Electromagnetic tensor

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For an explanation and meanings of the index notation in this article see, see Einstein notation and antisymmetric tensor.
Electromagnetism

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 of a physical system. The field tensor was first used after the 4-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.

SI units and the particle physicist's convention for the signature of Minkowski space (+,−,−,−), will be used throughout this article.

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\ {\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,

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

Relationship with the Classical Fields

The electromagnetic tensor is completely isomorphic to the electric and magnetic fields, though the electric and magnetic fields change with the choice of the reference frame, while the electromagnetic tensor does not. In general, the relationship is quite complicated, but in Cartesian coordinates, using the coordinate system's own reference frame, the relationship is very simple.

E_{i}=cF^{{i0}},

where c is the speed of light, and

B_{i}=-{\frac 12}\epsilon _{{ijk}}F^{{jk}},

where \epsilon _{{ijk}} is the Levi-Civita symbol. In contravariant matrix form,

{\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}}=F^{{\mu \nu }}.

The covariant form is given by index lowering,

F_{{\mu \nu }}=\eta _{{\mu \alpha }}\eta _{{\nu \beta }}F^{{\alpha \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 mixed form appears in the Lorentz force equation when using the contravariant four-velocity: {\frac {dp^{\mu }}{d\tau }}=qF_{{\nu }}^{{\mu }}u^{\nu }, where

F_{{\nu }}^{{\mu }}={\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}}.

From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is being assumed, and the electric and magnetic fields are with respect to coordinate system's own reference frame, as in the equations above.

Properties

The matrix form of the field tensor yields the following properties:[3]

  1. Antisymmetry:
    F^{{\mu \nu }}\,=-F^{{\nu \mu }}
    (hence the name bivector).
  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
    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 \scriptstyle (F^{{\mu \nu }}) with its dual tensor \scriptstyle (G^{{\mu \nu }}) gives the Lorentz invariant:
    G_{{\gamma \delta }}F^{{\gamma \delta }}={\frac {1}{2}}\epsilon _{{\alpha \beta \gamma \delta }}F^{{\alpha \beta }}F^{{\gamma \delta }}=-{\frac {4}{c}}\left({\mathbf B}\cdot {\mathbf E}\right)\,
    where \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 \epsilon _{{0123}}=+1.
  5. Determinant:
    \det \left(F\right)={\frac {1}{c^{2}}}\left({\mathbf B}\cdot {\mathbf E}\right)^{{2}}
    which is the square of the above invariant.

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:

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

and reduce to:

\partial _{{\alpha }}F^{{\alpha \beta }}=\mu _{0}J^{{\beta }}

where

J^{{\alpha }}=(c\rho ,{\mathbf {J}})

is the 4-current. In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

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

which reduce to Bianchi identity:

\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:

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

Relativity

Main article: Maxwell's equations in curved spacetime

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 (non-gravitational) physical laws being recognised after the advent of special relativity. This theory stipulated that all the (non-gravitational) 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 second equation above leads to the continuity equation:

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

where the semi-colon 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^{\alpha }{}_{{;\alpha }}\,=0

Lagrangian formulation of classical electromagnetism (no charges and currents)

See also: Classical field theory

When there are no electric charges (ρ = 0) and no electric currents (J = 0), Classical electromagnetism and Maxwell's equations can be derived from the action:

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

where

{\mathrm {d}}^{4}x\;   is over space and time.

This means the Lagrangian density is

{\begin{aligned}{\mathcal {L}}&=-{\frac {1}{4\mu _{0}}}F_{{\mu \nu }}F^{{\mu \nu }}\\&=-{\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)\\&=-{\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)\\\end{aligned}}

The two middle terms are the same, so the Lagrangian density is

{\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).

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

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

The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:

\partial _{\mu }\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right)=0.\,

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

\partial _{\mu }F^{{\mu \nu }}=0

That equation is another way of writing the two homogeneous Maxwell's equations, making the substitutions:

~E^{i}/c=-F^{{0i}}\,
\epsilon ^{{ijk}}B_{k}=-F^{{ij}}\,

where i, j, k take the values 1, 2, and 3.

When there are sources, the Lagrangian needs an extra term to account for the coupling between charges (currents) and the electromagnetic field:

J^{\mu }A_{\mu }.

In that case the Euler-Lagrange equation yields the inhomogeneous Maxwell's equations:

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

Quantum electrodynamics and field theory

Main articles: Quantum electrodynamics and quantum field theory

The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity, from {\mathcal {L}}={\bar \psi }(i\hbar c\,\gamma ^{\alpha }D_{\alpha }-mc^{2})\psi -{\frac {1}{4\mu _{0}}}F_{{\alpha \beta }}F^{{\alpha \beta }},  to incorporate the creation and annihilation of photons (and electrons).

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,
      T_{{[abc]}}={\frac {1}{3!}}(T_{{abc}}+T_{{bca}}+T_{{cab}}-T_{{acb}}-T_{{bac}}-T_{{cba}})

      So if

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

      then

      {\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}}

    See also

    References

    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 Edition). Pearson Education, Dorling Kindersley. ISBN 81-7758-293-3. 
    3. ↑ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0-7167-0344-0. 
    • Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4. 
    • Jackson, John D. (1999). Classical Electrodynamics. John Wiley & Sons, Inc. ISBN 0-471-30932-X. 
    • Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2. 
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