We study magnetohydrodynamic ( MHD ) effects arising in the collapse of magnetized , rotating , massive stellar cores to proto-neutron stars ( PNSs ) .
We perform axisymmetric numerical simulations in full general relativity with a hybrid equation of state .
The formation and early evolution of a PNS are followed with a grid of 2500 \times 2500 zones , which provides better resolution than in previous ( Newtonian ) studies .
We confirm that significant differential rotation results even when the rotation of the progenitor is initially uniform .
Consequently , the magnetic field is amplified both by magnetic winding and the magnetorotational instability ( MRI ) .
Even if the magnetic energy E _ { EM } is much smaller than the rotational kinetic energy T _ { rot } at the time of PNS formation , the ratio E _ { EM } / T _ { rot } increases to 0.1–0.2 by the magnetic winding .
Following PNS formation , MHD outflows lead to losses of rest mass , energy , and angular momentum from the system .
The earliest outflow is produced primarily by the increasing magnetic stress caused by magnetic winding .
The MRI amplifies the poloidal field and increases the magnetic stress , causing further angular momentum transport and helping to drive the outflow .
After the magnetic field saturates , a nearly stationary , collimated magnetic field forms near the rotation axis and a Blandford-Payne type outflow develops along the field lines .
These outflows remove angular momentum from the PNS at a rate given by \dot { J } \sim \eta E _ { EM } C _ { B } , where \eta is a constant of order \sim 0.1 and C _ { B } is a typical ratio of poloidal to toroidal field strength .
As a result , the rotation period quickly increases for a strongly magnetized PNS until the degree of differential rotation decreases .
Our simulations suggest that rapidly rotating , magnetized PNSs may not give rise to rapidly rotating neutron stars .