Information Conservation in a Noncommutative Quantum Black Hole Based on Canonical Typicality*

A method for calculating the radiation spectrum of an arbitrary black holes was recently proposed by Ma et al., [Europhys. Lett. 122 (2018) 30001] in which a non-thermal spectrum of a black hole can be obtained from its entropy using an approach based on canonical typicality. The non-thermal spectrum of a black hole enables a nonzero correlation between the black hole and its radiation, which can ensure that information is conserved during black hole evaporation. In this paper, by using the Kantowski-Sachs metric and Feynman-Hibbs procedure, the entropy of a noncommutative quantum black hole is calculated based on the Wheeler-DeWitt equation. Then, the radiation spectrum of the noncommutative quantum black hole is studied based on canonical typicality method. At last, the correlation between the radiation spectra is calculated. It is shown that the noncommutative effect increases the correlation among radiation and the information remains conserved for noncommutative quantum black holes.     

引力彩虹时空中Kerr黑洞的熵谱和面积谱

利用黑洞的绝热不变性,研究了引力彩虹时空中Kerr黑洞的熵谱和面积谱.首先,在引力彩虹时空背景下,计算了Kerr黑洞的绝热不变作用量,并将其与玻尔-索末菲量子化条件相结合,给出了黑洞的熵谱.得到的熵谱没有引力彩虹时空本身具有的粒子能量依赖性,且是与经典Kerr黑洞中原始贝肯斯坦熵谱相同的等间距熵谱.然后,根据黑洞热力学第一定律和黑洞熵谱,给出了与原始贝肯斯坦谱不同的面积谱.该面积谱是非等间距的,而且有对黑洞面积的依赖性,但不依赖于探测粒子的能量.面积谱表明,随着黑洞面积的减少,面积间隔逐步变小;当黑洞达到普朗克尺度时,面积量子可降为零.这表示黑洞面积不再减少,黑洞出现辐射剩余.而在忽略色散关系的修正效应或在大黑洞极限下,面积谱的修正项可以忽略,引力彩虹Kerr黑洞面积谱可以回归到原始贝肯斯坦谱.此外,对引力彩虹时空Kerr黑洞的熵进行了讨论,得到了带有面积倒数修正项的黑洞熵,分析了黑洞熵的量子修正与面积谱量子修正的一致性.     

Loop quantum gravity and black hole entropy quantization

Using the spin networks and the asymptotic quasinormal mode frequencies of black holes given by loop quantum gravity, the minimum horizon area gap is obtained. Then the quantum area spectrum of black holes is derived and the black hole entropy is a realized quantization. The results show that the black hole entropy given by loop quantum gravity is in full accord with the Bekenstein-Hawking entropy with a suitable Immirzi. Supported by the National Natural Science Foundation of China (Grant No. 10773002)     

Universal canonical entropy for gravitating systems

The thermodynamics of general relativistic systems with boundary, obeying a Hamiltonian constraint in the bulk, is determined solely by the boundary quantum dynamics, and hence by the area spectrum. Assuming, for large area of the boundary, (a) an area spectrum as determined by non-perturbative canonical quantum general relativity (NCQGR), (b) an energy spectrum that bears a power law relation to the area spectrum, (c) an area law for the leading order microcanonical entropy, leading thermal fluctuation corrections to the canonical entropy are shown to be logarithmic in area with a universal coefficient. Since the microcanonical entropy also has universal logarithmic corrections to the area law (from quantum space-time fluctuations, as found earlier) the canonical entropy then has a universal form including logarithmic corrections to the area law. This form is shown to be independent of the index appearing in assumption (b). The index, however, is crucial in ascertaining the domain of validity of our approach based on thermal equilibrium.     

Black hole spectroscopy via adiabatic invariance

During the last years, one had to combine the proposal about how quasinormal frequencies are related with black holes and the proposal about the adiabatic invariance of black holes in order to derive the quantized entropy spectrum and its minimum change for several black holes. In this Letter we exclusively utilize the statement that the black hole horizon area is an adiabatic invariant and derive an equally spaced entropy spectrum of a black hole with its quantum to be equal to the one given by Bekenstein. Interestingly, in our approach no concept of quasi-normal mode is needed.     

Entropy corrections to five-dimensional black holes and de Sitter spaces

It is shown that non-rotating black holes in three or four dimensions possess a canonical entropy. Recently study indicated that there were logarithmic corrections to Bekenstein–Hawking entropy in area with a uncertain coefficient which depends on specific models. In this paper, the thermal fluctuations on Bekenstein–Hawking entropy in five-dimensional topological AdS (TAds)-black holes and topological de Sitter (Tds) spaces will be considered based on a uniformly spaced area spectrum approach.     

