Covariant Electromagnetic Fluids

A relativistic plasma exchanges energy and momentum internally between charged matter and the electromagnetic field. Maxwell’s equations make that exchange exact: the divergence of the electromagnetic stress tensor is minus the Lorentz four-force on matter, so the sum of field and matter stress tensors is conserved. This covariant bookkeeping is the foundation on which ideal MHD, resistive MHD, and force-free electrodynamics impose different constitutive closures.

Required background. Hilbert Stress–Energy from Metric Variation supplies symmetric stress tensors and covariant conservation; Exterior Algebra and Differential Forms supplies the electromagnetic two-form and its homogeneous equation.

Helpful background. The Einstein Equation gives gravitational coupling; Relativistic Matter and Compact Objects places magnetofluids among the volume’s matter models; and Computational Relativity provides evolution formulations.

Maxwell fields and their sources

Use Heaviside–Lorentz electromagnetic units, so there is no $4\pi$ in Maxwell’s source equation. With $F=dA$ and the electric-field definition below, the antisymmetric field tensor $F_{ab}$ and current $J^a$ obey

$$\nabla_aF^{ba}=J^b, \qquad \nabla_{[a}F_{bc]}=0.$$

The order of indices in the source equation matters: equivalently, $\nabla_aF^{ab}=-J^b$. The homogeneous equation is $dF=0$ and locally permits $F=dA$. Taking a divergence of the source equation gives

$$\nabla_aJ^a=0,$$

because the double divergence of an antisymmetric tensor vanishes after commuting covariant derivatives. Charge conservation is therefore a compatibility condition of Maxwell’s equations, not an optional material equation.

For a unit timelike observer $n^a$, define measured electric and magnetic fields by

$$E^a=F^{ab}n_b, \qquad B^a=-{}^*F^{ab}n_b, \qquad n_aE^a=n_aB^a=0.$$

Here the Hodge dual is ${}^*F^{ab}=\tfrac12\epsilon^{abcd}F_{cd}$. The minus sign in $B^a$ follows from the site’s spacetime orientation, $\epsilon_{0123}=+\sqrt{-g}$, when the observer’s right-handed spatial basis has $\epsilon^{123}=+1$. In that orthonormal frame,

$$F^{0i}=E^i,\qquad F^{ij}=\epsilon^{ijk}B_k.$$

These components give $\boldsymbol\nabla\cdot\boldsymbol E=J^0$ and $\partial_t\boldsymbol E=\boldsymbol\nabla\times\boldsymbol B-\boldsymbol J$, fixing the convention without ambiguity. A reference that uses the opposite volume-form orientation may define $B^a$ with the opposite dual sign.

The invariant $F_{ab}F^{ab}=2(B^2-E^2)$ and the vanishing of $F\wedge F$ distinguish magnetic dominance and field degeneracy, respectively. In particular, $F\wedge F=0$ is equivalent to $E_aB^a=0$. Those classifications do not depend on the observer; see Gralla and Jacobson 2014, § 3.2.1, Eqs. (11)–(14).

Field stress and the Lorentz force

The electromagnetic stress–energy tensor is

$$T_{\rm EM}^{ab} =F^a{}_cF^{bc}-\frac14g^{ab}F_{cd}F^{cd}.$$

Using both Maxwell equations and antisymmetry gives

$$\nabla_aT_{\rm EM}^{ab}=-F^b{}_cJ^c.$$

Thus $f^b=F^b{}_cJ^c$ is the four-force density delivered to matter. If $T_{\rm m}^{ab}$ denotes all material stress, its equation is

$$\nabla_aT_{\rm m}^{ab}=F^b{}_cJ^c,$$

and hence

$$\nabla_a\!\left(T_{\rm m}^{ab}+T_{\rm EM}^{ab}\right)=0.$$

The sign can be checked in a local inertial frame: the temporal component transfers power $\boldsymbol J\cdot\boldsymbol E$ to matter, while the spatial component gives charge force plus current cross magnetic field. The field loses exactly what matter gains.

