Thermodynamics of Noncommutative Geometry Inspired Black Holes Based on Maxwell-Boltzmann Smeared Mass Distribution *

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Thermodynamics of noncommutative geometry inspired black holes based on Maxwell-Boltzmann smeared mass distribution *

Jun Liang1;1) Yan-Chun Liu2 Qiao Zhu1 1 College of Science, Shannxi University of Science and Technology, Xi’an 710021, China 2 Library, Shannxi University of Science and Technology, Xi’an 710021, China

Abstract: In order to further explore the eﬀects of non-Gaussian smeared mass distribution on the thermodynamical properties of noncommutative black holes, we construct noncommutative black holes based on Maxwell-Boltzmann smeared mass distribution in 3-dimensional spacetime. The thermodynamical properties of the black holes are investigated, including Hawking temperature, heat capacity, entropy and free energy. We ﬁnd that, there do not exist multiple black holes with the same temperature, while there exists a possible decay of the noncommutative black hole based on Maxwell-Boltzmann smeared mass distribution into the rotating commutative BTZ black hole.

Key words: Physics of black holes, Black hole thermodynamics, Noncommutative geometry PACS: 04.70.-s, 04.70.Dy, 02.40.Gh

1 Introduction the pointlike source term with a Gaussian distribution, however, the Gaussianity need not be required always, Black hole physics is one of the important aspects of we note that a Reyleigh smeared mass distribution is general relativity, and extensive studies in this field have introduced by Myung et al. in [17], they found that been done from theoretical view points. In recent years, there may exists a phase transition between the non- noncommutative geometry inspired black holes aroused commutative black hole based on Reyleigh distribution a lot of interest among researchers. Nicolini et al. firstly and the non-rotating BTZ black hole [20]. In order to found a noncommutative inspired Schwarzschild black further explore the effects of non-Gaussian smeared mass hole solution in 4 dimensions [1–3], and subsequently, distribution on the thermodynamical properties of non- the model was extended to include the electric charge commutative black holes, in this paper, we introduce the [4] and extra-spatial dimensions [5, 6]. Kim et al. in- noncommutativity by taking the mass density to be a vestigated thermodynamical similarity between the non- Maxwell-Boltzmann smeared mass distribution (see Ref. commutative Schwarzschild black hole and the Reissner- [21]) instead of the conventional Dirac delta function. Nordström black hole [7], Nozari and Mehdipour in- We deduce and discuss the thermodynamical quantities vestigated Parikh-Wilczek tunneling of noncommutative of black holes, including Hawking temperature, heat ca- black holes [8–11], Mann it et al. investigated the pair pacity, entropy and free energy. We find that there do creation of noncommutative black holes in a background not exist multiple black holes with the same temperature, with positive cosmological constant [12], Giri found out while there exists a possible decay of the noncommuta- and calculated the asymptotic quasinormal modes of a tive black hole based on Maxwell-Boltzmann smeared noncommutative geometry inspired Schwarzschild black mass distribution into the commutative BTZ black hole. hole [13], Ding et al. studied the influence of space- The outline of this paper is as follows: In section 2, time noncommutative parameter on the strong field grav- we construct a noncommutative inspired black hole so- itational lensing in the Reissner-Nordström black hole lution of Einstein equations in AdS3 spacetime using an spacetime [14], Mureika et al. analyzed comprehensively anisotropic perfect fluid. Then in section 3, we study a noncommutative 2-dimensional black hole [15], Myung the thermodynamics of this noncommutative black hole. et al. [16, 17] and Tejeiro et al. [18] studied black hole The final section is for conclusions. thermodynamics in noncommutative spaces. (For a com- prehensive review see Ref. [19].) In most of these works, the noncommutative smearing is obtained by replacing

Received ∗∗ March 2013 ∗ Supported by Natural Science Foundation of Education Department of Shannxi Provincial Government (12JK0954) and Doctorial Scientiﬁc Research Starting Fund of Shannxi University of Science and Technology (BJ12-02). 1) E-mail: [email protected] °c 2013 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

