Massive vector particles tunneling from black holes influenced by the generalized uncertainty principle

This study considers the generalized uncertainty principle, which incorporates the central idea of large extra dimensions, to investigate the processes involved when massive spin-1 particles tunnel from Reissner-Nordstrom and Kerr black holes under the effects of quantum gravity. For the black hole, the quantum gravity correction decelerates the increase in temperature. Up to $\mathcal{O}(\frac{1}{M_f^2})$, the corrected temperatures are affected by the mass and angular momentum of the emitted vector bosons. In addition, the temperature of the Kerr black hole becomes uneven due to rotation. When the mass of the black hole approaches the order of the higher dimensional Planck mass $M_f$, it stops radiating and yields a black hole remnant.


Introduction
Hawking stated that black holes can release radiation thermodynamically due to quantum vacuum fluctuation effects near the event horizon [1]. Subsequently, Hawking radiation has attracted much attention from theoretical physicists and various methods have been proposed for deriving Hawking radiation. In particular, a semiclassical derivation was developed that models Hawking radiation as a tunneling process, which includes the null geodesic method and Hamilton-Jacobi method. The null geodesic method was first proposed by Kraus and Wilczek [2,3], and then developed further by Parikh and Wilczek [4][5][6]. The Hamilton-Jacobi method was proposed by Angheben et al. [7] as an extension of Padmanabhan's methods [8,9]. Both approaches to tunneling rely on the fact that the tunneling probability for the classically forbidden trajectory from inside to outside the horizon is given by Γ = exp (−2ImI/ ), where I is the classical action of the trajectory. These two methods differ in how the imaginary part of the classical action is calculated. Many useful results have been obtained using the null geodesic and Hamilton-Jacobi methods .
A common feature of various quantum gravity theories, such as string theory, loop quantum gravity, and noncommutative geometry, is the existence of a minimum measurable length [31][32][33][34]. The generalized uncertainty principle (GUP) is a simple way of realizing this minimal length [35][36][37]. An effective model of the GUP in one-dimensional quantum mechanics, which incorporates the central idea of large extra dimensions, was given by [38] L f k(p) = tanh where the generators of the translations in space and time are the wave vector k and the frequency ω, and L f and M f are the higher dimensional minimal length and Planck mass, respectively. L f and M f satisfy L f M f = . The quantization in position representationx = x leads to Therefore, the low energy limit p ≪ M f including the order of (p/M f ) 3 gives where β = 1/(3M 2 f ). Then, the modified commutation relation is given by and the generalized uncertainty relation (GUR) is From Eqs (6) and (7), it can be concluded that the departure of GUP from the Heisenberg uncertainty principle increases with the momentum of the particle. We note that Eqs (4)- (7) only apply to particles in the low energy limit p ≪ M f , which is the specific case considered in the present study. In the low energy regime, the parameter β should be constrained in experiments designed to test the uncertainty principle, such as those by [39,40]. Other generalized uncertainty relations can be found in previous studies. A widely discussed relation, ∆x∆p ≥ 2 1 + l 2 ∆p 2 2 , was proposed based on some aspects of quantum gravity and string theory [35], where the cutoff l was selected as a string scale in the context of the perturbative string theory or Plank scale based on quantum gravity. Another interesting GUR was obtained by treating the mass source as a Gaussian wave function and the horizon as a horizon wave function [41], i.e., ∆r ≃ l p mp ∆p + γl p ∆p mp , where the first part represents the uncertainty of the radial size of the source and the second represents the horizon uncertainty, and γ is a parameter that represents the order of unity in the full quantum gravity regime, which becomes very small in the semiclassical regime.
Black holes are an important research area in the study of quantum gravity effects and many studies of black hole physics have incorporated the GUP. The thermodynamics of black holes have been investigated in the framework of GUP [42][43][44][45][46][47][48]. By combining the GUP with the tunneling method, Nozari and Mehdipour studied the modified tunneling rate of a Schwarzschild black hole [49]. The GUP-deformed Hamilton-Jacobi equation for fermions in curved spacetime was introduced and the corrected Hawking temperatures were derived for various types of spacetime in [50][51][52][53][54][55][56][57][58]. By studying the tunneling of fermions, it was found that the quantum gravity effects slowed down the increase in the Hawking temperatures, where this property naturally leads to a residual mass during black hole evaporation.
