A GRMHD primer

When one mentions GRMHD, the system under consideration involves electrically conducting fluids moving at high speeds in the presence of strong gravity, ie. we are looking at a combination of Euler's equations (for inviscid fluids), Maxwell's equations and Einstein's field equations. An added simplification is when the fluid is a perfect electrical conductor. This approximation is known as ideal MHD and can be equivalently described by the absence of electric fields in the frame of the fluid. We list the set of governing equations for GRMHD along with analogous non-relativistic equations,

Conservation Law	Equation	Non-relativistic equivalent law	Equation
Conservation of particle number		Conservation of mass
Conservation of energy and momentum		Conservation of momentum Conservation of energy
Maxwell's equation		Maxwell's equations (2)

where n is the particle number density, u^μ is the four-velocity of the fluid, T^μ_ν is the MHD stress-energy tensor for a perfect fluid (inviscid, no shear stresses and zero heat conductivity) and contains information on the fluid's momentum and energy, F^*μν is Maxwell's tensor whose components are the electric and magnetic field components (E and B respectively). The MHD stress-energy tensor for a perfect fluid takes the form,

where ρ is the fluid density in its rest frame (also called 'fluid frame'), u is the fluid internal energy, p is fluid/gas pressure, b^µ is the magnetic induction four vector defined as where ϵ is the Levi-Civita tensor, and . Physically, the spatial part of b^µ corresponds to the magnetic field in the fluid frame, Bⁱ.

Some comments about Maxwell's equations:

(i) The equation written above in the table above implies divergence-free nature of the magnetic fields, . This is because *F^μν is completely antisymmetric which implies *F⁰⁰ = 0, which in turn implies the no-monopoles requirment. This is a constraint equation and numerical schemes do not naturally enforce it. Hence, an additional component of GRMHD codes or MHD codes for that matter, is divergence cleaning. More about this here.

(ii) The remaining two Maxwell's equations allow computation of the four-current, . iharm uses a central difference scheme to compute the derivatives of the electromagnetic field tensor. It calculates the state of the primitives at half timestep to obtain the time derivatives. For this reason, it is not possible to compute j^µ solely from the primitives stored in the dump file and iharm outputs j^µ (jcon). current.c handles this.

To complete the set of equations above, one needs an additional equation that relates the state variables. This is known as the equation of state and iharm uses an ideal gas equation of state,