Frequent Links
Covariant formulation of classical electromagnetism
Electromagnetism 

Solenoid 
Covariant formulation 
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or nonrectilinear coordinate systems.
This article uses SI units for the purely spatial components of tensors (including vectors), the classical treatment of tensors and the Einstein summation convention throughout, and the Minkowski metric has the form diag (+1, −1, −1, −1). Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of total charge and current.
For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see Classical electromagnetism and special relativity.
Contents
Covariant objects
Preliminary 4vectors
For background purposes, we present here three other relevant fourvectors, which are not directly connected to electromagnetism, but which will be useful in this article:
 In meter, the "position" or "coordinate" fourvector is
 <math>x^\alpha = (ct, x, y, z) \,.</math>
 In meter·second^{−1}, the velocity fourvector (or fourvelocity) is
 <math>u^\alpha = \gamma(c,\bold{u}) \,</math>
 where γ(u) is the Lorentz factor at the 3velocity u.
 In kilogram·meter·second^{−1}, the fourmomentum (or momentum fourvector) of a particle is
 <math>p_\alpha = ( E/c,  \bold{p}) = mu_{\alpha} \,</math>
 where p is the 3momentum, E is the energy, and m is the particle's rest mass.
 In meter^{−1} the fourgradient is
 <math>\partial^{\nu} = \frac{\partial}{\partial x_{\nu}} = \left( \frac{1}{c} \frac{\partial}{\partial t},  \bold{\nabla} \right) \,,</math>
 In meter^{−2} the d'Alembertian operator is denoted: <math> \Box </math>.
The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is +, corresponding to the Minkowski metric tensor:
 <math>\eta^{\mu \nu}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\,</math>
Electromagnetic tensor
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor. In volt·seconds·meter^{−2}, the field strength tensor is written in terms of fields as:^{[1]}
 <math>F_{\alpha \beta} = \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)\,</math>
and the result of raising its indices is
 <math>F^{\mu \nu} \, \stackrel{\mathrm{def}}{=} \, \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \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)\,.</math>
where E is the electric field, B the magnetic field, and c the speed of light.
Fourcurrent
The fourcurrent is the contravariant fourvector which combines electric current density J and electric charge density ρ. In amperes·meter^{−2}, it is given by
 <math>J^{\alpha} = \, (c \rho, \bold{J} ) \,</math>
Fourpotential
In volt·seconds·meter^{−1}, the electromagnetic fourpotential is a covariant fourvector containing the electric potential (also called the scalar potential) φ and magnetic vector potential (or vector potential) A, as follows:
 <math>A_{\alpha} = \left(\phi/c,  \bold{A} \right)\,</math>
The relation between the electromagnetic potentials and the electromagnetic fields is given by the following equation:
 <math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta}  \partial_{\beta} A_{\alpha} \,</math>
Electromagnetic stress–energy tensor
The electromagnetic stress–energy tensor can be interpreted as the flux (density) of the momentum 4vector, and is a contravariant symmetric tensor which is the contribution of the electromagnetic fields to the overall stress–energy tensor. In joule·meter^{−3}, it is given by
 <math>T^{\alpha\beta} = \begin{pmatrix} \epsilon_{0}E^2/2 + B^2/2\mu_{0} & S_x/c & S_y/c & S_z/c \\ S_x/c & \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
S_y/c & \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ S_z/c & \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{pmatrix}\,</math>
where ε_{0} is the electric permittivity of vacuum, μ_{0} is the magnetic permeability of vacuum, the Poynting vector in watt·meter^{−2} is
 <math>\bold{S} = \frac{1}{\mu_{0}} \bold{E} \times \bold{B} </math>
and the Maxwell stress tensor in joule·meter^{−3} is given by
 <math>\sigma_{ij} = \epsilon_{0}E_{i}E_{j} + \frac{1}{\mu_{0}}B_{i}B_{j} 
\left(\frac12\epsilon_{0}E^2 + \frac{1}{2\mu_{0}}B^2\right)\delta_{ij} \,.</math>
The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:
 <math>T^{\alpha\beta} = \frac{1}{\mu_{0}} \left( \eta_{\gamma \nu}F^{\alpha \gamma}F^{\nu \beta}  \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)</math>
where η is the Minkowski metric tensor. Notice that we use the fact that
 <math>\epsilon_{0} \mu_{0} c^2 = 1\,</math>
which is predicted by Maxwell's equations.
