Remarks on Remnants by Fermions’ Tunnelling from Black Strings

Abstract

Hawking’s calculation is unable to predict the final stage of the black hole evaporation. When effects of quantum gravity are taken into account, there is a minimal observable length. In this paper, we investigate fermions’ tunnelling from the charged and rotating black strings. With the influence of the generalized uncertainty principle, the Hawking temperatures are not only determined by the rings, but also affected by the quantum numbers of the emitted fermions. Quantum gravity corrections slow down the increases of the temperatures, which naturally leads to remnants left in the evaporation.

1. Introduction

Hawking radiation is a quantum tunnelling phenomenon of particles across black holes’ horizons. To describe this phenomenon, the semiclassical tunnelling method, which relies on calculating the imaginary part of emission particle’s action, was put forward [1]. Adopting the WKB approximation, one can get the relationship between the tunnelling rate and the action of the classically forbidden trajectory of the particle. Here we adopt the canonically invariant expression [2–4] Γ∝exp−𝐼𝑚𝑝𝑑𝑟. (1) This canonically invariant relation was first derived in [5–7].

The null geodesic method and the Hamilton-Jacobi method are usual methods employed to derive the imaginary part [8–12]. In the null geodesic method [8], we should first perform the Painleve coordinate transformation on a metric and then use canonical momenta and Hamilton canonical equations to get the imaginary part. When the variable background spacetime is taken into account, the corrected Hawking temperature is higher than the standard one. Therefore, the variable background spacetime implies the accelerated evaporation. The equation of motion of a massive particle is different from that of the massless one. The former obeys de Broglie wave function relation. Therefore, the phase velocity of the particle was adopted to research the tunnelling radiation of massive particles in the subsequent investigations [13, 14]. The embryonic form of the Hamilton-Jacobi method [11] was first found in [9, 10]. In this method, the action satisfies the Hamilton-Jacobi equation. Taking into account the property of the spacetime, one carries out separation of variables on the action. Then, inserting the separated variables into the Hamilton-Jacobi equation and solving it, one gets the imaginary part. Extending this work to the tunnelling radiation of fermions, the standard Hawking temperatures of the spherically symmetric and charged black holes were recovered [15]. Other works about fermions’ tunnelling radiation are referred to in [16–24]. The standard temperatures were also recovered by anomaly cancellations [25–27].

Various theories of quantum gravity predict the existence of a minimal observable length [28–32]. This length can be implemented in the model of the generalized uncertainty principle (GUP) ℏΔ𝑥Δ𝑝≥21+𝛽Δ𝑝², (2) where 𝛽=𝛽₀(𝑙²_𝑝/ℏ²), 𝛽₀ is a dimensionless parameter, and 𝑙_𝑝 is the Planck length. The derivation of the GUP relies on the modified fundamental commutation relations. Kempf et al. first modified commutation relations [33] and got [𝑥_𝑖,𝑝_𝑗]=𝑖ℏ𝛿_𝑖𝑗[1+𝛽𝑝²], where 𝑥_𝑖 and 𝑝_𝑖 are operators of position and momentum defined by 𝑥_𝑖=𝑥_0𝑖,𝑝_𝑖=𝑝_0𝑖1+𝛽𝑝², (3) and 𝑥_0𝑖 and 𝑝_0𝑖 satisfy the canonical commutation relations [𝑥_0𝑖,𝑝_0𝑗]=𝑖ℏ𝛿_𝑖𝑗.

This modification plays an important role in quantum gravity. With considerations of modifications, the cosmological constant problem was discussed and the finiteness of the constant was derived in [34]. Using a new form of GUP, the Unruh effect has been analyzed in [35]. The quantum dynamics of the Friedmann-Robertson-Walker universe was gotten in [36]. The related predictions on postinflation preheating in the cosmology were derived in [37]. Using the modifications, the thermodynamics of the black holes were researched in [38–40] and the tunnelling radiation of scalar particles was investigated in [41]. In recent work [42], taking into account effects of quantum gravity, the authors modified the Dirac equation in curved spacetime and investigated fermions’ tunnelling from the Schwarzschild black hole. They derived that the quantum correction slows down the increase of the Hawking temperature, which leads to the remnant.

