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.
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