Subenoy Chakraborty, Nairwita Mazumder, Ritabrata Biswas
Using tunnelling approach, Hawking radiation is derived for a general class of non-static spherically symmetric space time. The standard tunnelling rate formula is obtained using the unified first law of thermodynamics on the trapping horizon and Hawking temperature is measured by an Kodama observer inside the horizon.
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Hawking radiation can usefully be viewed as a semi-classical tunnelling process that originates at the black hole horizon. The conservation of energy implies the effect of self-gravitation. For a static black hole, a generalized Painleve coordinate system is introduced, and Hawking radiation as tunnelling under the effect of self-gravitation is investigated. The corrected radiation is consistent with the underlying unitary theory.
Hawking radiation can usefully be viewed as a semi-classical tunnelling process that originates at the black hole horizon. For the stationary axisymmetric BTZ black hole, a generalized Painleve coordinate system (Painleve-BTZ coordinates) is introduced, and Hawking radiation as tunnelling under the effect of self-gravitation is investigated. The corrected radiation is obtained which is not precise thermal spectrum. The result is consistent with the underlying unitary theory. Moreover, Bekenstein-Hawking entropy of BTZ black hole is not necessarily corrected when we choose appropriate coordinat…
In this paper, we extend Parikh' work to the non-stationary black hole. As an example of the non-stationary black hole, we study the tunnelling effect and Hawking radiation from a Vaidya black hole whose Bondi mass is identical to its mass parameter. We view Hawking radiation as a tunnelling process across the event horizon and calculate the tunnelling probability. We find that the result is different from Parikh's work because $\frac{dr_{H}}{dv}$ is the function of Bondi mass m(v).
The paper deals with Hawking radiation from a general static black hole due to tunnelling of particle having non-zero mass. Due to complicated calculations, quantum corrections are calculated upto first order. The results are compared with those for massless particle near the horizon.
In Parikh and Wilczek's original works, the laws of black hole thermodynamics are not referred and it seems that there is no relation between Hawking radiation via tunnelling and the laws of black hole thermodynamics in their works. However, taking examples for the R-N black hole and the Kerr black hole, we find that they are correlated and even consistent if the tunnelling process is a reversible process.
The paper deals with Hawking radiation related to non-static spherically symmetric black hole. Quantum corrections are incorporated using Hamilton-Jacobi method beyond semi-classical approximation. It is found that different order correction terms satisfy identical differential equation as the semiclassical action and are solved by a typical technique. It has been shown that with proper choice of the proportionality factors, one loop back reaction effect in the space time can be obtained. Finally, using the law of black hole mechanics, a general modified form of the black hole entropy is obtai…
The complex path (or Hamilton-Jacobi) approach to Hawking radiation corresponds to the intuitive picture of particles tunnelling through the horizon and forming a thermal radiation. This method computes the tunnelling rate of a given particle from its equation of motion and equates it to the Boltzmann distribution of the radiation from which the Hawking temperature is identified. In agreement with the original derivation by Hawking and the other approaches, it has been checked case by case that the temperature is indeed universal for a number of backgrounds and the tunnelling of particles from…
The paper deals with Hawking radiation from both a general static black hole and a nonstatic spherically symmetric black hole. In case of static black hole, tunnelling of nonzero mass particles is considered and due to complicated calculations, quantum corrections are calculated only up to the first order. The results are compared with those for massless particles near the horizon. On the other hand, for dynamical black hole, quantum corrections are incorporated using the Hamilton-Jacobi method beyond semiclassical approximation. It is found that different order correction terms satisfy identi…
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.
The aim of this work is to review the tunnelling method as an alternative description of the quantum radiation from black holes and cosmological horizons. The method is first formulated and discussed for the case of stationary black holes, then a foundation is provided in terms of analytic continuation throughout complex space-time. The two principal implementations of the tunnelling approach, which are the null geodesic method and the Hamilton-Jacobi method, are shown to be equivalent in the stationary case. The Hamilton-Jacobi method is then extended to cover spherically symmetric dynamical …
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.
We investigate the Dirac particles tunnelling as a radiation of Warped AdS$_{3}$ black hole family in Topological Massive Gravity. Using the Hamilton-Jacobi method, we discuss tunnelling probability and Hawking temperature of the spin-1/2 particles for the black hole and its extremal cases. We observe that the Hawking temperature of the non-extremal black hole higher than the extremal black hole when $ω<\frac{2\ r_{0}}{3}$, because the non-extremal black hole become unstable in this case.
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.