In this paper, we extend this work to anti-de Sitter spacetime and investigate the tunnelling radiation of fermions from black strings, where effects of quantum gravity are taken into account. Black strings are cylindrically symmetric solutions of the Einstein-Maxwell equations with a negative cosmological constant. The solutions are asymptotically anti-de Sitter in transverse direction and along the axis. There are three Killing vectors, 𝜕_𝑡, 𝜕_𝜃, and 𝜕_𝑧, as the minimal symmetry. The AdS/CFT correspondence is an important topic in modern physics. Researches of anti-de Sitter spacetime are helpful to understand this correspondence. To incorporate effects of quantum gravity, we first modify the Dirac equation in curved spacetime by operators of position and momentum defined in [33] and then adopt the Hamilton-Jacobi method to get the imaginary parts of the action. The corrected Hawking temperatures are not only determined by the mass, charge, and angular momentum of the strings, but also affected by the quantum numbers (charge, angular momentum, mass, and energy) of the emitted fermions. Quantum gravity corrections slow down the increases of the Hawking temperatures. It is natural to lead to the remnants left in the evaporation.

The rest is organized as follows. In the next section, using operators of position and momentum defined in [33], we modify the Dirac equation in curved spacetime. In Section 3, we investigate the tunnelling radiation of charged fermions from the charged black string. The remnant is observed in the evaporation. In Section 4, the radiation of uncharged fermions in the rotating black string is discussed. Section 5 is devoted to our conclusion.

2. Generalized Dirac Equation

Here we adopt the modified fundamental commutation relation put forward in [33] to modify the Dirac equation in curved spacetime. Using (3), the square of momentum operators is gotten as 𝑝²=𝑝_𝑖𝑝^𝑖=−ℏ²1−𝛽ℏ²𝜕_𝑗𝜕^𝑗𝜕_𝑖⋅1−𝛽ℏ²𝜕^𝑗𝜕_𝑗𝜕^𝑖≃−ℏ²𝜕_𝑖𝜕^𝑖−2𝛽ℏ²𝜕^𝑗𝜕_𝑗𝜕^𝑖𝜕_𝑖. (4) The higher order terms of 𝛽 are neglected in the last step. In the theory of quantum gravity, the generalized frequency is found as [43] 𝜔=𝐸1−𝛽𝐸², (5) where 𝐸 is the energy operator and is defined as 𝐸=𝑖ℏ𝜕_𝑡. From the energy mass shell condition 𝑝²+𝑚²=𝐸², the generalized expression of the energy was derived [41, 43–45]. It is 𝐸=𝐸1−𝛽𝑝²+𝑚². (6) The generalized Dirac equation without considerations of electromagnetic effects in the flat spacetime has been derived in [44] by the consequence of the GUP. In curved spacetime, the Dirac equation with an electromagnetic field takes on the form 𝑖𝛾^𝜇𝜕_𝜇+Ω_𝜇+𝑖ℏ𝑒𝐴_𝜇𝑚Ψ+ℏΨ=0, (7) where Ω_𝜇≡(𝑖/2)𝜔_𝜇^𝑎𝑏Σ_𝑎𝑏, 𝜔_𝜇^𝑎𝑏 is the spin connection defined by the tetrad 𝑒^𝜆_𝑏 and ordinary connection 𝜔_𝜇^𝑎_𝑏=𝑒_𝜈^𝑎𝑒^𝜆_𝑏Γ^𝜈_𝜇𝜆−𝑒^𝜆_𝑏𝜕_𝜇𝑒_𝜆^𝑎. (8) 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 𝑔_𝜇𝜈=𝑒_𝜇^𝑎𝑒_𝜈^𝑏𝜂_𝑎𝑏,𝜂_𝑎𝑏=𝑔_𝜇𝜈𝑒^𝜇_𝑎𝑒^𝜈_𝑏,𝑒^𝜇_𝑎𝑒_𝜈^𝑎=𝛿^𝜇_𝜈,𝑒^𝜇_𝑎𝑒_𝜇^𝑏=𝛿^𝑏_𝑎. (9) In (7), Σ_𝑎𝑏 is the Lorentz spinor generators defined by Σ_𝑎𝑏=𝑖4𝛾^𝑎,𝛾^𝑏,𝛾^𝑎,𝛾^𝑏=2𝜂^𝑎𝑏. (10) Then one can construct 𝛾^𝜇’s in the curved spacetime as 𝛾^𝜇=𝑒^𝜇_𝑎𝛾^𝑎,𝛾^𝜇,𝛾^𝜈=2𝑔^𝜇𝜈. (11) To get the generalized Dirac equation in the curved spacetime, we rewrite (7) as −𝑖𝛾⁰𝜕₀Ψ=𝑖𝛾^𝑖𝜕_𝑖+𝑖𝛾^𝜇Ω_𝜇+𝑖𝛾^𝜇𝑖ℏ𝑒𝐴_𝜇+𝑚ℏΨ. (12) Using (4), (6), and (12) and neglecting the higher order terms of 𝛽, we get [41, 43–45] −𝑖𝛾⁰𝜕₀Ψ=𝑖𝛾^𝑖𝜕_𝑖+𝑖𝛾^𝜇Ω_𝜇+𝑖𝛾^𝜇𝑖ℏ𝑒𝐴_𝜇+𝑚ℏ×1+𝛽ℏ²𝜕_𝑗𝜕^𝑗−𝛽𝑚²Ψ, (13) which is rewritten as 𝑖𝛾⁰𝜕₀+𝑖𝛾^𝑖𝜕_𝑖1−𝛽𝑚²+𝑖𝛾^𝑖𝛽ℏ²𝜕_𝑗𝜕^𝑗𝜕_𝑖+𝑚ℏ1+𝛽ℏ²𝜕_𝑗𝜕^𝑗−𝛽𝑚²+𝑖𝛾^𝜇𝑖ℏ𝑒𝐴_𝜇1+𝛽ℏ²𝜕_𝑗𝜕^𝑗−𝛽𝑚²+𝑖𝛾^𝜇Ω_𝜇1+𝛽ℏ²𝜕_𝑗𝜕^𝑗−𝛽𝑚²Ψ=0. (14) Thus the generalized Dirac equation is derived. When 𝐴_𝜇=0, it describes an equation without electromagnetic fields. In the following sections, we adopt (14) to describe fermions tunnelling from the charged and rotating black strings.

3. Fermions’ Tunnelling from a Charged Black String

The 4-dimensional neutral black string solutions to Einstein-Maxwell equations with a negative cosmological constant were derived in [46]. Subsequently, the general solutions with electric charges were gotten [47]. In this section, we investigate charged fermions' tunnelling from a cylindrically symmetric black string. The black string solution is given by [47] 𝑑𝑠²=−𝑓(𝑟)𝑑𝑡²+1𝑔(𝑟)𝑑𝑟²+𝑟²𝑑𝜃²+𝛼²𝑟²𝑑𝑧², (15) with the electromagnetic potential 𝐴_𝜇=𝐴_𝑡,0,0,0=2𝑄,𝛼𝑟,0,0,0 (16) where 𝑓(𝑟)=𝑔(𝑟)=𝛼²𝑟²−(4𝑀/𝛼𝑟)+(4𝑄²/𝛼²𝑟²), 0≤𝜃≤2𝜋, 𝛼²=−Λ/3, and Λ is the negative cosmology constant. 𝑀 and 𝑄 are the ADM mass and charge per unit length in the z direction, respectively. The above spacetime is asymptotically anti-de Sitter in the transverse directions and string directions. The singularity at 𝑟=0 is enclosed by the horizon 𝑟₊ if the condition 𝑄²≤(3/4)𝑀^4/3 holds. The event horizon 𝑟₊ is located at 𝑟₊=12√2𝑅+−2𝑅+8𝑀𝛼³√,2𝑅 (17) where 𝑀𝑅=²𝛼⁶+𝑀²𝛼⁶²−4𝑄²3𝛼^431/21/3+𝑀²𝛼⁶−𝑀²𝛼⁶²−4𝑄²3𝛼^431/21/3. (18) The metric (15) describes a neutral black string solution when 𝑄=0.

