The Schwarzschild black hole’s remnant via the Bohr-Sommerfeld quantization rule

A d-dimensional Schwarzschild black hole is quantized by the action variable and the Bohr-Sommerfeld quantization rule in this paper. We find that the spectra of the horizon area and the entropy are evenly spaced. The black hole mass is also quantized and it’s spectrum spacing is proportional inversely to the mass. The ground state appears and has a constant entropy $\pi k_B$ . The ground state mass is shown to be the black hole remnant predicted by the generalized uncertainty principle and may be a candidate of dark matter.     

从Schwarzschild黑洞到极端Kerr-Newman黑洞

考虑Schwarzschild和Kerr-Newman黑洞的度规场和电磁场张量,研究了落入黑洞的物理粒子.得出结论:任何物理粒子落入Schwarzschild或Kerr-Newman黑洞的演化过程要满足δA≥0和δκ≤0两个条件,得出了不能通过有限次操作使Kerr-Newman黑洞的温度降低到绝对零度,从而变成极端Kerr-Newman黑洞,刚好与第三定律相符.还讨论了Kerr-Newman黑洞趋于极端情况时,其面积的变化趋势,提出了黑洞负热容的影响以及熵的重整化问题. 关键词：     

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.     

Charged Particle’s Tunneling in a Modified Reissner-Nordstrom Black Hole

In the tunneling framework of Hawking radiation, charged particle’s tunneling in the modified Reissner-Nordstrom black hole from gravity’s rainbow is investigated. To this end, following the Schwarzschild solution in gravity’s rainbow, the metric of the modified Reissner-Nordstrom black hole is given. In the tunneling process, the metric fluctuation is taken into account, due to not only the energy conservation and electric charge conservation, but also the spacetime quantum effects. The calculation shows out that the emission rate satisfies the first law of black hole thermodynamics and is consistent with an underlying unitary theory. In addition, it is found that the entropy of the modified black hole is different to the Benkestein-Hawking entropy and the quantum corrections of the entropy appears.     

Some aspects of virtual black holes

We first consider consistently third-quantize modified gravity. We then analyze certain aspects of virtual black holes in this third-quantized modified gravity. We see how a statistical mechanical origin for the Bekenstein-Hawking entropy naturally arises in this model. Furthermore, the area and hence the entropy of a real macroscopic black hole is quantized in this model. Virtual black holes cause a loss of quantum coherence, which gives an intrinsic entropy to all physical systems that can be used to define a direction of time and hence provide a solution to the problem of time.     

Black hole quantum tunnelling and black hole entropy correction

The Parikh–Wilczek tunnelling framework, which treats Hawking radiation as a tunnelling process, is investigated once more in this work. The first order correction, the log-corrected entropy-area relation, emerges naturally in the tunnelling picture if we consider the emission of a spherical shell. The second order correction to the emission rate for the Schwarzschild black hole is also calculated. At this level, the entropy of the black hole will contain three parts: the usual Bekenstein–Hawking entropy, a logarithmic term and an inverse area term. We find that the coefficient of the logarithmic term is −1. Thus, apart from a coefficient, our correction to the black hole entropy is consistent with that calculated in loop quantum gravity.     

Black hole entropy quantization

Ever since the pioneering works of Bekenstein and Hawking, black hole entropy has been known to have a quantum origin. Furthermore, it has long been argued by Bekenstein that entropy should be quantized in discrete (equidistant) steps given its identification with horizon area in (semi-)classical general relativity and the properties of area as an adiabatic invariant. This lead to the suggestion that the black hole area should also be quantized in equidistant steps to account for the discrete black hole entropy. Here we shall show that loop quantum gravity, in which area is not quantized in equidistant steps, can nevertheless be consistent with Bekenstein's equidistant entropy proposal in a subtle way. For that we perform a detailed analysis of the number of microstates compatible with a given area and show consistency with the Bekenstein framework when an oscillatory behavior in the entropy-area relation is properly interpreted.     

Existence of a semiclassical approximation in loop quantum gravity

We consider a spherically symmetric black hole in the Schwarzschild metric and apply Bohr-Sommerfeld quantization to determine the energy levels. The canonical partition function is then computed and we show that the entropy coincides with the Bekenstein-Hawking formula when the maximal number of states for the black hole is the same as computed in loop quantum gravity, proving in this case the existence of a semiclassical limit and obtaining an independent derivation of the Barbero-Immirzi parameter.     

Kerr Black Hole Entropy and its Quantization

By constructing the four-dimensional phase space based on the observable physical quantity of Kerr black hole and gauge transformation, the Kerr black hole entropy in the phase space was obtained. Then considering the corresponding mechanical quantities as operators and making the operators quantized, entropy spectrum of Kerr black hole was obtained. Our results show that the Kerr black hole has the entropy spectrum with equal intervals, which is in agreement with the idea of Bekenstein. In the limit of large event horizon, the area of the adjacent event horizon of the black hole have equal intervals. The results are in consistent with the results based on the loop quantum gravity theory by Dreyer et al.     

