Ideal Relativistic Magnetohydrodynamics

Ideal relativistic magnetohydrodynamics replaces microscopic charge motion by one conducting fluid and takes the conductivity to be large enough that the electric field vanishes in the fluid rest frame. Maxwell’s homogeneous equation then becomes an induction equation that transports magnetic flux with the flow. The electromagnetic field contributes magnetic pressure, directional tension, energy, and inertia to a single conserved stress tensor.

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

Helpful background. The Einstein Equation supplies dynamical spacetime coupling; Relativistic Matter and Compact Objects places RMHD in the matter hierarchy; and Computational Relativity develops conservative spacetime evolution.

Infinite conductivity and flux freezing

Use Heaviside–Lorentz units and the field conventions of Covariant Electromagnetic Fluids. The ideal condition is the invariant statement

$$F_{ab}u^b=0.$$

It says that the electric field measured by the fluid vanishes. In any local inertial laboratory frame it is exactly equivalent to $\boldsymbol E+\boldsymbol v\times\boldsymbol B=0$, including at relativistic speed. The temporal component also gives $\boldsymbol E\cdot\boldsymbol v=0$, which follows from that vector equation. Because the homogeneous Maxwell equation is $dF=0$, Cartan’s identity gives

$$\mathcal L_uF=d(u\mathbin{\lrcorner}F)+u\mathbin{\lrcorner}dF=0.$$

The electromagnetic two-form is Lie-dragged by the fluid. Magnetic flux through a material surface is therefore conserved as long as the solution remains smooth and the ideal condition holds. “Frozen-in field lines” is a visualization of this two-form statement; it is not a separate equation and can fail at singular current sheets or finite resistivity.

Define the magnetic four-vector measured by the fluid,

$$b^a=-{}^*F^{ab}u_b, \qquad b^a u_a=0.$$

The orthogonality follows immediately from antisymmetry. The dual sign follows the site’s volume-form orientation, as explained on the preceding page. Under the ideal condition the dual field is reconstructed as

$${}^*F^{ab}=b^a u^b-b^b u^a.$$

Substitution into $\nabla_a{}^*F^{ab}=0$, followed by multiplication by $-1$, gives the covariant induction equation

$$\nabla_a(u^a b^b-u^b b^a)=0.$$

The equivalent induction and total-stress equations are given in Martí and Müller 2015, § 3.1, Eqs. (1)–(7); the sign assigned to the dual must be translated with the volume-form orientation.

Magnetic stress and total conservation

Let $\varepsilon$ and $p$ be the material rest-frame energy density and pressure. Combining perfect-fluid and electromagnetic stresses gives

$$T^{ab}=(\varepsilon+p+b^2)u^a u^b +\left(p+\frac12b^2\right)g^{ab}-b^a b^b,$$

where $b^2=b^ab_a\geq0$. The terms have distinct roles. Magnetic energy adds $b^2/2$ to the comoving energy density; magnetic pressure adds $b^2/2$ transverse to the field; and $-b^ab^b$ supplies tension along it. The combination $\varepsilon+p+b^2$ is the total inertial enthalpy density.

The continuum system consists of

$$\nabla_a(n u^a)=0, \qquad \nabla_aT^{ab}=0, \qquad \nabla_a{}^*F^{ab}=0,$$

plus an equation of state and the ideal constraint. If composition or entropy evolves independently, its advection or reaction equation must also be supplied. Maxwell’s inhomogeneous equation determines the current needed to support the evolved field; it is not normally advanced as an independent ideal-MHD fluid variable.

In an Eulerian split the normal magnetic field must satisfy a divergence constraint. In flat space this is $\boldsymbol\nabla\cdot\boldsymbol B=0$, while the induction equation becomes

$$\partial_t\boldsymbol B =\boldsymbol\nabla\times(\boldsymbol v\times\boldsymbol B).$$

Analytically the constraint propagates. A discretization can nevertheless create divergence error, so constrained transport or a declared cleaning method is part of a numerical formulation, not merely an implementation detail.

