Fuel-Supply-limited Stellar Relaxation Oscillations: Application to Multiple Rings around Asymptotic Giant Branch Stars and Planetary Nebulae

We assemble the equations of general relativistic magnetohydrodynamics (MHD) in 3 + 1 form. These consist of the complete coupled set of Maxwellʹs equations for the electromagnetic field, Einsteinʹs 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.