Maxwell's equations in vacuum
In a vacuum (or for the microscopic equations, not including macroscopic material descriptions) Maxwell's equations can be written as two tensor equations.
The two inhomogeneous Maxwell's equations, Gauss's Law and Ampère's law (with Maxwell's correction) combine into (with + metric):^{[2]}
while the homogeneous equations  Faraday's law of induction and Gauss's law for magnetism combine to form:
Gauss–Faraday law (vacuum) <math>\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0 </math>
where F^{αβ} is the electromagnetic tensor, J^{α} is the 4current, ε^{αβγδ} is the LeviCivita symbol, and the indices behave according to the Einstein summation convention.
The first tensor equation corresponds to four scalar equations, one for each value of β. The second tensor equation actually corresponds to 4^{3} = 64 different scalar equations, but only four of these are independent. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with λ, μ, ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.
Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation can also be written more compactly as:
 <math> F_{[\alpha \beta , \gamma]} =0 </math>
In the absence of sources, Maxwell's equations reduce to:
 <math>\partial^{\nu} \partial_{\nu} F^{\alpha\beta} \,\ \stackrel{\mathrm{def}}{=}\ \, \Box F^{\alpha\beta} \,\ \stackrel{\mathrm{def}}{=}\ {1 \over c^2 } { \partial^2 F^{\alpha\beta} \over {\partial t }^2 }  \nabla^2 F^{\alpha\beta}= 0 \,,</math>
which is an electromagnetic wave equation in the field strength tensor.
Maxwell's equations in the Lorenz gauge
The Lorenz gauge condition is a Lorentzinvariant gauge condition. (This can be contrasted with other gauge conditions such as the Coulomb gauge; if it holds in one inertial frame it will generally not hold in any other.) It is expressed in terms of the fourpotential as follows:
 <math>\partial_{\alpha} A^{\alpha} = \partial^{\alpha} A_{\alpha}=0 \,.</math>
In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
 <math>\Box A^{\sigma} = \mu_{0} \, J^{\sigma}\,</math>
Lorentz force
Charged particle
Electromagnetic (EM) fields affect the motion of electrically charged matter: due to the Lorentz force. In this way, EM fields can be detected (with applications in particle physics, and natural occurrences such as in aurorae). In relativistic form, the Lorentz force (in newtons) uses the field strength tensor as follows.^{[3]}
Expressed in terms of coordinate time (not proper time) t in seconds, it is:
 <math> { d p_{\alpha} \over { d t } } = q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} \,</math>
where p_{α} is the fourmomentum (see above), q is the charge (in coulombs), and x^{β} is the position in meters.
In the comoving reference frame, this yields the socalled 4force
 <math> \frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta </math>
where u^{β} is the fourvelocity (see above), and τ is the particle's proper time which is related to coordinate time by dt = γdτ.
Charge continuum
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4vector, f_{μ}. The spatial part is the result of dividing the force on a small cell (in 3space) by the volume of that cell. The time component is 1/c times the power transferred to that cell divided by the volume of the cell. The density of Lorentz force is the part of the density of force due to electromagnetism. Its spatial part is
 <math>  \bold{f} =  (\rho \bold{E} + \bold{J} \times \bold{B})\,</math>.
In manifestly covariant notation it becomes:
 <math>f_{\alpha} = F_{\alpha\beta}J^{\beta} .\!</math>
The relationship between Lorentz force and electromagnetic stress–energy tensor is
 <math>f^{\alpha} =  {T^{\alpha\beta}}_{,\beta} \equiv  \frac{\partial T^{\alpha\beta}}{\partial x^\beta}.</math>
Conservation laws
Electric charge
The continuity equation:
 <math>{J^{\alpha}}_{,\alpha} \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} J^{\alpha} \, = \, 0 \,.</math>
expresses charge conservation.
