Stress–energy tensor

From Wikipedia, the free encyclopedia

 

Contravariant components of the stress–energy tensor.

The stress–energy tensor, sometimes stress–energy–momentum tensor or energy–momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Definition

The stress–energy tensor involves the use of superscripted variables (not exponents). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x⁰ = t, x¹ = x, x² = y, and x³ = z, where t is time in seconds, and x, y, and z are distances in meters.

The stress–energy tensor is defined as the tensor T^αβ of order two that gives the flux of the αth component of the momentum vector across a surface with constant x^β coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,

${\displaystyle T^{\alpha \beta }=T^{\beta \alpha }.}$

In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.

Identifying the components of the tensor

Because the stress–energy tensor is of order two, its components can be displayed in 4 × 4 matrix form:

${\displaystyle (T^{\mu \nu })_{\mu ,\nu =0,1,2,3}={\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}}.}$

In the following, i and k range from 1 through 3.

The time–time component is the density of relativistic mass, i.e. the energy density divided by the speed of light squared. Its components have a direct physical interpretation. In the case of a perfect fluid this component is

${\displaystyle T^{00}=\rho ,}$

where ${\displaystyle \rho }$ is the relativistic mass per unit volume, and for an electromagnetic field in otherwise empty space this component is

${\displaystyle T^{00}={1 \over c^{2}}\left({\frac {1}{2}}\epsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right),}$

where E and B are the electric and magnetic fields, respectively.

The flux of relativistic mass across the xⁱ surface is equivalent to the density of the ith component of linear momentum,

${\displaystyle T^{0i}=T^{i0}.}$

The components

${\displaystyle T^{ik}}$

represent flux of ith component of linear momentum across the x^k surface. In particular,

${\displaystyle T^{ii}}$

(not summed) represents normal stress, which is called pressure when it is independent of direction. The remaining components

${\displaystyle T^{ik}\quad i\neq k}$

represent shear stress (compare with the stress tensor).

In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress energy tensor in engineering differs from the stress–energy tensor here by a momentum convective term.

Covariant and mixed forms

In most of this article we work with the contravariant form, T^μν of the stress–energy tensor. However, it is often necessary to work with the covariant form,

${\displaystyle T_{\mu \nu }=T^{\alpha \beta }g_{\alpha \mu }g_{\beta \nu },}$

or the mixed form,

${\displaystyle T^{\mu }{}_{\nu }=T^{\mu \alpha }g_{\alpha \nu },}$

or as a mixed tensor density

${\displaystyle {\mathfrak {T}}^{\mu }{}_{\nu }=T^{\mu }{}_{\nu }{\sqrt {-g}}\,.}$

In this article we use the spacelike sign convention (−+++) for the metric signature.

Conservation law

In special relativity

The stress–energy tensor is the conserved Noether current associated with spacetime translations.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,

${\displaystyle 0=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }{}.\!}$

When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as

${\displaystyle 0=T^{\mu \nu }{}_{,\nu }=\partial _{\nu }T^{\mu \nu }.\!}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}T^{\mu \nu }\mathrm {d} ^{3}s_{\nu }\!}$

where N is any compact four-dimensional region of spacetime; ${\displaystyle \partial N}$ is its boundary, a three-dimensional hypersurface; and ${\displaystyle \mathrm {d} ^{3}s_{\nu }}$ is an element of the boundary regarded as the outward pointing normal.

In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum is also conserved:

${\displaystyle 0=(x^{\alpha }T^{\mu \nu }-x^{\mu }T^{\alpha \nu })_{,\nu }.\!}$

In general relativity

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes. But in this case, a coordinate free definition of the divergence is used which incorporates the covariant derivative

${\displaystyle 0=\operatorname {div} T=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }=T^{\mu \nu }{}_{,\nu }+\Gamma ^{\mu }{}_{\sigma \nu }T^{\sigma \nu }+\Gamma ^{\nu }{}_{\sigma \nu }T^{\mu \sigma }}$

where ${\displaystyle \Gamma ^{\mu }{}_{\sigma \nu }}$ is the Christoffel symbol which is the gravitational force field.

Consequently, if ${\displaystyle \xi ^{\mu }}$ is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as

${\displaystyle 0=\nabla _{\nu }(\xi ^{\mu }T_{\mu }^{\nu })={\frac {1}{\sqrt {-g}}}\partial _{\nu }({\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu })}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}{\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }=\int _{\partial N}\xi ^{\mu }{\mathfrak {T}}_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }}$

In general relativity

In general relativity, the symmetric stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation: energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the gravitational field energy and momentum densities. Any such stress–energy pseudotensor can be made to vanish locally by a coordinate transformation.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.

