In a new classification of merging binary neutron stars (NSs) we separate short gamma-ray bursts (GRBs) in two sub-classes. The ones with $E_{iso}lesssim10^{52}$ erg coalesce to form a massive NS and are indicated as short gamma-ray flashes (S-GRFs). The hardest, with $E_{iso}gtrsim10^{52}$ 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 $E_{iso}=(3.95pm0.21)times10^{52}$ erg and in the 0.1--100 GeV range $E_{LAT}=(5.78pm0.60)times10^{52}$ erg, the most energetic GeV emission ever observed in S-GRBs. The theoretical redshift $z_{th}=0.75pm0.17$ that we derive from the fireshell theory is consistent with the spectroscopic measurement $z=0.903pm0.003$, 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.
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