Hawking radiation from black hole horizon can be viewed as a quantum tunnelling process, and fermions via tunnelling can successfully recover Hawking temperature. In this paper, considering the tunnelling particles with spin 1/2 (namely, Dirac particles), we further improve Kerner and Man's fermion tunnelling method to study Hawking radiation via tunnelling from rotating black holes in de Sitter spaces, specifically including that from Kerr de Sitter black hole and Kerr-Newman de Sitter black hole. As a result, Hawking temperatures at the event horizon (EH) and the cosmological horizon (CH) are well described via Dirac particles tunnelling.
Recent research shows that Hawking radiation can be treated as a quantum tunnelling process, and Hawking temperature of Dirac particles across the horizon of a black hole can be correctly recovered via fermions tunnelling method. In this paper, motivated by fermions tunnelling method, we attempt to apply the analysis to derive Hawking radiation of Dirac particles via tunnelling from black ring solutions of 5-dimensional Einstein-Maxwell-dilaton gravity theory. Finally, it is interesting to find as in black hole cases, fermions tunnelling can also result in correct Hawking temperatures for the …
Thermal radiation of electrically charged fermions from rotating black hole with electric and magnetic charges in de Sitter space is considered. The tunnelling probabilities for outgoing and incoming particles are obtained and the Hawking temperature is calculated. The relation for the classical action for the particles in the black hole's background is also found.
It has recently been suggested that the attempt to understand Hawking radiation as tunnelling across black hole horizons produces a Hawking temperature double the standard value. It is explained here how one can obtain the standard value in the same tunnelling approach.
In this paper, with the Parikh-Wilczek tunnelling framework the positions of the event horizon of the Vaidya black hole and the Vaidya-Bonner black hole are calculated respectively. We find that the event horizon and the apparent horizon of these two black holes correspond respectively to the two turning points of the Hawking radiation tunnelling barrier. That is, the quantum ergosphere coincides with the tunnelling barrier. Our calculation also implies that the Hawking radiation comes from the apparent horizon.
Attempts to understand Hawking radiation as tunnelling across a black hole horizon require the consideration of singular integrals. Although Schwarzschild coordinates lead to the standard Hawking temperature, isotropic radial coordinates may appear to produce an incorrect value. It is demonstrated here how the proper regularization of singular integrals leads to the standard temperature for the isotropic radial coordinates as well as for other smooth transformations of the radial variable, which of course describe the same black hole.
Kerner and Mann's recent work shows that, for an uncharged and non-rotating black hole, its Hawking temperature can be exactly derived by fermions tunnelling from its horizons. In this paper, our main work is to improve the analysis to deal with charged fermion tunnelling from the general dilatonic black holes, specifically including the charged, spherically symmetric dilatonic black hole, the rotating Einstein-Maxwell-Dilaton-Axion (EMDA) black hole and the rotating Kaluza-Klein (KK) black hole. As a result, the correct Hawking temperatures are well recovered by charged fermions tunnelling fr…
Parikh-Wilczek tunnelling framework, which treats Hawking radiation as a tunnelling process, is investigated again. As the first order correction, the log-corrected entropy-area relation naturally emerges in the tunnelling picture if we consider the emission of a spherical shell. The second order correction of the emission rate for the Schwarzschild black hole is calculated too. In this level, the result is still in agreement with the unitary theory, however, the entropy of the black hole will contain three parts: the usual Bekenstein-Hawking entropy, the logarithmic term and the inverse area …
We apply the tunnelling method for the emission and absorption of Elko particles in the event horizon of a black string solution. We show that Elko particles are emitted at the expected Hawking temperature from black strings, but with a quite different signature with respect to the Dirac particles. We employ the Hamilton-Jacobi technique to black hole tunnelling, by applying the WKB approximation to the coupled system of Dirac-like equations governing the Elko particle dynamics. As a typical signature, different Elko particles are shown to produce the same standard Hawking temperature for blac…
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…
We give an extensive study of the tunnelling between arbitrary (anti-)de Sitter spacetimes separated by an infinitesimally thin relativistic shell in arbitrary spacetime dimensions. In particular, we find analytically an exact expression for the tunnelling amplitude. The detailed spacetime structures that can arise are discussed, together with an effective "regularization scheme" for "before tunnelling" configurations.
