  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. 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
timescale, magnetic braking and viscous damping of differential rotation will
ultimately alter the equilibrium structure, possibly leading to delayed
catastrophic collapse. Here we consider the slow-rotation, weak-magnetic field
limit in which E_rot << E_mag << W, where E_rot is the rotational kinetic
energy, E_mag is the magnetic energy, and W is the gravitational binding energy
of the star. We assume the system to be axisymmetric and solve the MHD
equations in both Newtonian gravitation and general relativity. 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 Alfven timescales ("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.
