General relativistic magnetohydrodynamics for the numerical construction of dynamical spacetimes

We assemble the equations of general relativistic magnetohydrodynamics (MHD) in 3 + 1 form. These consist of the complete coupled set of Maxwell's equations for the electromagnetic field, Einstein's equations for the gravitational field, and the equations of relativistic MHD for a perfectly conducting ideal gas. The adopted form of the equations is suitable for evolving numerically a relativistic MHD fluid in a dynamical spacetime characterized by a strong gravitational field.