Abstract. The standard Hawking formula predicts the complete evaporation of black holes.In this paper, we introduce effects of quantum gravity into fermions' tunneling from ReissnerNordstrom and Kerr black holes. The quantum gravity effects slow down the increase of Hawking temperatures. This property naturally leads to a residue mass in black hole evaporation. The corrected temperatures are affected by the quantum numbers of emitted fermions. Meanwhile, the temperature of the Kerr black hole is a function of θ due to the rotation.arXiv:1307.0172v2 [gr-qc]

In this review, we discuss effects of quantum gravity on black hole physics. After a brief review of the origin of the minimal observable length from various quantum gravity theories, we present the tunneling method. To incorporate quantum gravity effects, we modify the Klein-Gordon equation and Dirac equation by the modified fundamental commutation relations. Then we use the modified equations to discuss the tunneling radiation of scalar particles and fermions. The corrected Hawking temperatures are related to the quantum numbers of the emitted particles. Quantum gravity corrections slow down the increase of the temperatures. The remnants are observed as M Res. The mass is quantized by the modified Wheeler-DeWitt equation and is proportional to n in quantum gravity regime. The thermodynamical property of the black hole is studied by the influence of quantum gravity effects.

In this paper, using Hamilton-Jacobi method, we address the tunnelling of fermions in a 4dimensional Schwarzschild spacetime. Base on the generalized uncertainty principle, we introduce the influence of quantum gravity. After solving the equation of motion of the spin 1/2 field, we derive the corrected Hawking temperature. It turns out that the correction depends not only on the black hole's mass but also on the mass (energy) of emitted fermions. It is of interest that, in our calculation, the quantum gravity correction decelerates the temperature increase during the radiation explicitly. This observation then naturally leads to the remnants in black hole evaporation.Our calculation shows that the residue mass is M p /β 0 , where M p is the Planck mass and β 0 is a dimensionless parameter accounting for quantum gravity effects. The evaporation singularity is then avoided.

We investigate the thermodynamic behavior of nonlinear electrodynamics (NLED) black holes in an extended phase space, which includes the cosmological constant Λ = −3/l 2 and dimensionful couplings a in NLED as thermodynamic variables. For a generic NLED black hole with the charge Q, we find that the Smarr relation is satisfied in the extended phase space, and the state of equation can be written as T l =T (r + /l, Q/l, al −c ), where [a] = L c , and T and r + are the temperature and horizon radius of the black hole, respectively. For some values of Q/l and al −c , the phase structure of the black hole is uniquely determined. Focusing on Born-Infeld and iBorn-Infeld AdS black holes, we obtain the corresponding phase diagrams in the a/l 2 -Q/l plane, which provides a new viewpoint towards the black holes' phase structure and critical behavior. For Born-Infeld black holes, the critical line and the region, where a reentrant phase transition occurs, in the a/l 2 -Q/l plane are both finite and terminate at ã c ,Q c ≃ {0.069, 0.37}. However for iBorn-Infeld black holes, the critical line and the reentrant phase transition region in the a/l 2 -Q/l plane are semiinfinite and extend to Q/l = ∞. We also examine thermal and electrical stabilities of Born-Infeld and iBorn-Infeld AdS black holes. * Electronic address: pengw@scu.edu.cn † Electronic address: iverwu@scu.edu.cn ‡ Electronic address: hyanga@scu.edu.cn Recently, a new type of NLED black holes, namely iBorn-Infeld AdS (iBI-AdS) black holes, have been considered in [49] as holographic models behaving as prototypes of Mott insulators. The Lagrangian of the iBorn-Infeld field can be obtained from that of the Born-Infeld field by extending the BI parameter (a in eqn. (57)) to a negative real number. In [50], it showed that the nonlinearity correction tends to reduce/increase the strength of the repulsive force between two electrons for the Born-Infeld/iBorn-Infeld field. So it is natural to expect that the iBI-AdS black hole is dual to a theory with strong interactions between electrons, which could lead to Mott-like behavior. Moreover, negative magneto-resistance and the Mott insulator to metal transition induced by a magnetic field can be realized at low temperatures in the iBorn-Infeld holographic models. Compared to the Born-Infeld case, the iBorn-Infeld case leads to a much richer transport behavior in the dual theory.As shown in [49], iBI-AdS black holes satisfy the constraints to ensure consistency in the form of ghosty perturbations and/or gradient instabilities at the decoupling limit. However, the thermodynamic behavior and phase structure of iBI-AdS black holes have yet to be discussed.In this paper, we first investigate the thermodynamic behavior of generic NLED black holes in the extended phase space. Then, we turn to study the phase structure and critical behavior of BI-AdS and iBI-AdS black holes by studying the phase diagrams in the Q/l-a/l 2 plane. The rest of this paper is organized as follows. In section II, we derive the NLED black hole solution, com...