Electromagnetic stress–energy tensor

Articles about
Electromagnetism
Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss's law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday's law Jefimenko equations Larmor formula Lenz's law Liénard–Wiechert potential London equations Lorentz force Maxwell's equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e

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.

The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is^[1] $${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left\,,}$$ where ${\displaystyle F^{\mu \nu }}$ is the electromagnetic tensor and where ${\displaystyle \eta _{\mu \nu }}$ is the Minkowski metric tensor of metric signature (− + + +) and the Einstein summation convention over repeated indices is used.

Explicitly in matrix form: $${\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}},}$$ where $${\displaystyle u={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)}$$ is the volumetric energy density, $${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} }$$ is the Poynting vector, $${\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)\delta _{ij}}$$ is the Maxwell stress tensor, and ${\displaystyle c}$ is the speed of light. Thus, each component of ${\displaystyle T^{\mu \nu }}$ is dimensionally equivalent to pressure (with SI unit pascal).

The in the Gaussian system (shown here with a prime) that correspond to the permittivity of free space and permeability of free space are $${\displaystyle \epsilon _{0}'={\frac {1}{4\pi }},\quad \mu _{0}'=4\pi }$$ then: $${\displaystyle T^{\mu \nu }={\frac {1}{4\pi }}\left}$$ and in explicit matrix form: $${\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}}}$$ where the energy density becomes $${\displaystyle u={\frac {1}{8\pi }}\left(\mathbf {E} '^{2}+\mathbf {B} '^{2}\right)}$$ and the Poynting vector becomes $${\displaystyle \mathbf {S} ={\frac {c}{4\pi }}\mathbf {E} '\times \mathbf {B} '.}$$

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

The element ${\displaystyle T^{\mu \nu }}$ of the stress–energy tensor represents the flux of the component with index ${\displaystyle \mu }$ of the four-momentum of the electromagnetic field, ⁠${\displaystyle P^{\mu }}$⁠, going through a hyperplane. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity.

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]

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: $${\displaystyle \partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,}$$ where ${\displaystyle f_{\rho }}$ is the (4D) Lorentz force per unit volume on matter.

This equation is equivalent to the following 3D conservation laws $${\displaystyle {\begin{aligned}{\frac {\partial u_{\mathrm {em} }}{\partial t}}+\mathbf {\nabla } \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} &=0\\{\frac {\partial \mathbf {p} _{\mathrm {em} }}{\partial t}}-\mathbf {\nabla } \cdot \sigma +\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} &=0\ \Leftrightarrow \ \epsilon _{0}\mu _{0}{\frac {\partial \mathbf {S} }{\partial t}}-\nabla \cdot \mathbf {\sigma } +\mathbf {f} =0\end{aligned}}}$$ respectively describing the electromagnetic energy density $${\displaystyle u_{\mathrm {em} }={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)}$$ and electromagnetic momentum density $${\displaystyle \mathbf {p} _{\mathrm {em} }={\mathbf {S} \over {c^{2}}},}$$ where ${\displaystyle \mathbf {J} }$ is the electric current density, ${\displaystyle \rho }$ the electric charge density, and ${\displaystyle \mathbf {f} }$ is the Lorentz force density.

• 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