbf
{D} }{\partial t}}+\nabla \times \mathbf {H}
\right)\,,}
J
ν
bound
=
(
c
ρ
bound
,
J
bound
)
=
(
−
c
∇
⋅
P
,
∂
P
∂
t
+
∇
×
M
)
.
{\displaystyle {J^{\nu }}_{\text{bound}}=(c\rho
_{\text{bound}},\mathbf {J} _{\text{bound}})=\left(-\
c\nabla \cdot \mathbf {P} ,{\frac {\partial
\mathbf {P} }{\partial t}}+\nabla \times \mathbf
{M} \right)\,.}
Maxwell's macroscopic equations have been
used, in addition the definitions of the electric
displacement D and the magnetic intensity
H:
D
=
ϵ
0
E
+
P
{\displaystyle \mathbf {D} =\epsilon _{0}\mathbf
{E} +\mathbf {P} \,}
H
=
1
μ
0
B
−
M
.
{\displaystyle \mathbf {H} ={\frac {1}{\mu
_{0}}}\mathbf {B} -\mathbf {M} \,.}
where M is the magnetization and P the electric
polarization.
=== Magnetization-polarization tensor ===
The bound current is derived from the P and
M fields which form an antisymmetric contravariant
magnetization-polarization tensor
M
μ
ν
=
(
0
P
x
c
P
y
c
P
z
c
−
P
x
c
0
−
M
z
M
y
−
P
y
c
M
z
0
−
M
x
−
P
z
c
−
M
y
M
x
0
)
,
{\displaystyle {\mathcal {M}}^{\mu \nu }={\begin{pmatrix}0&P_{x}c&P_{y}c&P_{z}c\\-P_{x}c&0&-M_{z}&M_{y}\\-P_{y}c&M_{z}&0&-M_{x}\\-P_{z}c&-M_{y}&M_{x}&0\end{pmatrix}},}
which determines the bound current
J
ν
bound
=
∂
μ
M
μ
ν
.
{\displaystyle {J^{\nu }}_{\text{bound}}=\partial
_{\mu }{\mathcal {M}}^{\mu \nu }\,.}
=== Electric displacement tensor ===
If this is combined with Fμν we get the
antisymmetric contravariant electromagnetic
displacement tensor which combines the D and
H fields as follows:
D
μ
ν
=
(
0
−
D
x
c
−
D
y
c
−
D
z
c
D
x
c
0
−
H
z
H
y
D
y
c
H
z
0
−
H
x
D
z
c
−
H
y
H
x
0
)
.
{\displaystyle {\mathcal {D}}^{\mu \nu }={\begin{pmatrix}0&-D_{x}c&-D_{y}c&-D_{z}c\\D_{x}c&0&-H_{z}&H_{y}\\D_{y}c&H_{z}&0&-H_{x}\\D_{z}c&-H_{y}&H_{x}&0\end{pmatrix}}.}
The three field tensors are related by:
D
μ
ν
=
1
μ
0
F
μ
ν
−
M
μ
ν
{\displaystyle {\mathcal {D}}^{\mu \nu }={\frac
{1}{\mu _{0}}}F^{\mu \nu }-{\mathcal {M}}^{\mu
\nu }\,}
which is equivalent to the definitions of
the D and H fields given above.
== Maxwell's equations in matter ==
The result is that Ampère's law,
∇
×
H
=
J
free
+
∂
D
∂
t
{\displaystyle \mathbf {\nabla } \times \mathbf
{H} =\mathbf {J} _{\text{free}}+{\frac {\partial
\mathbf {D} }{\partial t}}}
,and Gauss's law,
∇
⋅
D
=
ρ
free
{\displaystyle \mathbf {\nabla } \cdot \mathbf
{D} =\rho _{\text{free}}}
,combine into one equation:
The bound current and free current as defined
above are automatically and separately conserved
∂
ν
J
ν
bound
=
0
{\displaystyle \partial _{\nu }{J^{\nu }}_{\text{bound}}=0\,}
∂
ν
J
ν
free
=
0
.
{\displaystyle \partial _{\nu }{J^{\nu }}_{\text{free}}=0\,.}
=== Constitutive equations ===
==== Vacuum ====
In vacuum, the constitutive relations between
the field tensor and displacement tensor are:
μ
0
D
μ
ν
=
η
μ
α
F
α
β
η
β
ν
.
