Electromagnetic stress–energy tensor

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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[edit]

SI units[edit]

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. 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 (−+++),

\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[edit]

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:

\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[edit]

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

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

Second, the tensor T^{\nu}_{\ \alpha} 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[edit]

Main article: 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} + \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 \,

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.

See also[edit]

References[edit]

  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. ^ however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)
  4. ^ Garg, Anupam. Classical Electromagnetism in a Nutshell, p. 564 (Princeton University Press, 2012).