Electromagnetic energy–momentum
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current fourvector
 <math>{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_{\beta} = 0</math>
or
 <math>\eta_{\alpha \nu} { T^{\nu \beta } }_{,\beta} + F_{\alpha \beta} J^{\beta} = 0,</math>
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
Covariant objects in matter
Free and bound 4currents
In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, J^{ν} Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations;
 <math>J^{\nu} = {J^{\nu}}_{\text{free}} + {J^{\nu}}_{\text{bound}} \,,</math>
where
 <math>{J^{\nu}}_{\text{free}}=(c\rho_{\text{free}},\mathbf{J}_{\text{free}})=\left(c \nabla \cdot \mathbf{D},  \ \frac{\partial \mathbf{D}}{\partial t}+\nabla\times\mathbf{H}\right) \,,</math>
 <math>{J^{\nu}}_{\text{bound}}=(c\rho_{\text{bound}},\mathbf{J}_{\text{bound}})=\left( \ c \nabla \cdot \mathbf{P}, \frac{\partial \mathbf{P}}{\partial t}+\nabla\times\mathbf{M}\right) \,.</math>
Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D (in coulomb·meter^{−1}) and the magnetic intensity H (in ampere·meter^{−1}):
 <math>\bold{D} = \epsilon_0 \bold{E} + \bold{P} \,</math>
 <math>\bold{H} = \frac{1}{\mu_{0}} \bold{B}  \bold{M} \,.</math>
where M is the magnetization (in ampere·meter^{−2}) and P the electric polarization (in coulomb·meter^{−2}).
Magnetizationpolarization tensor
The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetizationpolarization tensor (in ampere·meter^{2})^{[1]}
 <math>
\mathcal{M}^{\mu \nu} = \begin{pmatrix} 0 & P_xc & P_yc & P_zc \\  P_xc & 0 &  M_z & M_y \\  P_yc & M_z & 0 &  M_x \\  P_zc &  M_y & M_x & 0 \end{pmatrix},
</math>
which determines the bound current
 <math>{J^{\nu}}_{\text{bound}}=\partial_{\mu} \mathcal{M}^{\mu \nu} \,.</math>
Electric displacement tensor
If this is combined with F^{μν} we get the antisymmetric contravariant electromagnetic displacement tensor (in ampere·meter^{−1}) which combines the D and H fields as follows:
 <math>
\mathcal{D}^{\mu \nu} = \begin{pmatrix} 0 &  D_xc &  D_yc &  D_zc \\ D_xc & 0 &  H_z & H_y \\ D_yc & H_z & 0 &  H_x \\ D_zc &  H_y & H_x & 0 \end{pmatrix}.
</math>
The three field tensors are related by:
 <math>\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu}  \mathcal{M}^{\mu \nu} \,</math>
which is equivalent to the definitions of the D and H fields given above.
Maxwell's equations in matter
The result is that Ampère's law,
 <math>\bold{\nabla} \times \bold{H} = \bold{J}_{\text{free}} + \frac{\partial \bold{D}} {\partial t}</math>,
and Gauss's law,
 <math>\bold{\nabla} \cdot \bold{D} = \rho_{\text{free}}</math>,
combine into one equation:
Gauss–Ampère law (matter) <math>{J^{\nu
cellpadding border border colour = #0073CF background colour=#F5FFFA}}
The bound current and free current as defined above are automatically and separately conserved
 <math>\partial_{\nu} {J^{\nu}}_{\text{bound}} = 0 \,</math>
 <math>\partial_{\nu} {J^{\nu}}_{\text{free}} = 0 \,.</math>
Constitutive equations
Vacuum
In a vacuum, the constitutive relations between the field tensor and displacement tensor are:
 <math>\mu_0 \mathcal{D}^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,.</math>
Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define F^{μν} by
 <math>F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,</math>
the constitutive equations may, in a vacuum, be combined with GaussAmpère law to get:
 <math>\partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. \!</math>
The electromagnetic stress–energy tensor in terms of the displacement is:
 <math>T_\alpha^\pi = F_{\alpha\beta} \mathcal{D}^{\pi\beta}  \frac{1}{4} \delta_\alpha^\pi F_{\mu\nu} \mathcal{D}^{\mu\nu}</math>
where δ_{α}^{π} is the Kronecker delta. When the upper index is lowered with η, it becomes symmetric and is part of the source of the gravitational field.
Matter
Thus we have reduced the problem of modeling the current, J^{ν} to two (hopefully) easier problems — modeling the free current, J^{ν}_{free} and modeling the magnetization and polarization, <math> \mathcal{M}^{\mu\nu}</math>. For example, in the simplest materials at low frequencies, one has
 <math>\bold{J}_{\text{free}} = \sigma \bold{E} \,</math>
 <math>\bold{P} = \epsilon_0 \chi_e \bold{E} \,</math>
 <math>\bold{M} = \chi_m \bold{H} \,</math>
where one is in the instantaneously comoving inertial frame of the material, σ is its electrical conductivity, χ_{e} is its electric susceptibility, and χ_{m} is its magnetic susceptibility.
