Mathematical object that describes the electromagnetic field in spacetime
In
electromagnetism , the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor , Faraday tensor or Maxwell bivector ) is a mathematical object that describes the
electromagnetic field in spacetime. The field tensor was first used after the four-dimensional
tensor formulation of
special relativity was introduced by
Hermann Minkowski . The tensor allows related physical laws to be written concisely, and allows for the
quantization of the electromagnetic field by the Lagrangian formulation described
below .
Definition
The electromagnetic tensor, conventionally labelled F , is defined as the
exterior derivative of the
electromagnetic four-potential , A , a differential 1-form:
[1]
[2]
F
=
d
e
f
d
A
.
{\displaystyle F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.}
Therefore, F is a
differential 2-form — an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
.
{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.}
where
∂
{\displaystyle \partial }
is the
four-gradient and
A
{\displaystyle A}
is the
four-potential .
SI units for Maxwell's equations and the
particle physicist's sign convention for the
signature of
Minkowski space (+ − − −) , will be used throughout this article.
Relationship with the classical fields
The Faraday
differential 2-form is given by
F
=
(
E
x
/
c
)
d
x
∧
d
t
+
(
E
y
/
c
)
d
y
∧
d
t
+
(
E
z
/
c
)
d
z
∧
d
t
+
B
x
d
y
∧
d
z
+
B
y
d
z
∧
d
x
+
B
z
d
x
∧
d
y
,
{\displaystyle F=(E_{x}/c)\ dx\wedge dt+(E_{y}/c)\ dy\wedge dt+(E_{z}/c)\ dz\wedge dt+B_{x}\ dy\wedge dz+B_{y}\ dz\wedge dx+B_{z}\ dx\wedge dy,}
where
d
t
{\displaystyle dt}
is the time element times the speed of light
c
{\displaystyle c}
.
This is the
exterior derivative of its 1-form antiderivative
A
=
A
x
d
x
+
A
y
d
y
+
A
z
d
z
−
(
ϕ
/
c
)
d
t
{\displaystyle A=A_{x}\ dx+A_{y}\ dy+A_{z}\ dz-(\phi /c)\ dt}
,
where
ϕ
(
x
→
,
t
)
{\displaystyle \phi ({\vec {x}},t)}
has
−
∇
→
ϕ
=
E
→
{\displaystyle -{\vec {\nabla }}\phi ={\vec {E}}}
(
ϕ
{\displaystyle \phi }
is a scalar potential for the
irrotational/conservative vector field
E
→
{\displaystyle {\vec {E}}}
) and
A
→
(
x
→
,
t
)
{\displaystyle {\vec {A}}({\vec {x}},t)}
has
∇
→
×
A
→
=
B
→
{\displaystyle {\vec {\nabla }}\times {\vec {A}}={\vec {B}}}
(
A
→
{\displaystyle {\vec {A}}}
is a vector potential for the
solenoidal vector field
B
→
{\displaystyle {\vec {B}}}
).
Note that
{
d
F
=
0
⋆
d
⋆
F
=
J
{\displaystyle {\begin{cases}dF=0\\{\star }d{\star }F=J\end{cases}}}
where
d
{\displaystyle d}
is the exterior derivative,
⋆
{\displaystyle {\star }}
is the
Hodge star ,
J
=
−
J
x
d
x
−
J
y
d
y
−
J
z
d
z
+
ρ
d
t
{\displaystyle J=-J_{x}\ dx-J_{y}\ dy-J_{z}\ dz+\rho \ dt}
(where
J
→
{\displaystyle {\vec {J}}}
is the
electric current density , and
ρ
{\displaystyle \rho }
is the
electric charge density ) is the 4-current density 1-form, is the differential forms version of Maxwell's equations.
The
electric and
magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in
Cartesian coordinates :
E
i
=
c
F
0
i
,
{\displaystyle E_{i}=cF_{0i},}
where c is the speed of light, and
B
i
=
−
1
/
2
ϵ
i
j
k
F
j
k
,
{\displaystyle B_{i}=-1/2\epsilon _{ijk}F^{jk},}
where
ϵ
i
j
k
{\displaystyle \epsilon _{ijk}}
is the
Levi-Civita tensor . This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will
transform covariantly , and the fields in the new frame will be given by the new components.