We consider dynamical black hole formation from a collapsing fluid described by a symmetric and flat FRW metric. Using the Hamilton-Jacobi method the local Hawking temperature for the formed trapping/apparent horizon is calculated. The local Hawking temperature depends on the tunneling path, which we take to be along a null direction $(Δs=0)$. We find that the local Hawking temperature depends directly on the equation of state of the collapsing fluid. We argue that Hawking radiation by quantum tunnelling from future inner and future outer trapping horizons is possible. However, only radiation …
Tunnelling amplitudes and Hawking radiation from worldline QFT
本研究对比坍缩背景下的霍金辐射与电场中施温格对产生，用世界线方法构建两类情形下背景耦合波动方程的解，揭示二者隧穿过程鞍点的世界线性质差异。
Tunnelling mechanism is today considered a popular and widely used method in describing Hawking radiation. However, in relation to black hole (BH) emission, this mechanism is mostly used to obtain the Hawking temperature by comparing the probability of emission of an outgoing particle with the Boltzmann factor. On the other hand, Banerjee and Majhi reformulated the tunnelling framework deriving a black body spectrum through the density matrix for the outgoing modes for both the Bose-Einstein distribution and the Fermi-Dirac distribution. In contrast, Parikh and Wilczek introduced a correction …
Acoustic Hawking radiation as a tunnelling effect in Michel accretion
研究米歇尔吸积中声学黑洞的霍金辐射，通过分析稳态流入的欧拉扰动，发现声波在声障处发生隧穿效应，霍金温度与频率受时空几何增强。
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…
In this work, we study Hawking radiation from a general static black hole due to tunnelling of particle having nonzero mass. Hawking temperature has been calculated using both the tunnelling method and the Hamilton-Jacobi method and the results do no depend on the mass of the particle. Due to complicated form of the equations involved quantum corrections are evaluated only upto first order and it is possible to interprete the correction term as loop corrections of the back reaction effects in the space time. Finally, modified expressions for entropy and surface gravity has been evaluated and i…
In this paper we study Hawking-like radiation in a FRW-Universe using semi-classical tunnelling approach and the Hamilton-Jacobi method. Radial null geodesics are used to picture the process as a "tunnelling" of particles from behind the apparent horizon and Hawking -like temperature has been calculated. Quantum corrections have been evaluated in the Hamilton-Jacobi Method by solving Klein-Gordan wave equation and the temperature agrees at the semiclassical level. Also it is found that Hamilton-Jacobi formalism does not depend on the choice of the coordinate system. Finally, leading order corr…
In this study, the quantum gravity effect on the tunnelling radiation of charged massive spin-0 scalar particle from 2+1 dimensional charged rotating Banados-Teitelboim-Zanelli (BTZ) black hole is looked into by using the Hamilton-Jacobi approach. For this, we calculate the modified Hawking temperature of the black hole by using the modified Klein-Gordon equation based on the Generalized Uncertainty Principle (GUP), and we noticed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the angular momentum, energy, charge and mass of t…
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…
By studying the Hawking radiation of the most general static spherically symmetric black hole arising from scalar and Dirac particles tunnelling, we find the Hawking temperature is invariant in the general coordinate representation (\ref{arbitrary1}), which satisfies two conditions: a) its radial coordinate transformation is regular at the event horizon; and b) there is a time-like Killing vector.
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…
GUP effects on Hawking temperature in a hot NUT Kerr Newman Kasuya Anti de Sitter black hole
本文研究了量子引力效应下热NUT Kerr Newman Kasuya Anti de Sitter黑洞霍金温度的修改及其稳定性影响。
Following the initial work of Calcagni et al. on the black holes in multi-fractional theories, we focus on the Schwarzschild black hole in multi-fractional theory with q-derivatives. After presenting its Hawking and Hayward temperatures in detail, we verify these results by appealing to the well-known Hamilton-Jacobi and null geodesic methods of the tunnelling approach to Hawking radiation. A special emphasis is placed on the difference between the geometric and fractional frames.
We investigate the Hawking radiation of 3+1 and 4+1 dimensional black holes in the $z = 4$ Horava-Lifshitz gravity with fermion tunnelling. It turns out that the Hawking temperatures are recovered and are in consistence with those obtained by calculating surface gravity of the black holes. For the 3+1 dimensional black holes, the Hawking temperatures are related to the fundamental parameters of Horava-Lifshitz gravity.
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…
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…
In this paper, we investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole's mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.
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…
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.
