Go to previous page Go up Go to next page

3 Equations of General-Relativistic Ideal Magnetohydrodynamics

General-relativistic MHD is concerned with the dynamics of relativistic, electrically-conducting fluids (plasma) in the presence of magnetic fields. Here, we concentrate on purely ideal GRMHD, neglecting the presence of viscosity and heat conduction in the limit of infinite conductivity, i.e., the fluid is assumed to be a perfect conductor (see Section 3.3 for a brief discussion of the validity of this approximation). Like the GRHD equations discussed in the preceding Section 2, the GRMHD equations can also be cast in first-order, flux-conservative, hyperbolic form that is well suited to numerical work. Concerning this issue, the contribution of Anile [15Jump To The Next Citation Point] is remarkable (but see also [220] for an in-depth analysis of these equations), as he carried out a comprehensive study of the mathematical structure of the GRMHD equations. In order to analyze the hyperbolicity of the equations Anile found it convenient to write those equations using the following set of covariant variables V = (uμ,bμ,p, s), where bμ is the magnetic-field four-vector in the fluid rest frame and s is the specific entropy. With respect to these Anile variables, the system of GRMHD equations can be written as a quasi-linear system of the form μA B 𝒜 B ∇ μV = 0, where the indices A and B run from zero to nine, as the number of variables. The particular form of the 10 × 10 matrices 𝒜 μ can be found in [15Jump To The Next Citation Point]. The eigenvalue structure of the system can be obtained by considering a generic characteristic hypersurface of the previous quasi-linear system, μ φ (x ) = 0, such that it defines a characteristic matrix μ 𝒜 φ μ, whose determinant must vanish, with μ φμ ≡ ∂ φ∕∂x. If we now consider a wave propagating in an arbitrary direction x with a speed λ, the normal to the characteristic hypersurface is given by the (spacelike) 4-vector φμ = (− λ,1,0, 0), which can be substituted in the determinant of the characteristic matrix to obtain the corresponding characteristic polynomial, whose zeroes give the characteristic speed of the waves propagating in the x-direction. Three different kinds of waves can be obtained according to which factor in the equation resulting from the condition det(𝒜 μφμ ) = 0 vanishes, either entropic waves, Alfvén waves, or magnetosonic waves. It is worth noting that in Anile’s study [15Jump To The Next Citation Point] both the entropy and Alfvén waves appear as double roots of the characteristic polynomial, a direct result of working with an augmented system of equations. Therefore, those nonphysical waves must be removed from the wave decomposition when building up a numerical scheme based upon such wave structure to solve the GRMHD equations.

In recent years there has been intense research on formulating and solving numerically the MHD equations in general-relativistic spacetimes, either background or dynamic [196Jump To The Next Citation Point86Jump To The Next Citation Point40149Jump To The Next Citation Point20Jump To The Next Citation Point201Jump To The Next Citation Point108Jump To The Next Citation Point366Jump To The Next Citation Point24Jump To The Next Citation Point286Jump To The Next Citation Point265Jump To The Next Citation Point91Jump To The Next Citation Point]. Both, artificial viscosity and HRSC schemes have been developed and most of the astrophysical applications previously attempted with no magnetic fields included are currently being revisited, namely for studies of gravitational collapse, neutron-star dynamics, black-hole accretion, and jet formation. We note, however, that all HRSC numerical approaches employed, with the exception of [201Jump To The Next Citation Point24Jump To The Next Citation Point], are based on central schemes (or incomplete Riemann solvers), for which the use of the full wave decomposition is not needed (see below).

In terms of the (Faraday) electromagnetic tensor μν F, Maxwell’s equations read

∇ ν ∗F μν = 0, ∇ νF μν = 𝒥 μ, (46 )
where μν μ ν ν μ μνλδ F = U E − U E − η U λBδ, its dual ∗ μν 1 μνλδ F = 2η Fλδ, and μνλδ 1 η = √−g-[μν λδ], where [μνλ δ] is the completely antisymmetric Levi–Civita symbol. E μ and B μ stand for the electric and magnetic fields measured by an observer with four-velocity U μ, and 𝒥 μ is the electric four-current, μ μ μν 𝒥 = ρqu + σF uν, where ρq is the proper charge density and σ is the electric conductivity.

 3.1 Numerical approaches
  3.1.1 Nonconservative formulations
  3.1.2 Conservative formulations
 3.2 Recovery of primitive variables
 3.3 Going further

  Go to previous page Go up Go to next page