Electromagnetic tensor

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The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor or Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism.

Details

The electromagnetic tensor ${\displaystyle \,F_{ab}}$  is commonly written as a matrix:

${\displaystyle F_{ab}={\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$ 

where

E is the electric field

B the magnetic field and

c the speed of light. When using natural units, the speed of light is taken to equal 1.

From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties (Mathematical note: In this article, the abstract index notation will be used.):

• antisymmetry: ${\displaystyle F^{ab}\,=-F^{ba}}$  (hence the name bivector).
• zero trace.
• six independent components.

More formally, the electromagnetic tensor may be written in terms of the four-potential ${\displaystyle \,A^{a}}$ 

${\displaystyle F_{ab}\equiv {\frac {\partial A_{b}}{\partial x^{a}}}-{\frac {\partial A_{a}}{\partial x^{b}}}\equiv \partial _{a}A_{b}-\partial _{b}A_{a}\equiv A_{b,a}-A_{a,b}}$ 

where ${\displaystyle A^{a}=(\phi ,{\vec {A}}c)}$  and ${\displaystyle A_{a}\,=\eta _{ab}A^{b}}$  (${\displaystyle \,\eta }$  is the Minkowski metric).

Derivation

To derive all the elements in the electromagnetic tensor we need to define

${\displaystyle \partial _{a}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,}$ 

and

${\displaystyle A_{a}=\left({\frac {\Phi }{c}},-A_{x},-A_{y},-A_{z}\right)\,}$ 

where

${\displaystyle A\,}$  is the vector potential

${\displaystyle \Phi \,}$  is the scalar potential and

${\displaystyle c\,}$  is the speed of light

Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:

${\displaystyle {\vec {E}}=-{\frac {\partial {\vec {A}}}{\partial t}}-\nabla \Phi \,}$ 

${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}\,}$ 

As an example, the x components are just

${\displaystyle E_{x}=-{\frac {\partial A_{x}}{\partial t}}-{\frac {\partial \Phi }{\partial x}}\,}$ 

${\displaystyle B_{x}={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\,}$ 

Using the definitions we began with can re-write these two equations to look like:

${\displaystyle E_{x}=-c\left(\partial _{1}A_{0}-\partial _{0}A_{1}\right)\,}$ 

${\displaystyle B_{x}=\partial _{3}A_{2}-\partial _{2}A_{3}\,}$ 

Evaluating all the components results in a second-rank, antisymmetric and covariant tensor:

${\displaystyle F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}\,}$ 

Significance of the Field Tensor

Hidden beneath the surface of this overly complex mathematical equation is an ingenious unification of maxwell's equations for electromagnetism. Consider the electrostatic equation

${\displaystyle \nabla \cdot {\textbf {E}}={\frac {\rho }{\epsilon _{0}}}}$ 

which tells us that the divergence of the Electric field vector is equal to the charge density, and the electrodynamic equation

${\displaystyle \nabla \times {\textbf {B}}-{\frac {1}{c^{2}}}{\frac {\partial {\textbf {E}}}{\partial t}}=\mu _{0}{\textbf {J}}}$ 

that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative four pi times the current density.

These two equations for electricity reduce to

${\displaystyle \partial _{a}F^{ab}=\mu _{0}J^{b}}$ 

where

${\displaystyle J^{a}=(\rho ,J)\,}$  is the 4-current.

The same holds for magnetism. If we take the magnetostatic's equation

${\displaystyle \nabla \cdot {\textbf {B}}=0}$ 

which tells us that there are no "true" magnetic charges, and the magnetodynamics equation

${\displaystyle {\frac {\partial {\textbf {B}}}{\partial t}}+\nabla \times {\textbf {E}}=0}$ 

which tells us the change of the magnetic field with respect to time plus the curl of the Electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to

${\displaystyle F_{ab,c}+F_{bc,a}+F_{ca,b}=0.\,}$ 

The field tensor and 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 (non-gravitational) physical laws being recognised after the advent 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 elegant presentation of physical laws. For example, Maxwell's equations of electromagnetism may be written using the field tensor as:

${\displaystyle F_{[ab,c]}\,=0}$  and ${\displaystyle F^{ab}{}_{,b}\,=\mu _{0}J^{a}}$ 

where the comma indicates a partial derivative. The second equation implies conservation of charge:

${\displaystyle J^{a}{}_{,a}\,=0}$ 

In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way:

${\displaystyle F_{[ab;c]}\,=0}$  and ${\displaystyle F^{ab}{}_{;b}\,=\mu _{0}J^{a}}$ 

where the semi-colon represents a covariant derivative, as opposed to a partial derivative. The elegance of these equations stems from the simple replacing of partial with covariant derivatives, a practice sometimes referred to in the parlance of GR as 'replacing partial with covariant derivatives'. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):

${\displaystyle J^{a}{}_{;a}\,=0}$ 

Role in Quantum Electrodynamics and Field Theory

The Lagrangian of QED extends beyond the classical Lagrangian established in relativity from ${\displaystyle L=-{\frac {1}{4\pi }}F^{\mu \nu }F_{\mu \nu },}$  to incorporate the creation and annihilation of photons (and electrons).

In Quantum field theory, it is used for the template of the gauge field strength tensor. That is used in addition to the local interaction Lagrangian, nearly identical to its role in QED.

See also

• Application of tensor theory in physics
• Classification of electromagnetic fields

References

• . ISBN 0195146654. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)