We consider the tunnelling of charged spin-(1/2) fermions from a Kerr-Newman black hole and demonstrate that the expected Hawking temperature is recovered. We discuss certain technical subtleties related to the obtention of this result.
We apply the covariant anomaly cancellation method to compute the Hawking fluxes from the event and cosmic horizons of the Schwarzschild-de Sitter black hole. The derivation is new from the existing ones as we split the space in three different regions (near to and away from the event and cosmic horizons) and write down the covariant energy-momentum tensor using three step functions which covers the whole region leading elegantly to the conditions required to compute the Hawking fluxes from the event and cosmic horizons.
Instantons and their quantisation in pure Yang-Mills theory formulated in the background of de Sitter spacetime represented by spatially-closed ($k = 1$) Friedmann-Robertson-Walker metric are discussed. As for the classical treatment of the instanton physics, first, explicit instanton solutions are found and next, quantities like Pontryagin index and the semiclassical approximation to the inter-vacua tunnelling amplitude are evaluated. The Atiyah-Patodi-Singer index theorem is checked as well by constructing explicitly the normalizable fermion zero modes in this de Sitter spacetime instanton b…
We calculate corrections to the Bekenstein-Hawking entropy formula for the five-dimensional topological AdS (TAdS)-black holes and topological de Sitter (TdS) spaces due to thermal fluctuations. We can derive all thermal properties of the TdS spaces from those of the TAdS black holes by replacing $k$ by $-k$. Also we obtain the same correction to the Cardy-Verlinde formula for TAdS and TdS cases including the cosmological horizon of the Schwarzschild-de Sitter (SdS) black hole. Finally we discuss the AdS/CFT and dS/CFT correspondences and their dynamic correspondences.
We investigate the tunnelling of spin 1/2 particles through event horizons. We first apply the tunnelling method to Rindler spacetime and obtain the Unruh temperature. We then apply fermion tunnelling to a general non-rotating black hole metric and show that the Hawking temperature is recovered.
Recently, Hawking radiation has been treated, by Robinson and Wilczek, as a compensating flux of the energy momentum tensor required to cancel gavitational anomaly at the event horizon(EH) of a Schwarzschild-type black hole. In this paper, motivated by their work, Hawking radiation from the event horizon(EH) and the de Sitter cosmological horizon(CH) of black holes in de Sitter spaces, specifically including the purely de Sitter black hole, and the static, spherically symmetric Schwarzschild-de Sitter black hole as well as the rotating Kerr-de Sitter black hole, has been studied by anomalies. …
We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a $2+1$-dimensional spacetime, extending the work of Moretti and Pinamonti in a $3+1$-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the $3+1$-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state ex…
In an asymptotically flat spacetime of dimension d > 3 and with the Newtonian gravitational constant G, a spherical black hole of initial horizon radius r_h and mass M ~ r_h^{d-3}/G has a total decay time to Hawking emission of t_d ~ r_h^{d-1}/G ~ G^{2/(d-3)}M^{(d-1)/(d-3)} which grows without bound as the radius r_h and mass M are taken to infinity. However, in asymptotically anti-de Sitter spacetime with a length scale l and with absorbing boundary conditions at infinity, the total Hawking decay time does not diverge as the mass and radius go to infinity but instead remains bounded by a time…
In this study, the Generalized Uncertainty Principle (GUP) effect on the Hawking radiation formed by tunneling of a massive vector boson particle from the $2+1$ dimensional New-type Black Hole was investigated. We used modified massive vector boson equation based on the GUP. Then, the Hamilton-Jacobi quantum tunneling approach was used to work out the tunneling probability of the massive vector boson particle and, Hawking temperature of the black hole. Due to the GUP effect, the modified Hawking temperature was found to depend on the black hole properties, the AdS$_{3}$ radius, and on the ener…
Particle production from the vacuum is a remarkable aspect of particle physics. Prime examples are the Schwinger process of particle production in strong electric fields and the Hawking process of particle production from black holes. These processes can be viewed as quantum tunnelling of particles from the vacuum. The tunnelling approach, and the closely related instanton or complex path approaches, are reviewed here with emphasis on paths in the complex coordinate plane. The method is applied to particle production from a black hole in a magnetic field, where ultra-high energy charged partic…
