Electromagnetic tensor

The electromagnetic field tensor (also Faraday tensor or simply field strength tensor ) is a physical quantity that describes the electromagnetic field as a field in the space-time in electrodynamics. It was introduced in 1908 by Hermann Minkowski in the framework of the theory of relativity. Known from physics and engineering vectorial field quantities such as electric and magnetic field strength can be derived from the field strength tensor. The term tensor for the type of this size is an acronym, actually it is a tensor field, ie one -to-point varying from point tensor.

Definition

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

For example, with the classical vector potential

The above definition is also valid for quantum electrodynamics. There is just the vector potential operatorwertig. 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:

• Is antisymmetric:
• Disappearing track:
• 6 free components

Here are some frequently occurring contractions:

In the Lagrangian occurs on this Lorentz invariant term:

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

With the convention = 1.

In some invoices this size occurs:

The energy -momentum tensor of general relativity for the electromagnetic field is formed from:

View as matrix

The matrix representation of the Feldstärketensors is coordinate- dependent. In a flat space-time ( ie, with Minkowski metric ) and Cartesian coordinates, the contravariant field tensor can be written as:

( This matrix is sometimes also briefly referred to as tensor, but is not itself the tensor ). The covariance form of the matrix representation of the tensor is of the signature using ( , -, -, - ) in accordance with

Inhomogeneous Maxwell equations in a compact formulation

It is customary to define the dual electromagnetic field tensor:

Where the covariant field strength tensor is.

This allows both the homogeneous and the inhomogeneous Maxwell equations write compact:

With the following four-current was used:

Representation in differential form notation

Hereinafter the CGS system is used to work out the fundamental relationships more clearly.

The field strength tensor is a second-level differential form on the space-time. Maxwell's equations in differential form of writing and read with the magnetic flux density and the current density both as a one - turn shapes on the space-time.

Since it is assumed in the control of the absence of magnetic charges, and the field tensor can thus be represented as a derivative of a 1 -shape. corresponds to the space-time vector potential. In the presence of magnetic charges If we add another vector potential, the source of which is the magnetic flux density.

For example, the field strength tensor of a point charge with the distance from the charge in its rest frame ( conditions are uniform motion and absence of spatial curvature ).

4, the shape is the Lagrangian of the electromagnetic field.

Derivation of the vectorial field sizes

Relative to the motion of an observer through space and time, the field strength tensor can be decomposed into an electric and a magnetic component. The observer takes these shares as a true electric or magnetic field strength. Different observers moving with each other can therefore perceive different electric or magnetic field strengths.

For example, if in an electrical generator in a " magnetic " field is moved relative a wire, the Feldstärkentensor has relative to the wire movement, and thus from the perspective of the electrons contained in the wire and an electrical component which is responsible for the induction of the electric voltage on dissection.

In flat space-time ( Minkowski space ) can be the vector fields and from the coordinate representation of the Feldstärketensors read: you get the above matrix representation. A more general relation is derived from the decomposition, which correspond to a time-like and space-like vector fields.

Occurrence in quantum electrodynamics

The field strength tensor enters directly in the QED Lagrangian (here without gauge fixing terms ) to: