Fermion's tunnelling with effects of quantum gravity

In this paper, using Hamilton-Jacobi method, we address the tunnelling of fermions in a 4-dimensional 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 $\gtrsim M_p/\beta_0$, where $M_p$ is the Planck mass and $\beta_0$ is a dimensionless parameter accounting for quantum gravity effects. The evaporation singularity is then avoided.


I. Introduction 2
II. Generalized Dirac equation in curved spacetime 4 III. Fermion tunnelling with effects of quantum gravity 6 IV. Discussion and conclusion 10 Acknowledgments 10 References 10

I. INTRODUCTION
Hawking radiation is described as a quantum tunnelling effects of particles at horizons of black holes [1][2][3][4][5][6]. With the consideration of the background variation in black hole evaporation, Parikh and Wilczek studied the tunnelling behaviors of massless scalar particles [3]. They derived the modified emission spectra for spherically symmetric black holes. The leading corrections to the Hawking temperature are found to be also dependent on the energy of emitted particles. This work was extended to massive and charged scalar particles. The Hawking radiation of general black holes was studied [4,5]. For an outgoing massive particle, the equation of motion is different from that of a massless particle. The trajectory of massless particles is a null geodesic. While the massive particle's motion satisfies de Broglie wave and is the phase velocity of outgoing particles. Subsequently, the tunnelling behaviors of fermions were carefully investigated with Hamilton-Jacobi method by Kerner and Mann [7].
Various theories of quantum gravity predict the existence of a minimum measurable length [13][14][15][16][17]. This length can be approached from the generalized uncertainty principle (GUP). Through the modified fundamental commutation relation [18] [ the expression of GUP is derived as ∆x∆p ≥ 2 [1 + β(∆p) 2 ], where β = β 0 /M 2 p . M p is the Planck mass. β 0 is a dimensionless parameter. From simple electroweak consideration, it is readily to find an upper limit β 0 < 10 34 . x i and p i are defined by x i = x 0i and p i = p 0i (1 + βp 2 ), respectively. x 0i and p 0j satisfy the canonical commutation relations [x 0i , p 0j ] = i δ ij . The modification of the fundamental commutation relation is not unique.
These modifications are widely applied to gain some information about the quantum properties of gravity. Black holes are effective modes to explore effects of quantum gravity.
Incorporating effects of quantum gravity into black hole physics by GUP, some interesting implications and results were achieved [22][23][24][25][26][27][28]. It showed in [22] that a small black hole is unstable. Moreover, the constraint for a large black hole comparable to the size of the cavity in connection with the critical mass is needed. The characteristic size in the absorption process, represented by the black hole irreducible mass, was gotten in [24]. The remnant mass and corrections to the area law and heat capacity were obtained in [25]. In [26], following Parikh-Wilczek tunnelling method, based on GUP, the radiation of massless scalar particles in the Schwarzschild black hole was discussed. The commutation relation between the radial coordinate and the conjugate momentum are modified with GUP. The authors treat the natural cutoffs as a minimal length, a minimal momentum and a maximal momentum.
They addressed the tunnelling rate of black holes. The corrected Hawking temperature was obtained and related to the energy of emitted particles.
The purpose of this paper is to investigate fermions' tunnelling behavior cross the event horizon of a 4-dimensional Schwarzschild black hole, where effects of quantum gravity are taken into account. We first modify the Dirac equation in curved spacetime to reflect the influence of quantum gravity. The model we adopt is the generalized uncertainty principle.
We use the Hamilton-Jacob method to solve the equation of motion of the spinor field. Then the tunnelling rate and Hawking temperature are calculated. Our results show that the quantum correction to the Hawking temperature is dependent not only on the black hole's mass but also on the mass and energy of emitted fermions. Moreover, the correction slows down the temperature increase during the evaporation. This in turn leads to the remnants in black hole evaporation and prevents the existence of the thermodynamics singularity.
The rest is organized as follows. In Sect.2, taking into account effects of quantum gravity, we modify Dirac equation in curved spacetime by GUP and get a generalized Dirac equation.
In Sect.3, the fermion tunnelling behavior in the Schwarzschild black hole is addressed and the corrected Hawking temperature is derived. Sect.4 is devoted to our discussion and conclusion.

II. GENERALIZED DIRAC EQUATION IN CURVED SPACETIME
To take into account the effects of quantum gravity, we adopt the generalized commutation relation in [18] to modify the Dirac equation. In Eq. (1), the momentum operators are defined by (2) The square of momentum operators is where in the last step, we only keep the leading order term of β. To account for the effects from quantum gravity, the frequency is generalized as [29] with the energy operator E = i ∂ 0 . Substituting the mass shell condition p 2 + m 2 = E 2 , we get the generalized expression of energy [30,32] The tunnelling of massless scalar particles in the Schwarzschild black hole was studied in detail and the corrected Hawking temperature was derived in [26]. On the other hand, Dirac equation with the consequence of GUP in flat spacetime has been investigated in [30].
We start with the Dirac equation in curved spacetime, where ω µ ab is the spin connection defined by the tetrad e λ b and ordinary connection The Latin indices live in the flat metric η ab while Greek indices are raised and lowered by the curved metric g µν . The tetrad can be constructed from Back in equation (6), Σ ab 's are the Lorentz spinor generators defined by Then one can construct the γ µ 's in curved spacetime as To get the generalized Dirac equation in curved spacetime, we rewrite eq. (6) as where i = 1, 2, · · · denotes the spatial coordinates. The left hand-side of the equation above is related to the energy. Using the generalized expression of energy eqn. (5) and the square of momentum operators eqn. (3), only keeping the leading order term of β, we get Therefore, the generalized Dirac equation in curved spacetime can be written as This is the equation we are going to solve in the next section.