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2 Derivation of the noncommutative fHrL 2 black hole solution in 3-dimensions 1.5 Following [1], instead of the pointlike mass den- 1 M=0.002 sity described by a δ-function, we consider a Maxwell- 0.5 M=0.076 Boltzmann smeared mass density [21] r 0.5 1 1.5 2 2.5 3 3.5 -0.5 M=0.200 2 2 Mr − r ρ (r) = e 4θ , (1) θ 16πθ2 -1 Fig. 1. f vs r for θ = 0.1, l = 2 and J = 1. We where the total mass√ M is diffused throughout the re- obtain r0 ≈ 1.22 and M0 ≈ 0.076. gion of linear size θ and θ is a positive parameter. The smeared mass distribution is now given by From Fig. 1, we see that, for this NCBTZ spacetime, Z r h ³ 2 ´ 2 i there are two horizons, i.e., the inner(Cauchy) horizon rC 0 0 0 r − r M (r) = 2πr ρ (r )dr = M 1− +1 e 4θ . (2) and outer(event) horizon r , and the distance between θ θ 4θ H 0 the horizons will increase by increasing the black hole mass M. Fig. 1 also shows that there exists a extremal In the limit of √r → ∞, we get M (r) → M. θ θ mass of M = M below which no black hole can be found. In order to find a black hole solution in AdS space- 0 3 At the extremal mass M , the inner and outer horizons time, we introduce the Einstein equation (in c = G = 1 0 are met at the extremal horizon r (r ≤ r ≤ r ). The unit) 0 C 0 H horizon radius of the extremal black hole is determined 1 1 R − g = 8πT + g , (3) by the condition of f = 0 and ∂rf = 0, which gives µν 2 µν µν l2 µν ³ 2 ´ r2 3 r J2 − 0 where l is related with the cosmological constant by r 0 + e 4θ 2 0 l2 4r2 2r J 1 µν − 0 + 0 − = 0, (8) Λ = − 2 . The conservation law T ,ν = 0 tell us that h ³ ´ r2 i 2 3 l r2 0 l 2r 2 0 − 4θ 0 the energy-momentum tensor takes an anisotropic form 8θ 1− 4θ +1 e µ Tν = diag(−ρθ,pr,p⊥), where the radial and tangential pressure are given by pr = −ρθ and p⊥ = −ρθ − r∂rρθ, and then, the mass of the extremal black hole can be respectively. Then solving the Einstein equations of mo- written as tion, we obtain the line element as 2 2 r0 J 2 + 2 l 4r0 2 2 −1 2 2 ϕ 2 M0 = h ³ ´ 2 i . (9) ds = −f(r)dt +f (r)dr +r (dϕ+N dt ), (4) r2 r0 0 − 4θ 8 1− 4θ +1 e where As an example, we obtain r0 ≈ 1.22 and M0 ≈ 0.076 h ³ 2 ´ 2 i 2 2 numerically for θ = 0.1, l = 2 and J = 1. r − r r J f(r) = −8M 1− +1 e 4θ + + , (5) 4θ l2 4r2 3 Thermodynamics of the NCBTZ black and hole J N ϕ = − . (6) 2r2 One of the most interesting aspects of black hole Noticing that in the limit of √r → ∞, this solution re- physics is the thermodynamical properties. Let us now θ duces to the well known rotating BTZ black hole solution consider the Hawking temperature of the NCBTZ black with angular momentum J and total mass M, hole which is calculated as 1 ³ 2 2 ´ ³ 2 2 ´−1 r J r J TH = ∂rf|rH ds2 = − 8M + + dt2 + 8M + + dr2 4π 2 2 2 2 l 4r l 4r ³ ´ r2 ( J2l2 H ) 2 2 − 4θ 2 2 rH + 2 rH e rH J l 4rH 2 ϕ 2 = 1− − . 2 4 h ³ ´ r2 i + r (dϕ+N dt ); (7) 2πl 4r r2 H H 2 H − 4θ 16θ 1− 4θ +1 e therefore, for convenience, we call this black hole NCBTZ (10) black hole . The horizons of the element (4) can be obtained from the equation f(r) = 0. f as a function of r In Fig. 2, we plot the Hawking temperature of the for different mass M is plotted in Fig. 1. NCBTZ black hole as a function of rH (from (10)), and

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we see that, the case is diﬀerent from that of the regu- r0 numerically, which is the same as that obtained from lar black hole with the Gaussian matter source in AdS5 the condition of f = 0 and ∂rf = 0; this shows the con- background [22], TH is a monotonically increasing func- sistency of the approach. From Fig. 2(a), we see that tion of rH , which indicates that there do not exist mul- by increasing the parameter θ the value of r0 also in- tiple black holes with the same temperature. And the creases. And for the purpose of comparison, we also plot noncommutativity leads to the existence of a extremal the Hawking temperature of a rotating BTZ black hole horizon r0 which black hole (due to Hawking radiation as a function of the horizon radius. Fig. 2(b) show that and evaporation) can shrink to because physically TH the value of r0 is insensitive to the change in the value cannot be negative, and at this extremal horizon r0 the of l. From Fig. 2(c), we see that the value of r0 becomes black hole cools down to a zero temperature. By setting larger as J increases.

