  It has recently become clear that in both short and long gamma-ray bursts
(GRBs) it coexists a sequence of different events, each characterized by
specific physical processes and corresponding values of the Lorentz gamma
factors. The ultra-relativistic prompt emission (UPE) phase, with Lorentz
factor $\Gamma\leq10^4$, is followed by a mildly relativistic plateau-afterglow
phase with $\Gamma\lesssim2$. The GeV radiation, with $\Gamma\lesssim50$,
coexists with the above two phases. It is shown that: a) the GeV radiation
originates at the onset of the formation of a black hole (BH), b) its
luminosity follows specific power-law dependence when measured in the rest
frame of the source with a decay index $\gamma=-1.29\pm0.06$ in the case of the
short GRBs, and $\gamma=-1.20\pm0.04$ in the case of the long GRBs, c) these
energetics requirements are used to derive the mass and spin of the BH
originating this extended GeV emission. We present these conceptual results for
the case of short GRBs in this article and give the extended analysis for long
GRBs in a companion article. A direct astrophysical application of these
results is that the merger of binary neutron stars leading to BH formation
emits GeV radiation: the GeV emission is a necessary and sufficient condition
to indicate the creation of a BH in S-GRBs.
