We reformulate the Hamilton-Jacobi tunneling method for calculating Hawking radiation in static, spherically-symmetric spacetimes by explicitly incorporating a preferred family of frames. These frames correspond to a family of observers tied to a locally static timelike Killing vector of the spacetime. This formulation separates the role of the coordinates from the choice of vacuum and thus provides a coordinate-independent formulation of the tunneling method. In addition, it clarifies the nature of certain constants and their relation to these preferred observers in the calculation of horizon temperatures. We first use this formalism to obtain the expected temperature for a static observer at finite radius in the Schwarzschild spacetime. We then apply this formalism to the Schwarzschild-de Sitter spacetime, where there is no static observer with 4-velocity equal to the static timelike Killing vector. It is shown that a preferred static observer, one whose trajectory is geodesic, measures the lowest temperature from each horizon. Furthermore, this observer measures horizon temperatures corresponding to the well-known Bousso-Hawking normalization.
In the present paper, the Hawking radiation of massive bosons from 4-dimensional and 5-dimensional black strings are studied in quantum tunneling formalism. Firstly, we derive the Hamilton-Jacobi equation set via the Proca equation and WKB approximation. Then, the tunneling rates and Hawking temperature of the black strings are obtained. Our calculations show that the tunneling rates and Hawking temperatures are related to the properties of black strings' spacetime. When compare our results with those of scalars and fermions cases, it finds that they are the same.
We study the perception of Hawking radiation by different observers outside a black hole. The analysis is done in terms of an effective-temperature function that varies along the trajectory of each observer. The vacuum state of the radiation field is chosen to be non-stationary, so as to mimic the switching-on of Hawking radiation that would appear in a real black hole collapse. We analyse how this vacuum is perceived by observers staying at a fixed radius, by observers coming in free-fall from radial infinity at different times, and by observers in free-fall released from finite radial positi…
We investigated Dirac Particles' Hawking radiation from event horizon of Kerr black hole in terms of the tunneling formalism. Applying WKB approximation to the general covariant Dirac equation in Kerr spacetime background, we obtain the tunneling probability for fermions and Hawking temperature of Kerr black hole. The result obtained by taking the fermion tunneling into account is consistent with the previous literatures.
In this letter, we derive the Hawking temperature in the tunneling method by considering the system is open to the environment. This consideration yields the temperature which is higher than the standard one. The correction of temperatures depends on spectral density of bath oscillators which is responsible for dissipation effect in our analysis.
In this paper, we study Hawking radiation as a massless particles tunneling process across the event horizon from the Schwarzschild and Reissner-Nordström black holes pierced by an infinitely long spinning cosmic string and a global monopole. Applying the WKB approximation and using a generalized Painlevé line element for stationary axisymmetric spacetimes, also by taking into account that the ADM mass of the black hole decreases due to the presence of topological defects, it is shown that the Hawking temperature remains unchanged for these black holes. The tunneling of charged massive particl…
We use a method recently introduced in Barceló et al (2011 Phys. Rev. D 83 41501), to analyse Hawking radiation in a Schwarzschild black hole as perceived by different observers in the system. The method is based on the introduction of an 'effective temperature' function that varies with time. First we introduce a non-stationary vacuum state for a quantum scalar field, which interpolates between the Boulware vacuum state at early times and the Unruh vacuum state at late times. In this way we mimic the process of switching on Hawking radiation in realistic collapse scenarios. Then, we analyse t…
Recently, it has been suggested that Hawking radiation can be derived from quantum tunnelling methods. In this letter, we calculated Hawking temperature of dilatonic black holes from tunnelling formalism. The two semi-classical methods adopted here are: the null-geodesic method proposed by Parikh and Wilczek and the Hamilton-Jacobi method propsed by Angheben et al. We apply the two methods to anylysis the Hawking temperature of the static spherical symmetric dilatonic black hole, the rotating Kaluza-Klein black hole, and the rotating Kerr-Sen black hole.