The Einstein field equations

In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu },}$

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, ${\displaystyle R}$ is the Ricci scalar (the tensor contraction of the Ricci tensor), ${\displaystyle g_{\mu \nu }\,}$ the metric tensor, and ${\displaystyle G}$ is the universal gravitational constant.

Stress–energy in special situations

Isolated particle

In special relativity, the stress–energy of a non-interacting particle with mass m and trajectory ${\displaystyle \mathbf {x} _{\text{p}}(t)}$ is:

${\displaystyle T^{\alpha \beta }(\mathbf {x} ,t)={\frac {m\,v^{\alpha }(t)v^{\beta }(t)}{\sqrt {1-(v/c)^{2}}}}\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))={\frac {E}{c^{2}}}\;v^{\alpha }(t)v^{\beta }(t)\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))}$

where ${\displaystyle (v^{\alpha })_{\alpha =0,1,2,3}\!}$ is the velocity vector (which should not be confused with four-velocity, since it is missing a ${\displaystyle \gamma }$)

${\displaystyle (v^{\alpha })_{\alpha =0,1,2,3}=\left(1,{\frac {d\mathbf {x} _{\text{p}}}{dt}}(t)\right)\,,}$

δ is the Dirac delta function and ${\displaystyle E={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}}$ is the energy of the particle.

Stress–energy of a fluid in equilibrium

For a perfect fluid in thermodynamic equilibrium, the stress–energy tensor takes on a particularly simple form

${\displaystyle T^{\alpha \beta }\,=\left(\rho +{p \over c^{2}}\right)u^{\alpha }u^{\beta }+pg^{\alpha \beta }}$

where ${\displaystyle \rho }$ is the mass–energy density (kilograms per cubic meter), ${\displaystyle p}$ is the hydrostatic pressure (pascals), ${\displaystyle u^{\alpha }}$ is the fluid's four velocity, and ${\displaystyle g^{\alpha \beta }}$ is the reciprocal of the metric tensor. Therefore, the trace is given by

${\displaystyle T=3p-\rho c^{2}\,.}$

The four velocity satisfies

${\displaystyle u^{\alpha }u^{\beta }g_{\alpha \beta }=-c^{2}\,.}$

In an inertial frame of reference comoving with the fluid, better known as the fluid's proper frame of reference, the four velocity is

${\displaystyle (u^{\alpha })_{\alpha =0,1,2,3}=(1,0,0,0)\,,}$

the reciprocal of the metric tensor is simply

${\displaystyle (g^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}\,=\left({\begin{matrix}-c^{-2}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\,}$

and the stress–energy tensor is a diagonal matrix

${\displaystyle (T^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).}$

Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor of a source-free electromagnetic field is

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left(F^{\mu \alpha }g_{\alpha \beta }F^{\nu \beta }-{\frac {1}{4}}g^{\mu \nu }F_{\delta \gamma }F^{\delta \gamma }\right)}$

where ${\displaystyle F_{\mu \nu }}$ is the electromagnetic field tensor.

Scalar field

The stress–energy tensor for a scalar field ${\displaystyle \phi }$ which satisfies the Klein–Gordon equation is

${\displaystyle T^{\mu \nu }={\frac {\hbar ^{2}}{m}}(g^{\mu \alpha }g^{\nu \beta }+g^{\mu \beta }g^{\nu \alpha }-g^{\mu \nu }g^{\alpha \beta })\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi -g^{\mu \nu }mc^{2}{\bar {\phi }}\phi ,}$

and when the metric is flat (Minkowski) its components work out to be:

${\displaystyle {\begin{aligned}T^{00}&={\frac {\hbar ^{2}}{mc^{4}}}\left(\partial _{0}{\bar {\phi }}\partial _{0}\phi +c^{2}\partial _{k}{\bar {\phi }}\partial _{k}\phi \right)+m{\bar {\phi }}\phi ,\\T^{0i}=T^{i0}&=-{\frac {\hbar ^{2}}{mc^{2}}}\left(\partial _{0}{\bar {\phi }}\partial _{i}\phi +\partial _{i}{\bar {\phi }}\partial _{0}\phi \right),\ \mathrm {and} \\T^{ij}&={\frac {\hbar ^{2}}{m}}\left(\partial _{i}{\bar {\phi }}\partial _{j}\phi +\partial _{j}{\bar {\phi }}\partial _{i}\phi \right)-\delta _{ij}\left({\frac {\hbar ^{2}}{m}}\eta ^{\alpha \beta }\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi +mc^{2}{\bar {\phi }}\phi \right).\end{aligned}}}$