Discrete Spectra of Charged Black Holes

Bekenstein proposed that the spectrum of horizon area of quantized black holes must be discrete and uniformly spaced. We examine this proposal in the context of spherically symmetric charged black holes in a general class of gravity theories. By imposing suitable boundary conditions on the reduced phase space of the theory to incorporate the thermodynamic properties of these black holes and then performing a simplifying canonical transformation, we are able to quantize the system exactly. The resulting spectra of horizon area, as well as that of charge are indeed discrete. Within this quantization scheme, near-extremal black holes (of any mass) turn out to be highly quantum objects, whereas extremal black holes do not appear in the spectrum, a result that is consistent with the postulated third law of black hole thermodynamics.     

Information entropy for static spherically symmetric black holes

By using the new equation of state density derived from the generalized uncertainty relation, the number of the quantum states near event horizon is obtained, with which then the information entropy of static spherically symmetric black holes has been discussed. It is found that the divergent integral of quantum states near the event horizon can be naturally avoided if using the new equation of state density without introducing the ultraviolet cut-off. The information entropy of black holes can be obtained precisely by the residue theorem, which is shown to be proportional to the horizon area. The information entropy of black holes obtained agrees with the Bechenstein--Hawking entropy when the suitable cutoff factor is adopted.     

Stefan-Boltzmann law for black bodies and black holes

Since black holes radiate with a thermal spectrum and therefore possess a radiation pressure, Boltzmann's derivation of Stefan's Law can be applied to black holes. In order that the entropy be proportional to the surface area of the black hole, the pressure must be negative. If the second law is not to be violated, then the temperature must also be negative. This leads to a canonical formulation for fluctuations. A comparison with other approaches is given and doubts are raised concerning the validity of conventional black hole thermodynamics.     

Power-Law entropy corrected holographic dark energy model

Among various scenarios to explain the acceleration of the universe expansion, the holographic dark energy (HDE) model has got a lot of enthusiasm recently. In the derivation of holographic energy density, the area relation of the black hole entropy plays a crucial role. Indeed, the power-law corrections to entropy appear in dealing with the entanglement of quantum fields in and out the horizon. Inspired by the power-law corrected entropy, we propose the so-called “power-law entropy-corrected holographic dark energy” (PLECHDE) in this Letter. We investigate the cosmological implications of this model and calculate some relevant cosmological parameters and their evolution. We also briefly study the so-called “power-law entropy-corrected agegraphic dark energy” (PLECADE).     

Wave function of the quantum black hole

We show that the Wald Noether-charge entropy is canonically conjugate to the opening angle at the horizon. Using this canonical relation, we extend the Wheeler–DeWitt equation to a Schrödinger equation in the opening angle, following Carlip and Teitelboim. We solve the equation in the semiclassical approximation by using the correspondence principle and find that the solutions are minimal uncertainty wavefunctions with a continuous spectrum for the entropy and therefore also of the area of the black hole horizon. The fact that the opening angle fluctuates away from its classical value of 2π indicates that the quantum black hole is a superposition of horizonless states. The classical geometry with a horizon serves only to evaluate quantum expectation values in the strict classical limit.     

Black Holes as Atoms

Stationary spacetimes containing a black hole have several properties akin to those of atoms. For instance, such spacetimes have only three classical degrees of freedom, or observables, which may be taken to be the mass, the angular momentum, and the electric charge of the hole. There are several arguments supporting a proposal originally made by Bekenstein that quantization of these classical degrees of freedom gives an equal spacing for the horizon area spectrum of black holes. We review some of these arguments and introduce a specific Hamiltonian quantum theory of black holes. Our Hamiltonian quantum theory gives, among other things, a discrete spectrum for the classical observables, and it produces an area spectrum which is closely related to Bekenstein's proposal. We also present a foamlike model of horizons of spacetime. In our model spacetime horizon consists of microscopic Schwarzschild black holes. Applying our Hamiltonian approach to this model we find that the entropy of any horizon is one quarter of its area.     

探讨黑洞Hawking辐射的新方法——量子统计法

将黑洞看作由裸黑洞和二维热力学面(黑洞的视界)组成的正则系综,利用量子统计方法给出黑洞Hawking 辐射的能量谱.找到黑洞辐射温度与熵的关系. 关键词： Hawking辐射 正则系统 量子统计     

Quantum Statistic Entropy of Three-Dimensional BTZ Black Hole

Using the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity, we investigate entropy of a black line on the background of the three-dimensional BTZ. In our calculation, we need not introduce cutoff and can remove the divergent term in the original brick-wall method via the new equation of state density. And it is obtained that the entropy of the black line is proportional to the area of the horizon (perimeter). Further it is shown the entropy of black line is the entropy of quantum state on the surface of horizon (perimeter). The black line entropy is the intrinsic property of the black hole. The entropy is a quantum effect. By using quantum statistical method, we directly obtain the partition function of Bose field and fermi field on the background of the black line. The difficulty to solve wave equation of various particles is avoided. We offer a new simple and direct way for calculating the entropy of various spacetime black holes (black plane, black line and black column). PACS 04.20.Dw; 97.60.Lf     

Spectroscopy of a canonically quantized horizon

Deviations from Hawking's thermal black hole spectrum, observable for macroscopic black holes, are derived from a model of a quantum horizon in loop quantum gravity. These arise from additional area eigenstates present in quantum surfaces excluded by the classical isolated horizon boundary conditions. The complete spectrum of area unexpectedly exhibits evenly spaced symmetry. This leads to an enhancement of some spectral lines on top of the thermal spectrum. This can imprint characteristic features into the spectra of black hole systems. It most notably gives the signature of quantum gravity observability in radiation from primordial black holes, and makes it possible to test loop quantum gravity with black holes well above Planck scale.     