In contravariant
matrix form with metric signature (+,-,-,-),
F
μ
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 }={\begin{bmatrix}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{bmatrix}}.}
The covariant form is given by
index lowering ,
F
μ
ν
=
η
α
ν
F
β
α
η
μ
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_{\mu \nu }=\eta _{\alpha \nu }F^{\beta \alpha }\eta _{\mu \beta }={\begin{bmatrix}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{bmatrix}}.}
The Faraday tensor's
Hodge dual is
G
α
β
=
1
2
ϵ
α
β
γ
δ
F
γ
δ
=
0
−
B
x
−
B
y
−
B
z
B
x
0
E
z
/
c
−
E
y
/
c
B
y
−
E
z
/
c
0
E
x
/
c
B
z
E
y
/
c
−
E
x
/
c
0
{\displaystyle {G^{\alpha \beta }={\frac {1}{2}}\epsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }={\begin{bmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}}}}
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.
Properties
The matrix form of the field tensor yields the following properties:
[3]
Antisymmetry :
F
μ
ν
=
−
F
ν
μ
{\displaystyle F^{\mu \nu }=-F^{\nu \mu }}
Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (Ex , Ey , Ez ) and magnetic field (Bx , By , Bz ).
Inner product: If one forms an inner product of the field strength tensor a
Lorentz invariant is formed
F
μ
ν
F
μ
ν
=
−
2
(
E
2
c
2
−
B
2
)
{\displaystyle F_{\mu \nu }F^{\mu \nu }=-2\left({\frac {E^{2}}{c^{2}}}-B^{2}\right)}
meaning this number does not change from one
frame of reference to another.
Pseudoscalar invariant: The product of the tensor
F
μ
ν
{\displaystyle F^{\mu \nu }}
with its
Hodge dual
G
μ
ν
{\displaystyle G^{\mu \nu }}
gives a
Lorentz invariant :
G
γ
δ
F
γ
δ
=
1
2
ϵ
α
β
γ
δ
F
α
β
F
γ
δ
=
−
4
c
B
⋅
E
{\displaystyle G_{\gamma \delta }F^{\gamma \delta }={\frac {1}{2}}\epsilon _{\alpha \beta \gamma \delta }F^{\alpha \beta }F^{\gamma \delta }=-{\frac {4}{c}}\mathbf {B} \cdot \mathbf {E} \,}
where
ϵ
α
β
γ
δ
{\displaystyle \epsilon _{\alpha \beta \gamma \delta }}
is the rank-4
Levi-Civita symbol . The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is
ϵ
0123
=
−
1
{\displaystyle \epsilon _{0123}=-1}
.
Determinant :
det
(
F
)
=
1
c
2
(
B
⋅
E
)
2
{\displaystyle \det \left(F\right)={\frac {1}{c^{2}}}\left(\mathbf {B} \cdot \mathbf {E} \right)^{2}}
which is proportional to the square of the above invariant.
Trace :
F
=
F
μ
μ
=
0
{\displaystyle F={{F}^{\mu }}_{\mu }=0}
which is equal to zero.
Significance
This tensor simplifies and reduces
Maxwell's equations as four vector calculus equations into two tensor field equations. In
electrostatics and
electrodynamics ,
Gauss's law and
Ampère's circuital law are respectively:
∇
⋅
E
=
ρ
ϵ
0
,
∇
×
B
−
1
c
2
∂
E
∂
t
=
μ
0
J
{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}},\quad \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {J} }
and reduce to the inhomogeneous Maxwell equation:
∂
α
F
β
α
=
−
μ
0
J
β
{\displaystyle \partial _{\alpha }F^{\beta \alpha }=-\mu _{0}J^{\beta }}
, where
J
α
=
(
c
ρ
,
J
)
{\displaystyle J^{\alpha }=(c\rho ,\mathbf {J} )}
is the
four-current .