Variant definitions of stress–energy

There are a number of inequivalent definitions of non-gravitational stress–energy:

Hilbert stress–energy tensor

The Hilbert stress–energy tensor is defined as the functional derivative

${\displaystyle T_{\mu \nu }={\frac {-2}{\sqrt {-g}}}{\frac {\delta ({\mathcal {L}}_{\mathrm {matter} }{\sqrt {-g}})}{\delta g^{\mu \nu }}}=-2{\frac {\delta {\mathcal {L}}_{\mathrm {matter} }}{\delta g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} },}$

where ${\displaystyle {\mathcal {L}}_{\mathrm {matter} }}$ is the nongravitational part of the Lagrangian density of the action. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information.

Canonical stress–energy tensor

Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress–energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be gauge invariant because space-dependent gauge transformations do not commute with spatial translations.

In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudo-tensor.

Belinfante–Rosenfeld stress–energy tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.

Gravitational stress–energy

By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor.

In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.

Poynting vector

From Wikipedia, the free encyclopedia

 

Dipole radiation of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.

In physics, the Poynting vector represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m²). It is named after its discoverer John Henry Poynting who first derived it in 1884. Oliver Heaviside also discovered it independently.

Definition

In Poynting's original paper and in many textbooks, the Poynting vector is defined as

${\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} ,}$

where bold letters represent vectors and

• E is the electric field vector;
• H is the magnetic field's auxiliary field vector.

This expression is often called the Abraham form. The Poynting vector is usually denoted by S or N.

In the "microscopic" version of Maxwell's equations, this definition must be replaced by a definition in terms of the electric field E and the magnetic field B (it is described later in the article).

It is also possible to combine the electric displacement field D with the magnetic field B to get the Minkowski form of the Poynting vector, or use D and H to construct yet another version. The choice has been controversial: Pfeifer et al. summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms.

The Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for other types of energy as well, e.g., for mechanical energy. The Umov–Poynting vector discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

Interpretation

A DC circuit consisting of a battery (V) and resistor (R),

showing the direction of the Poynting vector (S, blue) in

the space surrounding it, along with the fields it is derived

from; the electric field (E, red) and the magnetic field

 (H, green). In the region around the battery the Poynting

vector is directed outward, indicating power flowing out

of the battery into the fields; in the region around the

resistor the vector is directed inward, indicating field power

flowing into the resistor. Across any plane P between the

battery and resistor, the Poynting flux is in the direction of

the resistor. The magnitudes (lengths) of the vectors are not

shown accurately; only the directions are significant.

The Poynting vector appears in Poynting's theorem, an energy-conservation law:

${\displaystyle {\frac {\partial u}{\partial t}}=-\mathbf {\nabla } \cdot \mathbf {S} -\mathbf {J_{\mathrm {f} }} \cdot \mathbf {E} ,}$

where J_f is the current density of free charges and u is the electromagnetic energy density for linear, nondispersive materials, given by

${\displaystyle u={\frac {1}{2}}\!\left(\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \right)\!,}$

where

• E is the electric field;
• D is the electric displacement field;
• B is the magnetic field;
• H is the magnetic auxiliary field.

The first term in the right-hand side represents the electromagnetic energy flow into a small volume, while the second term subtracts the work done by the field on free electrical currents, which thereby exits from electromagnetic energy as dissipation, heat, etc. In this definition, bound electrical currents are not included in this term, and instead contribute to S and u.

For linear, nondispersive and isotropic (for simplicity) materials, the constitutive relations can be written as

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} ,\quad \mathbf {H} ={\frac {1}{\mu }}\mathbf {B} ,}$

where

• ε is the permittivity of the material;
• μ is the permeability of the material.

Here ε and μ are scalar, real-valued constants independent of position, direction, and frequency.

In principle, this limits Poynting's theorem in this form to fields in vacuum and nondispersive linear materials. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms.

Formulation in terms of microscopic fields

The "microscopic" version of Maxwell's equations admits only the fundamental fields E and B, without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is no D or H. When this model is used, the Poynting vector is defined as

${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,}$

where

• μ₀ is the vacuum permeability;
• E is the electric field;
• B is the magnetic field.