For a spin-1/2 fermion, there are two states corresponding to spin-up and spin-down. Here we only investigate the state with spin-up. The investigation of the state with spin-down is parallel and the same result can be obtained. To describe the motion of a charge fermion, we suppose that the wave function takes on the form 𝐴0𝐵0𝑖Ψ=expℏ𝐼(𝑡,𝑟,𝜃,𝑧), (19) where 𝐴 and 𝐵 are functions of 𝑡,𝑟,𝜃, and 𝑧 and 𝐼 is the action of the fermion with spin-up state. To find gamma matrices, we should first construct a tetrad. It is straightforward to construct a tetrad from the metric (15). The tetrad is 𝑒_𝜇^𝑎=diag𝑓,1/√𝑔,𝑟,𝛼𝑟. (20) Then gamma matrices are gotten as 𝛾^𝑡=1𝑓(𝑟)𝑖00−𝑖,𝛾^𝜃=𝑔^𝜃𝜃0𝜎¹𝜎¹0,𝛾^𝑟=𝑔(𝑟)0𝜎³𝜎³0,𝛾^𝑧=𝑔^𝑧𝑧0𝜎²𝜎²0. (21) In the above equations, 𝑔^𝜃𝜃=1/𝑟 and 𝑔^𝑧𝑧=1/𝛼𝑟. To apply the WKB approximation, we insert the wave function and the gamma matrices into the generalized Dirac equation and then divide by the exponential term and multiply by ℏ. The resulting equation of leading order in ℏ is derived and decoupled into four equations 1−𝑖𝐴𝑓𝜕_𝑡𝐼−𝐵1−𝛽𝑚²√𝑔𝜕_𝑟𝐼−𝐴𝑚𝛽𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐵𝛽√𝑔𝜕_𝑟𝐼𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐴𝑚1−𝛽𝑚²−𝑖𝐴𝑒𝐴_𝑡𝑓×1−𝛽𝑚²−𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²1=0,𝑖𝐵𝑓𝜕_𝑡𝐼−𝐴1−𝛽𝑚²√𝑔𝜕_𝑟𝐼−𝐵𝑚𝛽𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐴𝛽√𝑔𝜕_𝑟𝐼𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐵𝑚1−𝛽𝑚²+𝑖𝐵𝑒𝐴_𝑡𝑓×1−𝛽𝑚²−𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²𝐴−=0,1−𝛽𝑚²𝑔^𝜃𝜃𝜕_𝜃𝐼+𝛽𝑔^𝜃𝜃𝜕_𝜃𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²−𝑖1−𝛽𝑚²𝑔^𝑧𝑧𝜕_𝑧𝐼+𝑖𝛽𝑔^𝑧𝑧𝜕_𝑧𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²𝐵−=0,1−𝛽𝑚²𝑔^𝜃𝜃𝜕_𝜃𝐼+𝛽𝑔^𝜃𝜃𝜕_𝜃𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²−𝑖1−𝛽𝑚²𝑔^𝑧𝑧𝜕_𝑧𝐼+𝑖𝛽𝑔^𝑧𝑧𝜕_𝑧𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜃𝜃𝜕_𝜃𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²=0. (22)(23) Obviously, it is difficult to get the solution of the action 𝐼 from the above equations. However, the action can be separated by the property of the black string. Considering the Killing vectors of the spacetime, the author separated the action as 𝐼=−𝜔𝑡+𝑊(𝑟)+𝑙𝜃+𝐽𝑧 [17], where 𝜔 is the energy of the emitted fermion. From the above four equations, we carry out separation of variables as 𝐼=−𝜔𝑡+𝑊(𝑟)+Θ(𝜃,𝑧). (24) We first observe (23) and find that they are irrelevant to 𝐴 and 𝐵 and can be reduced to the same equation. Inserting (24) into (23) yields 𝑔^𝜃𝜃𝜕_𝜃Θ+𝑖𝑔^𝑧𝑧𝜕_𝑧Θ×1−𝛽𝑚²−𝛽𝑔^𝑟𝑟𝜕_𝑟𝑊²−𝛽𝑔^𝜃𝜃𝜕_𝜃Θ²−𝛽𝑔^𝑧𝑧𝜕_𝑧Θ²=0. (25) In the above equation, the