Dong-Shan He^,?College of Physics and Electronic Engineering, Xianyang Normal University, Xianyang 712000, China

Corresponding authors: ?E-mail:hfrnsm@163.com

Received:2019-04-8Online:2019-08-1

Fund supported:

*Supported by the National Natural Science Foundation of China under Grant.11647060 Shaanxi Youth Outstanding Talent Support Plan, and the Fundamental Research Funds of Xianyang Normal University under Grant.15XSYK034 Shaanxi Youth Outstanding Talent Support Plan, and the Fundamental Research Funds of Xianyang Normal University under Grant.XSYGG201802

Abstract
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.
Keywords： information conservation;noncommutative black hole;canonical typicality


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Dong-Shan He. Information Conservation in a Noncommutative Quantum Black Hole Based on Canonical Typicality^*. [J], 2019, 71(8): 1007-1010 doi:10.1088/0253-6102/71/8/1007

1 Introduction

Hawking discovered that black holes radiate particles with a perfect thermal spectrum by adopting a quantum field theoretical approach in curved space.^[1-2] However, this important discovery has led to the black hole information loss problem, due to there is no correlation between pure thermal spectra. In 2000, Parikh and Wilczek^[3] demonstrated that the thermal spectrum of a black hole should be corrected through the quantum tunneling method, leading to the discovery of a non-thermal radiation spectrum. The presence of a non-thermal radiation spectrum indicates that there is correlation among emitted particles. Based on the non-thermal radiation spectrum, Zhang et al.^[4] proved that entropy is conserved by accounting for the correlation of black hole radiation. Their work gives a chain rule for conditional entropy and correlation information without detailed physical processes of correlation. Recently, He and Cai^[5-6] provided a semiclassical gravitational correlation explanation.

Noncommutative gravity has been considered in string theory.^[7] Moreover, noncommutative quantum cosmology was presented in Ref. [8], and has also been shown to be relevant in the black hole information paradox.^[9-11] The Schwarzschild metric can be mapped by the Kantowski-Sachs (KS) metric through a coordinate transformation. In Refs. [12-14], the thermodynamics of a noncommutative quantum black hole were studied via the Feynman-Hibbs procedure.^[15] Recently, a simple method for obtaining the radiation spectrum of a black hole was proposed by Ma et al.^[16] based on canonical typicality. In this way, Guo and Cai^[17] showed that entropy is conserved in a loop quantum black hole, and in Ref. [18], the correlation of the black hole radiation added by the dark energy was studied.

In this letter, we study a noncommutative quantum Schwarzchild black hole, the temperature and entropy corrections originate from the noncommutative quantum effect are obtained. Then, the non-thermal spectrum of Hawking radiation is derived with the general principle of canonical typicality, and the correlation between radiation particles is calculate. The correlation is shown to be increased by the noncommutative gravity effect, and the correlation between radiation particles ensures that information is conserved. This letter is organized as follows. In Sec. 2, we first review the Wheeler-DeWitt equation(WDWE) for a noncommutative Schwarzchild black hole using the KS metric. Then, using the Feynman-Hibbs procedure, we obtain the temperature and entropy of a noncommutative black hole. In Sec. 3, the non-thermal spectrum is obtained by using an approach based on canonical typicality. Then, utilizing this spectrum, we calculate the correlation between the black hole and its radiation. Finally, a conclusion is presented in Sec. 4.

2 Noncommutative Quantum Black Hole

The metric of a Schwarzschild black hole can be written as

(1)$\qquad\quad d s^2=-\Bigl(1-\frac{2M}{r}\Bigr)d t^2+\Bigl(1-\frac{2M}{r}\Bigr)^{-1} d r^2 \\ \qquad\quad + \, r^2(d\theta^2+\sin^2 \theta d\phi^2 )\,,$
where $M$ is the mass of the black hole. Inside a black hole ($r<2M$), since $g_{tt}>0$ and $g_{rr}<0$, $\partial_t$ becomes a spacelike vector. Thus, the time and space coordinates should be interchanged $t\leftrightarrow r$, and taking the transformation, we obtain^[12-13,19]