In
magnetostatics and magnetodynamics,
Gauss's law for magnetism and
Maxwell–Faraday equation are respectively:
∇
⋅
B
=
0
,
∂
B
∂
t
+
∇
×
E
=
0
{\displaystyle \nabla \cdot \mathbf {B} =0,\quad {\frac {\partial \mathbf {B} }{\partial t}}+\nabla \times \mathbf {E} =\mathbf {0} }
which reduce to the
Bianchi identity :
∂
γ
F
α
β
+
∂
α
F
β
γ
+
∂
β
F
γ
α
=
0
{\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0}
or using the
index notation with square brackets
[note 1] for the antisymmetric part of the tensor:
∂
α
F
β
γ
=
0
{\displaystyle \partial _{[\alpha }F_{\beta \gamma ]}=0}
Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically (
≡
0
{\displaystyle \equiv 0}
). The implication of that identity is far-reaching: it means that the EM field theory leaves no room for magnetic monopoles and currents of such.
Relativity
The field tensor derives its name from the fact that the electromagnetic field is found to obey the
tensor transformation law , this general property of physical laws being recognised after the advent of
special relativity . This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of
tensors . The tensor formalism also leads to a mathematically simpler presentation of physical laws.
The inhomogeneous Maxwell equation leads to the
continuity equation :
∂
α
J
α
=
J
α
,
α
=
0
{\displaystyle \partial _{\alpha }J^{\alpha }=J^{\alpha }{}_{,\alpha }=0}
implying
conservation of charge .
Maxwell's laws above can be generalised to
curved spacetime by simply replacing
partial derivatives with
covariant derivatives :
F
α
β
;
γ
=
0
{\displaystyle F_{[\alpha \beta ;\gamma ]}=0}
and
F
α
β
;
α
=
μ
0
J
β
{\displaystyle F^{\alpha \beta }{}_{;\alpha }=\mu _{0}J^{\beta }}
where the semi-colon
notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the
curved space Maxwell equations . Again, the second equation implies charge conservation (in curved spacetime):
J
α
;
α
=
0
{\displaystyle J^{\alpha }{}_{;\alpha }\,=0}
Lagrangian formulation of classical electromagnetism
Classical electromagnetism and
Maxwell's equations can be derived from the
action :
S
=
∫
(
−
1
4
μ
0
F
μ
ν
F
μ
ν
−
J
μ
A
μ
)
d
4
x
{\displaystyle {\mathcal {S}}=\int \left(-{\begin{matrix}{\frac {1}{4\mu _{0}}}\end{matrix}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right)\mathrm {d} ^{4}x\,}
where
d
4
x
{\displaystyle \mathrm {d} ^{4}x}
is over space and time.
This means the
Lagrangian density is
L
=
−
1
4
μ
0
F
μ
ν
F
μ
ν
−
J
μ
A
μ
=
−
1
4
μ
0
(
∂
μ
A
ν
−
∂
ν
A
μ
)
(
∂
μ
A
ν
−
∂
ν
A
μ
)
−
J
μ
A
μ
=
−
1
4
μ
0
(
∂
μ
A
ν
∂
μ
A
ν
−
∂
ν
A
μ
∂
μ
A
ν
−
∂
μ
A
ν
∂
ν
A
μ
+
∂
ν
A
μ
∂
ν
A
μ
)
−
J
μ
A
μ
{\displaystyle {\begin{aligned}{\mathcal {L}}&=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\\&=-{\frac {1}{4\mu _{0}}}\left(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\right)\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right)-J^{\mu }A_{\mu }\\&=-{\frac {1}{4\mu _{0}}}\left(\partial _{\mu }A_{\nu }\partial ^{\mu }A^{\nu }-\partial _{\nu }A_{\mu }\partial ^{\mu }A^{\nu }-\partial _{\mu }A_{\nu }\partial ^{\nu }A^{\mu }+\partial _{\nu }A_{\mu }\partial ^{\nu }A^{\mu }\right)-J^{\mu }A_{\mu }\\\end{aligned}}}
The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is
L
=
−
1
2
μ
0
(
∂
μ
A
ν
∂
μ
A
ν
−
∂
ν
A
μ
∂
μ
A
ν
)
−
J
μ
A
μ
.
{\displaystyle {\mathcal {L}}=-{\frac {1}{2\mu _{0}}}\left(\partial _{\mu }A_{\nu }\partial ^{\mu }A^{\nu }-\partial _{\nu }A_{\mu }\partial ^{\mu }A^{\nu }\right)-J^{\mu }A_{\mu }.}
Substituting this into the
Euler–Lagrange equation of motion for a field:
∂
μ
(
∂
L
∂
(
∂
μ
A
ν
)
)
−
∂
L
∂
A
ν
=
0
{\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }A_{\nu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\nu }}}=0}
So the Euler–Lagrange equation becomes:
−
∂
μ
1
μ
0
(
∂
μ
A
ν
−
∂
ν
A
μ
)
+
J
ν
=
0.
