Covariant formulation of classical electromagnetism

Electromagnetism

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Electrodynamics Lorentz force law Electromagnetic induction Faraday's law Lenz's law Displacement current Magnetic potential Maxwell's equations Electromagnetic field Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current London equations Mathematical descriptions of the electromagnetic field
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Covariant formulation Electromagnetic tensor (stress–energy tensor) Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted

Main article: Mathematical descriptions of the electromagnetic field

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 non-rectilinear coordinate systems.

This article uses the classical treatment of tensors and 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.

Covariant objects

Preliminary 4-vectors

Main article: Lorentz covariance

Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:

4-Displacement:

x^{\alpha }=(ct,{\mathbf {x}})=(ct,x,y,z)\,.

4-Velocity:

u^\alpha = \gamma(c,\bold{u}) \,

where γ(u) is the Lorentz factor at the 3-velocity u.

4-Momentum:

p^{\alpha }=(E/c,{\mathbf {p}})=m_{o}u^{\alpha }\,

where

{\mathbf {p}}

is 3-momentum,

E

is the kinetic energy, and

m_{o}

is rest mass.

4-Gradient

\partial^{\nu} = \frac{\partial}{\partial x_{\nu}} = \left( \frac{1}{c} \frac{\partial}{\partial t}, - \bold{\nabla} \right) \,,

The d'Alembertian operator is denoted: $\Box$ .

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:

\eta^{\mu \nu}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}\,

Electromagnetic tensor

Main article: Electromagnetic tensor

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities. ^[1]

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)\,

and the result of raising its indices is

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)\,.

where E is the electric field, B the magnetic field, and c the speed of light.

Four-current

Main article: Four-current

The four-current is the contravariant four-vector which combines electric current density j and electric charge density ρ.

J^{\alpha }=\,(c\rho ,{\mathbf {j}})\,

Four-potential

Main article: Four-potential

The electromagnetic four-potential is a covariant four-vector containing the electric potential (also called the scalar potential) ϕ and magnetic vector potential (or vector potential) a, as follows:

A^{\alpha }=\left(\phi /c,{\mathbf {A}}\right)\,.

The differential of the electromagnetic potential is

F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,.

Electromagnetic stress–energy tensor

Main article: Electromagnetic stress–energy tensor

The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum 4-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall stress–energy tensor:

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}}\,,

where ε₀ is the electric permittivity of vacuum, μ₀ is the magnetic permeability of vacuum, the Poynting vector is

\bold{S} = \frac{1}{\mu_{0}} \bold{E} \times \bold{B}

and the Maxwell stress tensor is given by

\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} \,.

The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:

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)

where η is the Minkowski metric tensor. Notice that we use the fact that

\epsilon _{0}\mu _{0}c^{2}=1\,,

which is predicted by Maxwell's equations.

Maxwell's equations in vacuo

Main article: Maxwell's equations

In vacuo (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]

Gauss–Ampère law

$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$

while the homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form:

Gauss–Faraday law

$\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0$

where F^αβ is the electromagnetic tensor, J^α is the 4-current, ε^αβγδ is the Levi-Civita 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 4³ = 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:

F_{[\alpha \beta ,\gamma ]}=0.

In the absence of sources, Maxwell's equations reduce to:

\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 \,,

which is an electromagnetic wave equation in the field strength tensor.

Maxwell's equations in the Lorenz gauge

Main article: Lorenz gauge condition

The Lorenz gauge condition is a Lorentz-invariant 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 four-potential as follows:

\partial_{\alpha} A^{\alpha} = \partial^{\alpha} A_{\alpha}=0 \,.

In the Lorenz gauge, the microscopic Maxwell's equations can be written as:

\Box A^{\sigma }=\mu _{0}\,J^{\sigma }\,.

Lorentz force

Main article: Lorentz force

Charged particle

Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.

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 uses the field strength tensor as follows.^[3]

Expressed in terms of coordinate time t, it is:

{ d p_{\alpha} \over { d t } } = q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} \,

where p_α is the four-momentum, q is the charge, and x^β is the position.

In the co-moving reference frame, this yields the 4-force

\frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta

where u^β is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by dt = γdτ.

Charge continuum

Lorentz force per spatial volume f on a continuous charge distribution (charge density ρ) in motion.

Conservation laws

Electric charge

The continuity equation:

{J^{\alpha}}_{,\alpha} \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} J^{\alpha} \, = \, 0 \,.

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 four-vector

{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_{\beta} = 0

\eta_{\alpha \nu} { T^{\nu \beta } }_{,\beta} + F_{\alpha \beta} J^{\beta} = 0,

which expresses the conservation of linear momentum and energy by electromagnetic interactions.