summation of factors in the square brackets cannot be zero. Therefore, it should be 𝑔^𝜃𝜃𝜕_𝜃Θ+𝑖𝑔^𝑧𝑧𝜕_𝑧Θ=0, (26) which yields a complex function solution (other than the trivial constant solution) of Θ. However, this solution has no contribution to the tunnelling rate. Therefore, we will not consider its contribution in the calculation. Another important relation predicted by (26) is 𝑔^𝜃𝜃(𝜕_𝜃Θ)²+𝑔^𝑧𝑧(𝜕_𝑧Θ)²=0. Now we focus our attention on the first two equations. Inserting (24) into (22) and canceling 𝐴 and 𝐵 yield 𝐴₆𝜕_𝑟𝑊⁶+𝐴₄𝜕_𝑟𝑊⁴+𝐴₂𝜕_𝑟𝑊²+𝐴₀=0, (27) where 𝐴₆=𝛽²𝑔³𝐴𝑓,₄=𝛽𝑔²𝑓𝑚²𝛽−2−𝛽²𝑔²𝑒²𝐴²_𝑡,𝐴₂=𝑔𝑓1−𝛽𝑚²1+𝛽𝑚²−2𝛽𝑔𝑒𝐴_𝑡𝜔−𝑒𝐴_𝑡1−𝛽𝑚²,𝐴₀=−𝑚²𝑓1−𝛽𝑚²²−𝜔−𝑒𝐴_𝑡1−𝛽𝑚²². (28) Neglect the higher order terms of 𝛽 and solve (27) at the event horizon. Thus the imaginary part of the radial action is 𝐼𝑚𝑊_±=±𝑑𝑟𝑔𝑓𝜔−𝑒𝐴_𝑡1−𝛽𝑚²²+𝑚²𝑓×1+𝛽𝑚²𝜔+𝛽²⁰−𝑒𝐴_𝑡𝜔₀𝑓=±𝜋𝜔−𝑒𝐴_𝑡+𝑓^′1+𝛽𝜉, (29) where +(−) denote the outgoing (ingoing) solutions, 𝑓^′=2𝛼²𝑟₊+(4𝑀/𝛼𝑟²₊)−(8𝑄²/𝛼²𝑟³₊), 𝜉=(3/2)𝑚²+((2𝑚²+1)/2𝜔₀)+(2𝑒𝐴_𝑡+/𝑓^′𝑟₊)−(2/3)(𝜔₀/𝑓^′𝑟₊), 𝜔₀=𝜔−𝑒𝐴_𝑡, 𝜔₀=𝜔−𝑒𝐴_𝑡+, and 𝐴_𝑡+=2𝑄/𝛼𝑟₊ is the electromagnetic potential at the event horizon. Using the relation of the roots of 𝑓=0, it is easily proved that 𝜉>0. The tunnelling rate of the charged fermion at the event horizon is Γ∝exp−𝐼𝑚𝑝_𝑟𝑝𝑑𝑟=exp−𝐼𝑚^out_𝑟𝑑𝑟−𝑝ⁱⁿ_𝑟𝑝𝑑𝑟=exp∓2𝐼𝑚^out,in_𝑟𝑑𝑟. (30) Here, 𝑝_𝑟=𝜕_𝑟𝑊, and out(in) correspond to +(−). Thus the tunnelling rate is gotten as Γ∝exp−2𝜋𝜔−𝑒𝐴_𝑡+𝑓^′1+𝛽𝜉. (31) However, the temporal contribution to the tunneling amplitude was missed in the above calculation [2, 5–7]. We use Kruskal coordinates (𝑇,𝑅) to find this temporal contribution. The region exterior is described by 𝑇=𝑒^𝜅𝑟∗sinh(𝜅𝑡),𝑅=𝑒^𝜅𝑟∗cosh(𝜅𝑡), (32) where 𝑟_∗=𝑟+(1/2𝜅)ln((𝑟−𝑟₊)/𝑟₊) is the tortoise coordinate and 𝜅 is the surface gravity. The interior region is given by 𝑇=𝑒^𝜅𝑟∗cosh(𝜅𝑡),𝑅=𝑒^𝜅𝑟∗sinh(𝜅𝑡). (33) To find the temporal contribution, we connect these two patches across the horizon. Rotate the time 𝑡 as 𝑡→𝑡−(𝜋/2)𝑖𝜅. As pointed out in [2–4], this rotation would lead to an additional imaginary contribution coming from the temporal part, namely, 𝐼𝑚[(𝜔−𝑒𝐴_𝑡+)Δ𝑡^out,in]=(1/2)𝜋(𝜔−𝑒𝐴_𝑡+)𝜅. So the total temporal contribution is 𝐼𝑚[(𝜔−𝑒𝐴_𝑡+)Δ𝑡]=𝜋(𝜔−𝑒𝐴_𝑡+)𝜅. Therefore, the tunnelling rate with the consideration of the temporal contribution is −1Γ∝expℏ𝐼𝑚𝜔−𝑒𝐴_𝑡+Δ𝑡+𝐼𝑚𝑝_𝑟𝑑𝑟=exp−4𝜋𝜔−𝑒𝐴_𝑡+𝑓^′11+2𝛽𝜉. (34) This is the Boltzmann factor with the Hawking temperature at the event horizon taking 𝑓𝑇=^′4𝜋1+𝛽𝜉=𝑇₀11−2𝛽𝜉, (35) where 𝑇₀=(1/2𝜋)(𝛼²𝑟₊+(2𝑀/𝛼𝑟²₊)−(4𝑄²/𝛼²𝑟³₊)) is the standard Hawking temperature of the black string.