The corresponding form of Poynting's theorem is

${\displaystyle {\frac {\partial u}{\partial t}}=-\nabla \cdot \mathbf {S} -\mathbf {J} \cdot \mathbf {E} ,}$

where J is the total current density and the energy density u is given by

${\displaystyle u={\frac {1}{2}}\!\left(\varepsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)\!,}$

where ε₀ is the vacuum permittivity. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.

The two alternative definitions of the Poynting vector are equal in vacuum or in non-magnetic materials, where B = μ₀H. In all other cases, they differ in that S = (1/μ₀) E × B and the corresponding u are purely radiative, since the dissipation term −J ⋅ E covers the total current, while the E × H definition has contributions from bound currents which are then excluded from the dissipation term.

Since only the microscopic fields E and B occur in the derivation of S = (1/μ₀) E × B, assumptions about any material present are completely avoided, and Poynting vector and theorem are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the "macroscopic" form E × H.

Time-averaged Poynting vector

The above form for the Poynting vector represents the instantaneous power flow due to instantaneous electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of sinusoidally varying fields at a specified frequency. The results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes.

We would thus not be considering the instantaneous E(t) and H(t) used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) using phasor notation. Note that these complex amplitude vectors are not functions of time, as they are understood to refer to oscillations over all time. A phasor such as ${\displaystyle \mathbf {E_{\mathrm {m} }} }$ is understood to signify a sinusoidally varying field whose instantaneous amplitude E(t) follows the real part of ${\displaystyle \mathbf {E_{\mathrm {m} }} e^{j\omega t}}$ where ω is the (radian) frequency of the sinusoidal wave being considered.

In the time domain it will be seen that the instantaneous power flow will be fluctuating at a frequency of 2ω. But what is normally of interest is the average power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over a full cycle ${\displaystyle T=2\pi /\omega }$. The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as:

${\displaystyle \mathbf {S} _{\mathrm {m} }={\tfrac {1}{2}}\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*},}$

where ^∗ denotes the complex conjugate. The time-averaged power flow (according to the instantaneous Poynting vector averaged over a full cycle, for instance) is then given by the real part of ${\displaystyle \mathbf {S} _{\mathrm {m} }}$. The imaginary part is usually ignored, however it signifies "reactive power" such as the interference due to a standing wave or the near field of an antenna. In a single electromagnetic plane wave (rather than a standing wave which can be described as two such waves travelling in opposite directions), E and H are exactly in phase, so ${\displaystyle \mathbf {S} _{\mathrm {m} }}$ is simply a real number according to the above definition.

The equivalence of ${\displaystyle \operatorname {Re} (\mathbf {S} _{\mathrm {m} })}$ to the time-average of the instantaneous Poynting vector S can be shown as follows.

${\displaystyle {\begin{aligned}\mathbf {S} (t)&=\mathbf {E} (t)\times \mathbf {H} (t)\\&=\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }e^{j\omega t}\right)\times \operatorname {Re} \!\left(\mathbf {H} _{\mathrm {m} }e^{j\omega t}\right)\\&={\tfrac {1}{2}}\!\left(\mathbf {E} _{\mathrm {m} }e^{j\omega t}+\mathbf {E} _{\mathrm {m} }^{*}e^{-j\omega t}\right)\times {\tfrac {1}{2}}\!\left(\mathbf {H} _{\mathrm {m} }e^{j\omega t}+\mathbf {H} _{\mathrm {m} }^{*}e^{-j\omega t}\right)\\&={\tfrac {1}{4}}\!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}+\mathbf {E} _{\mathrm {m} }^{*}\times \mathbf {H} _{\mathrm {m} }+\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }e^{2j\omega t}+\mathbf {E} _{\mathrm {m} }^{*}\times \mathbf {H} _{\mathrm {m} }^{*}e^{-2j\omega t}\right)\\&={\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}\right)+{\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }e^{2j\omega t}\right)\!.\end{aligned}}}$

The average of the instantaneous Poynting vector S over time is given by:

${\displaystyle \langle \mathbf {S} \rangle ={\frac {1}{T}}\int _{0}^{T}\mathbf {S} (t)\,dt={\frac {1}{T}}\int _{0}^{T}\!\left[{\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}\right)+{\tfrac {1}{2}}\operatorname {Re} \!\left({\mathbf {E} _{\mathrm {m} }}\times {\mathbf {H} _{\mathrm {m} }}e^{2j\omega t}\right)\right]dt.}$

The second term is the double-frequency component having an average value of zero, so we find:

${\displaystyle \langle \mathbf {S} \rangle =\operatorname {Re} \!\left({\tfrac {1}{2}}{\mathbf {E} _{\mathrm {m} }}\times \mathbf {H} _{\mathrm {m} }^{*}\right)=\operatorname {Re} \!\left(\mathbf {S} _{\mathrm {m} }\right)}$

According to some conventions the factor of 1/2 in the above definition may be left out.

