Lorentz Force and Electromagnetic Stress-Energy

The Lorentz force and electromagnetic stress-energy are two sides of one local exchange law. The field loses exactly the four-momentum gained by charged matter. Energy density, Poynting flux, radiation pressure, and mechanical work are therefore components of one symmetric tensor, not separate rules.

Required background. Special Relativity supplies four-momentum, four-velocity, Lorentz covariance, and index operations; Calculus and Linear Algebra supplies multivariable differentiation, integration, and tensor contractions; Classical Fields and Their Actions supplies field actions and variational reasoning. Helpful background. Ordinary Differential Equations supplies evolution and balance-law reasoning; Differential Forms and Integration supplies oriented flux integration; Hilbert Stress-Energy from Metric Variation develops the general metric-response construction.

Force and work on charge

For a particle of charge $q$ and four-velocity $u^\mu$,

$$\boxed{ \frac{dp^\mu}{d\tau} =qF^\mu{}_{\nu}u^\nu.}$$

With $F_{0i}=-E_i$ and $F_{ij}=\epsilon_{ijk}B^k$, its spatial components give

$$\frac{d\mathbf p}{dt} =q(\mathbf E+\mathbf v\times\mathbf B).$$

The time component gives the power delivered to the particle:

$$\frac{dE_{\mathrm{particle}}}{dt} =q\,\mathbf E\mathbin{\cdot}\mathbf v.$$

The magnetic term bends the momentum but does no instantaneous work because $\mathbf v\mathbin{\cdot}(\mathbf v\times\mathbf B)=0$. The covariant and three-vector force laws are derived in Tong 2015, §5.4.1, printed pp. 115–116, PDF.

For a continuous current $J^\mu=(\rho,\mathbf J)$, the four-force density transferred to matter is

$$f^\mu=F^\mu{}_{\nu}J^\nu.$$

Its components are

$$f^0=\mathbf J\mathbin{\cdot}\mathbf E, \qquad \mathbf f=\rho\mathbf E+\mathbf J\times\mathbf B.$$

Positive $\mathbf J\mathbin{\cdot}\mathbf E$ means matter gains energy locally.

Stress-energy from metric response

Temporarily allow an arbitrary background metric and consider

$$S_{\mathrm{EM}} =-\frac14\int d^4x\,\sqrt{-g}\, F_{\rho\sigma}F^{\rho\sigma}.$$

Define the Hilbert stress-energy tensor by varying the inverse metric:

$$T_{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm{EM}}}{\delta g^{\mu\nu}}.$$

When the potential is held fixed, $F_{\mu\nu}$ with both indices down is metric-independent. The two required variations are

$$\delta\sqrt{-g} =-\frac12\sqrt{-g}\,g_{\mu\nu}\delta g^{\mu\nu},$$

and

$$\delta(F_{\rho\sigma}F^{\rho\sigma}) =2F_{\mu\lambda}F_\nu{}^\lambda\delta g^{\mu\nu}.$$

Substitution gives

$$\boxed{ T_{\mu\nu} =F_{\mu\lambda}F_\nu{}^\lambda -\frac14g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}.}$$

Tong 2019, §§4.5.2–4.5.3, printed pp. 183–186, Eqs. (4.46) and (4.52), PDF gives both the inverse-metric definition and the Maxwell variation in the same metric and field-tensor conventions used here.

Energy, momentum, and spatial stress

Return to Cartesian Minkowski coordinates. Since

$$F_{\rho\sigma}F^{\rho\sigma}=2(B^2-E^2),$$

the components are

$$\boxed{T^{00}=u=\frac12(E^2+B^2),}$$ $$\boxed{T^{0i}=T^{i0}=S^i=(\mathbf E\times\mathbf B)^i,}$$

and

$$\boxed{ T^{ij} =-E^iE^j-B^iB^j +\frac12\delta^{ij}(E^2+B^2).}$$

Here $u$ is field energy density, $\mathbf S$ is both energy flux and momentum density in $c=1$ units, and $T^{ji}$ is the flux of $i$-momentum through a surface normal to the $j$ direction. Some continuum-mechanics texts instead call

$$\sigma^{ij}_{\mathrm{Maxwell}}=-T^{ij}$$

the Maxwell traction stress. Naming the sign convention prevents a tension from being mistaken for a pressure.

Metric variation makes $T_{\mu\nu}$ symmetric. In four spacetime dimensions its trace vanishes:

$$T^\mu{}_{\mu} =F_{\mu\lambda}F^{\mu\lambda} -\frac44F_{\rho\sigma}F^{\rho\sigma}=0.$$

The components, spatial stress, and Poynting vector are developed in Tong 2015, §§5.6.1–5.6.2, printed pp. 127–132, PDF, translated from Tong’s simultaneous unit and signature conventions.