III. FERMION TUNNELLING WITH EFFECTS OF QUANTUM GRAVITY
In this section, we address the tunnelling behavior of spin-1/2 fermions across the event horizon of the Schwarzschild black hole. Effects of quantum gravity are taken into account.
The metric is given by dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), with f (r) = g (r) = 1 − 2M r , and M is the black hole's mass. We have set G = c = 1. The event horizon is located at r h = 2M. The fermion's motion is determined by the generalized Dirac equation (13). For a spin-1/2 particle, there are two states corresponding respectively to spin up and spin down. Follow the standard ansatz, to describe the motion semi-classically, we assume the wave function of the spin up state as where A, B and I are functions of coordinates t, r, θ, φ, and I is the action of the emitted fermions. The process of spin down is the same as that of spin up. To solve eqn. (13), one should choose appropriate gamma matrices by exploiting eqn.s (8)- (10). It is straightforward to guess a tetrad for the metric (14) e µ a = diag √ f, 1/ √ g, r, r sin θ .
Then, our gamma matrices are given by with g θθ = 1 r , g φφ = 1 r sin θ . σ i 's are the Pauli matrices with i = 1, 2, 3.
Our task is to find the solutions of eqn. (13). First substitute the wave function eqn.
(15) and the matrices eqn. (16) in the generalized Dirac equation eqn. (13), and cancel the exponential factor. Since we are working with WKB approximation, the contributions from ∂A, ∂B and high orders of are neglected. We finally obtain decoupled four Hamilton-Jacobi To find the relevant solution, since the metric has a time-like killing vector, we perform the separation of variables as follows where ω turns out to be the energy of the emitted particle. We insert eqn. (21) into eqn.s (17)- (20) and first focus on the last two equations. They are identical after divided respectively by A and B and can be rewritten as follows In the equation above, the value in the square bracket can not vanish since β is a small quantity representing the effects from quantum gravity. Therefore, the expression in the round brackets is zero and yields the solution of Θ. In the previous work, though Θ has a complex solution (other than the trivial one Θ = constant) and gives rise to a contribution to the imaginary part of the action, it has no contribution to the tunnelling rate. Therefore eqn. (22) is simplified as After cancelling A and B, eqn. (17) and eqn. (18) are identical and give rise to with A 6 = β 2 g 3 f, Using eqn. (23), we find Q = 0. Neglecting the higher orders of β and solving the above equations at the event horizon yields [39], In the above equation, f = g = 1 − 2M r . The real part is irrelevant to the tunnelling rate. The +/− sign corresponds to outgoing/ingoing wave. Then the tunnelling rate [37] of the spin-1/2 fermion crossing the horizon is Γ = P (emission) P (absorption) = exp (−2 ImI + ) exp (−2 ImI − ) = exp (−2 ImW + − 2 ImΘ) exp (−2 ImW − − 2 ImΘ) This is the Boltzmann factor with Hawking temperature where T 0 = 1 8πM is the original Hawking temperature. It shows that there is a small correction to the Hawking temperature, and the correction value is dependent not only on the black hole's mass but also on the mass and energy of emitted fermions. This property has been obtained in literature. In [3], energy conservation is enforced by dynamical geometry and the tunnelling rate is found to be Γ = exp[−8πω(M − ω 2 )]. Then the corrected Hawking temperature is T = 1 8πM −4πω , where the leading correction to the Hawking temperature is related to the energy of emitted particles. To address effects of quantum gravity, the authors of [26,33] adopted the modified commutation relation between the radial coordinate and the conjugate momentum. They studied the quantum tunnelling of scalar particles in the Schwarzschild black hole. The tunnelling rate was derived as . Thus the correction to the Hawking temperature is also related to the black hole's mass and the particle's energy.
It is of interest to note that in eqn. (28), the quantum correction slows down the increase of the temperature during the radiation. This correction therefore causes the radiation ceased at some particular temperature, leaving the remnant mass. To estimate the residue mass, it is enough to consider massless particles. The temperature stops increasing when Then with the observation dM = ω and β = β 0 /M 2 p where M p is the Planck mass and β 0 < 10 34 [38] is a dimensionless parameter marking quantum gravity effects, we can get where we have assumed the maximal energy of the radiated particle is ω ≃ M p . This result is consistent with those obtained in [25,[34][35][36]. Compared with previous results, our calculation explicitly shows how the residue mass of black holes arises due to quantum gravity effects. The singularity of black hole evaporation is then prevented by the quantum gravity correction.

IV. DISCUSSION AND CONCLUSION
In this work, we modified the Dirac equation in curved spacetime to include the quantum gravity influence. To fulfill this purpose, we employed the generalized uncertainty principle model. This model is derived from the existence of minimal length which arises when combine quantum and gravity. We calculated the radiation of spin 1/2 particles in the 4-dimensional Schwarzschild spacetime with Hamilton-Jacob method. The tunnelling rate and Hawking temperature were presented.
We found that the quantum gravity correction is related not only to the black hole's mass but also to the mass (energy) of emitted fermions. More interestingly, our result shows that the quantum gravity correction explicitly retards the temperature rising in the black hole evaporation. Therefore, at some point during the evaporation, the quantum correction balances the traditional temperature rising tendency. This leads to the existence of the remnants. We showed that the remnants is M Res Mp β 0 , where M p is the Planck mass and β 0 < 10 34 from simple electroweak consideration. Therefore, the classical thermodynamics singularity can be avoided and a residue temperature T Res β 0 8πMp of black holes exists. We use the 4-dimensional Schwarzschild metric in this work. It is of interest to employ other geometries in the studies. In our calculation, we keep only the leading order of and β = β 0 /M 2 p . It is expected that higher orders of corrections may give more information in the future work.