Electromagnetic stress–energy tensor explained

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 negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.

Definition

ISQ convention

The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is^[1] $$T^\; =\; \backslash frac\; \backslash left[F^\{\backslash mu\; \backslash alpha\}F^\backslash nu\{\}\_\{\backslash alpha\}\; -\; \backslash frac\{1\}\{4\}\; \backslash eta^\{\backslash mu\backslash nu\}F\_\{\backslash alpha\backslash beta\}\; F^\{\backslash alpha\backslash beta\}\backslash right]\; \backslash ,,$$where

is the electromagnetic tensor and where is the Minkowski metric tensor of metric signature and the Einstein summation convention over repeated indices is used.

Explicitly in matrix form:where$$u\; =\; \backslash frac\backslash left(\backslash epsilon\_0\; E^2+\backslash fracB^2\backslash right)$$is the volumetric energy density,$$\backslash mathbf\; =\; \backslash frac\backslash mathbf\backslash times\backslash mathbf$$is the Poynting vector, $$\backslash sigma\_\; =\; \backslash epsilon\_0\; E\_i\; E\_j\; +\; \backslash fracB\_i\; B\_j\; -\; \backslash frac\; \backslash left(\backslash epsilon\_0\; E^2\; +\; \backslash fracB^2\; \backslash right)\backslash delta\; \_$$is the Maxwell stress tensor, and

is the speed of light. Thus, each component of is dimensionally equivalent to pressure (with SI unit pascal).

Gaussian CGS conventions

The permittivity of free space and permeability of free space in the Gaussian convention are$$\backslash epsilon\_0\; =\; \backslash frac,\backslash quad\; \backslash mu\_0\; =\; 4\backslash pi$$then:$$T^\; =\; \backslash frac\; \backslash left[F^\{\backslash mu\backslash alpha\}F^\{\backslash nu\}\{\}\_\{\backslash alpha\}\; -\; \backslash frac\{1\}\{4\}\; \backslash eta^\{\backslash mu\backslash nu\}F\_\{\backslash alpha\backslash beta\}F^\{\backslash alpha\backslash beta\}\backslash right]$$and in explicit matrix form:where the Poynting vector becomes: $$\backslash mathbf\; =\; \backslash frac\backslash mathbf\backslash times\backslash mathbf.$$

The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the Abraham–Minkowski controversy.^[2]

The element

of the stress–energy tensor represents the flux of the component with index μ of the four-momentum of the electromagnetic field,, going through a hyperplane. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity.

Algebraic properties

The electromagnetic stress–energy tensor has several algebraic properties:

The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy–momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon.^[3]

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:$$\backslash partial\_\backslash nu\; T^\; +\; \backslash eta^\; \backslash ,\; f\_\backslash rho\; =\; 0\; \backslash ,$$where

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

This equation is equivalent to the following 3D conservation lawsrespectively describing the flux of electromagnetic energy density$$u\_\backslash mathrm\; =\; \backslash fracE^2\; +\; \backslash fracB^2$$and electromagnetic momentum density $$\backslash mathbf\_\backslash mathrm\; =,$$where

is the electric current density, the electric charge density, and is the Lorentz force density.

See also

• Ricci calculus
• Covariant formulation of classical electromagnetism
• Mathematical descriptions of the electromagnetic field
• Maxwell's equations
• Maxwell's equations in curved spacetime
• General relativity
• Einstein field equations
• Magnetohydrodynamics
• Vector calculus

Notes and References

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