We recently improved the famous result of Parikh and Wilczek, who found a probability of emission of Hawking radiation which is compatible with a non-strictly thermal spectrum, showing that such a probability of emission is really associated to two non-strictly thermal distributions for boson and fermions. Here we finalize the model by finding the correct value of the pre-factor of the Parikh and Wilczek probability of emission. In fact, that expression has the "of order" sign instead of the equality. In general, in this kind of leading order tunnelling calculations, the exponent arises indeed…
Viscous dissipation, as a small perturbative effect about the Bondi flow, shrinks its sonic sphere. An Eulerian perturbation on the steady flow gives a wave equation and the corresponding dispersion relation. The perturbation is a high-frequency travelling acoustic wave, in which small dissipation is taken iteratively. The wave, propagating radially outwards against the bulk inflow, is blocked just within the sonic horizon, where the amplitude of the wave diverges because of viscosity. The blocked acoustic wave can still tunnel outward through the horizon with a viscosity-dependent decaying am…
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…
We work with the notion of apparent/trapping horizons for spherically symmetric, dynamical spacetimes: these are quasi-locally defined, simply based on the behaviour of congruence of light rays. We show that the sign of the dynamical Hayward-Kodama surface gravity is dictated by the inner/outer nature of the horizon. Using the tunneling method to compute Hawking Radiation, this surface gravity is then linked to a notion of temperature, up to a sign that is dictated by the future/past nature of the horizon. Therefore two sign effects are conspiring to give a positive temperature for the black h…
Tunneling of Hawking Radiation in Starobinsky-Bel-Robinson gravity
本文用隧穿方法研究 SBR 引力中 Schwarzschild 型黑洞的霍金辐射，发现修改经典几何产生的半经典温度修正与固定背景并加入超越半经典项的修正相当。
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…
We present a generalization of the Hawking effect for dynamical trapping horizons by calculating the tunneling rate in the Hamilton-Jacobi formalism. It turns out that all horizons classified by Hayward are subjected to thermal quantum effects. While the Hawking effect for future outer and past inner trapping horizons is given as a particle emission, we show that the Hawking effect for future inner and past outer trapping horizons translates to an absorption. The universality of the treatment allows a natural transfer to the static case.
Hawking radiation emanating from two-dimensional charged and uncharged dilatonic black holes - dimensionally reduced from (2+1) spinning and spinless, respectively, BTZ black holes - is viewed as a tunnelling process. Two dimensional dilatonic black holes (AdS(2) included) are treated as dynamical backgrounds in contrast to the standard methodology where the background geometry is fixed when evaluating Hawking radiation. This modification to the geometry gives rise to a nonthermal part in the radiation spectrum. Nonzero temperature of the extremal two-dimensional charged black hole is found. T…
We examine the possibility of artificial Hawking radiation by proposing a non-PT-symmetric weakly pseudo-Hermitian two-band model containing a tilting parameter by pursuing Weyl semimetal blackhole analogy. We determine the tunnelling probability using such a Hamiltonian through the event horizon that acts as a classically forbidden barrier.
We propose a model for the geometry of a dynamical spherical shell in which the metric is asymptotically Schwarzschild, but deviates from Ricci-flatness in a finite neighbourhood of the shell. Hence, the geometry corresponds to a `hairy' black hole, with the hair originating on the shell. The metric is regular for an infalling shell, but it bifurcates, leading to two disconnected Schwarzschild-like spacetime geometries. The shell is interpreted as either collapsing matter or as Hawking radiation, depending on whether or not the shell is infalling or outgoing. In this model, the Hawking radiati…
We discuss some of the drawbacks of using event horizons to define black holes. The reasons are both practical, physical and theoretical. We argue that locally defined trapping horizons can remedy many of these drawbacks. We examine of the question of whether black hole thermodynamics should be associated with event horizons or trapping horizons. To this end we discuss what role trapping horizons may play in black hole thermodynamics. In addition, we show how trapping horizons may give rise to Hawking radiation and discuss the issue of gravitational entropy.
Quasinormal Modes and Hawking Radiation of Black Holes with Primary Scalar Hair
研究超越Horndeski引力中原初标量毛黑洞的准正则模、灰体因子和霍金辐射，发现标量毛在环降特性、频率偏移和回波中留下特征印记，灰体因子呈现共振隧穿行为。
We regard the Parikh-Wilczek's tunnelling model of Hawking radiation as a quantum mechanical process of stimulated emission. The hypothesized microstates are found at the horizon with double degeneracy. A Jaynes-Cummings toy model for a black hole in the cavity is proposed to demonstrate how to write a qubit via the angular-dependent transition coupling, which might be related to the soft Goldstone hairs at analytic continuation. At last, we show how information is retained in the black hole by computing the time evolution of mutual entanglement entropy in the cavity-black holes system.
