Electromagnetic stress–energy tensor: Difference between revisions

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In physics, the electromagnetic stress–energy tensor is the portion of the stress–energy tensor due to the electromagnetic field.^[1]

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

where $F^{\mu \nu }$ is the electromagnetic tensor. This expression is when using a metric of signature (-,+,+,+). If 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_{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 $\eta _{\mu \nu }$ is the Minkowski metric tensor of metric signature (−+++),

{\mathbf {S}}={\frac {1}{\mu _{0}}}{\mathbf {E}}\times {\mathbf {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_{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 Poynting vector becomes:

{\mathbf {S}}={\frac {c}{4\pi }}{\mathbf {E}}\times {\mathbf {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 μ^th-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

This tensor has several noteworthy algebraic properties. First, it is a symmetric tensor:

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

Second, the tensor $T_{\ \alpha }^{\nu }$ is traceless:

T_{\ \alpha }^{\alpha }=0

.

Third, the energy density is positive-definite:

T^{00}>0

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.^[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 (3D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws

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

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

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

{\mathbf {p}}_{\mathrm {em} }={{\mathbf {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).

[1] Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0

[2] Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0

[3] wever see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)

[4] Garg, Anupam. Classical Electromagnetism in a Nutshell, p. 564 (Princeton University Press, 2012).

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@@ Line 26: / Line 26: @@
 is the [[Poynting vector]],
-:<math>\sigma_{ij} = \epsilon_0 E_i E_j   + \frac{1}
+:<math>\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}. </math>
-{{\mu _0 }}B_i B_j - \frac{1}{2} \left( \epsilon_0 E^2  + \frac{1}{\mu _0}B^2 \right)\delta _{ij}. </math>
-is the [[Maxwell stress tensor]], and ''c'' is the [[speed of light]].
+is the [[Maxwell stress tensor]], and ''c'' is the [[speed of light]]. Thus, <math>T^{\mu\nu}</math> is expressed and measured in SI pressure units ([[Pascal_(unit)|pascals]]).
 === CGS units ===