In the past, Hawking radiation was viewed as a tunneling process and the barrier was just created by the outgoing particle itself. In this paper, Parikh's recent work is extended to the case of massive particles' tunneling. We investigate the behavior of the tunneling massive particles from a particular black hole solution-G.H Dilaton black hole which is obtained from the string theory, and calculate the emission rate at which massive particles tunnel across the event horizon. We obtain that the result is also consistent with an underlying unitary theory. Furthermore, the result takes the same…
Hawking radiation from Unruh's and Canonical acoustic black hole is considered from viewpoint of anomaly cancellation method developed by Robinson and Wilczek. Thus, the physics near the horizon can be described using an infinite collection of massless two-dimensional scalar fields in the background of a dilaton and the gravitational anomaly is canceled by the flux of a 1 + 1 dimensional blackbody at the Hawking temperature of the space-time. Consequently, by this method, we can get the Hawking's temperature for Canonical and Unruh's acoustic black hole.
The discussions of Hawking radiation via tunneling method have been performed extensively in the case of scalar particles. Moreover, there are also several works in discussing the tunneling method for Hawking radiation by using higher spins, e.g. neutrino, photon, and gravitino, in the background of static black holes. Interestingly, it is found that the Hawking temperature for static black holes using the higher spins particles has no difference compared to the one computed using scalars. In this paper, we study the Hawking radiation for a spherically symmetric and time dependent black holes …
Planck scale corrections arising from deformed special relativity on Hawking radiation in Parikh and Wilczk's tunneling framework are studied. We calculate the emission rate of massless particles tunneling though the corrected horizon of modified black holes from gravity's rainbow. In the tunneling process, when a particle get across the quantum horizon, the metric fluctuation not only due to the energy conservation but also quantum effects of the space-time are taken into account. Our results show that, the emission rate is related to the changes of the black hole's quantum corrected entropy …
Given a field vacuum state in a black hole spacetime, this state can be analysed in terms of how it is perceived (in terms of particle content) by different observers. This can be done by means of the effective-temperature function introduced by Barceló et al. in [1]. In Barbado et al. [2], this function was analysed in a case by case basis for a number of interesting situations. In this work, we find a general analytic expression for the effective-temperature function which, apart from the vacuum state choice, depends on the position, the local velocity and the acceleration of the specific ob…
We study the behaviour of scalar fields on background geometries which undergo quantum tunneling. The two examples considered are a moving mirror in flat space which tunnels through a potential barrier, and a false vacuum bubble which tunnels to form a black hole. WKB approximations to the Schrodinger and Wheeler-DeWitt equations are made, leading one to solve field equations on the Euclidean metric solution interpolating between the classically allowed geometries. The state of the field after tunneling can then be determined using the method of non-unitary Bogolubov transformations developed …
Hawking radiation spectrum via fermions tunneling is investigated through horizon radii of Pleba$\acute{\textmd{n}}$ski-Demia$\acute{\textmd{n}}$ski family of black holes. To this end, we determine the tunneling probabilities for outgoing and incoming charged fermion particles and obtain their corresponding Hawking temperatures. The graphical behavior of Hawking temperatures and horizon radii (cosmological and event horizons) is also studied. We find consistent results with those already available in literature.
Massive particles are radiated from black holes through the Hawking mechanism together with the more familiar radiation of massless particles. For $E >= m$, the emission rate is identical to the massless case. But $E < m$ particles can also tunnel across the horizon. A study of the dispersion relation and wave packet simulations show that their classical trajectory is similar to that of a boomerang. The tunneling formalism is used to calculate the probability for detecting such $E < m$ particles, for a Schwarzschild black hole of astrophysical size or in an analogue gravity experiment, as a fu…
We study tunneling process through quantum horizon of a Schwarzschild black hole in noncommutative spacetime. This is done by considering the effect of smearing of the particle mass as a Gaussian profile in flat spacetime. We show that even in this noncommutative setup there will be no correlation between the different modes of radiation which reflects the fact that information doesn't come out continuously during the evaporation process at least at late-time. However, due to spacetime noncommutativity, information might be preserved by a stable black hole remnant.
