Relativistic Hydrodynamics, GRMHD, and Radiation

Relativistic matter computation couples conservative shocks, nonlinear primitive recovery, tabulated microphysics, magnetic divergence control, approximate radiation transport, and an artificial or alternative treatment of vacuum. Each layer can fail while the others appear stable. This chapter therefore proceeds from the fluid update outward through closure and regularization, using analytic waves, shock tubes, stationary stars, accretion flows, tori, and transport limits to isolate errors before a full astrophysical application.

Helpful background. Formulations of General Relativity supplies spacetime coupling, Ordinary Differential Equations supplies evolution and nonlinear solves, Relativistic Matter and Compact Objects supplies microphysics, Computational Relativity supplies shared evidence practices, and Fluid and Kinetic Theory supplies fluid, MHD, and kinetic balance laws.

Enter this chapter

The common evidence tuple records equations and variables, EOS or reaction data with versions and checksums, reconstruction and fluxes, nonlinear tolerances, floors and repairs, magnetic control, transport closure, grid and boundaries, environment, and code revision. Every nonconservative operation is accumulated in mass, energy, momentum, and composition budgets. Smooth convergence and discontinuous benchmarks are both necessary because a shock-capturing scheme has different accuracy regimes.

Font 2008, §§2–4 connects conservative relativistic hydrodynamics, primitive recovery, MHD, and standard tests. These pages narrow their claims to the declared numerical and microphysical ranges rather than treating one shock tube or equilibrium as universal validation.

Route through the six pages

Conservative Relativistic Hydrodynamics defines densitized balance laws, reconstruction, Riemann fluxes, geometric sources, shock tubes, and integrated budgets. Primitive-Variable Recovery then stress-tests the inverse map over Lorentz factor, magnetization, EOS range, valid states, and deliberately invalid conserved inputs.

Equations of State and Microphysics Interfaces verifies units, interpolation, derivatives, first-law and Maxwell identities, sound speed, phase boundaries, and range policy. Divergence Control in Relativistic MHD compares constrained transport, vector potentials, and cleaning with native and independent magnetic-divergence measures.

Radiation and Neutrino Transport compares leakage, moment, Monte Carlo, and hybrid strategies with a dated evidence cutoff of 2026-08-18 and a crossing-beam counterexample. Atmospheres, Vacuum Interfaces, and Tests closes the chapter by treating floor resets as measured source terms and requiring a joint floor–resolution limit for low-density observables.

Cross-layer benchmarks

A smooth sound or Alfvén wave tests designed convergence and EOS derivatives. Relativistic shock tubes test wave speeds, reconstruction, Riemann solvers, primitive recovery, and physical-state preservation. TOV stars test geometric sources, surfaces, atmosphere interaction, and long-term stationarity. Bondi flow and magnetized tori test boundaries, low-density funnels, accretion budgets, and divergence control.

Radiation adds diffusion, free streaming, moving matter, equilibrium, and crossing beams. Foucart 2023, §§4.1–4.3 and §5.2 explains why leakage, moments, and Monte Carlo have different advantages and limitations. Model spread is kept separate from grid, packet, and tolerance uncertainty.

Independent checks change mechanism: exact Riemann solutions, a forward primitive round trip, analytic EOS queries, a second divergence operator, Monte Carlo against a moment closure, or a floor-free method. Commands on child pages describe expected rerun interfaces; they do not claim an absent implementation executed.

What matter simulations can establish

Ideal GRMHD does not validate resistive, kinetic, or subgrid plasma physics. A table is valid only in its declared thermodynamic domain. M1 cannot resolve arbitrary beam crossings. An atmosphere sets a magnetization and ejecta sensitivity floor. A stable bulk remnant can coexist with unresolved composition, luminosity, or low-mass ejecta, so every claim names its observable.

Review the chapter

Diagnostic. A merger conserves baryon mass to $10^{-6}$ in the flux update, but atmosphere resets inject $10^{-3}$ of the reported ejecta mass and transport choices shift composition by twenty percent. Which result is validated?

Solution

The flux update’s internal conservation may be verified, but the problem does not state an ejecta or composition accuracy target. A reset contribution of $10^{-3}$ of the ejecta is a $0.1\%$ effect, which may or may not matter at that target; its indirect dynamical effect also needs a floor study. A twenty-percent transport dependence is a separate modeling sensitivity in composition. Quote each uncertainty against its own observable, include reset corrections, and separate numerical refinement from transport-model variation. Bulk baryon conservation alone validates neither ejecta classification nor composition.

Exercises

For an ideal gas with $\Gamma=5/3$, $\rho=1$, $p=0.2$, and vanishing magnetic field, calculate the specific internal energy, enthalpy, and relativistic adiabatic sound speed. Compare with the Newtonian sound-speed formula at the same pressure and density.

Solution

The EOS gives $\epsilon=p/[(\Gamma-1)\rho]=0.3$, and $h=1+\epsilon+p/\rho=1.5$. Hence $c_s^2=\Gamma p/(\rho h)=2/9$ and $c_s=\sqrt2/3\simeq0.4714$ in $c=1$ units. The Newtonian expression gives $c_{s,\mathrm N}^2=\Gamma p/\rho=1/3$. The difference comes from relativistic inertia: enthalpy includes internal energy and pressure. Agreement is recovered when both become small compared with rest-mass energy, so $h\to1$.

References

• Font, J. A. (2008). “Numerical hydrodynamics and magnetohydrodynamics in general relativity.” Living Reviews in Relativity 11, 7. DOI.
• Foucart, F. (2023). “Neutrino transport in general relativistic neutron star merger simulations.” Living Reviews in Computational Astrophysics 9, 1. DOI.