In a new classification of merging binary neutron stars ( NSs ) we separate short gamma-ray bursts ( GRBs ) in two sub-classes . The ones with E _ { iso } \lesssim 10 ^ { 52 } erg coalesce to form a massive NS and are indicated as short gamma-ray flashes ( S-GRFs ) . The hardest , with E _ { iso } \gtrsim 10 ^ { 52 } erg , coalesce to form a black hole ( BH ) and are indicated as genuine short-GRBs ( S-GRBs ) . Within the fireshell model , S-GRBs exhibit three different components : the P-GRB emission , observed at the transparency of a self-accelerating baryon- e ^ { + } e ^ { - } plasma ; the prompt emission , originating from the interaction of the accelerated baryons with the circumburst medium ; the high-energy ( GeV ) emission , observed after the P-GRB and indicating the formation of a BH . GRB 090510 gives the first evidence for the formation of a Kerr BH or , possibly , a Kerr-Newman BH . Its P-GRB spectrum can be fitted by a convolution of thermal spectra whose origin can be traced back to an axially symmetric dyadotorus . A large value of the angular momentum of the newborn BH is consistent with the large energetics of this S-GRB , which reach in the 1 – 10000 keV range E _ { iso } = ( 3.95 \pm 0.21 ) \times 10 ^ { 52 } erg and in the 0.1 – 100 GeV range E _ { LAT } = ( 5.78 \pm 0.60 ) \times 10 ^ { 52 } erg , the most energetic GeV emission ever observed in S-GRBs . The theoretical redshift z _ { th } = 0.75 \pm 0.17 that we derive from the fireshell theory is consistent with the spectroscopic measurement z = 0.903 \pm 0.003 , showing the self-consistency of the theoretical approach . All S-GRBs exhibit GeV emission , when inside the Fermi -LAT field of view , unlike S-GRFs , which never evidence it . The GeV emission appears to be the discriminant for the formation of a BH in GRBs , confirmed by their observed overall energetics .