We revisit the tunneling picture for the Hawking effect in light of the charged Nariai manifold, because this general relativistic solution, which displays two horizons, provides the bonus to allow the knowledge of exact solutions of the field equations. We first perform a revisitation of the tunneling ansatz in the framework of particle creation in external fields a la Nikishov, which corroborates the interpretation of the semiclassical emission rate Gamma as the conditional probability rate for the creation of a couple of particles from the vacuum. Then, particle creation associated with the…
In the paper [arXiv:0809.1554], the scalar particles' Hawking radiation from the apparent horizon of Friedmann-Robertson-Walker(FRW) universe was investigated by using the tunneling formalism. They obtained the Hawking temperature associated with the apparent horizon, which was extensively applied in investigating the relationship between the first law of thermodynamics and Friedmann equations. In this paper, we calculate Fermions' Hawking radiation from the apparent horizon of FRW universe via tunneling formalism. Applying WKB approximation to the general covariant Dirac equation in FRW space…
We study the Hawking thermal spectrum in dragging coordinate system and the tunneling radiation characteristics of hot NUT-Kerr-Newman-Kasuya spacetime. The tunneling rates at the event and cosmological horizon are found to be related to the change of Bekenstein-Hawking entropy. The radiation spectrum is not pure thermal and thus there is a correction to the Hawking thermal spectrum.
We have studied the Hawking radiation from {\it generalized} rotating and static $(2+1)$-dimensional BTZ black holes. In this regard, we have benefited the quantum tunneling approach with WKB approximation and obtained the tunneling rate of outgoing scalar and spinor particles across the horizons. We have also obtained the Hawking temperature at the horizons corresponding to the emission of these particles. It is shown that the tunneling rate and Hawking temperature of generalized $(2+1)$-dimensional BTZ black holes are different from ordinary $(2+1)$-dimensional BTZ black holes due to the Noe…
We study Hawking radiation as a phenomenon of tunneling through event horizons of charged torus-like as well as dilaton black holes involving cosmological constant based on Kerner and Mann's formulation. We obtain tunneling probabilities as well as Hawking's emission temperature of outgoing charged particles by applying the semiclassical WKB approximation to the general covariant Dirac equation. The graphical behavior of Hawking temperature and horizon radius is investigated. We find results consistent with those already given in literature.
We obtain, using a reformulation of the tunneling mechanism, the Hawking black body spectrum with the appropriate temperature for a black hole. This is a new result in the tunneling formalism of discussing Hawking effect. Our results are given for a spherically symmetric geometry that is asymptotically flat.
We study Hawking radiation of Dirac particles with spin-$1/2$ as a tunneling process from Schwarzschild-de Sitter and Reissner-Nordström-de Sitter black holes in background spacetimes with a spinning cosmic string and a global monopole. Solving Dirac's equation by employing the Hamilton-Jacobi method and WKB approximation we find the corresponding tunneling probabilities and the Hawking temperature. Furthermore, we show that the Hawking temperature of those black holes remains unchanged in presence of topological defects in both cases.
We study the perception of the radiation phenomena of Hawking radiation and Unruh effect by using two main tools: the Unruh-DeWitt detectors and the effective temperature function (ETF), this last tool based on Bogoliubov transformations. Using the Unruh-DeWitt detectors we find an adiabatic expansion of the detection properties along linear trajectories with slowly varying acceleration in Minkowski, which allows us to calculate the spectrum detected, finding the thermal spectrum as the zeroth order contribution. Using the ETF we study the perception of Hawking radiation by observers following…
We examine Hawking radiation from a Schwarzschild black hole in several reference frames using the quasi-classical tunneling picture. It is shown that when one uses, $Γ\propto \exp(Im [\oint p dr])$, rather than, $Γ\propto \exp(2 Im [\int p dr])$, for the tunneling probability/decay rate one obtains twice the original Hawking temperature. The former expression for $Γ$ is argued to be correct since $\oint p dr$ is invariant under canonical transformations, while $\int p dr$ is not. Thus, either the tunneling methods of calculating Hawking radiation are suspect or the Hawking temperature is twic…
In this paper we investigate vector particles' Hawking radiation from a BTZ black hole. By applying the WKB approximation and the Hamilton-Jacobi Ansatz to the Proca equation, we obtain the tunneling spectrum of vector particles. The expected Hawking temperature is recovered.