Covariant objects in matter

Free and bound 4-currents

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;

J^{\nu} = {J^{\nu}}_{\text{free}} + {J^{\nu}}_{\text{bound}} \,,

where

{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) \,,

{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) \,.

Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H:

\bold{D} = \epsilon_0 \bold{E} + \bold{P} \,

\bold{H} = \frac{1}{\mu_{0}} \bold{B} - \bold{M} \,.

where M is the magnetization and P the electric polarization.

Magnetization-polarization tensor

The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor ^[1]

\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},

which determines the bound current

{J^{\nu}}_{\text{bound}}=\partial_{\mu} \mathcal{M}^{\mu \nu} \,.

Electric displacement tensor

If this is combined with F^μν we get the antisymmetric contravariant electromagnetic displacement tensor which combines the D and H fields as follows:

\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}.

The three field tensors are related by:

\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} \,

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,

\bold{\nabla} \times \bold{H} = \bold{J}_{\text{free}} + \frac{\partial \bold{D}} {\partial t}

and Gauss's law,

\bold{\nabla} \cdot \bold{D} = \rho_{\text{free}}

combine into one equation:

Gauss–Ampère law (matter)

${J^{\nu}}_{\text{free}} = \partial_{\mu} \mathcal{D}^{\mu \nu}$

The bound current and free current as defined above are automatically and separately conserved

\partial_{\nu} {J^{\nu}}_{\text{bound}} = 0 \,

\partial_{\nu} {J^{\nu}}_{\text{free}} = 0 \,.

Constitutive equations

Main article: Constitutive equation

Vacuum

In a vacuum, the constitutive relations between the field tensor and displacement tensor are:

\mu_0 \mathcal{D}^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,.

Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define F^μν by

F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,

the constitutive equations may, in a vacuum, be combined with Gauss-Ampère law to get:

\partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. \!

The electromagnetic stress–energy tensor in terms of the displacement is:

T_\alpha^\pi = F_{\alpha\beta} \mathcal{D}^{\pi\beta} - \frac{1}{4} \delta_\alpha^\pi F_{\mu\nu} \mathcal{D}^{\mu\nu}

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.

Linear, nondispersive 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, $\mathcal{M}^{\mu\nu}$ . For example, in the simplest materials at low frequencies, one has

\bold{J}_{\text{free}} = \sigma \bold{E} \,

\bold{P} = \epsilon_0 \chi_e \bold{E} \,

\bold{M} = \chi_m \bold{H} \,

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 ${\mathcal {D}}$ and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are:^[4]

\mathcal{D}^{\mu\nu}u_\nu= c^2\epsilon F^{\mu\nu} u_\nu

{\star\mathcal{D}^{\mu\nu}}u_\nu= \frac{1}{\mu}{\star F^{\mu\nu}} u_\nu

where u is the 4-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), $\star$ and denotes the Hodge dual.

Lagrangian for classical electrodynamics

Vacuum

The Lagrangian density for classical electrodynamics is

\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} \,.

In the interaction term, the four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the four-current is not itself a fundamental field.

The Euler–Lagrange equation for the electromagnetic Lagrangian density $\mathcal{L}(A_{\alpha},\partial_{\beta}A_{\alpha})\,$ can be stated as follows:

\partial_{\beta}\left[\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})}\right] - \frac{\partial \mathcal{L}}{\partial A_{\alpha}}=0 \,.

Noting

F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}\,

the expression inside the square bracket is

\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}

The second term is

\frac{\partial \mathcal{L}}{\partial A_{\alpha}}= - J^{\alpha} \,.

Therefore, the electromagnetic field's equations of motion are

\frac{\partial F^{\beta\alpha}}{\partial x^{\beta}}=\mu_0 J^{\alpha} \,.

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:

\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} \,.

Using Euler–Lagrange equation, the equations of motion for $\mathcal{D}^{\mu\nu}$ can be derived.

The equivalent expression in non-relativistic vector notation is

\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} \,.

Notes and references

1 2 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 81-7758-293-3.

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Covariant formulation of classical electromagnetism

Covariant objects

Preliminary 4-vectors

Electromagnetic tensor

Four-current

Four-potential

Electromagnetic stress–energy tensor

Maxwell's equations in vacuo

Maxwell's equations in the Lorenz gauge

Lorentz force

Charged particle

Charge continuum

Conservation laws

Electric charge

Electromagnetic energy–momentum

Covariant objects in matter

Free and bound 4-currents

Magnetization-polarization tensor

Electric displacement tensor

Maxwell's equations in matter

Constitutive equations

Vacuum

Linear, nondispersive matter

Lagrangian for classical electrodynamics

Vacuum

Matter

See also

Notes and references

Further reading