Electromagnetic tensor

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

Definition

The electromagnetic tensor can be defined using the electromagnetic four-potential:

$A^{\mu} = \left( \frac{\phi}{c} , \bold A \right)$

and its covariant form is found by multiplying by the Minkowski metric η of signature (+,−,−,−) :

$A_{\mu} \, = \eta_{\mu\nu} A^{\nu} = \left( \frac{\phi}{c}, -\bold A \right)$

where A is the vector potential, ϕ is the scalar potential and c is the speed of light.

The Electric and magnetic fields can be expressed in terms of A and ϕ by:

$\bold{E} = -\frac{\partial \bold{A}}{\partial t} - \bold{\nabla} \phi \,$

$\bold{B} = \bold{\nabla} \times \bold{A} \,$

By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form $A$:

$F_{\mu\nu} \ \stackrel{\mathrm{def}}{=}\ (d A)_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu.$

therefore F is a differential 2-form on spacetime. In an inertial frame, the matrices of F read:

$F^{\mu\nu} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix}$

and by lowering indices

$F_{\mu\nu} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix}$

There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing E/c → B and B → −E/c, to get the dual tensor G^μν.

$G^{\mu\nu} = \begin{bmatrix} 0 & -B_x & -B_y & -B_z \\ B_x & 0 & E_z/c & - E_y/c \\ B_y & -E_z/c & 0 & E_x/c \\ B_z & E_y/c & -E_x/c & 0 \end{bmatrix}$

Properties

The matrix form of the field tensor yields the following properties:

1. Antisymmetry:

   $F^{\mu\nu} \, = - F^{\nu\mu}$

   (hence the name bivector).
2. Six independent components: which are the three spatial components of the electric field (E_x, E_y, E_z) and magnetic field (B_x, B_y, B_z).
3. Inner product and Lorentz covariance: 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:

   $G_{\gamma\delta}F^{\gamma\delta}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = -\frac{4}{c} \left( \bold B \cdot \bold E \right) \,$

   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:

   $\det \left( F \right) = \frac{1}{c^2} \left( \bold B \cdot \bold E \right) ^{2}$

   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:

$\bold{\nabla} \cdot \bold{E} = \frac{\rho}{\epsilon_0},\quad \bold{\nabla} \times \bold{B} - \frac{1}{c^2} \frac{ \partial \bold{E}}{\partial t} = \mu_0 \bold{J}$

and reduce to:

$\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}$

where

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

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

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

which reduce to:

$\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0$

or using the index notation with square brackets 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

$\begin{align} \mathcal{L} & = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} \\ & = - \frac{1}{4\mu_0} \left( \partial_\mu A_\nu - \partial_\nu A_\mu \right) \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) \\ & = -\frac{1}{4\mu_0} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\nu A^\mu \right)\\ \end{align}$

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:

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

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

$\partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) = 0. \,$

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

$\partial_\nu F^{\mu \nu} = 0$

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

$~E^i/c = -F^{0 i} \,$

$\epsilon^{ijk} B_k = -F^{ij} \,$

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

When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case $\partial_\nu F^{\mu \nu}$ is equal to the 4-current instead of zero.

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

   $\begin{align} 0 & = \begin{matrix} \frac{2}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha }) \\ & = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (2F_{ \alpha \beta }) + \partial_\alpha (2F_{ \beta \gamma }) + \partial_\beta (2F_{ \gamma \alpha }) \} \\ & = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (F_{ \alpha \beta } - F_{ \beta \alpha}) + \partial_\alpha (F_{ \beta \gamma } - F_{ \gamma \beta}) + \partial_\beta (F_{ \gamma \alpha } - F_{ \alpha \gamma}) \} \\ & = \begin{matrix} \frac{1}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } - \partial_\gamma F_{ \beta \alpha} - \partial_\alpha F_{ \gamma \beta} - \partial_\beta F_{ \alpha \gamma} ) \\ & = \partial_{[ \gamma} F_{ \alpha \beta ]} \end{align}$

See also

• Electromagnetic stress–energy tensor
• Application of tensor theory in physics
• Classification of electromagnetic fields
• Covariant formulation of classical electromagnetism
• Ricci calculus
• Riemann–Silberstein vector

References

• 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.