We investigate the probabilities of the tunneling and the radiation spectrums of the massive spin-1 particles from arbitrary dimensional Gauss-Bonnet-Axions (GBA) Anti-de Sitter (AdS) black brane, via using the WKB approximation to the Proca spin-1 field equation. The tunneling probabilities and Hawking temperature of the arbitrary dimensional GBA AdS black brane is calculated via the Hamilton-Jacobi approach. We also compute the Hawking temperature via Parikh-Wilczek tunneling approach. The results obtained from the two methods are consistent. In our setup, the Gauss-Bonnet (GB) coupling affe…
We study the Hawking radiation in a new class of black hole solutions in the Einstein-Gauss-Bonnet theory. The black hole has been argued to have vanishing mass and entropy, but finite Hawking temperature. To check if it really emits radiation, we analyse the Hawking radiation using the original method of quantization of scalar field in the black hole background and the quantum tunneling method, and confirm that it emits radiation at the Hawking temperature. A general formula is derived for the Hawking temperature and backreaction in the tunneling approach. Physical implications of these resul…
Applying Parikh's semi-classical tunneling method, we consider Hawking radiation of the charged massive particles as a tunneling process from the Reissner-Nordstrom-de Sitter black hole with a global monopole. The result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the radiant spectrum is not a pure thermal one, but is consistent with an underlying unitary theory.
In the context of a noncommutative model of coordinate coherent states, we present a Schwarzschild-like metric for a Vaidya solution instead of the standard Eddington-Finkelstein metric. This leads to the appearance of an exact $(t - r)$ dependent case of the metric. We analyze the resulting metric in three possible causal structures. In this setup, we find a zero remnant mass in the long-time limit, i.e. an instable black hole remnant. We also study the tunneling process across the quantum horizon of such a Vaidya black hole. The tunneling probability including the time-dependent part is obta…
Based on an observer-centric methodology, we pinpoint the basic origin of the spectral Planckianity of the asymptotic Hawking modes in the conventional treatments of the evaporating horizons. By considering an observer who analyzes a causal horizon in a generic spacetime, we first clarify how the asymptotic Planckian spectrum is imposed on the exponentially redshifted Hawking modes through a geometric dispersion mechanism developed by a semiclassical environment which is composed by all the modes that build up the curvature of the causal patch of the asymptotic observer. We also discuss the ac…
A white hole (WH) is a time-reversed black hole (BH) solution in General relativity with a spacetime region to which cannot be entered from the outside. Recently they have been proposed as a possible solution to the information loss problem [Haggard and Rovelli, 2015]. In particular it has been argued that the quantization of the gravitational field may prevent a BH from collapsing entirely with an exponential decay law associated to the black-hole-to-white-hole (BHWH) tunneling scenario [Barcelo, Carballo, and Garay, 2017]. During this period of BHWH transition the Hawking radiation should ta…
The boundary of any observer's spacetime is the boundary that divides what the observer can see from what they cannot see. The boundary of an observer's spacetime in the presence of a black hole is not the true (future event) horizon of the black hole, but rather the illusory horizon, the dimming, redshifting surface of the star that collapsed to the black hole long ago. The illusory horizon is the source of Hawking radiation seen by observers both outside and inside the true horizon. The perceived acceleration (gravity) on the illusory horizon sets the characteristic frequency scale of Hawkin…
The role of chirality is discussed in unifying the anomaly and the tunneling formalisms for deriving the Hawking effect. Using the chirality condition and starting from the familiar form of the trace anomaly, the chiral (gravitational) anomaly, manifested as a nonconservation of the stress tensor, near the horizon of a black hole, is derived. Solution of this equation yields the stress tensor whose asymptotic infinity limit gives the Hawking flux. Finally, use of the same chirality condition in the tunneling formalism gives the Hawking temperature that is compatible with the flux obtained by a…
We calculated the Dirac Particles' Hawking radiation from the outer horizon of BTZ black hole via tunneling formalism. Applying WKB approximation to the Dirac equation in (2+1) dimensional BTZ spacetime background, we obtain the radiation spectrum for fermions and Hawking temperature of BTZ black hole. The results obtained by taking the fermion tunneling into account is consistent with the previous literatures.
Hawking radiation is studied for arbitrary scalars, fermions, and spin-1 bosons, using a tunneling approach, to every order in $\hbar$ but ignoring back-reaction effects. It is shown that the additional quantum terms yield no new contribution to the Hawking temperature. Indeed, it is found that the limit of small $\hbar$ in the standard quantum WKB approximation is replaced by the near-horizon limit in the gravitational WKB approach.