(2)$N^2=\Bigl(\frac{2M}{t}-1 \Bigr)^{-1},\quad e^{2 \sqrt{3}\gamma}=\frac{2M}{t}-1,\\ e^{-2 \sqrt{3}\gamma}e^{-2 \sqrt{3}\Omega}=t^2\,.$
Then, the Schwarzschild black hole metric (1) can be rewritten as

(3)$\qquad d s^2=-N^2 d t^2+e^{2 \sqrt{3}\gamma}d r^2+e^{-2 \sqrt{3}\gamma} e^{-2 \sqrt{3}\Omega} \\ \qquad \times\,(d\theta^2+\sin^2 \theta d\phi^2 )\,.$
This is the Kantowski-Sachs metric,^[8] where $N$ is an arbitrary lapse function, $\Omega$ and $\gamma$ are scale factors for the black hole. Using the canonical quantization method, with $H \psi=0$, we could get

(4)$\quad e^{\sqrt{3}\gamma+2\sqrt{3}\Omega} [-P_{\Omega}^2 +P_{\gamma}^2-48e^{-2\sqrt{3}\Omega}]\psi (\Omega ,\gamma)=0\,.$
Taking $P_{\Omega}=-i ({\partial}/{\partial \Omega})$ and $P_{\gamma}=-i ({\partial}/{\partial \gamma})$, the corresponding Wheeler-DeWitt equation can be obtained as

(5)$\Bigl[-\frac{\partial^2}{\partial \Omega^2}+\frac{\partial^2} {\partial \gamma^2}+48 e^{-2\sqrt{3}\Omega}\Bigr]\psi (\Omega ,\gamma)=0\,.$
Assuming $\psi (\Omega ,\gamma)=\phi(\Omega) \chi(\gamma)$, the solutions of Eq. (5) are given by

(6)$\psi (\Omega ,\gamma)=e^{\pm i \sqrt{3}\nu \gamma} K_{i\nu}(4e^{-\sqrt{3}\Omega})\,,$
where $\nu$ is a separation constant and, $K_{i\nu}$ are the modified Bessel functions of the second kind.

For the proposed noncommutative quantum black hole, we assume that the coordinates $\Omega$ and $\gamma$ obey the commutation relation $[\Omega,\gamma]=i \theta$, similar to the behavior in noncommutative quantum mechanics. It is well known that, this noncommutativity can be reformulated by the Moyal star product $*$ of functions.^[20]

(7)$f(\Omega,\gamma)*g(\Omega,\gamma)=f(\Omega,\gamma)e^{({i \theta}/{2}) (\overleftarrow{\partial_\Omega}\overrightarrow{\partial_\gamma}-\overleftarrow{\partial_\gamma}\overrightarrow{\partial_\Omega})} g(\Omega,\gamma)\,.$
Then, according to Eq. (5), the noncommutative Wheeler-DeWitt equation can be simply written as

(8)$\Bigl[-\frac{\partial^2}{\partial \Omega^2}+\frac{\partial^2}{\partial \gamma^2}+48 e^{-2\sqrt{3}\Omega}\Bigr]*\psi (\Omega ,\gamma)=0\,.$
By introducing new variables, $\Omega\rightarrow \Omega-\theta P_{\gamma}/2$ and $\gamma\rightarrow \gamma-\theta P_{\Omega}/2$, the momenta remain the same, and the potential term is modified as

(9)$V(\Omega,\gamma)*\psi (\Omega ,\gamma)=V(\Omega-\theta P_{\gamma}/2, \gamma-\theta P_{\Omega}/2)\psi (\Omega ,\gamma)\,.$
Thus, the noncommutative Wheeler-DeWitt equation can be rewritten as

(10)$\Bigl[-\frac{\partial^2}{\partial \Omega^2}+\frac{\partial^2}{\partial \gamma^2}+48 e^{-2\sqrt{3}\Omega+\sqrt{3}\theta P_{\gamma}}\Bigr]\psi (\Omega ,\gamma)=0\,.$
We solve this equation by separation of variables with the following ansatz

