Characteristic Waves in Relativistic MHD

Ideal relativistic MHD carries seven physical characteristic families in one spatial direction: one material or entropy wave, two Alfvén waves, two slow magnetosonic waves, and two fast magnetosonic waves. Their speeds determine causal domains, Riemann-solver signal bounds, and which boundary data may be prescribed. Magnetic tension makes propagation depend on the angle between the wave normal and the field, while relativity confines every admissible characteristic to the metric light cone.

Required background. Hilbert Stress–Energy from Metric Variation supplies the conserved total stress; Exterior Algebra and Differential Forms supplies the induction equation and magnetic constraint.

Helpful background. The Einstein Equation gives curved-spacetime coupling; Relativistic Matter and Compact Objects gives the matter context; and Computational Relativity uses characteristic speeds in numerical fluxes and boundaries.

Linear modes in the fluid frame

Linearize a homogeneous ideal-RMHD state in a local orthonormal frame comoving with the fluid. This is also the local short-wavelength characteristic calculation in curved spacetime: background gradients do not enter the principal symbol. Let $w=\varepsilon+p$ be material enthalpy density and $b^2$ the comoving magnetic magnitude. Define

$$v_A^2=\frac{b^2}{w+b^2}, \qquad c_s^2=\left(\frac{\partial p}{\partial\varepsilon}\right)_{s,Y_i}.$$

Here $s$ is entropy per baryon and $Y_i$ are composition fractions held fixed during the perturbation. Positive $w$ makes $0\leq v_A^2\lt1$ for finite field strength. This normalization is dimensionless because $b^2$ and $w$ are both energy densities. For a plane perturbation $e^{i\boldsymbol k\cdot\boldsymbol x-i\omega t}$, let $\vartheta$ be the angle between $\boldsymbol k$ and the magnetic field, and set $c=\omega/|\boldsymbol k|$. Alfvén waves are transverse disturbances restored by field-line tension; their signed normal phase speeds are $\pm v_A\cos\vartheta$.

The compressive waves couple pressure to magnetic pressure and tension. Their dispersion polynomial becomes a quadratic in the squared speed $z=c^2$:

$$z^2-Az+c_s^2v_A^2\cos^2\vartheta=0, \qquad A=c_s^2+v_A^2-c_s^2v_A^2\sin^2\vartheta.$$

Thus the slow and fast squared speeds are

$$c_{\rm slow,fast}^2 =\frac12\left[A\mp\sqrt{A^2-4c_s^2v_A^2\cos^2\vartheta}\right].$$

One way to obtain the polynomial is to substitute $u^a=(1,0,0,0)$ and wave normal $\phi_a=(-c,\hat k_i)$ into the covariant magnetosonic determinant. Its three contractions are $u^a\phi_a=-c$, $b^a\phi_a=b\cos\vartheta$, and $\phi_a\phi^a=1-c^2$. Multiplying by $c_s^2$ and dividing by $w+b^2$ gives the expression above, which remains well defined in the cold limit $c_s\to0$. The determinant and physical wave ordering are given in Martí and Müller 2015, § 8.2.1, Eqs. (127)–(136).

The entropy family has zero speed in the comoving frame. It carries changes in density or entropy compatible with unchanged pressure and normal velocity. For a boost along the wave normal with background speed $v$, transforming the wave covector gives

$$\lambda=\frac{v+c}{1+vc}.$$

This form makes causal preservation explicit when $|v|\lt1$ and $|c|\leq1$. For a general boost with transverse flow, the wave normal also changes; one must transform the full covector or solve the covariant determinant, rather than apply this one-dimensional formula to an unchanged field angle.

Parallel and perpendicular degeneracies

Special directions provide stronger checks than a generic numerical eigenvalue list. For propagation parallel to the magnetic field, transverse Alfvén and longitudinal acoustic disturbances decouple. The compressive branches reduce to speeds $c_s$ and $v_A$, with “fast” assigned to the larger and “slow” to the smaller. At $c_s=v_A$ eigenvalues coincide and eigenvectors require a regularized basis.

