Covariant formulation of classical electromagnetism

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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 which is "manifestly covariant" (i.e. in terms of covariant four-vectors and tensors), in the formalism of special relativity. 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.

Most of this article is in cgs units, with the metric (-+++). This article uses abstract index notation and the Einstein summation convention throughout.

For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see the article: Classical electromagnetism and special relativity.

Contents

[edit] Covariant objects

[edit] Electromagnetic tensor

Main article: Electromagnetic tensor

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor. In cgs units, with the metric (-+++), the field strength tensor is written in terms of fields as:

F^{\alpha\beta} = \left( \begin{matrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{matrix} \right) .

[edit] Four-Current

Main article: Four-current

The four-current is the four-vector which combines electric current and electric charge density. It is given by

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

where ρ is the charge density, \vec{J} is the current density, and c is the speed of light.

[edit] Four-potential

Main article: Four-potential

The electromagnetic four-potential is a four-vector containing the electric potential and magnetic vector potential, as follows:

A^{\alpha} = \left(\phi, \vec{A} \right)

where φ is the scalar potential and \vec{A} is the vector potential.

The relation between the electromagnetic potentials and the electromagnetic fields is given by the following equation:

F^{\alpha\beta} = \partial^{\alpha} A^{\beta} - \partial^{\beta} A^{\alpha} \,\!

where F is the electromagnetic tensor.

[edit] Electromagnetic stress-energy tensor

Main article: Electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor is a covariant tensor which corresponds to the contribution to the overall stress-energy tensor due to electromagnetic fields. It is given (in cgs, with the tensor -+++) by

T^{\alpha\beta} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) & -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{bmatrix}

where

\vec{S}=\frac{c}{4\pi}\vec{E}\times\vec{B} is the Poynting vector,
g_{\alpha\beta}\! is the Minkowski metric tensor, and
the Maxwell stress tensor is given by
\sigma_{ij}=\frac{1}{4\pi}\bigl(E_{i}E_{j}+B_{i}B_{j}- \tfrac{1}{2}(E^2+B^2)\delta_{ij}\bigr)

The electromagnetic stress-energy tensor is related to the electromagnetic field tensor by the equation (in cgs):

T^{\alpha\beta} = \frac{1}{4\pi} [ -F^{\alpha \gamma}F_{\gamma}{}^{\beta} - \frac{1}{4}g^{\alpha\beta}F_{\gamma\delta}F^{\gamma\delta}].

where g is the Minkowski metric tensor.

[edit] Other, non-electromagnetic objects

For background purposes, we present here three other relevant four-vectors, which are not directly connected to electromagnetism, but which will be useful in this article:

  • The "position" or "coordinate" four-vector is
xα = (ct,x,y,z).
u^\alpha = \gamma(c,\vec{u})
where \vec{u} is the (three-vector) velocity and γ is the Lorentz factor associated with \vec{u}.
p^\alpha = (E/c,\vec{p}) = mu^\alpha
where \vec{p} is the (three-vector) momentum, E is the relativistic kinetic energy, and m is the particle's rest mass.

[edit] Maxwell's equations

Maxwell's equations (in cgs) can be written as two tensor equations

\frac{\partial F^{\alpha\beta}}{\partial x^\alpha}=\frac{4\pi}{c}J^\beta \qquad\hbox{and}\qquad 0=\epsilon^{\alpha\beta\gamma\delta}\frac{\partial F_{\alpha\beta}}{\partial x^\gamma}

where F αβ is the electromagnetic tensor, J α is the 4-current, є αβγδ is the Levi-Civita symbol (a mathematical construct), and the indices behave according to the Einstein summation convention.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's Law and Ampere's Law (with Maxwell's correction). The second equation is an expression of the homogenous equations, Faraday's law of induction and Gauss's law for magnetism.

Maxwell's equations, in the absence of sources, reduce to a wave equation in the field strength:

\partial_{\gamma} \partial^{\gamma} F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ \Box F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ \nabla^2 F^{\alpha\beta} - {1 \over c^2 } { \partial^2 F^{\alpha\beta} \over {\partial t }^2 }= 0.