In this letter, we use the Parikh-Wilczek tunneling model of Hawking radiation to illustrate that a reformulation of Verlinde's entropic gravity is needed to derive the Newton's law for a temperature-varying screen, demanded by the conservation of energy. Furthermore, the entropy stored in the holographic screen is shown to be additive and its temperature dependence can be obtained.
It was found that, in an isotropic coordinate system, the tunneling approach brings a factor of 1/2 for the Hawking temperature of a Schwarzschild black hole. In this paper, we address this kind of problem by studying the relation between the Hawking temperature and the deformation of integral contour for the scalar and Dirac particles tunneling. We find that correct Hawking temperature can be obtained exactly as long as the integral contour deformed corresponding to the radial coordinate transform if the transformation is a non-regular or zero function at the event horizon.
We present a derivation of Hawking radiation through tunneling mechanism for a general class of asymptotically flat, spherically symmetric spacetimes. The tunneling rate $Γ\sim \exp{(ΔS)}$ arises as a consequence of the first law of thermodynamics, TdS=dE + PdV. Therefore, this approach demonstrates how tunneling is intimately connected with the first law of thermodynamics through the principle of conservation of energy. The analysis is also generally applicable to any reasonable theory of gravity so long as the first law of thermodynamics for horizons holds in the form, TdS=dE + PdV.
Recently, Angheben et.al. [hep-th/0503081] have presented a refined method for calculating the (tree-level) black hole temperature by way of the tunneling paradigm. In the current letter, we demonstrate how their formalism can be suitably adapted to accommodate the (higher-order) effects of the gravitational back-reaction.
We study time-dependant Hawking-like radiation as seen by an infalling observer during gravitational collapse of a thin shell. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in Eddington-Finkelstein coordinates. We solve the equations for the whole process from the beginning of the collapse till the moment when the collapsing shell reaches zero radius. The radiation distribution is not thermal in the whole frequency regime, but it is approximately thermal for the wavelengths of the order of the Schwarzschild radius of …
We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is not precisely thermal. We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor.
A real tunneling solution is an instanton for the Hartle-Hawking path integral with vanishing extrinsic curvature (vanishing ``momentum'') at the boundary. Since the final momentum is fixed, its conjugate cannot be specified freely; consequently, such an instanton will contribute to the wave function at only one or a few isolated spatial geometries. I show that these geometries are the extrema of the Hartle-Hawking wave function in the semiclassical approximation, and provide some evidence that with a suitable choice of time parameter, these extrema are the maxima of the wave function at a fix…
We use the tunneling formalism to calculate the Hawking radiation of massive particles. For E>=m, we recover the traditional result, identical to the massless case. But E<m particles can also tunnel across the horizon in a Hawking process. We study the probability for detecting such E<m particles as a function of the distance from the horizon and the energy of the particle in the tunneling formalism. We derive a general formula and obtain simple approximations in the near-horizon limit and in the limit of large radii.
We discuss Hawking radiation from a five-dimensional squashed Kaluza-Klein black hole on the basis of the tunneling mechanism. A simple manner, which was recently suggested by Umetsu, is possible to extend the original derivation by Parikh and Wilczek to various black holes. That is, we use the two-dimensional effective metric, which is obtained by the dimensional reduction near the horizon, as the background metric. By using same manner, we derive both the desired result of the Hawking temperature and the effect of the back reaction associated with the radiation in the squashed Kaluza-Klein b…
We consider coordinate systems adapted to accelerated observers, construct a generalised Rindler metric and then adopt a specific form of it to compute the semiclassical Unruh temperature, using a WKB approximation within a Hamilton Jacobi analysis in the tunneling picture. Corrections to this temperature are next computed by going beyond the usual WKB approximation. A connection of the corrected Unruh temperature with the Hawking temperature is established. This connection is also explained through the method of Global Embedding Minkowski Spacetime (GEMS).
Hawking radiation (HR) of massive vector particles from a rotating Warped Anti-de Sitter black hole in 2+1 dimensions (WAdS$_{\text{3}}$BH) is studied in detail. The quantum tunneling approach with the Hamilton-Jacobi method (HJM) is applied in the Proca equation (PE), and we show that the radial function yields the tunneling rate of the outgoing particles. Comparing the result obtained with the Boltzmann factor, we satisfactorily reproduce the Hawking temperature (HT) of the WAdS$_{\text{3}}$BH.