For propagation perpendicular to the field, magnetic tension has no component along the wave normal. The Alfvén normal speed and slow speed vanish, while the fast speed is

$$c_f^2=c_s^2+v_A^2-c_s^2v_A^2.$$

This expression contains two useful limits. If $b^2\to0$, then $v_A\to0$ and $c_f\to c_s$, recovering relativistic hydrodynamics. If $c_s\to0$, then $c_f\to v_A$, recovering a cold magnetic compression. Because

$$1-c_f^2=(1-c_s^2)(1-v_A^2),$$

causal material sound and Alfvén speeds imply $c_f\leq1$ in the perpendicular case.

These are called Type I (perpendicular) and Type II (parallel) degeneracies. A repeated eigenvalue need not mean that physical hyperbolicity has failed: generic eigenvector formulae can become zero or singular even when a regular basis exists. The constrained seven-field system must be distinguished from augmented systems carrying extra constraint modes; see Martí and Müller 2015, §§ 3.2.2 and 8.2.2–8.2.3.

Hyperbolicity and causal bounds

For a thermodynamically admissible state with $0\lt c_s^2\lt1$ and positive enthalpy, the physical ideal-RMHD system has real characteristics and a complete eigenbasis, including appropriately regularized magnetic degeneracies. The pressureless endpoint $c_s=0$ needs separate care: coincident speeds alone do not prove strong hyperbolicity there.

The full angular polynomial supplies a useful causal check. Its value at $z=1$ is $(1-c_s^2)(1-v_A^2)\geq0$, and its vertex lies at $A/2\leq1$. Its two real, nonnegative roots therefore lie at or below one. Evaluating it at $z=v_A^2\cos^2\vartheta$ gives

$$-v_A^4(1-c_s^2)\sin^2\vartheta\cos^2\vartheta\leq0,$$

so the Alfvén squared normal speed lies between the slow and fast roots. A superluminal computed root can signal an acausal equation-of-state derivative, a primitive-state error, or an inconsistent formula.

Hyperbolicity is local. It does not say that a smooth solution remains smooth: shocks and current sheets still form. It also does not ensure that the ideal closure is valid below a collision length or inside a reconnection layer. Finite conductivity and kinetic effects can add relaxation or dispersive modes absent from ideal RMHD.

The characteristic normal is itself part of the input. Boundary classification uses speeds relative to the boundary normal; rotating the normal changes the field angle and therefore the incoming families. At a sub-fast outflow, some characteristics can enter the domain and require physical boundary data. Prescribing every primitive variable there overdetermines the continuum problem.

Numerical use and failure modes

Approximate Riemann solvers often need only lower and upper bounds on the fast speeds, while characteristic reconstruction needs eigenvectors. Bounding speeds too narrowly violates the domain of dependence and can destabilize the scheme; bounds that are too wide add diffusion and require a correspondingly smaller Courant-limited timestep. Degenerate states expose formulae containing divisions by a vanishing tangential field or eigenvalue separation.

Independent implementation checks should sweep field angle from parallel to perpendicular and field strength from hydrodynamic to magnetically dominated. At each state, compare polynomial residuals, verify the symmetry of opposite waves in the comoving frame, order slow below Alfvén below fast, and check $|\lambda|\leq1$. Continuity through degeneracies is more informative than agreement at one generic state. Martí and Müller 2015, §§ 8.2.3–8.2.4 gives regular eigenvectors.

Exercises

Use the perpendicular fast-speed formula to prove that the mode is causal whenever $0\leq c_s^2\leq1$ and $0\leq v_A^2\leq1$.

Solution

Rewrite the complement as

$$1-c_f^2=1-c_s^2-v_A^2+c_s^2v_A^2 =(1-c_s^2)(1-v_A^2).$$

Both factors are nonnegative, so $c_f^2\leq1$. The original expression is also nonnegative, hence $0\leq c_f^2\leq1$.

References

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