The constitutive relations between the <math>\mathcal{D}</math> and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are:^{[4]}
 <math>\mathcal{D}^{\mu\nu}u_\nu= c^2\epsilon F^{\mu\nu} u_\nu</math>
 <math>{\star\mathcal{D}^{\mu\nu}}u_\nu= \frac{1}{\mu}{\star F^{\mu\nu}} u_\nu</math>
where u is the 4velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), <math>\star</math> and denotes the Hodge dual.
Lagrangian for classical electrodynamics
Vacuum
The Lagrangian density for classical electrodynamics (in joule·meter^{−3}) is
 <math> \mathcal{L} \, = \, \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{int}} =  \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta}  A_{\alpha} J^{\alpha} \,.</math>
In the interaction term, the fourcurrent should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the fourcurrent is not itself a fundamental field.
The Euler–Lagrange equation for the electromagnetic Lagrangian density <math> \mathcal{L}(A_{\alpha},\partial_{\beta}A_{\alpha})\,</math> can be stated as follows:
 <math> \partial_{\beta}\left[\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})}\right]  \frac{\partial \mathcal{L}}{\partial A_{\alpha}}=0 \,.</math>
Noting
 <math>F_{\mu \nu}=\partial_{\mu} A_{\nu}  \partial_{\nu} A_{\mu}\,</math>,
the expression inside the square bracket is
 <math>\begin{align}\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})} & =  \ \frac{1}{4 \mu_0}\ \frac{\partial (F_{\mu \nu}\eta^{\mu\lambda}\eta^{\nu\sigma}F_{\lambda \sigma})}{\partial (\partial_{\beta}A_{\alpha})} \\
& =  \ \frac{1}{4 \mu_0}\ \eta^{\mu\lambda}\eta^{\nu\sigma}
\left(F_{\lambda\sigma}(\delta^{\beta}_{\mu}\delta^{\alpha}_{\nu}  \delta^{\beta}_{\nu}\delta^{\alpha}_{\mu}) +F_{\mu\nu}(\delta^{\beta}_{\lambda}\delta^{\alpha}_{\sigma}  \delta^{\beta}_{\sigma}\delta^{\alpha}_{\lambda}) \right) \\
& =  \ \frac{F^{\beta\alpha}}{\mu_0}\,.
\end{align}</math>
The second term is
 <math>\frac{\partial \mathcal{L}}{\partial A_{\alpha}}=  J^{\alpha} \,.</math>
Therefore, the electromagnetic field's equations of motion are
 <math> \frac{\partial F^{\beta\alpha}}{\partial x^{\beta}}=\mu_0 J^{\alpha} \,.</math>
which is one of the Maxwell equations above.
Matter
Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
 <math> \mathcal{L} \, = \,  \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta}  A_{\alpha} J^{\alpha}_{\text{free}} + \frac12 F_{\alpha \beta} \mathcal{M}^{\alpha \beta} \,.</math>
Using Euler–Lagrange equation, the equations of motion for <math> \mathcal{D}^{\mu\nu}</math> can be derived.
The equivalent expression in nonrelativistic vector notation is
 <math> \mathcal{L} \, = \, \frac12 \left(\epsilon_{0} E^2  \frac{1}{\mu_{0}} B^2\right)  \phi \, \rho_{\text{free}} + \bold{A} \cdot \bold{J}_{\text{free}} + \bold{E} \cdot \bold{P} + \bold{B} \cdot \bold{M} \,.</math>
See also
 Relativistic electromagnetism
 Electromagnetic wave equation
 Liénard–Wiechert potential for a charge in arbitrary motion
 Nonhomogeneous electromagnetic wave equation
 Moving magnet and conductor problem
 Electromagnetic tensor
 Proca action
 Stueckelberg action
 Quantum electrodynamics
 Wheeler–Feynman absorber theory
Notes and references
 ^ ^{a} ^{b} Vanderlinde, Jack (2004), classical electromagnetic theory, Springer, pp. 313–328, ISBN 9781402026997
 ^ Classical Electrodynamics by Jackson, 3rd Edition, Chapter 11 Special Theory of Relativity
 ^ The assumption is made that no forces other than those originating in E and B are present, that is, no gravitational, weak or strong forces.
 ^ D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Dorling Kindersley. p. 563. ISBN 8177582933.
Further reading
 Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0517029618.
 Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0716703440.
 Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0080181767.
 R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. AddisonWesley. ISBN 0201627345.

