The Bekenstein-Hawking Corpuscular Cascading from the Backreacted Black Hole

Exciting peculiarities of Planck-scale physics have immediate effects on the Bekenstein-Hawking radiation emitted from black holes (BHs). In this paper, using the tunneling formalism, we determine the Bekenstein-Hawking temperature for the vector particles from a backreacted black hole (BBH) constructed from a conformal scalar field surrounded by a BTZ (Banados-Teitelboim-Zanelli) BH. Then, under the effect of the generalized uncertainty principle, we extend our calculations for scalar particles to understand the effects of quantum gravity. Then, we calculate an evaporation time for the BBH, the total number of Bekenstein-Hawking particles, and the quantum corrections of the number. We observe that remnants of the BH evaporation occur and that they affect the Bekenstein-Hawking temperature of the BBH as well as the total number of Bekenstein-Hawking particles.


I. INTRODUCTION
It is a remarkable fact that according to the seminal works of Hawking [15,16], BHs are not entirely black.That was the surprising claim made by Hawking over forty years ago.Examining the behavior of quantum fluctuations around the event horizon of a BH, Hawking substantiated the theory that BHs radiate thermal radiation, with a constant temperature so-called Hawking temperature directly proportional to the surface gravity κ, which is the gravitational acceleration experienced at the BH's horizon [16,45]: When black holes evaporates, theirs temperature is increased adiabatically as a function of the remaining mass.Quantum fluctuations create a virtual particle pair near the BH horizon.While the particle with negative energy tunnels into the horizon (absorption), the other one having positive energy flies off into spatial infinity (emission) and produces the HR.There are various methods to calculate Hawking radiation, such that two kinds of tunneling methods, null geodesic method and Hamilton-Jacobi (HJ) method, are the most popular ones [2, 8, 12, 14, 17, 21, 23-25, 27-30, 33, 34, 40, 42-44, 46-59].Both of these two approaches to tunneling use the fact that by applying the WKB approximation, the emission and absorption probabilities of the tunneling particles give the tunneling rate Γ as [10,11,[18][19][20]41] where S is the action of the classically forbidden trajectory of the tunneling particle, which has a net energy * ali.ovgun@emu.edu.trE net and temperature T .One of the methods for finding S is to use the HJ method.This method is generally performed by substituting a suitable ansatz, considering the symmetries of the spacetime, into the relativistic HJ equation [4,20,29,30,39,40,54].The resulting radial integral always possesses a pole located at the event horizon.However, using the residue theory the associated pole can be analytically evaded.
Here we use the backreacted black hole (BBH) which is a BTZ black hole surrounded by the conformal scalar field [9,26,36,37].Our motivation to work on 3 dimensions is to make the problem much much easier.Firstly, there are no propagating degrees of freedom.A quantum gravity in 3 dimensions is renormalizable and finite.Let us find out what will happen with Hawking temperature of scalar and vector particles of the BBH.
This paper is organized as follows.Sec. 2 we introduces the geometrical and thermodynamical features of the BBH spacetime.In Sec. 3, we study the Proca equation for a massive boson in this geometry.Then, we employ the HJ method with the separation of variables technique to obtain the HR of the BBH.Then, in Sec. 4, we repeat the calculations for the radiating scalar particles under the effect of the quantum gravity.We compute the corrected Hawking temperature.Last but not least, in Sec. 5, we obtain the total number of outgoing Hawking particles and the paper ends with our conclusion in Sec. 6.

II. BACKREACTED BLACK HOLE (BBH)
The BBH is constructed from the conformal scalar field surrounded BTZ black hole.For this reason, one calculated the stress-energy tensor < T µν > by considering the transparent boundary conditions at infinity [26,36,37].The semiclassical equations are used to find the O( ) correction to the BBH geometry.
The exact solution metric for BBH in the presence of a conformally coupled scalar field in three dimensions can be expressed as [36]: where Note that F (M ) is [36]: where δ is an arbitrary phase and the cosmological constant Λ = 1 l 2 .It is noted that the M is mass of the BBH, G = 1/8, l p = /8 is the length of the Planck, the Plank mass m p = /l p = 8.The BTZ BH is recovered where F (M ) = 0 at M ≫ 1.Furthermore, the metric has an event horizon which is located at

III. HR OF VECTOR PARTICLES FROM BBH
To calculate the Hawking radiation of vector particles tunneling from BBH, the Proca equation is used on the BBH geometry.The massive vector particles is described by the Proca equation with the wave function ψ is given by [29,30,[47][48][49][50], in which It is assumed that the vector function is with the action Using the WKB approximation, the action can be choosen as Here the energy is defined by E and the angular momentum of the spin-1 vector particles is defined by J, furthermore k is a constant.Now Eqs. ( 15), ( 16), (17), and ( 18) are substituted into Eq.( 14) and considering the leading order in .Then 3 × 3 matrix (let us say matrix) is obtained: (c 1 , c 2 , c 3 ) T = 0.It is noted that the superscript T means the transition to the transposed vector.So, the non-zero components of the matrix of are calculated as follows The condition of the finding nontrivial solutions of any linear equations ( det = 0 ) gives (15) Then the solution of W (r) yields It is noted that the W + (r) and W − (r) shows that the vector particles move away from the BBH and move towards to the BBH.Moreover, there are poles located at horizon and the imaginary part of W ± (r) can be calculated by using the complex path integration method.[50][51][52] Then, the integral becomes and so the vector particles tunnels through the horizon out/in with the probabilities of The ingoing vector particles must have the P absorption = 1 which means that their chance to fall inside is 100% in agreement with the definition of the BH [49][50][51].Consequently, we can choose the Imk = −ImW − , then it becomes W + = −W − and we can calculate the tunneling rate of the vector particles as (20) Note that the Hawking temperature of the BBH is recovered by using relation between the tunneling rate and the Boltzman factor e − E T .Hence, the Hawking temperature of BBH is [26,36,37]

