We investigate the acceleration of electrons and positrons by magnetic-field-aligned electric fields in the polar funnel of an accreting black hole (BH). Applying the pulsar outer-gap theory to BH magnetospheres, we find that such a lepton accelerator arises in the immediate vicinity of the event horizon due to frame-dragging, and that their gamma-ray luminosity increases with decreasing accretion rate. Furthermore, we demonstrate that the gamma-ray flux is enhanced along the rotation axis by more than an order of magnitude if the BH spin increases from $a=0.90M$ to $a=0.9999M$. As a result, if a ten-solar-mass, almost-maximally rotating BH is located within 3 kpc, when its accretion rate is between 0.005% and 0.01% of the Eddington rate, its high-energy flare becomes detectable with the Fermi/Large Area Telescope, provided that the flare lasts longer than 1.2 months and that we view the source nearly along the rotation axis. In addition, its very-high-energy flux is marginally detectable with the Cherenkov Telescope Array, provided that the flare lasts longer than a night and that our viewing angle is about 45 degrees with respect to the rotation axis.
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