We study of the collapse of a magnetized spherical star to a black hole in general relativity theory. The matter and gravitational fields are described by the exact Oppenheimer-Snyder solution for the collapse of a spherical, homogeneous dust ball. We adopt a ''dynamical Cowling approximation'' whereby the matter and the geometry (metric), while highly dynamical, are unaffected by the electromagnetic fields. The matter is assumed to be perfectly conducting and threaded by a dipole magnetic field at the onset of collapse. We determine the subsequent evolution of the magnetic and electric fields without approximation; the fields are determined analytically in the matter interior and numerically in the vacuum exterior. We apply junction conditions to match the electromagnetic fields across the stellar surface. We use this model to experiment with several coordinate gauge choices for handling spacetime evolution characterized by the formation of a black hole and the associated appearance of singularities. These choices range from ''singularity avoiding'' time coordinates to ''horizon penetrating'' time coordinates accompanied by black hole excision. The later choice enables us to integrate the electromagnetic fields arbitrarily far into the future. At late times the longitudinal magnetic field in the exterior has been transformed into a transverse electromagnetic wave; part of the electromagnetic radiation is captured by the hole and the rest propagates outward to large distances. The solution we present for our simple scenario can be used to test codes designed to treat more general evolutions of relativistic MHD fluids flowing in strong gravitational fields in dynamical spacetimes.
We have solved for the evolution in general relativity of the electromagnetic field of a magnetic star that collapses from rest to a Schwarzschild black hole. We adopted a ''dynamical Cowling'' model by which the matter and gravitational field are prescribed by the Oppenheimer-Snyder solution for spherical collapse. Starting from Maxwell's equations in 3+1 form, we solved for the magnetic and electric fields both in the vacuum exterior and the stellar interior. We assumed that the initial star was threaded by a dipole magnetic field and that the interior behaves as an MHD fluid. We showed how the electromagnetic junction conditions could be applied at the stellar surface in order that the fields in the MHD interior, which were determined analytically, could be matched to the fields in the vacuum exterior. The exterior fields were evolved numerically.
To gain experience with solving Maxwell's equations numerically in a dynamical spacetime containing both matter and a black hole, we solved the above problem in three different coordinate gauges: Schwarzschild, maximal and Kerr-Schild. While the first two coordinate systems have singularity avoiding properties, they both lead to growing numerical inaccuracies in the exterior as the surface approaches the horizon (in the case of Schwarzschild time slicing) or a limit surface inside the horizon (in the case of maximal slicing) at late times. Kerr-Schild slicing is horizon penetrating and does not suffer from grid stretching; accordingly this slicing enables us to apply excision boundary conditions once the star collapses inside the horizon. Adopting Kerr-Schild slicing coordinates with black hole excision allowed us to integrate Maxwell's equations to arbitrary late times. We followed the transition in the exterior of a quasi-static, longitudinal magnetic field, which evolved during the collapse in to match the growing frozen-in interior field, to a transverse electromagnetic wave. By the end of the simulation, the electromagnetic field energy in the near zone outside the black hole had either been captured by the hole or radiated away to large distances, in accord with the ``no-hair theorem''. At late times we recovered the expected power-law decay of the dipole fields with time, as well as the expected wavelength of the ringdown radiation.
There exist a number of very important astrophysical problems which require the solution of the coupled Maxwell-Einstein-MHD equations in a strong, dynamical gravitational field. For example, learning the outcome of rotating stellar core or supermassive star collapse, the mechanism for gamma-ray bursts, and/or the fate of the remnant of a binary neutron star merger may all require solving this coupled system of equations. The solution we have presented for our simple collapse scenario should serve as a preliminary guide to the design and testing of more sophisticated codes capable of handling these more complicated scenarios.