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Electromagnetic stress–energy tensor

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.

Contents

DefinitionEdit

SI unitsEdit

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

 

where   is the electromagnetic tensor and where   is the Minkowski metric tensor of metric signature (−+++). When using the metric with signature (+−−−), the expression for   will have opposite sign.

Explicitly in matrix form:

 

where

 

is the Poynting vector,

 

is the Maxwell stress tensor, and c is the speed of light. Thus,   is expressed and measured in SI pressure units (pascals).

CGS unitsEdit

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

 

then:

 

and in explicit matrix form:

 

where Poynting vector becomes:

 

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   of the stress–energy tensor represents the flux of the μth-component of the four-momentum of the electromagnetic field,  , going through a hyperplane (  is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity.

Algebraic propertiesEdit

The electromagnetic stress–energy tensor has several algebraic properties:

 
  • The tensor   is traceless:
 .
 

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 lawsEdit

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:

 

where   is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 4D conservation laws

 
  (or equivalently   with   being the Lorentz force density),

respectively describing the flux of electromagnetic energy density

 

and electromagnetic momentum density

 

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

See alsoEdit

ReferencesEdit

  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).