The observer projections provide a second check. Write $\gamma^{ab}=g^{ab}+n^an^b$ for the spatial projector and $\epsilon^{abc}=n_d\epsilon^{dabc}$ for the observer’s spatial volume tensor. In the $n^a$ frame, field energy density, momentum density, and spatial stress are

$$U_{\rm EM}=\frac12(E^2+B^2), \qquad S^a=\epsilon^{abc}E_bB_c,$$ $$S_{\rm EM}^{ab}=\frac12(E^2+B^2)\gamma^{ab}-E^aE^b-B^aB^b.$$

The stress is tensile along magnetic field lines and pressurized transverse to them. Its trace satisfies $T_{{\rm EM} a}{}^a=0$, consistent with the absence of a classical electromagnetic mass scale in four spacetime dimensions.

Fluid current and closure choices

Relative to a material four-velocity $u^a$, split the current into

$$J^a=q u^a+j^a, \qquad u_a j^a=0,$$

where $q=-u_aJ^a$ is comoving charge density and $j^a$ is conduction current. Maxwell’s equations relate the field derivatives to the total current, but do not close the joint field–matter evolution. Matter conservation, an equation of state, and a constitutive relation for $j^a$ are needed; Gauss’s law remains a constraint on any proposed charge density.

Ideal MHD imposes vanishing comoving electric field $F_{ab}u^b=0$. Resistive MHD permits a finite conductivity; its simplest Ohm law is algebraic, while more detailed models evolve the conduction current with a relaxation time. Force-free electrodynamics removes material inertia from the leading momentum balance and demands $F^b{}_cJ^c=0$. These are different limits of a plasma, not interchangeable algebraic rewritings. The ideal stress and conservation laws appear in Martí and Müller 2015, § 3.1, Eqs. (1)–(7).

Charge neutrality also needs care. A plasma may be quasineutral on macroscopic scales while carrying a nonzero conduction current because positive and negative charges drift oppositely. Setting $q=0$ does not imply $J^a=0$. Conversely, a one-fluid current closure is trustworthy only when unresolved charge-separation and kinetic scales remain small compared with the scales being modeled.

Limits and independent checks

With $J^a=0$, electromagnetic stress is conserved separately. With $F_{ab}=0$, the matter equation reduces to ordinary fluid conservation. In a local frame with $E^a=0$, the Poynting vector vanishes for the observer comoving with that frame and the magnetic energy is $B^2/2$; a different observer generally sees an electric field and Poynting flux. The invariant statement is not “electric field zero for everyone” but the existence of a timelike frame in which it vanishes.

The decomposition assumes classical electromagnetism and minimal coupling. Polarization, magnetization, quantum electrodynamic vacuum effects, and non-Abelian fields alter the constitutive stress or field equations. An algebraic scalar Ohm law coupled to the full Maxwell equations can already have causal field propagation: finite conductivity by itself does not make Maxwell’s equations parabolic. Magnetic diffusion arises only after eliminating the fast electric-field dynamics, as the resistive-MHD page demonstrates.

Exercises

In a flat-space region with a spatially uniform electric field $E_x(t)$, zero magnetic field, and conductivity $\sigma\gt0$, derive the decay of field energy and the matter heating rate. Explain how this checks both the Maxwell-source sign and total stress conservation.

Solution

Ampère’s equation and $J_x=\sigma E_x$ give $\dot E_x=-\sigma E_x$, hence $E_x(t)=E_x(0)e^{-\sigma t}$. The field energy satisfies

$$\frac{d}{dt}\frac{E_x^2}{2}=-\sigma E_x^2, \qquad f^0=\boldsymbol J\cdot\boldsymbol E=\sigma E_x^2.$$

Thus the field loses exactly the heat gained by matter. More generally,

$$\nabla_aT_{\rm m}^{ab}+\nabla_aT_{\rm EM}^{ab} =F^b{}_cJ^c-F^b{}_cJ^c=0.$$

The opposite signs describe internal exchange. Positive work on matter is an equal loss from the electromagnetic field, so no external four-momentum is created.

References

• Gralla, Samuel E., and Ted Jacobson. “Spacetime Approach to Force-Free Magnetospheres.” Monthly Notices of the Royal Astronomical Society 445, 2500–2534 (2014). Open preprint.
• Martí, José María, and Ewald Müller. “Grid-Based Methods in Relativistic Hydrodynamics and Magnetohydrodynamics.” Living Reviews in Computational Astrophysics 1, 3 (2015). DOI.