It is shown that the corrected temperature appears and is lower than the standard one. The correction is not only determined by the mass and charge of the black string, but also affected by the quantum number (mass, charge, and energy) of the emitted fermion. Quantum gravity correction slows down the increase of the Hawking temperature caused by the evaporation. Finally, the black string is in a balance state. At this state, the evaporation stops and the remnant is produced.

It is of interest to discuss the corrected area entropy. The entropy can be derived by the first law of thermodynamics with the corrected temperature (35). However, the expression is complicated, so we do not write it here. The corrected temperatures were also gotten in [48–53]. When 𝛽=0, the standard Hawking temperature is recovered [17, 47].

4. Fermions’ Tunnelling from a Rotating Black String

In this section, we investigate uncharged fermions’ tunnelling from the event horizon of a rotating black string. Therefore, effects of the electromagnetic field in the generalized Dirac equation are not taken into account here. The rotating black string solution in a spacetime asymptotically anti-de Sitter in the radial direction was derived by Lemos and Zanchin [54]. The solution is 𝑑𝑠²=−𝛼²𝑟²−4𝑀1−𝑎²𝛼²/2𝛼𝑟𝑑𝑡²+𝛼²𝑟²−4𝑀1−(3/2)𝑎²𝛼²𝛼r⁻¹𝑑𝑟²−8𝑀𝑎1−𝑎²𝛼²/2+𝑟𝛼𝑟𝑑𝑡𝑑𝜑₂−4𝑀𝑎²𝛼𝑟𝑑𝜑²+𝛼²𝑟²𝑑𝑧², (36) where 𝛼²=−Λ/3, Λ is the negative cosmological constant, and 𝑎 is the angular momentum per unit mass. It is defined that 𝑎²𝛼²=1−𝜖/𝑀 and 𝜖=𝑀²−8𝐽²𝛼²/9. 𝑀 and 𝐽 are the mass and angular momentum line densities of the spacetime, respectively. The relation between 𝐽 and 𝑎 is given by 𝐽=(3/2)𝑀𝑎1−𝑎²𝛼²/2. For convenience of the investigation, the metric (36) is rewritten as 𝑑𝑠²𝛿=−Δ𝛾𝑑𝑡−𝛼²𝑑𝜑²+𝑟²𝛾𝑑𝜑−𝛿𝑑𝑡²+𝑑𝑟²Δ+𝛼²𝑟²𝑑𝑧², (37) where Δ=𝛼²𝑟²−𝑏𝛼𝑟,𝑏=4𝑀1−3𝑎²𝛼²2,𝛾=2−𝑎²𝛼²2−3𝑎²𝛼²,𝛿=𝑎𝛼²1−(3/2)𝑎²𝛼². (38) The event horizon is located at 𝑟₊=𝛼⁻¹𝑏^1/3 which is given for Δ=0. To describe the fermion's tunnelling from the event horizon, one can directly construct a tetrad and gamma matrices from the metric (37). For simplicity, to construct the tetrad and gamma matrices, we perform the dragging coordinate transformation 𝜑=𝜙+Ω𝑡,Ω=−Δ𝛾𝛿𝛼²+𝑟²𝛾𝛿𝛼⁴−Δ𝛿²+𝑟²𝛾²𝛼⁴, (39) on the metric (37) and get 𝑑𝑠²=−𝐹(𝑟)𝑑𝑡²+1𝐺(𝑟)𝑑𝑟²+𝑔_𝜙𝜙𝑑𝜙²+𝑔_𝑧𝑧𝑑𝑧²=−Δ𝑟²𝛼²𝛾²−𝛿²²−Δ𝛿²+𝛼⁴𝑟²𝛾²𝑑𝑡²+1Δ𝑑𝑟²+−Δ𝛿⁴𝛼⁴+𝑟²𝛾²𝑑𝜙²+𝛼²𝑟²𝑑𝑧². (40) Here we