Force-Free Electrodynamics

Force-free electrodynamics describes a plasma whose charges and currents are sufficient to support an electromagnetic field but whose material inertia and pressure are negligible in the leading momentum balance. The Lorentz four-force must then vanish. Together with Maxwell’s equations, field degeneracy, and magnetic dominance, that condition closes a hyperbolic electromagnetic system used for relativistic magnetospheres. The approximation fails precisely where particle inertia, resistivity, or electric dominance becomes dynamically important.

Required background. Hilbert Stress–Energy from Metric Variation supplies electromagnetic stress and conservation; Exterior Algebra and Differential Forms supplies $F$, $dF=0$, and field degeneracy.

Helpful background. The Einstein Equation supplies black-hole and stellar spacetime coupling; Relativistic Matter and Compact Objects locates the plasma limit; and Computational Relativity supplies constrained hyperbolic evolution.

Vanishing Lorentz force

Use Heaviside–Lorentz units and the observer-field conventions of Covariant Electromagnetic Fluids. With $\nabla_aF^{ba}=J^b$, Maxwell’s equations imply

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

If material stress is negligible compared with electromagnetic stress, consistency requires

$$F^b{}_cJ^c=0.$$

This is the force-free condition. It does not say that $J^a$ vanishes; the current is generally essential to maintaining the field. Contracting its spatial part in an observer frame gives

$$q\boldsymbol E+\boldsymbol J\times\boldsymbol B=0,$$

while its temporal part gives $\boldsymbol J\cdot\boldsymbol E=0$. The electromagnetic stress is therefore conserved at leading order, and field energy is transported rather than locally deposited as Joule heat within the ideal approximation.

A nonzero current in the kernel of $F_{ab}$ requires a degenerate field,

$$F\wedge F=0 \quad\Longleftrightarrow\quad \boldsymbol E\cdot\boldsymbol B=0.$$

Magnetic dominance adds

$$B^2-E^2>0.$$

These two invariant conditions guarantee the local existence of a timelike frame in which the electric field vanishes. Gralla and Jacobson derive the kernel, degeneracy, and geometric field-sheet picture in Gralla and Jacobson 2014, §§ 2–3.

Drift and current closure

In a chosen observer frame, the perpendicular force balance determines a drift velocity

$$\boldsymbol v_D=\frac{\boldsymbol E\times\boldsymbol B}{B^2}.$$

Because $E\cdot B=0$, its squared magnitude is $v_D^2=E^2/B^2<1$ exactly when the field is magnetically dominated. Motion parallel to $\boldsymbol B$ is not fixed by the electromagnetic drift alone; charges of opposite sign can supply a parallel current without changing the perpendicular bulk drift.

In flat spacetime, preserving $\boldsymbol E\cdot\boldsymbol B=0$ while using Maxwell’s equations yields the standard current closure

$$\boldsymbol J =q\frac{\boldsymbol E\times\boldsymbol B}{B^2} +\frac{\boldsymbol B\cdot(\boldsymbol\nabla\times\boldsymbol B) -\boldsymbol E\cdot(\boldsymbol\nabla\times\boldsymbol E)}{B^2}\boldsymbol B,$$

with $q=\boldsymbol\nabla\cdot\boldsymbol E$. Here $q$ is charge density in the chosen observer frame, not necessarily the comoving charge density used on the fluid page. To derive the parallel term, differentiate the constraint:

$$\begin{aligned} 0&=\partial_t(\boldsymbol E\cdot\boldsymbol B)\\ &=\boldsymbol B\cdot(\boldsymbol\nabla\times\boldsymbol B) -\boldsymbol J\cdot\boldsymbol B -\boldsymbol E\cdot(\boldsymbol\nabla\times\boldsymbol E). \end{aligned}$$

This determines $\boldsymbol J\cdot\boldsymbol B$; the perpendicular part already follows from force balance. The derivation and closure appear in Gralla and Jacobson 2014, § 3.1, Eqs. (3)–(10).

The formula divides by $B^2$, so it is singular at a magnetic null. Loss of magnetic dominance at $E^2=B^2\neq0$ is a different failure: the displayed current can remain finite, but the zero-electric-field frame reaches light speed and the magnetically dominated evolution loses its regular causal structure. These should not be confused.

The remaining Maxwell constraints are $\boldsymbol\nabla\cdot\boldsymbol B=0$ and Gauss’s law. Analytically, compatible initial data preserve them. Numerically, divergence error and drift in $E\cdot B$ must be controlled separately; projecting $\boldsymbol E$ after every step changes energy and should be reported as a modeled dissipation, not treated as invisible arithmetic.

Characteristics and energy flow

Force-free electrodynamics has fast modes that propagate on the light cone and Alfvén-like modes tied to the magnetic field sheets. In a local frame with $E=0$, disturbances communicate along the two null directions associated with each timelike field sheet. Magnetic dominance is what permits those sheets to have a causal time direction. The physical characteristic modes are derived in Komissarov 2002, § 3. A well-posed numerical evolution also needs an appropriate treatment of the constraint modes; the naive unconstrained six-component evolution is not automatically strongly hyperbolic simply because its physical modes are causal, as discussed in Gralla and Jacobson 2014, § 3.1.

The Poynting flux is

$$\boldsymbol S=\boldsymbol E\times\boldsymbol B=B^2\boldsymbol v_D.$$

Thus the electromagnetic energy flux follows the perpendicular drift, while tension transmits stress along the field. In stationary rotating magnetospheres, global regularity at light surfaces constrains currents and field-line rotation; local force-free equations alone do not determine those boundary data. Gralla and Jacobson 2014, §§ 4–7 develops the conserved currents, field sheets, and stationary magnetosphere structure.

Breakdown and matching to plasma physics

Force-free ordering requires material enthalpy and pressure to remain small compared with $B^2$. It can fail in current sheets, where the field reverses and $B^2$ becomes small; in gaps, where too few charges exist to screen the parallel electric field; at shocks or mass-loaded regions; and where radiation reaction changes the current carriers. The condition $E\cdot B=0$ can also fail during reconnection, allowing field energy to accelerate particles.

If a nonzero field reaches $B^2-E^2\leq0$, no subluminal drift frame can remove the electric field. Enforcing $E\lt B$ by clipping is a numerical prescription for unresolved dissipation, not a derivation from force-free theory. A physical continuation may require resistive electrodynamics, relativistic MHD, or kinetic plasma dynamics, with matching conditions that conserve charge and total energy–momentum.

The approximation is therefore strongest in magnetically dominated exterior regions and weakest exactly where dissipation and particle acceleration occur. Its predictive power comes from keeping that boundary explicit.

Exercises

Assume $\boldsymbol E\cdot\boldsymbol B=0$. Show that the drift velocity is subluminal if and only if the field is magnetically dominated, and verify that a boost by this drift removes the perpendicular electric field.

Solution

Orthogonality gives

$$v_D^2=\frac{|\boldsymbol E\times\boldsymbol B|^2}{B^4} =\frac{E^2}{B^2}.$$

Hence $v_D<1$ exactly when $B^2-E^2>0$. The nonrelativistic-looking transformation direction follows from $\boldsymbol E+\boldsymbol v_D\times\boldsymbol B=0$; the exact Lorentz transformation has the same cancellation because the fields are perpendicular and the boost factor is finite. A remaining boost parallel to $\boldsymbol B$ leaves the zero-electric-frame property intact.

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.
• Komissarov, Serguei S. “Time-Dependent, Force-Free, Degenerate Electrodynamics.” Monthly Notices of the Royal Astronomical Society 336, 759–766 (2002). DOI.