Local exchange and finite-volume balance

With these conventions, the inhomogeneous Maxwell equation is

$$\partial_\mu F^{\mu\nu}=-J^\nu.$$

Differentiate $T^{\mu\nu}$. The sourced term is

$$(\partial_\mu F^\mu{}_{\lambda})F^{\nu\lambda} =-F^\nu{}_{\lambda}J^\lambda,$$

while the remaining terms cancel by the Bianchi identity:

$$F^\mu{}_{\lambda}\partial_\mu F^{\nu\lambda} -\frac14\partial^\nu(F_{\rho\sigma}F^{\rho\sigma})=0.$$

Therefore

$$\boxed{ \partial_\mu T_{\mathrm{EM}}^{\mu\nu} =-F^\nu{}_{\lambda}J^\lambda=-f^\nu.}$$

This exchange derivation appears in Tong 2015, §5.6.2, printed pp. 130–131, PDF. If the matter model obeys $\partial_\mu T_{\mathrm{matter}}^{\mu\nu}=f^\nu$, then

$$\partial_\mu \left(T_{\mathrm{EM}}^{\mu\nu} +T_{\mathrm{matter}}^{\mu\nu}\right)=0.$$

The energy component is Poynting’s theorem:

$$\boxed{ \partial_tu +\boldsymbol\nabla\mathbin{\cdot}\mathbf S =-\mathbf J\mathbin{\cdot}\mathbf E.}$$

The rate of decrease of field energy equals outward Poynting flux plus work done on charges. For a fixed spatial volume $V$,

$$\boxed{ \frac{d}{dt}\int_Vu\,d^3x =-\int_{\partial V}\mathbf S\mathbin{\cdot}\mathbf n\,dA -\int_V\mathbf J\mathbin{\cdot}\mathbf E\,d^3x.}$$

Vanishing boundary flux does not conserve electromagnetic field energy when $\mathbf J\mathbin{\cdot}\mathbf E\ne0$. In a closed coupled system, the matter energy changes by the opposite bulk-work term. If $J^\mu$ is prescribed externally and no matter stress tensor is modeled, electromagnetic stress-energy alone is intentionally not conserved.

The momentum component is

$$\partial_tS^i+\partial_jT^{ji} =-(\rho\mathbf E+\mathbf J\times\mathbf B)^i.$$

Field momentum changes through spatial momentum flux and through the Lorentz force transferred to matter.

Common pitfalls

Conserving the field while charges absorb power. Field energy alone is not conserved when $\mathbf J\mathbin{\cdot}\mathbf E$ is nonzero. The conserved object includes matter.

Reversing the Poynting accounting. Outward flux and positive work on charges both decrease the field energy inside a fixed volume.

Mixing spatial-stress sign conventions. State whether $T^{ij}$ is momentum flux or whether $-T^{ij}$ is being called the Maxwell traction stress.

Exercises

Plane-wave stress-energy

At one event, let $\mathbf E=E_{\mathrm w}\widehat{\mathbf x}$ and $\mathbf B=E_{\mathrm w}\widehat{\mathbf y}$. Find the energy density, energy flux, and longitudinal momentum flux.

Solution

The energy density is

$$u=T^{00} =\frac12(E_{\mathrm w}^2+E_{\mathrm w}^2) =E_{\mathrm w}^2.$$

The Poynting vector is

$$\mathbf S=\mathbf E\times\mathbf B =E_{\mathrm w}^2\widehat{\mathbf z},$$

so $T^{0z}=u$. Because $E_z=B_z=0$,

$$T^{zz}=\frac12(E^2+B^2)=u.$$

Meanwhile $T^{xx}=T^{yy}=0$. Thus energy density, energy flux, and longitudinal momentum flux are equal in $c=1$ units, and $T^\mu{}_{\mu}=-u+u=0$. This local algebra is characteristic of a null wave but does not by itself prove that the fields solve the wave equation everywhere.

A closed box with conducting matter

Suppose a fixed box has no matter-energy flux through its wall, and its Poynting flux vanishes at some instant. Let $\mathbf J\mathbin{\cdot}\mathbf E\ge0$ everywhere inside and be strictly positive on a region of nonzero volume. What happens to field and matter energy at that instant?

Solution

The finite-volume law reduces to

$$\frac{dU_{\mathrm{EM}}}{dt} =-\int_V\mathbf J\mathbin{\cdot}\mathbf E\,d^3x\lt0.$$

The electromagnetic field loses energy. In a closed coupled model, matter gains the same amount:

$$\frac{dU_{\mathrm{matter}}}{dt} =\int_V\mathbf J\mathbin{\cdot}\mathbf E\,d^3x.$$

Zero boundary flux closes only the surface channel; it does not remove local field-to-matter transfer. Positivity in just one part of the box would not suffice: a negative work density elsewhere could reverse the sign of the volume integral.

Continue to Symmetry, Conservation, and Stress-Energy to compare Hilbert stress-energy with Noether currents and extend the exchange logic to other fields and matter models.

References

• David Tong, Electromagnetism (PDF), §5.4.1 and §§5.6.1–5.6.2, printed pp. 115–116 and 127–132, University of Cambridge Mathematical Tripos lecture notes, Lent Term 2015.
• David Tong, General Relativity (PDF), §§4.5.2–4.5.3, printed pp. 183–186, University of Cambridge lecture notes, Michaelmas Term 2019.