Keywords: Riemann invariants, black holes. We discuss the Kretschmann, Chern-Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman-Penrose formalism and in the framework of gravitoelectromagnetism, using the Kerr-Newman geometry as an example. An analogy with electromagnetic invariants leads to the definition of regions of gravitoelectric or gravitomagnetic dominance.

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We propose that the inner engine of a type I binary-driven hypernova (BdHN) is composed of a Kerr black hole (BH) in a non-stationary state, embedded in a uniform magnetic field B 0 aligned with the BH rotation axis, and surrounded by an ionized plasma of extremely low density of 10 −14 g cm −3 . Using GRB 130427A as a prototype we show that this inner engine acts in a sequence of elementary impulses. Electrons are accelerated to ultra-relativistic energy near the BH horizon and, propagating along the polar axis, θ = 0, they can reach energies of ∼ 10 18 eV, and partially contribute to ultra-high energy cosmic rays (UHECRs). When propagating with θ = 0 through the magnetic field B 0 they give origin by synchrotron emission to GeV and TeV radiation. The mass of BH, M = 2.3M , its spin, α = 0.47, and the value of magnetic field B 0 = 3.48 × 10 10 G, are determined self-consistently in order to fulfill the energetic and the transparency requirement. The repetition time of each elementary impulse of energy E ∼ 10 37 erg, is ∼ 10 −14 s at the beginning of the process, then slowly increasing with time evolution. In principle, this "inner engine" can operate in a GRB for thousands of years. By scaling the BH mass and the magnetic field the same "inner engine" can describe active galactic nuclei (AGN).