still only investigate the state with spin-up. Assume that the wave function of the fermion with spin-up state shares the same expression as (19), namely, 𝐴0𝐵0𝑖Ψ=expℏ𝐼𝑡,𝑟,𝜙,𝑧. (41) The tetrad is easily constructed as 𝑒_𝜇^𝑎=diag√𝐹,1/√𝐺,𝑔_𝜙𝜙,√𝑔_𝑧𝑧. (42) Now gamma matrices take on the form 𝛾^𝑡=1𝐹(𝑟)𝑖00−𝑖,𝛾^𝜙=𝑔^𝜙𝜙0𝜎¹𝜎¹0,𝛾^𝑟=𝐺(𝑟)0𝜎³𝜎³0,𝛾^𝑧=𝑔^𝑧𝑧0𝜎²𝜎²0. (43) In the above equations, 𝑔^𝜙𝜙=𝛼⁴/(−Δ𝛿⁴+𝛼⁴𝑟²𝛾²), 𝑔^𝑧𝑧=1/𝛼²𝑟². Inserting the wave function and the gamma matrices into the generalized Dirac equation and adopting the same process as the above section, we get 1−𝑖𝐴√𝐹𝜕_𝑡𝐼−𝐵1−𝛽𝑚²√𝐺𝜕_𝑟𝐼−𝐴𝑚𝛽𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐵𝛽√𝐺𝜕_𝑟𝐼𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐴𝑚1−𝛽𝑚²1=0,𝑖𝐵√𝐹𝜕_𝑡𝐼−𝐴1−𝛽𝑚²√𝐺𝜕_𝑟𝐼−𝐵𝑚𝛽𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐴𝛽√𝐺𝜕_𝑟𝐼𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²+𝐵𝑚1−𝛽𝑚²𝐴−=0,1−𝛽𝑚²𝑔^𝜙𝜙𝜕_𝜙𝐼+𝛽𝑔^𝜙𝜙𝜕_𝜙𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²−𝑖1−𝛽𝑚²𝑔^𝑧𝑧𝜕_𝑧𝐼+𝑖𝛽𝑔^𝑧𝑧𝜕_𝑧𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²𝐵−=0,1−𝛽𝑚²𝑔^𝜙𝜙𝜕_𝜙𝐼+𝛽𝑔^𝜙𝜙𝜕_𝜙𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²−𝑖1−𝛽𝑚²𝑔^𝑧𝑧𝜕_𝑧𝐼+𝑖𝛽𝑔^𝑧𝑧𝜕_𝑧𝐼×𝑔^𝑟𝑟𝜕_𝑟𝐼²+𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²=0. (44)(45) It is also difficult to solve the action 𝐼 from the above equations. We first observe the last two equations. They can be reduced into the same equation and yield 𝑔^𝜙𝜙𝜕_𝜙𝐼+𝑖𝑔^𝑧𝑧𝜕_𝑧𝐼=0. This implies 𝑔^𝜙𝜙𝜕_𝜙𝐼²+𝑔^𝑧𝑧𝜕_𝑧𝐼²=0. (46) Now our interest is the first two equations which determine the Hawking temperature of the black string. Considering the properties of the metrics (36) and (40), we carry out separation of variables as 𝐼=−𝜔−𝑗Ω𝑡+𝑊(𝑟,𝑧)+𝑗𝜙, (47) where 𝜔 and 𝑗 are the energy and angular momentum of the emitted fermion, respectively. Inserting (47) into (44) and canceling 𝐴 and 𝐵 yield 𝐵₆𝜕_𝑟𝑊⁶+𝐵₄𝜕_𝑟𝑊⁴+𝐵₂𝜕_𝑟𝑊²+𝐵₀=0, (48) where 𝐵₆=𝛽²𝐺³𝐵𝐹,₄=𝛽𝐺²𝐹𝑚²,𝐵𝛽−2₂=𝐺𝐹1−𝛽𝑚²²+2𝛽𝑚²1−𝑚²𝛽,𝐵₀=−𝑚²1−𝛽𝑚²²𝐹−𝜔−𝑗Ω². (49) Neglecting the higher order terms of 𝛽 and solving (48) at the event horizon, we get the solution of 𝑊. Thus the imaginary part of 𝑊 is 𝐼𝑚𝑊_±=±𝑑𝑟𝑚²𝐹+𝜔−𝑗Ω²×𝐺𝐹1+𝛽𝑚²+𝛽𝜔−𝑗Ω²𝐹=±𝜋𝜔−𝑗Ω₊𝛼⁴𝑟²₊𝛾2𝛼⁴𝑟³₊+𝑏𝛼²𝛾²−𝛿²1+𝛽𝜒, (50) where +(−) are the outgoing (ingoing) solutions, 𝜒=(3/2)𝑚²+3𝑗(𝜔−𝑗Ω₊)𝛿/(𝛼²𝛾²−𝛿²)𝛾𝑟²₊−(3/2)((𝜔−𝑗Ω₊)²𝛿²/(𝛼²𝛾²−𝛿²)𝑟²₊), and Ω₊=𝛿/𝛾 is the angular velocity at the event horizon. It is not difficult to prove that 𝜒>0.