The remnants are investigated by fermions' tunnelling from a 4-dimensional charged dilatonic black hole and a 5-dimensional black string. Based on the generalized uncertainty principle, effects of quantum gravity are taken into account. The quantum numbers of the emitted fermions affects the Hawking temperatures. For the black hole, the quantum gravity correction slows down the increase of the temperature, which leads to the remnant left in the evaporation. For the black string, the existence of the remnant is determined by the quantum gravity correction and effects of the extra compact dimens…
Hawking radiation of black ring solutions to 5-dimensional Einstein-Maxwell-dilaton gravity theory is analyzed by use of the Parikh-Wilczek tunnelling method. To get the correct tunnelling amplitude and emission rate, we adopted and developed the Angheben-Nadalini-Vanzo-Zerbini covariant approach to cover the effects of rotation and electronic discharge all at once, and the effect of back reaction is also taken into account. This constitute a unified approach to the tunnelling problem. Provided the first law of thermodynamics for black rings holds, the emission rate is proportional to the expo…
We revisit the Schwinger effect in de Sitter, anti-de Sitter spaces and charged black holes, and explore the interplay between quantum electrodynamics and the quantum gravity effect at one-loop level. We then advance a thermal interpretation of the Schwinger effect in curved spacetimes. Finally, we show that the Schwinger effect in a near-extremal black hole differs from Hawking radiation of charged particles in a non-extremal black hole and is factorized into those in an anti-de Sitter space and a Rindler space with the surface gravity for acceleration.
Recent work that treats the Hawking radiation as a semi-classical tunnelling process from the four-dimensional Schwarzschild and Reissner-Nordstrom black holes is extended to the case of higher dimensional Reissner-Nordstrom-de Sitter black holes. The result shows that the tunnelling rate is related to the change of Bekenstein-Hawking entropy and the exact radiant spectrum is no longer precisely thermal after considering the black hole background as dynamical and incorporating the self-gravitation effect of the emitted particles when the energy conservation and electric charge conservation are…
We extend the the concept of Hawking-Moss, or up-tunnelling, transitions in the early universe to include black hole seeds. The black hole greatly enhances the decay amplitude, however, order to have physically consistent results, we need to impose a new condition (automatically satisfied for the original Hawking-Moss instanton) that the cosmological horizon area should not increase during tunnelling. We motivate this conjecture physically in two ways. First, we look at the energetics of the process, using the formalism of extended black hole thermodynamics; secondly, we extend the stochastic …
We investigate the Generalized Uncertainty Principle (GUP) effect on the Hawking radiation of the 2+1 dimensional Martinez-Zanelli black hole by using the Hamilton-Jacobi method. In this connection, we discuss the tunnelling probabilities and Hawking temperature of the spin-1/2 and spin-0 particles for the black hole. Therefore, we use the modified Klein-Gordon and Dirac equations based on the GUP. Then, we observe that the Hawking temperature of the scalar and Dirac particles depend on not only the black hole properties, but also the properties of the tunnelling particle, such as angular mome…
The Hawking radiation spectrum of Kerr-Sen axion-dilaton black holes is derived, in the context of dark spinors tunnelling across the horizon. Since a black hole has a well defined temperature, it should radiate in principle all the standard model particles, similar to a black body at that temperature. We investigate the tunnelling of mass dimension one spin-1/2 dark fermions, that are beyond the standard model and are prime candidates to the dark matter. Their interactions with the standard model matter and gauge fields are suppressed by at least one power of unification scale, being restrict…
Using Proca equation and WKB approximation, we investigate Hawking radiation of vector particles via tunneling from 4-dimensional Kerr-de Sitter black hole and 5-dimensional Schwarzschild-Tangherlini black hole. The results show that the tunneling rates and Hawking temperatures are depended on the properties of spacetime (event horizon, mass and angular momentum). Besides, our results are the same as scalars and fermions tunneling from 4-dimensional Kerr-de Sitter black hole and 5-dimensional Schwarzschild-Tangherlini black hole.
