  Short Gamma-Ray Bursts (SGRBs) are among the most luminous explosions in the
universe, releasing in less than one second the energy emitted by our Galaxy
over one year. Despite decades of observations, the nature of their
"central-engine" remains unknown. Considering a binary of magnetized neutron
stars and solving Einstein equations, we show that their merger results in a
rapidly spinning black hole surrounded by a hot and highly magnetized torus.
Lasting over 35 ms and much longer than previous simulations, our study reveals
that magnetohydrodynamical instabilities amplify an initially turbulent
magnetic field of ~ 10^{12} G to produce an ordered poloidal field of ~ 10^{15}
G along the black-hole spin-axis, within a half-opening angle of ~ 30 deg,
which may naturally launch a relativistic jet. The broad consistency of our
ab-initio calculations with SGRB observations shows that the merger of
magnetized neutron stars can provide the basic physical conditions for the
central-engine of SGRBs.
