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).
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[edit] The equations
Maxwell's equations can be written as two tensor equations
- <math>\frac{\partial F^{\alpha\beta}}{\partial x^\alpha}=\mu_0J^\beta
\qquad\hbox{and}\qquad 0=\epsilon^{\alpha\beta\gamma\delta}\frac{\partial F_{\alpha\beta}}{\partial x^\gamma}</math>
where <math>F^{\alpha\beta}</math> is the field strength tensor (which incorporates the electric and magnetic fields), <math>J^{\alpha}</math> is the 4-current (incorporating both charge density and current density), <math>\epsilon^{\alpha\beta\gamma\delta}</math> 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 the absence of magnetic monopoles.
[edit] Other notation
Without the summation convention or the Levi-Civita symbol, the equations would be written
- <math>\sum_{\alpha=ct,x,y,z}{\partial F^{\alpha\beta}\over\partial x^\alpha}=\mu_0J^\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}
</math> where all indices range from 0 to 3 (or, more descriptively, over the set {ct,x,y,z}). The first tensor equation corresponds to four scalar equations, one for each value of <math>\beta</math>. The second tensor equation actually corresponds to <math>4^3=64</math> 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,
- <math>{\partial F^{\alpha\beta}\over \partial x^\gamma}\equiv \partial_\gamma F^{\alpha\beta}\equiv {F^{\alpha\beta}}_{,\gamma}</math>
Using the latter notation, Maxwell's equations can be written as <math> {F^{\alpha\beta}}_{,\alpha}=\mu_0 J^\beta</math> and <math>\epsilon^{\alpha\beta\gamma\delta} {F_{\alpha\beta,\gamma}}=0 </math>
[edit] Charge conservation
The 4-current is a contravariant vector given by:
- <math>J^{\alpha} = \, (c \rho, \vec{J} ) </math>
where <math> \rho </math> is the charge density and <math> \vec{J} </math> is the current density.
The 4-current satisfies the continuity equation
- <math>J^{\alpha}_{,\alpha} \, \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} J^{\alpha} \, = 0</math>
[edit] Field strength tensor and the 4-potential
The field strength tensor, an antisymmetric tensor, can be written:
- <math>F^{\alpha\beta} = \partial^{\alpha} A^{\beta} - \partial^{\beta} A^{\alpha} \,\!</math>
where
- <math>A^{\alpha} = \left(\frac{\phi}{c}, \vec{A} \right)</math>
is the 4-potential, φ is the scalar potential and <math> \vec{A} </math> is the vector potential.
When using metric (-+++), the field strength tensor is written in terms of fields as:
- <math>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) .</math>
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:
- <math> \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</math>.
Here, <math>\partial_{\alpha} \partial^{\alpha} </math> 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 <math>\, F_{ab}</math> is related to contravariant version <math>\, F^{ab}</math> by the Minkowski metric tensor <math>\eta</math>
- <math> F_{\alpha\beta} =\, \eta_{\alpha\gamma} \eta_{\beta\delta} F^{\gamma\delta} = F^{\alpha\beta} </math>.
[edit] 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
- <math> m c { d u^{\alpha} \over { d \tau } } = { {} \over {} }F^{\alpha \beta} q u_{\beta} </math>
where m is the particle mass, q is the charge, and
- <math> u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau} } </math>
is the 4-velocity of the particle. Here, <math> \tau </math> 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
- <math> \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} </math>
[edit] Electromagnetic stress-energy tensor
The electromagnetic stress-energy tensor is related to the field strength tensor by:
- <math> { T^{\alpha \beta } }_{,\beta} = { {} \over {} }F^{\alpha \beta} J_{\beta} </math>
- <math> { 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 ) </math>
[edit] See also
- Relativistic electromagnetism
- Electromagnetic wave equation
- Liénard–Wiechert potential for a charge in arbitrary motion
- Nonhomogeneous electromagnetic wave equation
- Maxwell's equations in curved spacetime
- Moving magnet and conductor problem
- Electromagnetic tensor
- Proca action
- Stueckelberg action
- Quantum electrodynamics
[edit] References
- [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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