In this study, we investigate massive spin-1 particles (W ± , Z 0 ) tunneling across the horizons of black holes using the Hamilton-Jacobi method, which incorporates the minimal length effect via Eqs (4) and (5). Our calculations show that the quantum gravity correction is related to the black hole's mass as well as to the mass and angular momentum of the emitted vector bosons. Furthermore, the quantum gravity correction explicitly retards the increase in temperature during the black hole evaporation process. As a result, the quantum correction will balance the traditional tendency for a temperature increase at some point during the evaporation, which leads to the existence of remnants.
The remainder of this paper is organized as follows. In Section 2, based on the GUP-corrected Lagrangian of the massive vector field, we derive the equation of motion for the vector bosons in curved spacetime. In Section 3, by incorporating GUP, we investigate the tunneling of charged massive bosons in a Reissner-Nordstrom black hole. The tunneling of massive bosons in a Kerr black hole is also studied and the remnants are derived in Section 4. Section 5 provides some discussion and the conclusions of this study. We use the spacelike metric signature convention (−, +, +, +) in this study.

Generalized field equations for massive vector bosons
We start from the kinetic term of the uncharged vector boson field in flat spacetime within the framework of GUP, 1 2 B µν B µν , where the modified field strength tensor is given by It should be noted that additional derivative terms exist. Next, we generalize this to the case of a charged vector boson field (W ± ) in charged black hole spacetime. Considering the gauge principle, the additional derivatives also act on the local unitary transformation operator U(x), so they must also be replaced by covariant derivatives [59]: where D ± µ = ∇ µ ± i eA µ with ∇ µ is the geometrically covariant derivative, A µ is the electromagnetic field of the black hole, and e denotes the charge of the W + boson. The difference in signs of the O(β) terms in Eqs (9) and (10) is attributable to the fact that g 00 always shares different signs with g ii .
By defining Accordingly, the corresponding generalized action should be This action is invariant under a local U (1) gauge transformation, which does not refer to spacetime transformation. By varying the action (12) with respect to the fields W − and W + , it follows immediately that Then, by substituting the GUP Lagrangian (11) in (13), we obtain This is the equation of motion for the W + boson field and we can repeat the same procedure to obtain the equation of motion for the W − boson field. By setting e to 0, we obtain the field equation for massive bosons in the case of uncharged bosons or in an uncharged spacetime background 3 Massive vector particles tunneling from a Reissner-Nordstrom black hole The Reissner-Nordstrom black hole describes a spherically symmetric static spacetime with charge Q. The metric is given by with the electromagnetic potential where and r ± = M ± M 2 − Q 2 represents the locations of the outer horizon and the inner horizon, respectively. In this study, without any loss of generality, we only consider the tunneling process for W + bosons. The calculation is similar for the W − case.
According to the WKB approximation, W + µ has the form of where S is defined as By substituting Eqs (20), (21), and the Reissner-Nordstrom metric (17) into Equation (15), and keeping only the lowest order in , we obtain the equations for the coefficients C µ where the P µ s are defined as Considering the property of Reissner-Nordstrom spacetime and the question that we aim to address, then following the standard process, we separate the variables where E is the energy of the emitted vector particles. By inserting Eq. (27) into Eqs (22)- (25), we can obtain a matrix equation where K is a 4×4 matrix, the elements of which are Eq. (28) has a nontrivial solution if the determinant of the matrix K equals zero. detK = 0 should yield the following equation (please refer to Appendix A for the definition of A i s). By neglecting the higher order terms of β and solving Eq. (30), we obtain the solution to the derivative of the radial action where Integrating Eq. (31) around the pole at the outer horizon r + = M + M 2 − Q 2 yields the solution of the radial action. The particle's tunneling rate is determined by the imaginary part of the action, where Ξ = 6m 2 + 6 It is quite clear that Ξ > 0. We note that W + represents the radial function for the outgoing particles and W − is for the ingoing particles. Thus, the tunneling rate of W + bosons at the outer event horizon is If we set = 1, then the effective Hawking temperature is deduced as where T 0 = r + −r − 4πr 2 + is the original Hawking temperature of a Reissner-Nordstrom black hole. From Eq. (36), it can be inferred that the corrected temperature relies on the quantum numbers (mass and angular momentum) of the emitted vector bosons. Moreover, the quantum effects explicitly counteract the temperature increase during evaporation, which will cancel it out at some point. Naturally, black hole remnants will be left.