To find the temporal contribution, we use the Kruskal coordinates (𝑇,𝑅). The region exterior to the string (𝑟>𝑟₊) is described by 𝑇=𝑒^𝜅+𝑟∗sinh𝜅₊𝑡,𝑅=𝑒^𝜅+𝑟∗cosh𝜅₊𝑡, (51) where 𝑟_∗=𝑟+(1/2𝜅₊)ln((𝑟−𝑟₊)/𝑟₊) and 𝜅₊ denote the surface gravity. The interior region is 𝑇=𝑒^𝜅+𝑟∗cosh𝜅₊𝑡,𝑅=𝑒^𝜅+𝑟∗sinh𝜅₊𝑡. (52) Adopting the same process as the above section, we get that the total temporal contribution is 𝐼𝑚[(𝜔−𝑗Ω₊)Δ𝑡]=𝜋(𝜔−𝑗Ω₊)𝜅₊. Therefore, the tunnelling rate is −1Γ∝expℏ𝐼𝑚𝜔−𝑗Ω₊Δ𝑡+𝐼𝑚𝑝_𝑟−𝑑𝑟=exp4𝜋𝜔−𝑗Ω₊𝛼⁴𝑟²₊𝛾2𝛼⁴𝑟³₊+𝑏𝛼²𝛾²−𝛿²11+2𝛽𝜒. (53) Equation (53) is the Boltzmann factor of the Hawking temperature at the event horizon taking 𝑇=2𝛼⁴𝑟³₊+𝑏𝛼²𝛾²−𝛿²4𝜋𝛼⁴𝑟²₊𝛾1+(1/2)𝛽𝜒=𝑇₀11−2𝛽𝜒, (54) where 𝑇₀=(2𝛼⁴𝑟³₊+𝑏)(𝛼²𝛾²−𝛿²)/4𝜋𝛼⁴𝑟²₊𝛾 is the standard Hawking temperature. Obviously, the corrected Hawking temperature is lower than the standard one. The correction is related not only to the mass and angular momentum of the black string but also to the quantum number (mass, angular momentum, and energy) of the emitted fermion. Due to 𝜒>0, there is a balance point. At this point, the evaporation stops and the remnant is left.

5. Conclusion

In this paper, taking into account the influence of quantum gravity, we modified the Dirac equation in curved spacetime by the modified fundamental commutation relations put forward in [33]. Then the tunnelling radiation of fermions from the event horizons of the charged and rotating black strings was investigated. The corrected Hawking temperatures were gotten. In the charged spacetime, the correction is related not only to the mass and charge of the black string but also to the quantum number (mass, charge, and energy) of the emitted fermion. In the rotating spacetime, the quantum number (mass, angular momentum, and energy) of the emitted fermion and the mass and angular momentum of the black string affect the Hawking temperature at the same time. Due to the quantum gravity corrections, the evaporation of the black strings slows down. Finally, the evaporation stops and the remnants are left. The remnants in the final state were also discussed in [55–58]. It is of interest to discuss remnants. A review of this topic can be found in [59].

When 𝛽=0, the standard Hawking temperatures are recovered.

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural Science Foundation of China with Grant no. 11205125.