Spectrum of a quantized black hole,correspondence principle,and holographic bound

An equidistant spectrum of the horizon area of a quantized black hole does not follow from the correspondence principle or from general statistical arguments. On the other hand, such a spectrum obtained in loop quantum gravity (LQG) either does not comply with the holographic bound or requires a special choice of the Barbero-Immirzi parameter for the horizon surface, distinct from its value for other quantized surfaces. The problem of distinguishability of the edges in LQG is discussed, with the following conclusion: Only under the assumption of partial distinguishability of the edges can the microcanonical entropy of a black hole be made both proportional to the horizon area and satisfying the holographic bound.     

Quantum corrections to the entropy of Einstein–Maxwell dilaton–axion black holes

We study corrections to the entropy of Einstein–Maxwell dilaton–axion black holes beyond semiclassical approximations. We consider the entropy of the black hole as a state variable and derive these corrections using the exactness criteria of the first law of thermodynamics. We note that from this general frame-work the entropy corrections for “simpler” black holes like Schwarzschild, Reissner–Nordström and anti-de Sitter–Schwarzschild black holes follow easily. This procedure gives us the modified area law as well.     

Thermodynamics of Hořava–Lifshitz black holes

We study black holes in the Ho?ava–Lifshitz gravity with a parameter λ. For 1/3≤λ<3, the black holes behave the Lifshitz black holes with dynamical exponent 0<z≤4, while for λ>3, the black holes behave the Reissner–Nordström type black hole in asymptotically flat spacetimes. Hence, these all are quite different from the Schwarzschild–AdS black hole of Einstein gravity. The temperature, mass, entropy, and heat capacity are derived for investigating thermodynamic properties of these black holes.     

Black Hole Entropy with Modified Dispersion Relations

There is much interest in resolving the quantum corrections to Bekenstein-Hawking entropy with a large length scale limit. The leading correction term is given by the logarithm of black hole area with a model-dependent coefficient. Recently the research for quantum gravity implies the emergence of a modification of theenergy-momentum dispersion relation (MDR), which plays an importantrole in the modified black hole thermodynamics. In this paper, we investigate the quantum corrections to Bekenstein-Hawking entropy in four-dimensional Schwarzschild black hole and Reissner-Nordström black hole respectively based on MDR.     

High dimensional AdS-like black hole and phase transition in Einstein-bumblebee gravity

In this study, we obtained an exact high dimensional anti-de Sitter (AdS) black hole solution in Einstein-bumblebee gravity theory. This AdS-like black hole can only exist with a linear functional potential of the bumblebee field. We found that the Smarr formula and the first law of black hole thermodynamics can still be constructed in this Lorentz symmetry breaking black hole spacetime, but the conceptions of the black hole horizon area/entropy and the volume inside the horizon should be renewed due to its anisotropy. We also found that two types of phase transition exist: small-large black hole phase transition and Hawking-Page phase transition, like those of the Schwarzschild AdS black hole. After Lorentz symmetry breaking, the black hole mass at the divergent point of heat capacity becomes small, and the Gibbs free energy of the meta-stable large black hole is also smaller, showing that the large stable black hole can be more easily formed.     

量子史瓦茨黑洞和暗物质

引入Sommerfeld作用量量子化条件来处理Schwarzschild黑洞的量子化问题. 发现此类量子化黑洞存在一个质量为m_G=123m_p的基态，处于基态的量子Schwarzschild黑洞不再存在Hawking蒸发和任何其他辐射，可名之曰暗星. 它的存在不仅可以解决信息丢失的疑难，而且极可能是构成暗物质的主要候选者. 关键词： 量子史瓦茨黑洞 暗物质     

de Sitter Tunneling,Emission Spectrum and Entropy/Area Quantum

The Banerjee-Majhi's recent work shows that the Hawking radiation and entropy/area quantum of the black hole horizon (EH) can be well described in the tunneling picture. In this paper, we develop this idea to the case of a de Sitter tunneling from the cosmological horizon (CH), and obtain the Hawking emission spectrum and entropy/area spectroscopy from the CH of the purely de Sitter black hole as well as the Schwarzschild-de Sitter black hole. It is interestingly found that the area of the CH is quantized by Δ A=4l_pl², as was given by Hod for the area quantum of -the EH by considering the Heisenberg uncertainty principle and Schwinger-type emission process. Also, we conclude from our derivation that the entropy/area quantum of the CH is universal in the sense that it is independent of the black hole parameters. This realization implies that, (at least) at a semiclassical level, the de Sitter gravity shares the similar quantum behavior as the usual gravity without presence of a cosmological constant.     

de Sitter Tunneling,Emission Spectrum and Entropy/Area Quantum

The Banerjee-Majhi's recent work shows that the Hawking radiation and entropy/area quantum of the black hole horizon (EH) can be well described in the tunneling picture. In this paper, we develop this idea to the case o a de Sitter tunneling from the cosmological horizon (CH), and obtain the Hawking emission spectrum and entropy/area spectroscopy from the CH of the purely de Sitter black hole as well as the Schwarzschild-de Sitter black hole. It i interestingly found that the area of the CH is quantized by A = 4l 2 pl , as was given by Hod for the area quantum of -the EH by considering the Heisenberg uncertainty principle and Schwinger-type emission process. Also, we conclude from our derivation that the entropy/area quantum of the CH is universal in the sense that it is independent of the black hole parameters. This realization implies that, (at least) at a semiclassical level, the de Sitter gravity shares the similar quantum behavior as the usual gravity without presence of a cosmological constant.