{\displaystyle \mu _{0}{\mathcal {D}}^{\mu
\nu }=\eta ^{\mu \alpha }F_{\alpha \beta }\eta
^{\beta \nu }\,.}
Antisymmetry reduces these 16 equations to
just six independent equations. Because it
is usual to define Fμν by
F
μ
ν
=
η
μ
α
F
α
β
η
β
ν
,
{\displaystyle F^{\mu \nu }=\eta ^{\mu \alpha
}F_{\alpha \beta }\eta ^{\beta \nu },}
the constitutive equations may, in vacuum,
be combined with the Gauss–Ampère law to
get:
∂
β
F
α
β
=
μ
0
J
α
.
{\displaystyle \partial _{\beta }F^{\alpha
\beta }=\mu _{0}J^{\alpha }.\!}
The electromagnetic stress–energy tensor
in terms of the displacement is:
T
α
π
=
F
α
β
D
π
β
−
1
4
δ
α
π
F
μ
ν
D
μ
ν
,
{\displaystyle T_{\alpha }{}^{\pi }=F_{\alpha
\beta }{\mathcal {D}}^{\pi \beta }-{\frac
{1}{4}}\delta _{\alpha }^{\pi }F_{\mu \nu
}{\mathcal {D}}^{\mu \nu },}
where δαπ is the Kronecker delta. When
the upper index is lowered with η, it becomes
symmetric and is part of the source of the
gravitational field.
==== Linear, nondispersive matter ====
Thus we have reduced the problem of modeling
the current, Jν to two (hopefully) easier
problems — modeling the free current, Jνfree
and modeling the magnetization and polarization,
M
μ
ν
{\displaystyle {\mathcal {M}}^{\mu \nu }}
. For example, in the simplest materials at
low frequencies, one has
J
free
=
σ
E
{\displaystyle \mathbf {J} _{\text{free}}=\sigma
\mathbf {E} \,}
P
=
ϵ
0
χ
e
E
{\displaystyle \mathbf {P} =\epsilon _{0}\chi
_{e}\mathbf {E} \,}
M
=
χ
m
H
{\displaystyle \mathbf {M} =\chi _{m}\mathbf
{H} \,}
where one is in the instantaneously comoving
inertial frame of the material, σ is its
electrical conductivity, χe is its electric
susceptibility, and χm is its magnetic susceptibility.
The constitutive relations between the
D
{\displaystyle {\mathcal {D}}}
and F tensors, proposed by Minkowski for a
linear materials (that is, E is proportional
to D and B proportional to H), are:
D
μ
ν
u
ν
=
c
2
ϵ
F
μ
ν
u
ν
{\displaystyle {\mathcal {D}}^{\mu \nu }u_{\nu
}=c^{2}\epsilon F^{\mu \nu }u_{\nu }}
⋆
D
μ
ν
u
ν
=
1
μ
⋆
F
μ
ν
u
ν
{\displaystyle {\star {\mathcal {D}}^{\mu
\nu }}u_{\nu }={\frac {1}{\mu }}{\star F^{\mu
\nu }}u_{\nu }}
where u is the 4-velocity of material, ε
and μ are respectively the proper permittivity
and permeability of the material (i.e. in
rest frame of material),
⋆
{\displaystyle \star }
and denotes the Hodge dual.
== Lagrangian for classical electrodynamics
==
=== Vacuum ===
The Lagrangian density for classical electrodynamics
is
L
=
L
f
i
e
l
d
+
L
i
n
t
=
−
1
4
μ
0
F
α
β
F
α
β
−
A
α
J
α
.
{\displaystyle {\mathcal {L}}\,=\,{\mathcal
{L}}_{\mathrm {field} }+{\mathcal {L}}_{\mathrm
{int} }=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta
}F_{\alpha \beta }-A_{\alpha }J^{\alpha }\,.}
In the interaction term, the four-current
should be understood as an abbreviation of
many terms expressing the electric currents
of other charged fields in terms of their
variables; the four-current is not itself
a fundamental field.
The Euler–Lagrange equation for the electromagnetic
Lagrangian density
L
(
A
α
,
∂
β
A
α
)
{\displaystyle {\mathcal {L}}(A_{\alpha },\partial
_{\beta }A_{\alpha })\,}
can be stated as follows:
∂
β
[
∂
L
∂
(
∂
β
A
α
)
]
−
∂
L
∂
A
α
=
0
.
{\displaystyle \partial _{\beta }\left[{\frac
{\partial {\mathcal {L}}}{\partial (\partial
_{\beta }A_{\alpha })}}\right]-{\frac {\partial
{\mathcal {L}}}{\partial A_{\alpha }}}=0\,.}
Noting
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F_{\mu \nu }=\partial _{\mu
}A_{\nu }-\partial _{\nu }A_{\mu }\,}
,the expression inside the square bracket
is
∂
L
∂
(
∂
β
A
α
)
=
−
1
4
μ
0
∂
(
F
μ
ν
η
μ
λ
η
ν
σ
F
λ
σ
)
∂
(
∂
β
A
α
)
=
−
1
4
μ
0
η
μ
λ
η
ν
σ
(
F
λ
σ
(
δ
μ
β
δ
ν
α
−
δ
ν
β
δ
μ
α
)
+
F
μ
ν
(
δ
λ
β
δ
σ
α
−
δ
σ
β
δ
λ
α
)
)
=
−
F
β
α
μ
0
.
