View source for Electromagnetic tensor

{{Short description|Mathematical object that describes the electromagnetic field in spacetime}}
{{for|an explanation and meanings of the index notation in this article|Einstein notation|antisymmetric tensor}}
{{distinguish-redirect|Electromagnetic field strength|Electric field strength|Magnetic field strength}}
{{electromagnetism|expanded=Covariance}}

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]] in spacetime.  The field tensor was first used after the four-dimensional [[tensor]] formulation of [[special relativity]] was introduced by [[Hermann Minkowski]]. The tensor allows related physical laws to be written very concisely, and allows for the [[quantization of the electromagnetic field]] by Lagrangian formulation described [[Electromagnetic tensor#Quantum electrodynamics and field theory|below]].

==Definition==

The electromagnetic tensor, conventionally labelled ''F'', is defined as the [[Exterior derivative#Exterior derivative of a k-form|exterior derivative]] of the [[electromagnetic four-potential]], ''A'', a differential 1-form:<ref>{{cite book |author1=J. A. Wheeler |author2=C. Misner |author3=K. S. Thorne | title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref><ref>{{cite book | author=D. J. Griffiths| title=Introduction to Electrodynamics |edition=3rd| publisher=Pearson Education, Dorling Kindersley| year=2007 | isbn=978-81-7758-293-2}}</ref>

:<math>F \ \stackrel{\mathrm{def}}{=}\ \mathrm{d}A.</math>

Therefore, ''F'' is a [[differential form|differential 2-form]]—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

:<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.</math>

where <math>\partial</math> is the [[four-gradient]] and <math>A</math> is the [[electromagnetic four-potential|four-potential]].

[[Maxwell's equations#Conventional formulation in SI units|SI units for Maxwell's equations]] and the [[sign convention#Metric signature|particle physicist's sign convention]] for the [[Metric signature|signature]] of [[Minkowski space]] {{nowrap|(+ − − −)}}, will be used throughout this article.

===Relationship with the classical fields===
The Faraday [[Differential form|differential 2-form]] is given by

:<math>
  F = (E_x/c)\ dx \wedge dt + (E_y/c)\ dy \wedge dt + (E_z/c)\ dz \wedge dt + B_x\ dy \wedge dz + B_y\ dz \wedge dx + B_z\ dx \wedge dy,
</math>

where <math> dt </math> is the time element times the speed of light <math> c </math>.

This is the [[exterior derivative]] of its 1-form antiderivative

:<math> A = A_x\ dx + A_y\ dy + A_z\ dz - (\phi/c)\ dt </math>,

where <math> \phi(\vec{x},t) </math> has <math> -\vec{\nabla}\phi = \vec{E} </math> (<math> \phi </math> is a scalar potential for the [[Conservative vector field|irrotational/conservative vector field]] <math> \vec{E} </math>) and <math> \vec{A}(\vec{x},t) </math> has <math> \vec{\nabla} \times \vec{A} = \vec{B} </math> (<math> \vec{A} </math> is a vector potential for the [[solenoidal vector field]] <math> \vec{B} </math>).

Note that

:<math> \begin{cases} dF = 0 \\ {\star}d{\star}F = J \end{cases} </math>

where <math> d </math> is the exterior derivative, <math>{\star}</math> is the [[Hodge star operator|Hodge star]], <math> J = -J_x\ dx - J_y\ dy - J_z\ dz + \rho\ dt </math> (where <math> \vec{J} </math> is the [[Current density|electric current density]], and <math> \rho </math> is the [[Charge density|electric charge density]]) is the 4-current density 1-form, is the differential forms version of Maxwell's equations.

The [[Electric field|electric]] and [[magnetic field]]s can be obtained from the components of the electromagnetic tensor. The relationship is simplest in [[Cartesian coordinate system|Cartesian coordinates]]:

:<math>E_i = c F_{0i},</math>
where ''c'' is the speed of light, and
:<math>B_i = -1/2\epsilon_{ijk} F^{jk},</math>
where <math>\epsilon_{ijk}</math> is the [[Levi-Civita tensor]]. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will [[covariant transformation|transform covariantly]], and the fields in the new frame will be given by the new components.
  
