Electromagnetic field strength tensor

The electromagnetic field strength tensor (also Faraday tensor or simply field strength tensor ) is a physical quantity that describes the electromagnetic field as a field in space-time in electrodynamics . It was introduced in 1908 by Hermann Minkowski as part of the theory of relativity . The vector field variables known from physics and technology, such as electric and magnetic field strength, can be derived from the field strength tensor. The term tensor for the nature of this size is an acronym, actually it's a tensor field , so a varying from point to point tensor .

definition

The electromagnetic field strength tensor is usually defined by the vector potential :

${\displaystyle \,F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }}$

z. B. with the classic vector potential

${\displaystyle A^{\mu }=\left(\phi /c,{\vec {A}}\right)}$

This definition is also valid for quantum electrodynamics . There just the vector potential is operator-valued. It is a special case of the field strength tensor definition of a general gauge theory .

Properties and formulas

The field strength tensor has the following properties:

• ${\displaystyle F^{\mu \nu }}$ is antisymmetric: ${\displaystyle F^{\mu \nu }=-F^{\nu \mu }}$
• Disappearing trace : ${\displaystyle F^{\mu \mu }=0}$
• Due to the antisymmetry, only 6 of the 16 components are independent

Here are some common contractions:

This Lorentz invariant term occurs in the Lagrangian :

${\displaystyle F_{\alpha \beta }F^{\alpha \beta }=\ 2\left(B^{2}-{\frac {E^{2}}{c^{2}}}\right)}$

Also of interest is the pseudoscalar invariant formed with the Levi-Civita symbol :

${\displaystyle \epsilon _{\alpha \beta \gamma \delta }F^{\alpha \beta }F^{\gamma \delta }=-{\frac {8}{c}}\left({\vec {B}}\cdot {\vec {E}}\right)}$

With the convention . ${\displaystyle \ \epsilon _{0123}=-1}$

This quantity also appears in some invoices:

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

The energy-momentum tensor of the general theory of relativity for the electromagnetic field is formed from : ${\displaystyle T^{\,\mu \nu }}$ ${\displaystyle F^{\,\alpha \beta }}$

${\displaystyle T^{\alpha \beta }=F^{\alpha \gamma }F_{\gamma }^{\;\;\beta }-{\frac {1}{4}}g^{\alpha \beta }F_{\mu \nu }F^{\nu \mu }}$

Representation as a matrix

The matrix representation of the field strength tensor depends on the coordinates . In a flat spacetime (i.e. with the Minkowski metric ) and Cartesian coordinates, the contravariant field strength tensor can be written as:

${\displaystyle F^{\mu \nu }=\left({\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\\\end{matrix}}\right)}$

(This matrix is ​​sometimes also called the tensor for short, but it is not the tensor itself). The covariant form of the matrix representation of the tensor reads accordingly when using the signature (+, -, -, -)

${\displaystyle F_{\mu \nu }=\left({\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\\\end{matrix}}\right)~.}$

Inhomogeneous Maxwell equations in a compact formulation

It is common to also define the dual electromagnetic field strength tensor:

${\displaystyle {\tilde {F}}^{\mu \nu }:={\frac {1}{2}}\,\varepsilon ^{\mu \nu \alpha \beta }\,F_{\alpha \beta }\quad \Rightarrow \quad {\tilde {F}}^{\mu \nu }=\left[{\begin{matrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\\\end{matrix}}\right]}$

where is the covariant field strength tensor. ${\displaystyle \,F_{\alpha \beta }}$

This allows both the homogeneous and the inhomogeneous Maxwell equations to be written down in compact form:

${\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}j^{\nu }\qquad \partial _{\mu }{\tilde {F}}^{\mu \nu }=0}$

where the following stream of four was used:

${\displaystyle j^{\mu }=\left(c\rho ,{\vec {j}}\right)}$

Representation in differential form notation

In the following, the CGS system is used to work out the fundamental relationships more clearly.

The field strength tensor is a second order differential form on spacetime. The Maxwell's equations are denominated in differential form of writing and having the magnetic current density and the electric current density , both as a 1-forms again on the space-time. ${\displaystyle F}$ ${\displaystyle \ast \mathrm {d} F=j_{\mathrm {mag} }}$ ${\displaystyle \ast \mathrm {d} \ast \mathrm {F} =j}$ ${\displaystyle j_{\mathrm {mag} }}$ ${\displaystyle j}$

Since the absence of magnetic charges is generally assumed, is , and the field strength tensor can thus be represented as a derivative of a 1-form . corresponds to the spatiotemporal vector potential. If magnetic charges are present, a further vector potential is added, the source of which is the magnetic current density. ${\displaystyle \mathrm {d} F=0}$ ${\displaystyle F=\mathrm {d} A}$ ${\displaystyle A}$ ${\displaystyle A}$

Example: The field strength tensor of a point charge at rest is with the distance . A corresponding Lorentz transformation provides the field strength tensor of a uniformly moving charge. ${\displaystyle q}$ ${\displaystyle F=-q\mathrm {d} t\land \mathrm {d} {\frac {1}{r}}}$ ${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$

The 4-form is the Lagrangian density of the electromagnetic field. ${\displaystyle {\tfrac {1}{2}}F\land *F}$

Derivation of the vector field quantities

In relation to the movement of an observer through space and time, the field strength tensor can be broken down into an electrical and a magnetic component. The observer perceives these components as an electric or magnetic field strength . Different observers moving towards one another can therefore perceive different electrical or magnetic field strengths.

Example: If a wire is moved in an electrical generator relative to a "magnetic" field, the field strength tensor, when broken down, has an electrical component relative to the movement of the wire and thus, from the point of view of the electrons contained in the wire, an electrical component that is responsible for the induction of the electrical voltage .

In flat space-time ( Minkowski space ), the vector fields and can be read from the coordinate representation of the field strength tensor: the above matrix representation is obtained. A allgemeingültigere relationship follows from the decomposition , where a time-like and , space-like vector fields, respectively. ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle F={\tfrac {1}{2}}F_{\mu \nu }\mathrm {d} x^{\mu }\land \mathrm {d} x^{\nu }}$ ${\displaystyle F=u\land E+\ast (u\land B)}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle B}$

Occurrence in quantum electrodynamics

The field strength tensor appears directly in the QED Lagrangian (here without calibration fixation terms):

${\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}\left[i\gamma _{\mu }D^{\mu }-m\right]\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }}$

literature

• JD Jackson: Classical Electrodynamics . de Gruyter, 2002, ISBN 3-11-016502-3 .
• C. Misner, KS Thorne, JA Wheeler: Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 .
• Torsten Fließbach : General Theory of Relativity . Spectrum Academic Publishing House, 2003, ISBN 3-8274-1356-7 .

Individual evidence

1. ^ Sylvan A. Jacques: Relativistic Field Theory of Fluids . arxiv : physics / 0411237