$\psi (\Omega ,\gamma)= e^{i\sqrt{3}\nu \gamma}\phi(\Omega)$,^[13] where $\sqrt{3}\nu$ is the eigenvalue of $P_\gamma$. Thus, $\psi(\Omega)$ satisfies the following equation

(11)$\frac{d^2 \phi(\Omega)}{d\Omega^2}-(48e^{-2\sqrt{3}\Omega+3\nu\theta}-3\nu^2)\phi(\Omega)=0\,.$
We perform a series expansion of the potential $48e^{-2\sqrt{3}\Omega}$ at $\Omega \rightarrow 0$, and maintain terms up to the second order of $\Omega$. After performing a change of variable $x=\sqrt[4]{4\pi/3} (2\sqrt{3}\Omega-1) l_p$, then Eq. (11) becomes

(12)$[-\frac{1}{2}l_p^2 E_p \frac{d^2}{d x^2}+\frac{3E_p}{4\pi l_p^2} x^2 e^{3\theta \nu} \phi(x) \\ \quad =\sqrt{\frac{3}{16\pi}} E_p [\frac{\nu^2}{4}-2 e^{3\theta \nu} \phi(x)\,.$
This equation corresponds to a harmonic oscillator with potential $V_{\rm NC}(x)=({3E_p}/{4\pi l_p^2}) x^2e^{3\theta \nu}$. Using the Feynman-Hibbs procedure,^[14-15] one can introduce a quantum correction to the partition function through the potential, the quantum correction to the potential $V_{\rm NC}(x)$ is given by

(13)$V^{\rm NC}_{\rm QFH}=\frac{\beta l_p^2 E_p V_{\rm NC}^{\prime\prime}(x)}{24} =\frac{ \beta E_p^2 e^{3\theta \nu}}{16\pi}\,.$
Then, the corrected partition function is

(14)$Z_{Q}^{\rm NC}=\frac{1}{\sqrt{24\beta V^{\rm NC}_{\rm QFH}}}e^{-\beta V^{\rm NC}_{\rm QFH}} \\ =\sqrt{\frac{2\pi}{3}}\frac{\exp[{-\frac{\beta^2 E_p^2e^{3\theta \nu}}{16\pi} -\frac{3\theta \nu}{2}}]}{\beta E_p}\,.$
The energy of the black hole $\bar{E}=M$ can be obtained from the partition function

(15)$\bar{E}=-\frac{\partial \ln Z^{\rm NC}_Q}{\partial \beta} =\frac{\beta E_p^2 e^{3\theta\nu}}{8\pi}+\frac{1}{\beta}\,.$
From Eq. (15), the corrected temperature can be obtained as

(16)$\beta_{\rm NC} \simeq \frac{8\pi M c^2}{E_p^2 e^{-3\theta \nu }}-\frac{1}{Mc^2} \\ = \beta_{H}^{\rm NC} \Bigl( 1-\frac{1}{\beta_H^{\rm NC} M c^2} \Bigr)\,,$
where $\beta_H^{\rm NC}=8\pi M e^{3\theta \nu }/E_p^2 $ is the noncommutative Hawking temperature.Moreover the entropy of the noncommutative quantum black hole can be obtained as

(17)$2S_{\rm NC}= k_B \ln Z_Q^{\rm NC}+k_B\frac{\bar{E}}{\beta_{\rm NC}}\\= S_{\rm BH}^{\rm NC}-\frac{1}{2}k_B \ln(24S_{\rm BH}^{\rm NC}/k_B)+O(S_{\rm BH}^{{\rm NC}-1})\,,$
where we define $S_{\rm BH}^{\rm NC}=4\pi M^2 e^{3\theta \nu }$ as the noncommutativeBekenstein-Hawking entropy. By setting $\theta=0$, we can obtain the entropy of thequantum black hole as $S_{N}=S_{\rm BH}-({1}/{2})k_B \ln(24S_{\rm BH}/k_B)+O(S_{\rm BH}^{-1})$.This logarithmic corrected term is consistent with that obtained in loop quantumgravity and string theory. Therefore, it can be seen that in Eq. (17),the logarithmic term arises from the quantum gravitational correction andthe noncommutative correction is related to $\theta$.

