In
relativistic physics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
, the electromagnetic stress–energy tensor is the contribution to the
stress–energy tensor due to the
electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. The electromagnetic stress–energy tensor contains the negative of the classical
Maxwell stress tensor
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as ...
that governs the electromagnetic interactions.
Definition
SI units
In free space and flat space–time, the electromagnetic stress–energy
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
in
SI units is
:
where
is the
electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
and where
is the
Minkowski metric tensor of
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
. When using the metric with signature , the expression on the right of the equals sign will have opposite sign.
Explicitly in matrix form:
:
where
:
is the
Poynting vector,
:
is the
Maxwell stress tensor
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as ...
, and ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. Thus,
is expressed and measured in SI pressure units (
pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
).
CGS unit conventions
The
permittivity of free space and
permeability of free space
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...
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
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
medium is less well understood and is the subject of the unresolved
Abraham–Minkowski controversy
The Abraham–Minkowski controversy is a physics debate concerning Electromagnetism, electromagnetic momentum within dielectric media. Two equations were first suggested by Hermann Minkowski (1908)
:* Wikisource translationThe Fundamental Equation ...
.
The element
of the stress–energy tensor represents the flux of the ''μ''th-component of the
four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. The trace of the energy–momentum tensor is a
Lorentz scalar
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
; 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
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
.
[Garg, Anupam. ''Classical Electromagnetism in a Nutshell'', p. 564 (Princeton University Press, 2012).]
Conservation laws
The electromagnetic stress–energy tensor allows a compact way of writing the
conservation laws
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
of linear
momentum and
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
in electromagnetism. The divergence of the stress–energy tensor is:
:
where
is the (4D)
Lorentz force per unit volume on
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
.
This equation is equivalent to the following 3D conservation laws
:
respectively describing the flux of electromagnetic energy density
:
and electromagnetic momentum density
:
where J is the
electric current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
, ''ρ'' the
electric charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
, and
is the Lorentz force density.
See also
*
Ricci calculus
*
Covariant formulation of classical electromagnetism
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformatio ...
*
Mathematical descriptions of the electromagnetic field
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equation ...
*
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
*
Maxwell's equations in curved spacetime
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate ...
*
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
*
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
*
Magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
*
Vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
References
{{DEFAULTSORT:Electromagnetic stress-energy tensor
Tensor physical quantities
Electromagnetism