Conserved and primitive variables

A finite-volume method evolves rest mass, momentum, energy, and magnetic field. To see the map concretely, work in a local Minkowski frame and let $\rho$ be rest-mass density, $W=(1-v^2)^{-1/2}$, and $w=\varepsilon+p$. The laboratory magnetic field and the comoving four-vector are related by

$$b^0=W\boldsymbol v\cdot\boldsymbol B, \qquad b^i=\frac{B^i}{W}+b^0v^i, \qquad b^2=\frac{B^2}{W^2}+(\boldsymbol v\cdot\boldsymbol B)^2.$$

Taking the temporal row of the mass current and stress tensor gives

$$\begin{aligned} D&=\rho W,\\ S_i&=(w+b^2)W^2v_i-b^0b_i,\\ \tau&=(w+b^2)W^2-\left(p+\frac12b^2\right)-(b^0)^2-D. \end{aligned}$$

Here $\tau=T^{00}-D$ subtracts rest-mass energy; a code evolving $T^{00}$ instead uses a different energy variable. Curved-coordinate formulations also include the appropriate lapse, shift, spatial metric, and volume density. The primitive variables $(\rho,p,v^i,B^i)$ determine these conserved variables explicitly once an equation of state supplies $\varepsilon$. Recovering primitives is nonlinear because $W$ and $b^a$ depend on the unknown velocity. The flat-space map and recovery equations appear in Martí and Müller 2015, § 3.1, Eqs. (11)–(13), and § 5.8.

This distinction matters physically. A conservative update preserves cell-integrated currents, but numerical error may move the conserved variables outside the set that admits $\rho\gt0$, $p\gt0$, and $v^2\lt1$. Magnetically dominated cells are especially ill-conditioned: the small material internal energy is recovered by subtracting large magnetic and kinetic contributions, so a small relative error in total energy can become a large relative error in temperature or pressure.

Checks and domain of validity

Setting $b^a\to0$ recovers perfect-fluid stress and relativistic Euler evolution. Taking weak fields and $v\ll1$ recovers the classical ideal-MHD induction equation and Maxwell stress. In the comoving orthonormal frame, $u^{\hat a}=(1,0,0,0)$ and $b^{\hat a}=(0,\boldsymbol B)$; the energy density is $\varepsilon+B^2/2$, transverse stress is $p+B^2/2$, and parallel stress is $p-B^2/2$. These components independently verify the signs in $T^{ab}$.

Ideal RMHD assumes microscopic scales permit one bulk velocity, displacement-current and charge-separation effects are consistently incorporated in the relativistic field formulation, and resistive dissipation is negligible on resolved scales. It does not describe collisionless pressure anisotropy, reconnection-layer kinetics, Hall drift, or radiation reaction unless those effects are added through a controlled closure. Infinite conductivity is a limit of timescales, not a claim that real plasma conductivity is literally infinite.

Exercises

In the fluid rest frame with magnetic field along the $z$ direction, compute the three diagonal spatial stresses and identify magnetic pressure and tension.

Solution

Set $u^{\hat a}=(1,0,0,0)$ and $b^{\hat a}=(0,0,0,B)$. The spatial part of the total stress is

$$T^{\hat i\hat j}=\left(p+\frac12B^2\right)\delta^{ij}-b^{\hat i}b^{\hat j}.$$

Hence $T^{\hat x\hat x}=T^{\hat y\hat y}=p+B^2/2$ and $T^{\hat z\hat z}=p-B^2/2$. The field pushes transversely and pulls along its direction.

References

• Martí, José María, and Ewald Müller. “Grid-Based Methods in Relativistic Hydrodynamics and Magnetohydrodynamics.” Living Reviews in Computational Astrophysics 1, 3 (2015). DOI.