Our universe is born of a tunnelling from nothing in quantum cosmology. Nothing here can be interpreted as a state with zero entropy. As a reliable modification of the Hartle-Hawking wave function of the universe, the improved Hartle-Hawking wave function proposed by Firouzjahi, Sarangi and Tye gives many interesting observational consequences which we explore in this paper. Fruitful observations are obtained for chaotic inflation, including a detectable spatial curvature and a negligible tunnelling probability for eternal chaotic inflation. And we find that the tensor-scalar ratio and the spa…
We study the geometry and dynamics of both isolated and dynamical trapping horizons by considering the allowed variations of their foliating two-surfaces. This provides a common framework that may be used to consider both their possible evolutions and their deformations as well as derive the well-known flux laws. Using this framework, we unify much of what is already known about these objects as well as derive some new results. In particular we characterize and study the "almost-isolated" trapping horizons known as slowly evolving horizons. It is for these horizons that a dynamical first law h…
Understood in terms of pure states evolving into mixed states, the possibility of information loss in black holes is closely related to the global causal structure of spacetime, as is the existence of event horizons. However, black holes need not be defined by event horizons, and in fact we argue that in order to have a fully unitary evolution for black holes, they should be defined in terms of something else, such as a trapping horizon. The Misner-Sharp mass in spherical symmetry shows very simply how trapping horizons can give rise to black hole thermodynamics, Hawking radiation and singular…
The Hawking radiation of static, spherically symmetric, asymptotically flat solutions in quadratic gravity, is here explored. The emitted fermions are governed by the Dirac equation in the generalized uncertainty principle (GUP) context, that probes the spacetime (quantum) nature. The Hawking thermal spectrum is scrutinized out of the tunnelling method and the WKB procedure. The Hawking flux is shown to vanish for small black holes, for a precise combination of the GUP parameter and the coupling constants that drive the gravitational interaction in quadratic gravity, yielding absolutely stable…
Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory discovering interesting interplay with the Hawking radiation. A more precise and general account of that phenomenon has been subsequently given within the framework of QFT in curved spacetime by two of the authors of the present paper. In particular, it has been shown that, in the limit of sharp localization on opposite sides of a Killing horizon, the quantum correlation functions of a scalar field appear to have thermal nature, and the tunnelling probability is proportional to $\exp…
We study various derivations of Hawking radiation in conformally rescaled metrics. We focus on two important properties, the location of the horizon under a conformal transformation and its associated temperature. We find that the production of Hawking radiation cannot be associated in all cases to the trapping horizon because its location is not invariant under a conformal transformation. We also find evidence that the temperature of the Hawking radiation should transform simply under a conformal transformation, being invariant for asymptotic observers in the limit that the conformal transfor…
Nested Apparent Horizons and Quantized Separation from Intense Hawking Backreaction
研究强霍金辐射下黑洞视界嵌套现象，揭示量子反作用导致的视界分离量子化特征，探索量子引力离散性几何路径。
We present an exact solution of the Maxwell-Einstein equations, which describes the exterior of a charged spherical mass collapsing into its own trapping horizon and then bouncing back from an anti-trapping horizon at the same space location of the same asymptotic region. The solution is locally but not globally isometric to the maximally extended Reissner-Nordström metric and depends on seven parameters. It is regular, and defined everywhere except for a small region, where quantum tunnelling is expected. This region lies outside the mass: the mass-bounce and its near exterior are governed by…
We discuss some of the drawbacks of using event horizons to define black holes and suggest ways in which black holes can be described without event horizons, using trapping horizons. We show that these trapping horizons give rise to thermodynamic behavior and possibly Hawking radiation too. This raises the issue of whether the event horizon or the trapping horizon should be seen as the true boundary of a black hole. This difference is important if we believe that quantum gravity will resolve the central singularity of the black hole and clarifies several of the issues associated with black hol…
A local Hawking temperature was recently derived for any future outer trapping horizon in spherical symmetry, using a Hamilton-Jacobi tunneling method, and is given by a dynamical surface gravity as defined geometrically. Descriptions are given of the operational meaning of the temperature, in terms of what observers measure, and its relation to the usual Hawking temperature for static black holes. Implications for the final fate of an evaporating black hole are discussed.
Trapped string states in AdS$_5$ black hole geometry: A path toward Hawking radiation
该研究通过正则量子化闭弦，发现AdS$_5$-Schwarzschild黑洞外部存在束缚半径，弦在其上产生量子化振荡，类似拉伸视界，可发射黑体辐射，为霍金辐射的量子起源提供新路径。
The notion of the black hole singularity and the proof of the singularity theorem were considered great successes in classical general relativity. Singularities had presented deep puzzles to physicists. Conceptual challenges were set up by the intractability of the singularity. The existence of black hole horizons which cover up the interior, including the singularity of the black hole from outside observers, builds an information curtain, further hindering physicists from understanding the nature of the singularity and the interior structure of black holes. The regular black hole is a concept…
Hawking Temperatures of Dynamical Black Holes from the RVB--Residue Method:Vaidya and Kinnersley Geometries
本文发展了基于局部留数的RVB方法，用于计算动态黑洞的霍金温度，并在Vaidya和Kinnersley黑洞中验证了其有效性。
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