In contravariant [[matrix (mathematics)|matrix]] form with metric signature (+,-,-,-),
:<math>
  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}.
</math>

The covariant form is given by [[Raising and lowering indices#Order-2|index lowering]],

:<math>
  F_{\mu\nu} = \eta_{\alpha\nu}F^{\beta\alpha}\eta_{\mu\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}.
</math>
The Faraday tensor's [[Hodge star operator|Hodge dual]] is 
:<math>
{ G^{\alpha\beta} = \frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}=\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}
}
</math>

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

===Properties===

The matrix form of the field tensor yields the following properties:<ref>{{cite book |author1=J. A. Wheeler |author2=C. Misner |author3=K. S. Thorne | title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>

#'''[[Skew-symmetric matrix|Antisymmetry]]:''' <math display="block">F^{\mu\nu} = - F^{\nu\mu}</math>
#'''Six independent components:''' In Cartesian coordinates, these are simply the three spatial components of the electric field (''E<sub>x</sub>, E<sub>y</sub>, E<sub>z</sub>'') and magnetic field (''B<sub>x</sub>, B<sub>y</sub>, B<sub>z</sub>'').
#'''Inner product:''' If one forms an inner product of the field strength tensor a [[Lorentz invariant]] is formed <math display="block">F_{\mu\nu} F^{\mu\nu} = -2 \left( \frac{E^2}{c^2} - B^2 \right)</math> meaning this number does not change from one [[frame of reference]] to another.
#'''[[Pseudoscalar]] invariant:''' The product of the tensor <math>F^{\mu\nu}</math> with its [[Hodge dual]] <math>G^{\mu\nu}</math> gives a [[Lorentz invariant]]: <math display="block">G_{\gamma\delta}F^{\gamma\delta} = \frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = -\frac{4}{c} \mathbf{B} \cdot \mathbf{E} \,</math> where <math>\epsilon_{\alpha\beta\gamma\delta}</math> 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 <math> \epsilon_{0123} = -1 </math>.
#'''[[Determinant]]:''' <math display="block">\det \left( F \right) = \frac{1}{c^2} \left( \mathbf{B} \cdot \mathbf{E} \right)^2</math> which is proportional to the square of the above invariant.
#'''[[Trace (linear algebra)|Trace]]:''' <math display="block">F={{F}^{\mu }}_{\mu }=0</math> which is equal to zero.

===Significance===

This tensor simplifies and reduces [[Maxwell's equations]] as four vector calculus equations into two tensor field equations. In [[electrostatic]]s and [[electrodynamic]]s, [[Gauss's law]] and [[Ampère's circuital law]] are respectively:

:<math>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0},\quad \nabla \times \mathbf{B} - \frac{1}{c^2} \frac{ \partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J} </math>

and reduce to the inhomogeneous Maxwell equation:

:<math>\partial_{\alpha} F^{\beta\alpha} = - \mu_0 J^{\beta}</math>,    where  <math>J^{\alpha} = ( c\rho, \mathbf{J} ) </math>   is the [[four-current]].

In [[magnetostatic]]s and magnetodynamics, [[Gauss's law for magnetism]] and [[Faraday's law of induction|Maxwell–Faraday equation]] are respectively:

:<math>\nabla \cdot \mathbf{B} = 0,\quad \frac{ \partial \mathbf{B}}{ \partial t } + \nabla \times \mathbf{E} = \mathbf{0} </math>

which reduce to [[Bianchi identity]]:

:<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0 </math>

or using the [[Ricci calculus#Symmetric and antisymmetric parts|index notation with square brackets]]{{ref|antisymmetric|[note 1]}} for the antisymmetric part of the tensor:

:<math> \partial_{ [ \alpha } F_{ \beta \gamma ] } = 0 </math>
Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically (<math>\equiv 0</math>). The implication of that identity is far-reaching: it means that the EM field theory leaves no room for magnetic monopoles and currents of such.

==Relativity==

{{main|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  physical laws being recognised after the advent of [[special relativity]]. This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of [[tensor]]s. The tensor formalism also leads to a mathematically simpler presentation of physical laws.

The inhomogeneous Maxwell equation leads to the [[continuity equation]]:

:<math>\partial_\alpha J^\alpha = J^\alpha{}_{,\alpha} = 0</math>

implying [[conservation of charge]].

