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| {{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}}
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| In [[physics]], the '''electromagnetic stress–energy tensor''' is the portion of the [[stress–energy tensor]] due to the [[electromagnetic field]].<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| == Definition ==
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| === SI units ===
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| In free space and flat space-time, the electromagnetic stress–energy [[tensor]] in [[SI units]] is<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| :<math>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] \,.</math> | |
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| where <math>F^{\mu\nu}</math> is the [[electromagnetic tensor]]. This expression is when using a metric of signature (-,+,+,+). If using the metric with signature (+,-,-,-), the expression for <math>T^{\mu \nu}</math> will have opposite sign.
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| Explicitly in matrix form:
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| :<math>T^{\mu\nu} =\begin{bmatrix} \frac{1}{2}\left(\epsilon_0 E^2+\frac{1}{\mu_0}B^2\right) & S_x/c & S_y/c & S_z/c \\
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| S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\
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| S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\
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| S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix},</math>
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| where <math>\eta_{\mu\nu}</math> is the [[Metric_tensor_(general_relativity)#Flat_spacetime|Minkowski metric tensor]] of [[metric signature]] (−+++),
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| :<math>\bold{S}=\frac{1}{\mu_0}\bold{E}\times\bold{B},</math> | |
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| is the [[Poynting vector]],
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| :<math>\sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}
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| {{\mu _0 }}B_i B_j - \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu _0}B^2 \right)\delta _{ij}. </math>
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| is the [[Maxwell stress tensor]], and ''c'' is the [[speed of light]].
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| === CGS units ===
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| The [[permittivity of free space]] and [[permeability of free space]] in [[Gaussian units|cgs-Gaussian units]] are
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| :<math>\epsilon_0=\frac{1}{4\pi},\quad \mu_0=4\pi\,</math> | |
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| then:
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| :<math>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}] \,.</math>
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| and in explicit matrix form:
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| :<math>T^{\mu\nu} =\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} \\
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| S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\
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| S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}</math>
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| where [[Poynting vector]] becomes:
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| :<math>\bold{S}=\frac{c}{4\pi}\bold{E}\times\bold{B}. </math>
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| 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]].<ref>however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)</ref>
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| The element <math>T^{\mu\nu}\!</math> of the stress–energy tensor represents the flux of the μ<sup>th</sup>-component of the [[four-momentum]] of the electromagnetic field, <math>P^{\mu}\!</math>, going through a [[hyperplane]] (<math> x^{\nu}</math> is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space-time) in [[general relativity]].
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| ==Algebraic properties==
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| This tensor has several noteworthy algebraic properties. First, it is a [[symmetric tensor]]:
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| :<math>T^{\mu\nu}=T^{\nu\mu}</math>
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| Second, the tensor <math>T^{\nu}_{\ \alpha}</math> is [[Trace (linear algebra)|traceless]]:
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| :<math>T^{\alpha}_{\ \alpha}= 0</math>.
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| Third, the energy density is [[Positive-definite function|positive-definite]]:
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| :<math>T^{00}>0</math>
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| These three algebraic properties have varying importance in the context of modern physics, and they remove or reduce ambiguity of the definition of the electromagnetic stress-energy tensor. The symmetry of the tensor is important in [[General Relativity]], because the [[Einstein tensor]] is symmetric. The tracelessness is regarded as important for the masslessness of the [[photon]].<ref>Garg, Anupam. ''Classical Electromagnetism in a Nutshell'', p. 564 (Princeton University Press, 2012).</ref>
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| == Conservation laws == | |
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| {{main|Conservation laws}}
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| 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:
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| :<math>\partial_\nu T^{\mu \nu} + \eta^{\mu \rho} \, f_\rho = 0 \,</math> | |
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| where <math>f_\rho</math> is the (3D) [[Lorentz force]] per unit volume on [[matter]].
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| This equation is equivalent to the following 3D conservation laws
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| :<math>\frac{\partial u_\mathrm{em}}{\partial t} + \bold{\nabla} \cdot \bold{S} + \bold{J} \cdot \bold{E} = 0 \,</math>
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| :<math>\frac{\partial \bold{p}_\mathrm{em}}{\partial t} - \bold{\nabla}\cdot \sigma + \rho \bold{E} + \bold{J} \times \bold{B} = 0 \,</math>
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| respectively describing the flux of electromagnetic energy density
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| :<math>u_\mathrm{em} = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 \,</math>
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| and electromagnetic momentum density
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| :<math>\bold{p}_\mathrm{em} = {\bold{S} \over {c^2}} </math>
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| where '''J''' is the [[electric current density]] and ''ρ'' the [[electric charge density]].
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| ==See also== | |
| *[[Ricci calculus]]
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| *[[Covariant formulation of classical electromagnetism]]
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| *[[Mathematical descriptions of the electromagnetic field]]
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| *[[Maxwell's equations]]
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| *[[Maxwell's equations in curved spacetime]]
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| *[[General relativity]]
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| *[[Einstein field equations]]
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| *[[Magnetohydrodynamics]]
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| *[[vector calculus]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Electromagnetic stress-energy tensor}}
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| [[Category:Tensors]]
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| [[Category:Electromagnetism]]
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