Multiplication by 1/2 is required to properly describe the power flow since the magnitudes of ${\displaystyle \mathbf {E} _{\mathrm {m} }}$ and ${\displaystyle \mathbf {H} _{\mathrm {m} }}$ refer to the peak fields of the oscillating quantities. If rather the fields are described in terms of their root mean square (rms) values (which are each smaller by the factor ${\displaystyle {\sqrt {2}}/2}$), then the correct average power flow is obtained without multiplication by 1/2.

Examples and applications

Coaxial cable

Poynting vector in a coaxial cable, shown in red.

For example, the Poynting vector within the dielectric insulator of a coaxial cable is nearly parallel to the wire axis (assuming no fields outside the cable and a wavelength longer than the diameter of the cable, including DC). Electrical energy delivered to the load is flowing entirely through the dielectric between the conductors. Very little energy flows in the conductors themselves, since the electric field strength is nearly zero. The energy flowing in the conductors flows radially into the conductors and accounts for energy lost to resistive heating of the conductor. No energy flows outside the cable, either, since there the magnetic fields of inner and outer conductors cancel to zero.

Resistive dissipation

If a conductor has significant resistance, then, near the surface of that conductor, the Poynting vector would be tilted toward and impinge upon the conductor. Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface. This is a consequence of Snell's law and the very slow speed of light inside a conductor. The definition and computation of the speed of light in a conductor can be given. Inside the conductor, the Poynting vector represents energy flow from the electromagnetic field into the wire, producing resistive Joule heating in the wire. For a derivation that starts with Snell's law see Reitz page 454.

Plane waves

In a propagating sinusoidal linearly polarized electromagnetic plane wave of a fixed frequency, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is found as above to be:

${\displaystyle \langle S\rangle ={\frac {1}{2\eta }}|E_{\mathrm {m} }|^{2}}$

where E_m is the complex amplitude of the electric field and η is the characteristic impedance of the transmission medium, or just η₀ ${\displaystyle \approx }$ 377Ω for a plane wave in free space. This directly follows from the above expression for the average Poynting vector using phasor quantities, and the fact that in a plane wave the magnetic field ${\displaystyle H_{\mathrm {m} }}$ is equal to the electric field ${\displaystyle E_{\mathrm {m} }}$ divided by η (and thus exactly in phase).

In optics, the time-averaged value of the radiated flux is technically known as the irradiance, more often simply referred to as the intensity.

Radiation pressure

The density of the linear momentum of the electromagnetic field is S/c² where S is the magnitude of the Poynting vector and c is the speed of light in free space. The radiation pressure exerted by an electromagnetic wave on the surface of a target is given by

${\displaystyle P_{\mathrm {rad} }={\frac {\langle S\rangle }{\mathrm {c} }}.}$

Static fields

Poynting vector in a static field, where E is the electric field, H the magnetic field, and S the Poynting vector.

The consideration of the Poynting vector in static fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of the Lorentz force, q(v × B). To illustrate, the accompanying picture is considered, which describes the Poynting vector in a cylindrical capacitor, which is located in an H field (pointing into the page) generated by a permanent magnet. Although there are only static electric and magnetic fields, the calculation of the Poynting vector produces a clockwise circular flow of electromagnetic energy, with no beginning or end.

While the circulating energy flow may seem nonsensical or paradoxical, it is necessary to maintain conservation of momentum. Momentum density is proportional to energy flow density, so the circulating flow of energy contains an angular momentum. This is the cause of the magnetic component of the Lorentz force which occurs when the capacitor is discharged. During discharge, the angular momentum contained in the energy flow is depleted as it is transferred to the charges of the discharge current crossing the magnetic field.

Adding the curl of a vector field

The Poynting vector occurs in Poynting's theorem only through its divergence ∇ ⋅ S, that is, it is only required that the surface integral of the Poynting vector around a closed surface describe the net flow of electromagnetic energy into or out of the enclosed volume. This means that adding a field to S which has zero divergence will result in a field which satisfies this required property of a Poynting vector field according to Poynting's theorem. Since the divergence of any curl is zero, one can add the curl of any vector field to the Poynting vector and the resulting vector field S' will still satisfy Poynting's theorem.

However the theory of special relativity, in which energy and momentum are defined locally and invariantly via the stress–energy tensor, shows that the above given expression for the Poynting vector is unique.