Maxwell's laws above can be generalised to [[curved spacetime]] by simply replacing [[partial derivative]]s with [[covariant derivative]]s:

:<math>F_{[\alpha\beta;\gamma]} = 0</math> and  <math>F^{\alpha\beta}{}_{;\alpha} = \mu_0 J^{\beta}</math>

where the semi-colon [[Ricci calculus|notation]] represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the [[Maxwell's equations in curved spacetime|curved space Maxwell equations]]. Again, the second equation implies charge conservation (in curved spacetime):

:<math>J^\alpha{}_{;\alpha} \, = 0</math>

==Lagrangian formulation of classical electromagnetism==

{{see also|Classical field theory}}

[[Classical electromagnetism]] and [[Maxwell's equations]] can be derived from the [[action (physics)|action]]:
<math display="block">\mathcal{S} = \int \left( -\begin{matrix} \frac{1}{4 \mu_0} \end{matrix} F_{\mu\nu} F^{\mu\nu} - J^\mu A_\mu \right) \mathrm{d}^4 x \,</math>
where <math>\mathrm{d}^4 x</math> is over space and time.

This means the [[Lagrangian (field theory)|Lagrangian]] density is

:<math>\begin{align}
  \mathcal{L} &= -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - J^\mu A_\mu \\
              &= -\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) - J^\mu A_\mu \\
              &= -\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) - J^\mu A_\mu \\
\end{align}</math>

The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is

:<math>\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) - J^\mu A_\mu.</math>

Substituting this into the [[Euler–Lagrange equation]] of motion for a field:

:<math> \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu A_\nu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 </math>

So the Euler–Lagrange equation becomes:

:<math> - \partial_\mu \frac{1}{\mu_0} \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) + J^\nu = 0. \,</math>

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

:<math> \partial_\mu F^{\mu \nu} = \mu_0 J^\nu </math>

That equation is another way of writing the two inhomogeneous [[Maxwell's equations]] (namely,  [[Gauss's law]] and [[Ampère's circuital law]]) using the substitutions:

:<math>\begin{align}
      \frac{1}{c}E^i &= -F^{0 i} \\
  \epsilon^{ijk} B_k &= -F^{ij}
\end{align}</math>

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

===Hamiltonian form===

The [[Hamiltonian field theory|Hamiltonian]] density can be obtained with the usual relation,

:<math>\mathcal{H}(\phi^i,\pi_i) = \pi_i \dot{\phi}^i(\phi^i,\pi_i) - \mathcal{L}</math>.

===Quantum electrodynamics and field theory===

{{main|Quantum electrodynamics|quantum field theory}}

The [[Lagrangian (field theory)|Lagrangian]] of [[quantum electrodynamics]] extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):

:<math>\mathcal{L} = \bar\psi \left(i\hbar c \, \gamma^\alpha D_\alpha - mc^2\right) \psi - \frac{1}{4\mu_0} F_{\alpha\beta} F^{\alpha\beta},</math>

where the first part in the right hand side, containing the [[Dirac spinor]] <math>\psi</math>, represents the [[Dirac field]]. 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.

==See also==
* [[Classification of electromagnetic fields]]
* [[Covariant formulation of classical electromagnetism]]
* [[Electromagnetic stress–energy tensor]]
* [[Gluon field strength tensor]]
* [[Ricci calculus]]
* [[Riemann–Silberstein vector]]

==Notes==

{{Reflist|group="note"}}

{{ordered list
|1={{note|antisymmetric}} By definition,

:<math> T_{[abc]} = \frac{1}{3!}(T_{abc} + T_{bca} + T_{cab} - T_{acb} - T_{bac} - T_{cba})</math>
So if
:<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0</math>
then
:<math>\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}</math>
}}

{{reflist}}

==References==
*{{cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=[[Oxford University Press]] | year=2004 | isbn=0-19-514665-4}}
*{{cite book | author=Jackson, John D. | title=Classical Electrodynamics | url=https://archive.org/details/classicalelectro0000jack_e8g9 | url-access=registration | publisher=[[John Wiley & Sons, Inc.]] | year=1999 | isbn=0-471-30932-X}}
*{{cite book | author1=Peskin, Michael E. | author2=Schroeder, Daniel V. | title=An Introduction to Quantum Field Theory | publisher=Perseus Publishing | year=1995 | isbn=0-201-50397-2 | url-access=registration | url=https://archive.org/details/introductiontoqu0000pesk }}

{{tensors}}

[[Category:Electromagnetism]]
[[Category:Minkowski spacetime]]
[[Category:Theory of relativity]]
[[Category:Tensor physical quantities]]
[[Category:Tensors in general relativity]]