3 Information Conservation Based on Canonical Typicality

In recent research,^[16] Ma et al. obtained the non-thermal blackhole radiation spectrum based on a canonical typicality method. The authorsconsidered a system $B$ consisting of a black hole $B'$ and a radiation particle $R$,the reduced density matrix of $R$ can be written as

(18)$\rho_R=\sum_{r} \frac{\Omega_B'(E-E_r)}{\Omega_B(E)}|r\rangle \langle r|\,,$
where $\Omega_B'(E-E_r)$ is the number of micro-states for theblack hole $B'$ with energy $E-E_r$ and $\Omega_B(E)$ is the totalnumber of micro-states for the system $B$. With the Boltzmann entropyof $B$ as $S_B=\ln \Omega_B(E)$, Eq. (18) can be written as

(19)$\rho_R=\sum_{r} e^{-\Delta S_{BB'}}|r\rangle \langle r|\,.$
According to Eq. (17), the entropy of the noncommutative quantum black hole with mass $M$ is

(20)$S_{\rm NC}(M)=4\pi M^2e^{3\theta \nu }-\frac{1}{2}k_B \ln(96\pi M^2 e^{3\theta \nu })\,.$
When a particle with energy $\omega$ escapes from the black hole, the entropy of the remaining black hole can be written as

(21)$S_{\rm NC}(M-\omega)=4\pi (M-\omega)^2e^{3\theta \nu }-\frac{1}{2}k_B\\ \quad\times\ln[96\pi (M-\omega)^2 e^{3\theta \nu }]\,.$
Thus, we find

(22)$\Delta S_{\rm NC}(\omega)=8\pi \omega (M-\omega/2)e^{3\theta \nu }-k_B \ln\Bigl[\frac{M}{M-\omega}\Bigr]\,.$
By expanding $\Delta S_{\rm NC}$ at $\theta\rightarrow 0$ and maintaining terms to the first order of $\theta$, we obtain

(23)$\Delta S_{\rm NC}(\omega)\approx 8\pi \omega (M-\omega/2)+24\pi\theta \nu \omega (M-\omega/2) \\ - k_B \ln[\frac{M}{M-\omega}]\,.$
It is obvious that the radiation spectrum is non-thermal. According to Eq.(19), the distribution probability of the particle with energy $\omega_1$ escaping from the black hole $M$ is given by

(24)$P(\omega_1)=e^{-\Delta S_{\rm NC}(\omega_1)}\,.$
Thus, the probability for the second particle is conditional probability

(25)$P(\omega_2|\omega_1)=e^{S_{\rm NC}(M-\omega_1-\omega_2)-S_{\rm NC}(M-\omega_1)}\,,$
and the joint possibility is given by

(26)$P(\omega_1,\omega_2)=P(\omega_1)P(\omega_2|\omega_1)\,.$
It is straightforward to verify that $P(\omega_2,\omega_1)= P(\omega_2+\omega_1)$.

The correlation between these two particles is

(27)$C(\omega_1,\omega_2)= \ln P(\omega_1+\omega_2)-\ln[P(\omega_1)P(\omega_2)]\\= 8\pi \omega_1 \omega_2 e^{3\theta \nu}+\ln\frac{(M-\omega_1)(M-\omega_2)}{M(M-\omega_1-\omega_2)} \\ \approx 8\pi \omega_1 \omega_2 +24\pi \theta \nu \omega_1 \omega_2+\frac{\omega_1 \omega_2}{M^2}\,.$
The first term of Eq.(27) agrees with result of Ref.[4],and this correlation arises from the classical gravity correlation,^[5-6]the second term denotes a correlation due to the noncommutative effect, and the lastterm represents the correlation arising from quantum gravity.