Quantum Statistical Entropy of Six Dimensional Horowitz-Strominger Black Holes

Using the membrane model which is developed from the brick-wall model, we calculated the bosonic and fermionic entropy of 6-d Horowitz-Strominger black holes. The result shows the quantum entropy of such black holes is still proportional to the area of the event horizon only if the cut-off is properly chosen. As for the extreme black holes,the entropy approaches zero.     

Logarithmic corrections to $${\mathcal{N} = 2}$$ black hole entropy: an infrared window into the microstates

Logarithmic corrections to the extremal black hole entropy can be computed purely in terms of the low energy data—the spectrum of massless fields and their interaction. The demand of reproducing these corrections provides a strong constraint on any microscopic theory of quantum gravity that attempts to explain the black hole entropy. Using quantum entropy function formalism we compute logarithmic corrections to the entropy of half BPS black holes in N=2{{\mathcal N}=2} supersymmetric string theories. Our results allow us to test various proposals for the measure in the OSV formula, and we find agreement with the measure proposed by Denef and Moore if we assume their result to be valid at weak topological string coupling. Our analysis also gives the logarithmic corrections to the entropy of extremal Reissner–Nordstrom black holes in ordinary Einstein–Maxwell theory.     

f(T) cosmology via Noether symmetry

Using the modified Kunstatter method, which employs as proper frequency the imaginary part instead of the real part of the quasinormal modes, the entropy spectrum and area spectrum of the modified Schwarzschild black holes in gravity??s rainbow are investigated. In the current study, two cases of modified dispersion relations concerning energy dependent and energy independent speed of light are considered. The entropy spectra with equal spacing are derived in these two cases. Furthermore, the obtained entropy spectra are independent of the energy of a test particle and are the same as the one of the usual Schwarzschild black hole. Also, the same area spectrum formulas are obtained in these different dispersion relations. However, due to the quantum effect of spacetime, the obtained area spectra are not equally spaced and are different from the one of the usual Schwarzschild black hole. Besides, in these two cases, the same black hole entropy formulas with logarithmic correction to the standard Bekenstein?CHawking area formula are obtained by the adiabatic invariant. The form of area spacing formulas and entropy formulas are independent of the particle??s energy, but the area spacing and entropy can have energy dependence through the area.     

Quantum gravity effects on spectroscopy of Kerr-Newman black hole in gravity’s rainbow

The effects of quantum gravity on spectroscopy for the charged rotating gravity’s rainbow are investigated in this paper. By utilizing an action invariant obtained from particles tunneling through the event horizon, the entropy and area spectrum for the modified Kerr-Newman black hole are derived. The equally spaced entropy spectrum characteristic of Bekenstein’s original derivation is recovered. And, the entropy spectrum is independent of the energy of the test particles, although the gravity’s rainbow itself is the energy dependent. Such that, the quantum gravity effects of gravity’s rainbow has no influence on the entropy spectrum. On the other hand, due to the spacetime quantum effects, the obtained area spectrum is different from the original Bekenstein spectrum. It is not equidistant and is dependent on the horizon area. And that, by analyzing the area spectrum from a specific rainbow function, a minimum area with a Planck scale is derived for the event horizon. At this point, the area quantum is zero and the black hole radiation stops. Thus, the black hole remnant for the gravity’s rainbow is obtained from the area quantization. In addition, the entropy for the modified Kerr-Newman black hole is calculated and the quantum correction to the area law is obtained and discussed.     

The uncertainty principle and Bekenstein–Hawking entropy corrections

There is much attention on the corrections to Bekenstein–Hawking entropy in area with a model-dependent coefficient. The corrections are generally composed of two parts: quantum corrections and thermal corrections. The generalized uncertainty principle (GUP), which will reduce to the conventional Heisenberg relation in situations of weak gravity, is one of the candidates to be utilized to obtain the quantum corrections to the Bekenstein–Hawking entropy. Recently the extended uncertainty principle (EUP) and generalized extended uncertainty principle (GEUP) are introduced to calculate entropy corrections with large length scales limit. In this paper, we obtain the quantum corrections to Bekenstein–Hawking entropy in four-dimensional Schwarzschild black holes based on the EUP and GEUP. Some attractive results are derived.