We investigate the Hawking radiation of Schwarzschild-de Sitter (SdS) black hole by massive particles tunneling method. We consider the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles and show that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum when energy and angular momentum are conserved. Our result is also in accordance with Parikh and Wilczek\rq s opinion and gives a correction to the Hawking radiation of SdS black hole.
We investigate Hawking radiation from black holes in (d+1)-dimensional anti-de Sitter space. We focus on s-waves, make use of the geometrical optics approximation, and follow three approaches to analyze the radiation. First, we compute a Bogoliubov transformation between Kruskal and asymptotic coordinates and compare the different vacua. Second, following a method due to Kraus, Parikh, and Wilczek, we view Hawking radiation as a tunneling process across the horizon and compute the tunneling probablility. This approach uses an anti-de Sitter version of a metric originally introduced by Painleve…
I consider the problem of reconciling "tunneling" approaches to black-hole radiation with the treatment by quantum field theory in curved space-time. It is not possible to do this completely, but using what appears to be the most direct and natural correspondence, the simplest case of such an approach does not describe a tunneling process but rather a WKB approximation to a two-point function. What it computes is a sort of rescaled Unruh temperature associated with world-lines tracing Killing trajectories near the horizon. This temperature is numerically equal to the Hawking temperature, but a…
Hawking radiation viewed as a semiclassical tunneling process from the event horizon of the (2 + 1)-dimensional rotating BTZ black hole is carefully reexamined by taking into account not only the energy conservation but also the conservation of angular momentum when the effect of the emitted particle's self-gravitation is incorporated. In contrast to previous analysis of this issue in the literature, our result obtained here fits well to the Kraus-Parikh-Wilczek's universal conclusion without any modification to the Bekenstein-Hawking area-entropy formulae of the BTZ black hole.
In the frame of Hamilton-Jacobi method, the back-reactions of the radiating particles together with the total entropy change of the whole system are investigated. The emission probability from this process is found to be equivalent to the null geodesic method. However its physical picture is more clear: the negative energy one of a virtual particle pair is absorbed by the black hole, resulting in the temperature, electric potential and angular velocity increase; then the black hole amount of heat, electric charge and angular momentum can spontaneously transfer to the positive energy particle; …
We study the Hawking radiation (HR) of scalar and Dirac particles (fermions) emitted from a rotating scalar hair black hole (RSHBH) within the context of three dimensional ($3D$) Einstein gravity using non-minimally coupled scalar field theory. Amalgamating the quantum tunneling approach with the Wentzel--Kramers--Brillouin (WKB) approximation, we obtain the tunneling rates of the outgoing particles across the event horizon. Inserting the resultant tunneling rates into the Boltzmann formula, we then obtain the Hawking temperature ($T_{H}$) of the $3D$ RSHBH.
Recent work, which treats the Hawking radiation as a semi-classical tunneling process at the horizon of the Schwarzschild and Reissner-Nordstrom spacetimes, indicates that the exact radiant spectrum is no longer pure thermal after considering the black hole background as dynamical and the conservation of energy. In this paper, we extend the method to investigate Hawking radiation as massless particles tunneling across the event horizon of the Kerr black hole and that of charged particles from the Kerr-Newman black hole by taking into account the energy conservation, the angular momentum conser…
We examine the possibility of artificial Hawking radiation by proposing a non-PT - symmetric weakly pseudo-Hermitian two band model containing a tilting parameter. We also determine the tunneling probability using our Hamiltonian through the event horizon that acts as a classically forbidden barrier.
Applying the Hamilton--Jacobi method we investigate the tunneling of photon across the event horizon of a static spherically symmetric black hole. The necessity of the gauge condition on the photon field, to derive the semiclassical Hawking temperature, is explicitly shown. Also, the tunneling of photon and gravitino beyond this semiclassical approximation are presented separately. Quantum corrections of the action for both cases are found to be proportional to the semiclassical contribution. Modifications to the Hawking temperature and Bekenstein-Hawking area law are thereby obtained. Using t…