Massive vector particles tunneling from a Kerr black hole
In this section, we investigate the tunneling of massive vector particles at the outer event horizon of a Kerr black hole where we consider the GUP. For simplicity, we suppose that the emitted vector particles are uncharged, so the motion of the vector field is described by Eq. (16). The line element within Kerr spacetime is given by where ρ 2 = r 2 + a 2 cos 2 θ, ∆ = r 2 − 2Mr + a 2 , M is the black hole mass, and a is the angular momentum per unit mass. To ensure that the event horizon coincides with the infinite red-shift surface, we introduce a new coordinate χ = ϕ − Ωt with Ω = 2M ra (r 2 +a 2 ) 2 −∆a 2 sin 2 θ , and thus the metric (37) becomes where Σ(r, θ) = (r 2 + a 2 ) 2 − ∆a 2 sin 2 θ. According to the WKB approximation, B µ has the form of where S is defined as By substituting Eqs (39), (40), and the Kerr metric (38) into (16), and keeping only the lowest order in , we obtain the equations for the coefficients C µ where the P µ s are defined as Considering the properties of Kerr spacetime, we separate the variables as where E and j denote the energy and angular momentum of the emitted particle, respectively. By inserting Eq. (46) into Eqs (41)-(44), we can obtain a matrix equation K (C 0 , C 1 , C 2 , C 3 ) T = 0 and the elements of K are expressed as where J θ is identified as ∂ θ S 0 . The determination of the coefficient matrix should be equal to zero to ensure that Eqs.
(please refer to Appendix A for definitions of the B i s). By neglecting the higher order terms of β and solving Eq. (48), we obtain the solution to the derivative of the radial action where Integrating Eq. (49) around the pole at the outer horizon r + = M + √ M 2 − a 2 yields the solution for the radial action. The particle's tunneling rate is determined by the imaginary part of the action, where Π = 6m 2 + 6 r 2 + +a 2 cos 2 θ (J 2 θ + j 2 csc 2 θ). It is obvious that Π > 0. The tunneling rate of the vector bosons at the outer event horizon is If we set = 1, then the effective Hawking temperature is deduced as where is the original Hawking temperature of a Kerr black hole. Similar to the results obtained for a Reissner-Nordstrom black hole, the corrected temperature is lower than the original Hawking temperature, where it is related to the black hole's mass and angular momentum, as well as to the mass and angular momentum of the emitted vector bosons. It should be noted that due to the quantum gravity effect, the corrected Hawking temperature of a Kerr black hole become uneven since Π is a function of θ.
Dimensional reduction near the horizon can be used to study the standard processes of particle tunneling [60,61], which is attributable to the fact that all large non-extremal black holes basically resemble Rindler space. For the standard Hawing radiation, all species of particles located very close to the horizon are effectively massless when considering infinite blueshift, so the Hawing temperatures of all particles are the same. However, our calculations also show that quantum gravity effects should make particles with different identities or quantum numbers differ in terms of their effective Hawking temperatures. When the particles approach the horizon, they can never be infinite-blueshifted because of the existence of minimal length.
When the quantum gravity effects are neglected, i.e., β = 0, the standard Hawking temperatures of Reissner-Nordstrom and Kerr black holes can be recovered by Eqs (36) and (54), respectively. To estimate the residual masses of the black holes at the level of the order of the magnitude, we consider the charge-free (Q = 0) and non-rotating (a = 0) case, i.e., Schwarzschild spacetime, as a special case of both Reissner-Nordstrom and Kerr black holes. Then, the corrected Hawking temperature bosons. In addition, we found that the GUP-corrected Hawking temperature is smaller than the original case, where it stops increasing when the mass of the black hole reaches the minimal value M Res , which is in the order of the higher dimensional Planck mass M f .
In this study, we employed the GUP framework given by Eqs (1) and (2), but several alternative forms of the GUP can be employed to study the tunneling of particles from a black hole horizon. For example, p i = p 0i (1 + β 0 M 2 p p 2 ) was employed by [51-53, 56, 58]. Different GUP forms may yield different results and further research is needed to clarify this issue.