In the context of black hole chemistry, we study the thermodynamics of asymptotically de Sitter black holes with conformal scalar hair in Einstein gravity. The hair parameter allows us to attain thermodynamic equilibrium between the event horizon and the cosmological horizon. We find that the system of the black hole and the de Sitter space surrounding it undergo a Reverse Hawking-Page phase transition provided we consider the grand-canonical ensemble.
This is my contribution to the Festschrift honoring Stephen Hawking on his 60th birthday. Twenty-five years ago, Gibbons and Hawking laid out the semi-classical properties of de Sitter space. After a summary of their main results, I discuss some further quantum aspects that have since been understood. The largest de Sitter black hole displays an intriguing pattern of instabilities, which can render the boundary structure arbitrarily complicated. I review entropy bounds specific to de Sitter space and outline a few of the strategies and problems in the search for a full quantum theory of the sp…
Hawking radiation is nowadays being understood as tunnelling through black hole horizons. Here, the extension of the Hamilton-Jacobi approach to tunnelling for non-rotating and rotating black holes in different non-singular coordinate systems not only confirms this quantum emission from black holes but also reveals the new phenomenon of absorption into white holes by quantum mechanical tunnelling. The role of a boundary condition of total absorption or emission is also clarified.
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.
We calculate the total flux of Hawking radiation from Kerr-(anti)de Sitter black holes by using gravitational anomaly method developed in gr-qc/0502074. We consider the general Kerr-(anti)de Sitter black holes in arbitrary $D$ dimensions with the maximal number [D/2] of independent rotating parameters. We find that the physics near the horizon can be described by an infinite collection of $(1+1)$-dimensional quantum fields coupled to a set of gauge fields with charges proportional to the azimuthal angular momentums $m_i$. With the requirement of anomaly cancellation and regularity at the horiz…
Incorporating Parikh and Wilczek's opinion to the Kerr de-Sitter (KdS) black hole Hawking non-thermal and purely thermal radiations have been investigated using Hamilton-Jacobi method. We have taken the background spacetime of KdS black hole as dynamical, involving the self-gravitation effect of the emitted particles, energy and angular momentum has been taken as conserved 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. The explored results gives a correction to the Hawking radiat…
We study the greybody factor and power spectra of Hawking radiation for the minimally or nonminimally coupled scalar field with exact numerical method in spherically symmetric Einstein-Gauss-Bonnet-de Sitter black hole spacetime. The effects of scalar coupling constant, angular momentum number of scalar, spacetime dimension, cosmological constant and Gauss-Bonnet coupling constant on the Hawking radiation are studied in detail. Specifically, the Gauss-Bonnet coupling constant always increases the greybody factor in the entire energy regime. Different from the case of Schwarzschild-de Sitter bl…
A modern and largely used approach to obtain Hawking radiation is the tunnelling mechanism. However, in various papers in the literature, the analysis concerned almost only to obtain the Hawking temperature through a comparison of the probability of emission of an outgoing particle with the Boltzmann factor. In a interesting and well written paper, Banerjee and Majhi improved the approach, by explicitly finding a black body spectrum associated with black holes. On the other hand, this result, which has been obtained by using a reformulation of the tunnelling mechanism, is in contrast which the…
The constant curvature (CC) black holes are higher dimensional generalizations of BTZ black holes. It is known that these black holes have the unusual topology of ${\cal M}_{D-1}\times S^1$, where $D$ is the spacetime dimension and ${\cal M}_{D-1}$ stands for a conformal Minkowski spacetime in $D-1$ dimensions. The unusual topology and time-dependence for the exterior of these black holes cause some difficulties to derive their thermodynamic quantities. In this work, by using globally embedding approach, we obtain the Hawking temperature of the CC black holes. We find that the Hawking temperat…
The interplay between a dS black hole and cosmological horizons introduces distinctive thermodynamic behavior in a dS spacetime (for example the well-known upper bounds of mass and entropy in Class. Quant. Grav. 37 (2020) 5). Based on this point, we present the Hawking-Page (HP) phase transition of the four-dimensional dS spacetime with non-linear charge correction when the effective pressure is fixed, and analyze the effects of different effective pressures and non-linear charge corrections on HP phase transition. The evolution of this dS spacetime undergoing the HP phase transition is also i…