{\displaystyle -\partial _{\mu }{\frac {1}{\mu _{0}}}\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right)+J^{\nu }=0.\,}
The quantity in parentheses above is just the field tensor, so this finally simplifies to
∂
μ
F
μ
ν
=
μ
0
J
ν
{\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}
That equation is another way of writing the two inhomogeneous
Maxwell's equations (namely,
Gauss's law and
Ampère's circuital law ) using the substitutions:
1
c
E
i
=
−
F
0
i
ϵ
i
j
k
B
k
=
−
F
i
j
{\displaystyle {\begin{aligned}{\frac {1}{c}}E^{i}&=-F^{0i}\\\epsilon ^{ijk}B_{k}&=-F^{ij}\end{aligned}}}
where i, j, k take the values 1, 2, and 3.
Hamiltonian form
The
Hamiltonian density can be obtained with the usual relation,
H
(
ϕ
i
,
π
i
)
=
π
i
ϕ
˙
i
(
ϕ
i
,
π
i
)
−
L
{\displaystyle {\mathcal {H}}(\phi ^{i},\pi _{i})=\pi _{i}{\dot {\phi }}^{i}(\phi ^{i},\pi _{i})-{\mathcal {L}}}
.
Quantum electrodynamics and field theory
The
Lagrangian of
quantum electrodynamics extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):
L
=
ψ
¯
(
i
ℏ
c
γ
α
D
α
−
m
c
2
)
ψ
−
1
4
μ
0
F
α
β
F
α
β
,
{\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\alpha }D_{\alpha }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\alpha \beta }F^{\alpha \beta },}
where the first part in the right hand side, containing the
Dirac spinor
ψ
{\displaystyle \psi }
, represents the
Dirac field . In
quantum field theory it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.
See also
Notes
^ By definition,
T
a
b
c
=
1
3
!
(
T
a
b
c
+
T
b
c
a
+
T
c
a
b
−
T
a
c
b
−
T
b
a
c
−
T
c
b
a
)
{\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}+T_{bca}+T_{cab}-T_{acb}-T_{bac}-T_{cba})}
So if
∂
γ
F
α
β
+
∂
α
F
β
γ
+
∂
β
F
γ
α
=
0
{\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0}
then
0
=
2
6
(
∂
γ
F
α
β
+
∂
α
F
β
γ
+
∂
β
F
γ
α
)
=
1
6
{
∂
γ
(
2
F
α
β
)
+
∂
α
(
2
F
β
γ
)
+
∂
β
(
2
F
γ
α
)
}
=
1
6
{
∂
γ
(
F
α
β
−
F
β
α
)
+
∂
α
(
F
β
γ
−
F
γ
β
)
+
∂
β
(
F
γ
α
−
F
α
γ
)
}
=
1
6
(
∂
γ
F
α
β
+
∂
α
F
β
γ
+
∂
β
F
γ
α
−
∂
γ
F
β
α
−
∂
α
F
γ
β
−
∂
β
F
α
γ
)
=
∂
γ
F
α
β
{\displaystyle {\begin{aligned}0&={\begin{matrix}{\frac {2}{6}}\end{matrix}}(\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha })\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}\{\partial _{\gamma }(2F_{\alpha \beta })+\partial _{\alpha }(2F_{\beta \gamma })+\partial _{\beta }(2F_{\gamma \alpha })\}\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}\{\partial _{\gamma }(F_{\alpha \beta }-F_{\beta \alpha })+\partial _{\alpha }(F_{\beta \gamma }-F_{\gamma \beta })+\partial _{\beta }(F_{\gamma \alpha }-F_{\alpha \gamma })\}\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}(\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }-\partial _{\gamma }F_{\beta \alpha }-\partial _{\alpha }F_{\gamma \beta }-\partial _{\beta }F_{\alpha \gamma })\\&=\partial _{[\gamma }F_{\alpha \beta ]}\end{aligned}}}
References