Electromagnetic tensor

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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 very concisely.

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]

${\displaystyle F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.}$

Therefore, F is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.}$

where ${\displaystyle \partial }$ is the four-gradient and ${\displaystyle A}$ 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

${\displaystyle F=(E_{x}/c)\ dx\wedge dt+(E_{y}/c)\ dy\wedge dt+(E_{z}/c)\ dz\wedge dt+B_{x}\ dy\wedge dz+B_{y}\ dz\wedge dx+B_{z}\ dx\wedge dy}$

This is the exterior derivative of its 1-form antiderivative

${\displaystyle A=A_{x}\ dx+A_{y}\ dy+A_{z}\ dz-(\phi /c)\ dt}$,

where ${\displaystyle \phi ({\vec {x}},t)}$ has ${\displaystyle -{\vec {\nabla }}\phi ={\vec {E}}}$ (${\displaystyle \phi }$ is a scalar potential for the irrotational/conservative vector field ${\displaystyle {\vec {E}}}$) and ${\displaystyle {\vec {A}}({\vec {x}},t)}$ has ${\displaystyle {\vec {\nabla }}\times {\vec {A}}={\vec {B}}}$ (${\displaystyle {\vec {A}}}$ is a vector potential for the solenoidal vector field ${\displaystyle {\vec {B}}}$).

Here is a video series by Michael Penn explaining the Faraday 2-form and its relations to Maxwell's equations.

Note that

${\displaystyle {\begin{cases}dF=0\\{}^{*}d^{*}F=J\end{cases}}}$

where ${\displaystyle d}$ is the exterior derivative, ${\displaystyle ^{*}}$ is the Hodge star, ${\displaystyle J=-J_{x}\ dx-J_{y}\ dy-J_{z}\ dz+\rho \ dt}$ (where ${\displaystyle {\vec {J}}}$ is the electric current density, and ${\displaystyle \rho }$ 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:

${\displaystyle E_{i}=cF_{0i},}$

where c is the speed of light, and

${\displaystyle B_{i}=-{\frac {1}{2}}\epsilon _{ijk}F^{jk},}$

where ${\displaystyle \epsilon _{ijk}}$ 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,

${\displaystyle F^{\mu \nu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}.}$

The covariant form is given by index lowering,

${\displaystyle F_{\mu \nu }=\eta _{\alpha \nu }F^{\beta \alpha }\eta _{\mu \beta }={\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}.}$

The Faraday tensor's Hodge dual is

${\displaystyle {G^{\alpha \beta }={\frac {1}{2}}\epsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }={\begin{bmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}}}}$

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]

1. Antisymmetry:
${\displaystyle F^{\mu \nu }=-F^{\nu \mu }}$
2. 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).
3. Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed
${\displaystyle F_{\mu \nu }F^{\mu \nu }=2\left(B^{2}-{\frac {E^{2}}{c^{2}}}\right)}$
meaning this number does not change from one frame of reference to another.
4. Pseudoscalar invariant: The product of the tensor ${\displaystyle F^{\mu \nu }}$ with its Hodge dual ${\displaystyle G^{\mu \nu }}$ gives a Lorentz invariant:
${\displaystyle G_{\gamma \delta }F^{\gamma \delta }={\frac {1}{2}}\epsilon _{\alpha \beta \gamma \delta }F^{\alpha \beta }F^{\gamma \delta }=-{\frac {4}{c}}\mathbf {B} \cdot \mathbf {E} \,}$
where ${\displaystyle \epsilon _{\alpha \beta \gamma \delta }}$ 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 ${\displaystyle \epsilon _{0123}=-1}$.
5. Determinant:
${\displaystyle \det \left(F\right)={\frac {1}{c^{2}}}\left(\mathbf {B} \cdot \mathbf {E} \right)^{2}}$
which is proportional to the square of the above invariant.
6. Trace:
${\displaystyle F={{F}^{\mu }}_{\mu }=0}$
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:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}},\quad \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {J} }$

and reduce to the inhomogeneous Maxwell equation:

${\displaystyle \partial _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }}$, where ${\displaystyle J^{\alpha }=(c\rho ,\mathbf {J} )}$ is the four-current.

In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

${\displaystyle \nabla \cdot \mathbf {B} =0,\quad {\frac {\partial \mathbf {B} }{\partial t}}+\nabla \times \mathbf {E} =0}$

which reduce to Bianchi identity:

${\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0}$

or using the index notation with square brackets [note 1] for the antisymmetric part of the tensor:

${\displaystyle \partial _{[\alpha }F_{\beta \gamma ]}=0}$

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:

${\displaystyle \partial _{\alpha }J^{\alpha }=J^{\alpha }{}_{,\alpha }=0}$

implying conservation of charge.

Maxwell's laws above can be generalised to curved spacetime by simply replacing partial derivatives with covariant derivatives:

${\displaystyle F_{[\alpha \beta$ ;\gamma ]}=0} and ${\displaystyle F^{\alpha \beta }{}_{;\alpha }=\mu _{0}J^{\beta }}$

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):

${\displaystyle J^{\alpha }{}_{;\alpha }\,=0}$

Lagrangian formulation of classical electromagnetism

Classical electromagnetism and Maxwell's equations can be derived from the action:

${\displaystyle {\mathcal {S}}=\int \left(-{\begin{matrix}{\frac {1}{4\mu _{0}}}\end{matrix}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right)\mathrm {d} ^{4}x\,}$

where ${\displaystyle \mathrm {d} ^{4}x}$ is over space and time.

This means the Lagrangian density is

{\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

${\displaystyle {\mathcal {L}}=-{\frac {1}{2\mu _{0}}}\left(\partial _{\mu }A_{\nu }\partial ^{\mu }A^{\nu }-\partial _{\nu }A_{\mu }\partial ^{\mu }A^{\nu }\right)-J^{\mu }A_{\mu }.}$

Substituting this into the Euler–Lagrange equation of motion for a field:

${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }A_{\nu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\nu }}}=0}$

So the Euler–Lagrange equation becomes:

${\displaystyle -\partial _{\mu }{\frac {1}{\mu _{0}}}\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right)+J^{\nu }=0.\,}$

The quantity in parentheses above is just the field tensor, so this finally simplifies to

${\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}$

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:

{\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,

${\displaystyle {\mathcal {H}}(\phi ^{i},\pi _{i})=\pi _{i}{\dot {\phi }}^{i}(\phi ^{i},\pi _{i})-{\mathcal {L}}}$.

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):

${\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\alpha }D_{\alpha }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\alpha \beta }F^{\alpha \beta },}$

where the first part in the right hand side, containing the Dirac spinor ${\displaystyle \psi }$, 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.

Notes

1. ^ By definition,
${\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}+T_{bca}+T_{cab}-T_{acb}-T_{bac}-T_{cba})}$

So if

${\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0}$

then

{\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}}}
1. J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation . W.H. Freeman & Co. ISBN   0-7167-0344-0.
2. D. J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN   978-81-7758-293-2.
3. J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation . W.H. Freeman & Co. ISBN   0-7167-0344-0.

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