This paper describes a phenomenon in which all but a tiny fraction of the fundamental holographic degrees of the SYK theory are confined (as in quark confinement) in the double-scaled infinite temperature limit. The mechanism for confinement is an essential ingredient in the duality between DSSYK and de Sitter space. The mechanism, which removes almost all states from the physical spectrum of the bulk de Sitter theory applies to configurations of a small number of fermions which would be expected to comprise Hawking radiation in de Sitter space. Without confinement there would be far too many …
We analyze, in the paradigm of open quantum systems, the reduced dynamics of a freely-falling two-level detector in de Sitter space-time in weak interaction with a reservoir of fluctuating quantized conformal scalar fields in the de Sitter invariant vacuum. We find that the detector is asymptotically driven to a thermal state at the Gibbons-Hawking temperature, regardless of its initial state. Our discussion therefore shows that the Gibbons-Hawking effect of de Sitter space-time can be understood as a manifestation of thermalization phenomena that involves decoherence and dissipation in open q…
We propose an experimental scheme to observe the Gibbons-Hawking effect in the acoustic analog of a 1+1-dimensional de Sitter universe, produced in an expanding, cigar-shaped Bose-Einstein condensate. It is shown that a two-level system created at the center of the trap, an atomic quantum dot interacting with phonons, observes a thermal Bose distribution at the de Sitter temperature.
We consider particle production in $1+1$ dimensional thermal Anti-de Sitter space under the influence of a constant electric field. The vacuum-persistence amplitude is given by a non-relativistic tunnelling instanton once we interpret the system as being governed by an "equivalent" non-relativistic Schrödinger equation. Working in the WKB approximation, we calculate the tunnelling rate in anti de Sitter space at finite temperature and observe that the particle production rate is enhanced. Additionally, it is observed that there is a critical temperature beyond which the production rate is affe…
Considering generalised action for dilaton coupled Maxwell-Einstein theory in four dimensions, Gao and Zhang obtained black holes solutions for asymptotically anti de Sitter (Ads) and de Sitter (ds) spacetimes. We study the Hawking radiation in Parikh-Wilczek's tunneling formalism as well as using Bogoliubov transformations. We compare the expression of the Hawking temperature obtained from these two different approaches. Stability and the extremality conditions for such black holes are discussed. The exact dependences of the Hawking temperature and flux on the dilaton coupling parameter are d…
The massive particles tunneling method has been used to investigate the Hawking non-thermal and purely thermal radiations of Schwarzschild Anti-de Sitter (SAdS) black hole. Considering the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles the imaginary part of the action has been derived from Hamilton-Jacobi equation. Using the conservation laws of energy and angular momentum we have showed that the non-thermal and purely thermal tunneling rates are related to the change of Bekenstein-Hawking entropy and the derived emission spectrum deviate…
We state and explain the total rate at which a charged non-rotating black hole emits charged particles, taking into account both Hawking radiation and Schwinger pair production simultaneously. We give concrete formulae for this emission rate in certain limits, with the greatest simplification occurring when the black hole is much larger than the particle's Compton wavelength. We provide an interpretation of the result in terms of a tunnelling process, both through the black hole horizon and the surrounding electric field, and comment on how suppression due to tunnelling modifies the emission s…
We substantiate the Hawking radiation as quantum tunneling of fields or particles crossing the horizon by using the Rindler coordinate. The thermal spectrum detected by an accelerated particle is interpreted as quantum tunneling in the Rindler spacetime. Representing the spacetime near the horizon locally as a Rindler spacetime, we find the emission rate by tunneling, which is expressed as a contour integral and gives the correct Boltzmann factor. We apply the method to non-extremal black holes such as a Schwarzschild black hole, a non-extremal Reissner-Nordström black hole, a charged Kerr bla…
We discuss the quantization of linearized gravity in the background de Sitter spacetime using a gauge-invariant formalism to write the perturbed gravitational field in the static patch. This field is quantized after fixing the gauge completely. The response rate of this field to monochromatic multipole sources is then computed in the thermal equilibrium state with the well known Gibbons-Hawking temperature. We compare this response rate with the one obtained in the Bunch-Davies-like vacuum state defined in the Poincaré patch. These response rates are found to be the same as expected. This agre…
It is shown that, for a de Sitter Universe, the Hartle-Hawking (HH) wave function can be obtained in a simple way starting from the Friedmann-Lemaitre-Robertson-Walker (FLRW) line element of cosmological equations. An oscillator having imaginary time is indeed derived starting from the Hamiltonian obtaining the HH condition. This proposes again some crucial matter on the meaning of complex time in cosmology. In order to overcome such difficulties, we propose an interpretation of the HH framework based on de Sitter Projective Holography.