IV. HR OF SCALAR PARTICLES FROM BBH WITH THE EFFECT OF THE QUANTUM GRAVITY
One common feature among various quantum gravity theories, such as string theory, loop quantum gravity, and noncommutative geometry, is the existence of a minimum measurable length [1,3,6,7,13,22,31,32,35].An effective model to realize the minimal length is the generalized uncertainty principle (GUP), based on which the first generalized uncertainty relation was proposed by [22] .
In this section, our aim is to find the effect of quantum gravity on the hawking radiation by using the GUP which describes the minimum measurable length.Firstly, the commutation relation is modified by using the quantum gravity [1,13,22,41] and the GUP is derived as follows It is noted that α = α 0 /(m 2 p ) = l 2 p / 2 is a small value, m p = lp is the Planck mass, l p is the Planck length (∼ 10 −35 m) and α 0 < 10 34 is a dimensionless parameter.
Quantum gravity effects the KG equation which is the relativistic wave equation for the scalar particles, because the position, momentum and energy operators are modified due to the GUP respectively as follows and Furthermore, the frequency is also generalized as with the energy operator E = i ∂ 0 .One can calculate the square of momentum operators upto order α 2 as ), (28) where in the last step, we only keep the leading order term of α.Therefore, the generalized KG equation with the wave function Ψ can be written as [41] Herein, we substitute the ansatz for the semiclassical wave function Ψ of the scalar particles Ψ = Ce i S(t,r,θ) (30) where C is the constant, into the generalized KG equation (Eq.29) with the BBH metric Eq.( 4) which is the background of scalar particle motion.Then, the differential equation for the action S is calculated as follows The separation of variables are used to solve the generalized KG equation after the Eq.( 31) is expanded into the lowest order of where ̺is the constant.Then we substitute Eqn.(32) into Eq.( 31) to solve for the W(r).Then the radial part of the scalar wave function is found that where the positive and negative ± signatures are for the outgoing and ingoing scalar particles.To solve this integral, after using the residue method around the pole at the horizon we obtain the solution Herein, similarly to the previous section, we use the fact that the probability of ingoing particles to 100% (P absorption = 1).Thus, the tunneling rate is calculated for the scalar particles with the effect of the quantum gravity as Now, it is easy to recover Hawking temperature for the scalar particles with the effect of quantum gravity It is easilty observed that when we choose α = 0, it is equel to the original result of Hawking Temperature.Hence, the Hawking radiation of BBH with the effect of the quantum gravity has remnants.

V. TOTAL NUMBER OF TUNNELING HAWKING PARTICLES
In this section, we calculate the estimation of the total number of quanta emitted by the BBH.One shows that the total number of quanta emited by the BBH is proportional to the square of the BBH's initial mass in Planck units.Firstly we introduce the Planck's law of black-body radiation to calculate the spectral luminosity density of an ideal black body as follows ( = G = k B = c = 1.) [5,38] Note that E, A and T are the energy, surface area and the temperature, respectively.Here the number of radiation degrees of freedom can be choosen as g = 1 for a scalar spin-0 particles and g = 2 for spin-1 photons or spin-2 gravitons.For g = 2 for the photons, the result of the integration of the Eq.( 37) is the Stefan-Boltzmann law which is stated that the power emited per unit area of the surface of a black hole is directly proportional to the 4th power of its temperature [5,38].After we take integral of the Eq.( 37) , the luminosity is found as Then the emission rate of the emitted quanta is obtained as which ζ(x) stands for the Riemann zeta function.Now, we recall the Hawking temperature of the BBH and the area of the BBH as Once shows that the mass loss rate is related with the luminosity as follows For simplicity we choose F (M ) = 0 which reduces the problem to the BTZ geometry and then the evaporation time of the BTZ BH is obtained as After that we calculate the emission rate of the Hawking particles The total number of the outgoing Hawking particles are obtained by following relation [5,38] and it is found as Note that it does not depend on the spin of the particles so both vector and scalar particles radiating from bh with the same number of particles [5,38].
For the case of scalar particles with the effect of quantum gravity, the total number of the outgoing Hawking particles is The total number of Hawking particles (N ) increases with the effect of the quantum gravity constant α.However, at a some point N becomes zero and no particles are emitted.

VI. CONLUSION
In this paper, firstly by using the generalized Klein-Gordon equation and the Proca equation we investigated the scalar/vector particle's tunneling from BBH.The generalized uncertainity principle and application on the fields are used to derive corrected Hawking radiation with the help of Hamilton-Jacobi method.Scalar and vector particles radiate from the BBH with an equavelent energy.Thus the corrected temperature with the effect of the quantum gravity decreases and at some points rem-nants are left.Then we calculate the total number of emitted Hawking particles from the special case of BBH which is BTZ BH.Also we check the quantum gravity effects on the total number of the emitted particles from the BH.It is shown that the emitted Hawking particles are an information-carrying units.This indicates that it is a corpuscular interpretation instead of an undulatory one and when black hole collapses unitarity is preserved, and followingly evaporates.Hence, the effect of the quantum gravity balances the classical temperature rising tendency and there exists the remnants.