Here, \partial_{\alpha} \partial^{\alpha} is the d'Alembertian operator.

Different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation).

The covariant version of the field strength tensor \, F_{ab} is related to contravariant version \, F^{ab} by the Minkowski metric tensor η

F_{\alpha\beta} =\, \eta_{\alpha\gamma} \eta_{\beta\delta} F^{\gamma\delta} = F^{\alpha\beta} .

[edit] Other notation

Without the summation convention or the Levi-Civita symbol, the equations would be written

\sum_{\alpha=ct,x,y,z}{\partial F^{\alpha\beta}\over\partial x^\alpha}=\frac{4\pi}{c}J^\beta \qquad\hbox{and}\qquad 0={\partial F_{\alpha\beta}\over\partial x^\gamma} +{\partial F_{\beta\gamma}\over\partial x^\alpha} +{\partial F_{\gamma\alpha}\over\partial x^\beta}

where all indices range from 0 to 3 (or, more descriptively, over the set {ct,x,y,z}), where c is the speed of light in free space. 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.

For convenience, professionals often write the 4-gradient (that is, the derivative with respect to x) using abbreviated notations; for instance,

{\partial F^{\alpha\beta}\over \partial x^\gamma}\equiv \partial_\gamma F^{\alpha\beta}\equiv {F^{\alpha\beta}}_{,\gamma}

Using the latter notation, Maxwell's equations can be written as {F^{\alpha\beta}}_{,\alpha}=\mu_0 J^\beta and \epsilon^{\alpha\beta\gamma\delta} {F_{\alpha\beta,\gamma}}=0\ .

[edit] Continuity equation

The continuity equation (an expression of charge conservation, has the following expression in terms of the four-current:

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

[edit] Lorentz force

Main article: Lorentz force

Fields are detected by their effect on the motion of matter. Electromagnetic fields affect the motion of particles through the Lorentz force. Using the Lorentz force, Newton's law of motion can be written in relativistic form using the field strength tensor as[1]

m c { d u^{\alpha} \over { d \tau } } = qu_\beta F^{\alpha \beta}

where m is the particle rest mass, q is the charge, u is the four-velocity (see above), and τ is the particle's proper time. This can also be written as

c\frac{d p^\alpha}{d \tau} = q u_\beta F^{\alpha \beta}

where p is the four-momentum (see above). In terms of (normal) time instead of proper time, the equation is

m c \gamma { d u^{\alpha} \over { dt } } = qu_\beta F^{\alpha \beta}

The relativistic version of Newton's law of motion differs from its nonrelativistic counterpart. The relativistic version emerges in order to maintain consistency of the transformation of forces as required by Maxwell's equations. Einstein used the well known problem of moving magnets and conductors to motivate the changes to the Lorentz force. See Einstein On the electrodynamics of moving bodies §6

[edit] Lagrangian for classical electrodynamics

The Lagrangian for classical electrodynamics (in SI) is

\mathcal{L} = \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{int}} = -\frac{1}{4 \mu_0} F^{\alpha\beta} F_{\alpha\beta} - J^{\alpha}A_{\alpha}

[edit] Differential equation for electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector (in SI):[citation needed]

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

[edit] Lorenz gauge condition

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, which can hold in one inertial frame but not another.) It is expressed in terms of the four-potential as follows:

\partial_\alpha A^\alpha = 0 \!

[edit] Maxwell's equations in the Lorenz gauge

In the Lorenz gauge, Maxwell's equations can be written as (in cgs):

\Box^2 A^\mu = -\frac{4\pi}{c} J^\mu

where \Box^2 denotes the d'Alembertian.

[edit] See also

[edit] Notes and references

  1. ^ The assumption is made that no forces other than those originating in E and B are present, for example, no gravitational or electroweak forces.

[edit] Further reading

[1] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8. 
[2] Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0. 
[3] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7. 
[4] R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5. 
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