The standard Hawking formula predicts the complete evaporation of black holes. Taking into account effects of quantum gravity, we investigate fermions' tunnelling from a 5-dimensional rotating black string. The temperature is determined not only by the string, but also affected by the quantum number of the emitted fermion and the effect of the extra spatial dimension. The quantum correction slows down the increase of the temperature, which naturally leads to the remnant in the evaporation.
Robinson-Wilczek's recent work, which treats Hawking radiation as a compensating flux to cancel gravitational anomaly at the horizon of a Schwarzschild-type black hole, is extended to study Hawking radiation of rotating black holes in anti-de Sitter spaces, especially that in dragging coordinate system, via gauge and gravitational anomalies. The results show that in order to restore gauge invariance and general coordinate covariance at the quantum level in the effective field theory, the charge and energy flux by requiring to cancel gauge and gravitational anomalies at the horizon, must have a…
We study whether the Hawking-Page phase transition may occur in topological de Sitter spaces (TdS) and Schwarzschild-de Sitter black hole (SdS). We show that at the critical temperature $T=T_1$, TdS with hyperbolic cosmological horizon can make the Hawking-Page transition from the zero mass de Sitter space to TdS. It is also shown that there is no Hawking-Page transition for TdS with Ricci-flat and spherical horizons, when the zero mass de Sitter space is taken as the thermal background. Also we find that the SdS undergoes a different phase transition at T=0 which the Nariai black hole is form…
We study quantum fields on spacetimes having a bifurcate Killing horizon by allowing the possibility that left- and right- (in-going and out-going) modes have different temperatures. We consider in particular the Rindler for both massless and massive fields, the static de Sitter and Schwarzschild black hole backgrounds for massive fields. We find that in all three cases, when any of the temperatures is different from the canonical one (Unruh, Hawking and Gibbons--Hawking, correspondingly) the correlation functions have extra singularities at the horizon.
Motivated by the success of the recently proposed method of anomaly cancellation to derive Hawking fluxes from black hole horizons of spacetimes in various dimensions, we have further extended the covariant anomaly cancellation method shortly simplified by Banerjee and Kulkarni to explore the Hawking radiation of the (3+1)-dimensional charged rotating black strings and their higher dimensional extensions in anti-de Sitter spacetimes, whose horizons are not spherical but can be toroidal, cylindrical or planar, according to their global identifications. It should be emphasized that our analysis …
The paper deals with non-thermal radiation spectrum by tunnelling mechanism with correction due to the generalized uncertainty principle (GUP) in the background of non-commutative geometry. Considering the reformulation of the tunnelling mechanism by Banerjee and Majhi, the Hawking radiation spectrum is evaluated through the density matrix for the outgoing modes. The GUP corrected effective temperature and the corresponding GUP corrected effective metric in non-commutative geometry are determined using Hawking's periodicity arguments. Thus, we obtain further corrections to the non-strictly the…
