Magnetic braking in differentially rotating, relativistic stars

We study the magnetic braking and viscous damping of differential rotation in incompressible, uniform density stars in general relativity. Differentially rotating stars can support significantly more mass in equilibrium than nonrotating or uniformly rotating stars, according to general relativity. The remnant of a binary neutron star merger or supernova core collapse may produce such a “hypermassive” neutron star. Although a hypermassive neutron star may be stable on a dynamical time scale, magnetic braking and viscous damping of differential rotation will ultimately alter the equilibrium structure, possibly leading to delayed catastrophic collapse. Here we treat the slow-rotation, weak-magnetic field limit in which [Formula Presented] where [Formula Presented] is the rotational kinetic energy, [Formula Presented] is the magnetic energy, and W is the gravitational binding energy of the star. We assume the system to be axisymmetric and solve the magnetohydrodynamics (MHD) equations in both Newtonian gravitation and general relativity. For an initially uniform magnetic field parallel to the rotation axis in which we neglect viscosity, the Newtonian case can be solved analytically, but the other cases we consider require a numerical integration. Toroidal magnetic fields are generated whenever the angular velocity varies along the initial poloidal field lines. We find that the toroidal fields and angular velocities oscillate independently along each poloidal field line, which enables us to transform the original 2+1 equations into 1+1 form and solve them along each field line independently. The incoherent oscillations on different field lines stir up turbulent-like motion in tens of Alfvén time scales (“phase mixing”). In the presence of viscosity, the stars eventually are driven to uniform rotation, with the energy contained in the initial differential rotation going into heat. Our evolution calculations serve as qualitative guides and benchmarks for future, more realistic MHD simulations in full 3+1 general relativity.