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{{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}}
==
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]]— 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>
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>
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>
===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( B^2-\frac{E^2}{c^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>. This and the previous Lorentz invariant vanish in the crossed field case.
#'''[[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>
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 the [[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>). This tensor equation reproduces the homogeneous Maxwell's equations.
==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 [[Covariant derivative#Notation|semicolon 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>
The stress-energy tensor of electromagnetism
:<math>T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,,</math>
satisfies
:<math>{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_\beta = 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>
Here <math>\phi^i=A^{i}</math> are the fields and the momentum density of the EM field is
:<math>\pi_i= T_{0i}=\frac{1}{\mu_0} F_0{}^{ \alpha}F_{i\alpha}=\frac{1}{\mu_0 c} \mathbf{E}\times\mathbf{B} \,.</math>
such that the conserved quantity associated with translation from [[Noether's theorem]] is the total momentum
:<math>\mathbf{P}= \sum_{\alpha} m_\alpha \dot{\mathbf{x}}_{\alpha} + \frac{1}{\mu_0 c}\int_{\mathcal{V}} \mathrm{d}^3 x\, \mathbf{E}\times\mathbf{B} \,.</math>
The Hamiltonian density for the electromagnetic field is related to the [[electromagnetic stress-energy tensor]]
:<math>T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,.</math>
as
:<math>\mathcal{H} = T_{0 0} = \frac{1}{2}\left(\epsilon_0 \mathbf{E}^2+\frac{1}{\mu_0}\mathbf{B}^2\right) = \frac{1}{8\pi}\left(\mathbf{E}^2+\mathbf{B}^2\right)\,.</math>
where we have neglected the [[perfect fluid|energy density of matter]], assuming only the EM field, and the last equality assumes the CGS system. The momentum of nonrelativistic charges interacting with the EM field in the [[Coulomb gauge]] (<math>\nabla\cdot \mathbf{A}=\nabla_i A^i = 0</math>) is
:<math>\mathbf{p}_\alpha = m_\alpha \dot{\mathbf{x}}_{\alpha} + \frac{q_{\alpha}}{c} \mathbf{A}(\mathbf{x}_\alpha) \,.</math>
The total Hamiltonian of the matter + EM field system is
:<math>H = \int_\mathcal{V} d^3 x \,T_{00} = H_{\rm mat} + H_{\rm em} \,.</math>
where for nonrelativistic point particles in the Coulomb gauge
:<math>H_{\rm mat} = \sum_\alpha m_{\alpha} |\dot{\mathbf{x}}_{\alpha}|^2+ \sum_{\alpha<\beta} \frac{q_{\alpha}q_{\beta}}{|\mathbf{x}_{\alpha} - \mathbf{x}_{\beta}| } =
\sum_\alpha \frac{1}{2m_\alpha} \left[\mathbf{p}_{\alpha} - \frac{q_{\alpha}}{c} \mathbf{A}(\mathbf{x}_\alpha)\right]^2 + \sum_{\alpha<\beta} \frac{q_{\alpha}q_{\beta}}{|\mathbf{x}_{\alpha} - \mathbf{x}_{\beta}| } \,.</math>
where the last term is identically <math>\frac{1}{8\pi} \int_\mathcal{V} d^3 x \mathbf{E}_{\parallel}^2</math> where <math>{E}_{\parallel i} = {\nabla_i}A_0</math>
and
:<math>H_{\rm em} = \frac{1}{8\pi} \int_\mathcal{V} d^3 x \left(\mathbf{E}_{\perp}^2+\mathbf{B}^2\right) \,.</math>
where and <math>{E}_{\perp i} = -\frac{1}{c}\partial_0 A_i</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]]
==
{{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]]
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