Electromagnetic stress–energy tensor

Electromagnetism

Electricity Magnetism
Electrostatics Electric charge Static electricity Electric field Conductor Insulator Triboelectricity Electrostatic discharge Induction Coulomb's law Gauss's law Electric flux / potential energy Electric dipole moment Polarization density
Magnetostatics Ampère's law Magnetic field Magnetization Magnetic flux Biot–Savart law Magnetic dipole moment Gauss's law for magnetism
Electrodynamics Lorentz force law Electromagnetic induction Faraday's law Lenz's law Displacement current Magnetic potential Maxwell's equations Electromagnetic field Electromagnetic pulse Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current London equations Mathematical descriptions of the electromagnetic field
Electrical network Electric current Electric potential Voltage Resistance Ohm's law Series circuit Parallel circuit Direct current Alternating current Electromotive force Capacitance Inductance Impedance Resonant cavities Waveguides
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

In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field.^[1] The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the classical Maxwell stress tensor that governs the electromagnetic interactions.

Definition

SI units

In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is^[2]

T^{\mu\nu} = \frac{1}{\mu_0} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right] \,.

where $F^{\mu\nu}$ is the electromagnetic tensor and where $\eta_{\mu\nu}$ is the Minkowski metric tensor of metric signature (−+++). When using the metric with signature (+−−−), the expression for $T^{\mu \nu}$ will have opposite sign.

Explicitly in matrix form:

T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}\left(\epsilon_0 E^2+\frac{1}{\mu_0}B^2\right) & S_\text{x}/c & S_\text{y}/c & S_\text{z}/c \\ S_\text{x}/c & -\sigma_{xx} & -\sigma_\text{xy} & -\sigma_\text{xz} \\ S_\text{y}/c & -\sigma_{yx} & -\sigma_\text{yy} & -\sigma_\text{yz} \\ S_\text{z}/c & -\sigma_{zx} & -\sigma_\text{zy} & -\sigma_\text{zz} \end{bmatrix},

where

\bold{S}=\frac{1}{\mu_0}\bold{E}\times\bold{B},

is the Poynting vector,

\sigma _{{ij}}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{{\mu _{0}}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)\delta _{{ij}}

is the Maxwell stress tensor, and c is the speed of light. Thus, $T^{\mu\nu}$ is expressed and measured in SI pressure units (pascals).

CGS units

The permittivity of free space and permeability of free space in cgs-Gaussian units are

\epsilon_0=\frac{1}{4\pi},\quad \mu_0=4\pi\,

then:

T^{\mu\nu} = \frac{1}{4\pi} [ F^{\mu\alpha}F^{\nu}{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}] \,.

and in explicit matrix form:

T^{\mu\nu} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) & S_\text{x}/c & S_\text{y}/c & S_\text{z}/c \\ S_x/c & -\sigma_\text{xx} & -\sigma_\text{xy} & -\sigma_\text{xz} \\ S_\text{y}/c & -\sigma_\text{yx} & -\sigma_\text{yy} & -\sigma_\text{yz} \\ S_\text{z}/c & -\sigma_\text{zx} & -\sigma_\text{zy} & -\sigma_\text{zz} \end{bmatrix}

where Poynting vector becomes:

\bold{S}=\frac{c}{4\pi}\bold{E}\times\bold{B}.

The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the unresolved Abraham–Minkowski controversy.^[3]

The element $T^{\mu\nu}\!$ of the stress–energy tensor represents the flux of the μth-component of the four-momentum of the electromagnetic field, $P^{\mu}\!$ , going through a hyperplane ( $x^{\nu}$ is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity.

Algebraic properties

The electromagnetic stress–energy tensor has several algebraic properties:

It is a symmetric tensor:

T^{\mu\nu}=T^{\nu\mu}

The tensor $T^{\nu}{}_{\alpha}$ is traceless:

T^{\alpha}{}_{\alpha}= 0

The energy density is positive-definite:

T^{00}\ge 0

The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The tracelessness relates to the masslessness of the photon.^[4]

Conservation laws

The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is:

\partial_\nu T^{\mu \nu} + \eta^{\mu \rho} \, f_\rho = 0 \,

where $f_\rho$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 4D conservation laws

\frac{\partial u_\mathrm{em}}{\partial t} + \bold{\nabla} \cdot \bold{S} + \bold{J} \cdot \bold{E} = 0 \,

\frac{\partial \bold{p}_\mathrm{em}}{\partial t} - \bold{\nabla}\cdot \sigma + \rho \bold{E} + \bold{J} \times \bold{B} = 0 \,

(or equivalently

{\mathbf {f}}+\epsilon _{0}\mu _{0}{\frac {\partial {\mathbf {S}}}{\partial t}}\,=\nabla \cdot {\mathbf {\sigma }}

with

\mathbf{f}

being the Lorentz force density),

respectively describing the flux of electromagnetic energy density

u_\mathrm{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \,

and electromagnetic momentum density

\bold{p}_\mathrm{em} = {\bold{S} \over {c^2}}

where J is the electric current density and ρ the electric charge density.

References

↑ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
↑ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
↑ however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)
↑ Garg, Anupam. Classical Electromagnetism in a Nutshell, p. 564 (Princeton University Press, 2012).

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.