Generalized uncertainty principles, effective Newton constant and the regular black hole

In this paper, the quantum spacetime with a running gravitational coupling is explored. Analyzing the gravity-induced quantum interference pattern and the Gedanken for weighting a photon, we find that the running Newton constant can be inspired by the generalized uncertainty principles. A characteristic momentum associated with the tidal effect is suggested, which incorporates the quantum effect with the geometric nature of gravity. When the simplest generalized uncertainty principle is considered, the minimal model of the regular black holes is reproduced by the effective Newton constant. The black hole's tunneling probability, accurate to the second order correction, is carefully analyzed. We find that the tunneling probability is regularized by the size of the black hole remnant. Moreover, for a given initial black hole, the remnant is the final state of a transition process that the probability is minimal. We also suggest a theory of modified gravity, by substituting the effective Newton constant into the Hilbert–Einstein action.