{\displaystyle {\begin{aligned}{\frac {\partial
{\mathcal {L}}}{\partial (\partial _{\beta
}A_{\alpha })}}&=-\ {\frac {1}{4\mu _{0}}}\
{\frac {\partial (F_{\mu \nu }\eta ^{\mu \lambda
}\eta ^{\nu \sigma }F_{\lambda \sigma })}{\partial
(\partial _{\beta }A_{\alpha })}}\\&=-\ {\frac
{1}{4\mu _{0}}}\ \eta ^{\mu \lambda }\eta
^{\nu \sigma }\left(F_{\lambda \sigma }(\delta
_{\mu }^{\beta }\delta _{\nu }^{\alpha }-\delta
_{\nu }^{\beta }\delta _{\mu }^{\alpha })+F_{\mu
\nu }(\delta _{\lambda }^{\beta }\delta _{\sigma
}^{\alpha }-\delta _{\sigma }^{\beta }\delta
_{\lambda }^{\alpha })\right)\\&=-\ {\frac
{F^{\beta \alpha }}{\mu _{0}}}\,.\end{aligned}}}
The second term is
∂
L
∂
A
α
=
−
J
α
.
{\displaystyle {\frac {\partial {\mathcal
{L}}}{\partial A_{\alpha }}}=-J^{\alpha }\,.}
Therefore, the electromagnetic field's equations
of motion are
∂
F
β
α
∂
x
β
=
μ
0
J
α
.
{\displaystyle {\frac {\partial F^{\beta \alpha
}}{\partial x^{\beta }}}=\mu _{0}J^{\alpha
}\,.}
which is one of the Maxwell equations above.
=== Matter ===
Separating the free currents from the bound
currents, another way to write the Lagrangian
density is as follows:
L
=
−
1
4
μ
0
F
α
β
F
α
β
−
A
α
J
free
α
+
1
2
F
α
β
M
α
β
.
{\displaystyle {\mathcal {L}}\,=\,-{\frac
{1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha
\beta }-A_{\alpha }J_{\text{free}}^{\alpha
}+{\frac {1}{2}}F_{\alpha \beta }{\mathcal
{M}}^{\alpha \beta }\,.}
Using Euler–Lagrange equation, the equations
of motion for
D
μ
ν
{\displaystyle {\mathcal {D}}^{\mu \nu }}
can be derived.
The equivalent expression in non-relativistic
vector notation is
L
=
1
2
(
ϵ
0
E
2
−
1
μ
0
B
2
)
−
ϕ
ρ
free
+
A
⋅
J
free
+
E
⋅
P
+
B
⋅
M
.
{\displaystyle {\mathcal {L}}\,=\,{\frac {1}{2}}\left(\epsilon
_{0}E^{2}-{\frac {1}{\mu _{0}}}B^{2}\right)-\phi
\,\rho _{\text{free}}+\mathbf {A} \cdot \mathbf
{J} _{\text{free}}+\mathbf {E} \cdot \mathbf
{P} +\mathbf {B} \cdot \mathbf {M} \,.}
== See also ==
Covariant classical field theory
Electromagnetic tensor
Electromagnetic wave equation
Liénard–Wiechert potential for a charge
in arbitrary motion
Moving magnet and conductor problem
Nonhomogeneous electromagnetic wave equation
Proca action
Quantum electrodynamics
Relativistic electromagnetism
Stueckelberg action
Wheeler–Feynman absorber theory
== Notes and references ==
== Further reading ==
Einstein, A. (1961). Relativity: The Special
and General Theory. New York: Crown. ISBN
0-517-02961-8.
Misner, Charles; Thorne, Kip S.; Wheeler,
John Archibald (1973). Gravitation. San Francisco:
W. H. Freeman. ISBN 0-7167-0344-0.
Landau, L. D.; Lifshitz, E. M. (1975). Classical
Theory of Fields (Fourth Revised English Edition).
Oxford: Pergamon. ISBN 0-08-018176-7.
R. P. Feynman; F. B. Moringo; W. G. Wagner
(1995). Feynman Lectures on Gravitation. Addison-Wesley.
ISBN 0-201-62734-5.
