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We investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.

The classical theory of black holes predicts that nothing, including light, could escape from the black holes. However, Stephen Hawking first showed that quantum effects could allow black holes to emit particles. The formula of Hawking temperature was first given in the frame of quantum field theory [

On the other hand, various theories of quantum gravity, such as string theory, loop quantum gravity, and quantum geometry, predict the existence of a minimal length [

An effective model of the GUP in one-dimensional quantum mechanics is given by [

The black hole is a suitable venue to discuss the effects of quantum gravity. Incorporating GUP into black holes has been discussed in a lot of papers [

In this paper, we investigate scalars and fermions tunneling across the horizons of black holes using the deformed Hamilton-Jacobi method which incorporates the minimal length via (

The organization of this paper is as follows. In Section

To be generic, we will consider a spherically symmetric background metric of the form

In the

Similarly, the deformed Dirac equation for a spin-

In order to generalize the deformed Hamilton-Jacobi equation, (

In this section, we investigate the particles’ tunneling at the event horizon

However, when one tries to calculate the tunneling rate

For the standard Hawing radiation, all particles very close to the horizon are effectively massless on account of infinite blueshift. Thus, the conformal invariance of the horizon make Hawing temperatures of all particles the same. The mass, angular momentum, and identity of the particles are only relevant when they escape the potential barrier. However, if quantum gravity effects are considered, behaviors of particles near the horizon could be different. For example, if we send a wave packet which is governed by a subluminal dispersion relation backwards in time toward the horizon, it reaches a minimum distance of approach and then reverses direction and propagate back away from the horizon, instead of getting unlimited blueshift toward the horizon [

For simplicity, we consider the Schwarzschild metric with

For particles emitted in a wave mode labelled by energy

The GUP is closely related to noncommutative geometry. In fact, when the GUP is investigated in more than one dimension, a noncommutative geometric generalization of position space always appears naturally [

In this paper, incorporating effects of the minimal length, we derived the deformed Hamilton-Jacobi equations for both scalars and fermions in curved spacetime based on the modified fundamental commutation relations. We investigated the particles’ tunneling in the background of a spherically symmetric black hole. In this spacetime configurations, we showed that the corrected Hawking temperature is not only determined by the properties of the black holes, but also dependent on the angular momentum and mass of the emitted particles. Finally, we studied how a Schwarzschild black hole evaporates in our model. We found that the black hole evaporates to zero mass in infinite time.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge useful discussions with Y. He, Z. Sun, and H. W. Wu. This work is supported in part by NSFC (Grant nos. 11005016, 11175039, and 11375121) and SYSTF (Grant no. 2012JQ0039).