Formulation of Maxwell's equations in special relativity

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The formulation of Maxwell's equations in special relativity refers to ways of writing Maxwell's equations of electromagnetism in the formalism of special relativity. In order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units).

1 The equations
2 Charge conservation
3 Field strength tensor and the 4-potential
4 Lorentz force
5 Lagrangian for Classical Electrodynamics
6 Electromagnetic stress-energy tensor
7 See also
8 References

[edit] The equations

$\mu_0 J^{\beta} = {\partial F^{\alpha\beta} \over {\partial x^{\alpha}} } \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ {F^{\alpha\beta}}_{,\alpha} \,\!$

and

$0 = \partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma} \ \stackrel{\mathrm{def}}{=}\ {F_{\alpha\beta,\gamma}} + {F_{\gamma\alpha,\beta}} +{F_{\beta\gamma,\alpha}} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\epsilon^{\alpha\beta\gamma\delta} {F_{\alpha\beta,\gamma}}$

where $\, J^{\alpha}$ is the 4-current, $\, F^{\alpha\beta}$ is the field strength tensor, $\, \epsilon_{\alpha\beta\gamma\delta}$ is the Levi-Civita symbol, and

${ \partial \over { \partial x^{\alpha} } } \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} \ \stackrel{\mathrm{def}}{=}\ {}_{,\alpha} \ \stackrel{\mathrm{def}}{=}\ \left(\frac{\partial}{\partial ct}, \nabla\right)$

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' 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 the absence of magnetic monopoles.

[edit] Charge conservation

The 4-current is a contravariant vector given by:

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

where $ρ$ is the charge density and $\vec{J}$ is the current density.

The 4-current satisfies the continuity equation

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

[edit] Field strength tensor and the 4-potential

The field strength tensor, an antisymmetric tensor, can be written:

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

where

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

is the 4-potential, φ is the scalar potential and $\vec{A}$ is the vector potential.

When using metric (-+++), the field strength tensor is written in terms of fields as:

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

The fact that both electric and magnetic fields are combined into a single tensor expresses the fact that, according to relativity, both of these are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame can appear as a magnetic field in another frame, and vice versa.

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] 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 equation. The Lorentz force equation can be written in terms of the field strength tensor as

$m c { d u^{\alpha} \over { d \tau } } = { {} \over {} }F^{\alpha \beta} q u_{\beta}$

where m is the particle mass, q is the charge, and

$u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau} }$

is the 4-velocity of the particle. Here, $τ$ is c times the proper time of the particle.

The relativistic version of the Lorentz force equation 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.[1]

[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] Electromagnetic stress-energy tensor

The electromagnetic stress-energy tensor is related to the field strength tensor by:

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

${ T_{\alpha \beta } } = { 1 \over { 4 \pi } } \left ( F_{\alpha \mu} F_{\beta}^{ \mu} + {1 \over 4} F_{\mu \nu} F^{ \mu \nu} \eta_{\alpha \beta} \right )$