General-Relativistic Magnetohydrodynamic Equations: the bare essential

TL;DR

This paper provides a concise overview of the fundamental equations of general relativistic magnetohydrodynamics used in numerical simulations of extreme astrophysical phenomena like neutron star mergers and black hole accretion.

Contribution

It offers a clear, comprehensive summary of the essential equations and their conservative formulation in a 3+1 spacetime split for modern numerical codes.

Findings

Summarizes the set of equations used in GRMHD simulations.

Highlights the importance of conservative formulation in numerical codes.

Facilitates understanding of plasma dynamics in curved spacetimes.

Abstract

Recent years have seen a significant progress in the development of general relativistic codes for the numerical solution of the equations of magnetohydrodynamics in spacetimes with high and dynamical curvature. These codes are valuable tools to explore the large-scale plasma dynamics such as that takes place when two neutron stars collide or when matter accretes onto a supermassive black hole. This chapter is meant to provide a very brief but complete overview of the set of equations that are normally solved in modern numerical codes after they are cast into a conservative formulation within a 3+1 split of spacetime.