By the chain rule,^[22] the probability for the specific microstate $(\omega_1, \omega_2,\ldots, \omega_n)$to occur is given by

(28)$P(\omega_1,\omega_2,\cdots,\omega_n)=P(\omega_1)P(\omega_2|\omega_1)\cdots P(\omega_n|\omega_1,\omega_2,\ldots,\omega_{n-1})\\ =\exp[{S_{\rm NC}(M-\omega_1-\omega_2-\cdots-\omega_n)-S_{\rm NC}(M)}]\\ = P(\omega_1+\omega_2+\cdots+\omega_n)\,.$
If $\omega_1+\omega_2+\cdots+\omega_n=M$, we have $ P(\omega_1,\omega_2,\ldots,\omega_n)=e^{-S_{\rm NC}(M)}$. This shows that the total entropy carried away by all emissions isprecisely equal to the entropy in the original black hole, which shows that theentropy is conserved in Hawking radiation.

4 Conclusion

In this paper, based on the Kantowski-Sachs metric and the noncommutative WDWE, thetemperature and entropy of a phase-space noncommutative Schwarzschild black hole were computed.We applied the technique of Ma et al.to noncommutative quantum black hole, and the non-thermal radiationspectrum was obtained. This finding agrees with the result given by the tunneling method.^[3]This non-thermal radiation spectrum shows that an information correlation exists among black holeradiation. This correlation consists of three parts: a classical gravitationalcorrelation,^[5-6] a non-commutative correlation, and a quantum effect-induced correlation.It is shown that the noncommutative and quantum effects increase the correlation among radiation,and the information conservation is still hold in noncommutative quantum black holes.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1] S. W. Hawking , Nature (London) 284 ( 1974) 30.
[Cited within: 1]

[2] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.
[Cited within: 1]

[3] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042.
[Cited within: 2]

[4] B. C. Zhang, Q. Y. Cai, L. You, et al., Phys. Lett. B 675 (2009) 98.
[Cited within: 2]

[5] D. S. He and Q. Y. Cai , Sci. China Ser. G 60 ( 2017) 040011.
[Cited within: 3]

[6] D. S. He and Q. Y. Cai, Chin. Sci. Bull. 63 (2018) 3089.
[Cited within: 3]

[7] N. Seiberg and E. Witten , J. High Energy Phys. 1999 (1999) 032.
[Cited within: 1]

[8] H. Garcia-Compean, O. Obregon, C. Ramirez, Phys. Rev. Lett. 88 (2002) 161301.
[Cited within: 2]

[9] A. Yakir, A. Casher, S. Nussinov, Phys. Lett. B 191 (1987) 51.
[Cited within: 1]

[10] P. Chen, Y. C. Ong, D. H. Yeom, Phys. Rep. 603 (2015) 1.


[11] W. G. Unruh and R. M. Wald, Rep. Prog. Phys. 80 (2017) 092002.
[Cited within: 1]

[12] C. Bastos, O. Bertolami, N. C. Dias, J. N. Prata, Phys. Rev. D 80 (2009) 124038.
[Cited within: 2]

[13] J. C. Lopez-Dominguez, O. Obregon, C. Ramirez , et al., J. Phys. Conf. Ser. 91 (2007) 012010.
[Cited within: 2]

[14] C. Bastos, O. Bertolami, N. C. Dias , et al., J. Phys. Conf. Ser. 222 (2010) 012033.
[Cited within: 2]

[15] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).
[Cited within: 2]

[16] Y. H. Ma, Q. Y. Cai, H. Dong, et al., Europhys. Lett. 122 (2018) 30001.
[Cited within: 2]

[17] X. K. Guo and Q. Y. Cai , Mod. Phys. Lett. A 33 (2018) 1850103.
[Cited within: 1]

[18] Y. H. Ma, J. F. Chen, C. P. Sun, Nucl. Phys. B 931 (2018) 418.
[Cited within: 1]

[19] C. Bastos, O. Bertolami, N. C. Dias, et al., Phys. Rev. D 82 (2010) 041502.
[Cited within: 1]

[20] M. Aguero, J. A. S. Aguilar, C. Ortiz , et al., Int. J. Theor. Phys. 46 (2007) 2928.
[Cited within: 1]

[21] M. Blaschke, Z. Stuchlík, F. Blaschke, et al., Phys. Rev. D 96 (2017) 104012.


[22] Q. Y. Cai, C. P. Sun, L. You, Nucl. Phys. B 905 (2016) 327.
[Cited within: 1]