TH = 0, for given θ, l and J, we can obtain the value of

Th Th Th 0.03 HcL 0.04 HbL 0.04 HaL 0.025

0.03 0.02 0.03 0.015 0.02 0.02 0.01 0.01 0.01 0.005

r r r 2 4 6 8 h 2 4 6 8 h 1 2 3 4 5 h

Fig. 2. TH as a function of rH . (a) The solid and dashed lines are for the NCBTZ black hole with θ = 0.5 and θ = 0.1, respectively, while the dotted line is for the rotating BTZ black hole. We have taken l = 5 and J = 0.05. (b) The solid and dashed line are for the NCBTZ black hole with l = 10 and l = 5, respectively. In both cases we have taken θ = 0.1 and J = 0.05. (c) The solid and dashed lines are for the NCBTZ black hole with J = 1 and J = 0.5, respectively. In both cases we have taken θ = 0.1 and l = 5.

In order to check the stability of the NCBTZ black Fig. 3(a)). The capacity is positive for rH > r0, therefore, hole, we calculate the heat capacity this NCBTZ black hole is stable. (The heat capacity will

become negative for rH < r0. However, this region is un- ∂M ∂M 1 C = = . (11) physical, it is not allowed by the deﬁnition of the horizon ∂TH ∂TH ∂rH radii.) And for the purpose of comparison, in Fig. 3(a), ∂rH we also plot the heat capacity of a rotating BTZ black

The heat capacity as a function of rH is shown in Fig. 3. hole as a function of the horizon radius. Fig. 3(b) show For instance, our numerical calculation shows that the that the heat capacity of the NCBTZ black hole is in- heat capacity vanishes at the extremal horizon r0 ≈ 0.885 sensitive to the change in the value of l. The behavior of (for θ = 0.1, l = 10 and J = 0.03) and r0 ≈ 1.898 the heat capacity of the NCBTZ black hole for diﬀerent (for θ = 0.5, l = 10 and J = 0.03), respectively (see J is shown in Fig. 3(c).

C C C 10 5 10

8 HaL 4 HbL 8 HcL

6 3 6

4 2 4

2 1 2

r r r 2 4 6 8 h 0.5 1 1.5 2 2.5 3 h 2 4 6 8 h

Fig. 3. The heat capacity as a function of rH . (a) The solid and dashed lines are for the NCBTZ black hole with θ = 0.5 and θ = 0.1, respectively, while the dotted line is for the rotating BTZ black hole. We have taken l = 10 and J = 0.03. (b) The solid and dashed line are for the NCBTZ black hole with l = 20 and l = 2, respectively. In both cases we have taken θ = 0.1 and J = 0.03. (c) The solid and dashed lines are for the NCBTZ black hole with J = 0.3 and J = 0.03, respectively. In both cases we have taken θ = 0.1 and l = 10.

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Our next step is to calculate the entropy of the based on Maxwell-Boltzmann distribution into the rotat- NCBTZ black hole. We require that the first law be sat- ing BTZ black hole [20]. isfied with the NCBTZ black hole, dM = TH dSH +ΩdJ, where Ω is the angular velocity of the black hole. On F the other hand, as physical quantities are evaluated on 0.001 the event horizon, M can be expressed as M = M(rH ,J), r ∂M ∂M 0.5 1 1.5 2 2.5 h and dM can be expressed as dM = ∂ drH+ ∂J dJ. From rH -0.001 the two different forms of dM one finds the expression 1 ∂M -0.002 for black hole entropy dSH = drH . Defining the TH ∂rH entropy of the NCBTZ black hole as follows [23] -0.003 Z -0.004 rH 1 ∂M S = drH , (12) r TH ∂rH 0 Fig. 4. Free energy F vs rH . The solid and dashed line show the free energy of the NCBTZ black hole where we find S = 0 for the extremal horizon radius r0, which is a reasonable choice because this choice automat- with θ = 0.1 and that of the rotating BTZ black ically guarantees vanishing thermodynamical entropy at hole, respectively. In both cases we have taken l = 10 and J = 0.03. absolute zero, as it is required by the third law of thermodynamics. Finally, we get Z π rH dr 4 Conclusions S = H . (13) ³ 2 ´ r2 2 r − H r0 1− H +1 e 4θ 4θ In this paper, we investigate the Hawing temperature, heat capacity, entropy and free energy of noncommuta- However, this entropy does not satisfy the area law. tive geometry inspired black holes based on Maxwell- As is well known, phase transition is one of important Boltzmann smeared mass distribution in 3-dimensional aspects in the study of thermodynamics of black holes, spacetime. The noncommutativity leads to the existence and the free energy plays a crucial role to test the phase of a extremal horizon radius, at which the black hole transition. In order to analyze the possibility of phase cools down to an absolute zero temperature. The tem- transition, let’s define the on-shell free energy as [17] perature is a monotonically increasing function of the event horizon, which indicates that there do not exist F (rH ,θ) = M(rH ,θ)−M0(θ)−TH (rH ,θ)S(rH ,θ). (14) multiple black holes with the same temperature. We use Here the extremal black hole is used as the ground state the first law of thermodynamics to derive the entropy, [24], As an example, we show the free energy the NCBTZ and it further confirms the incompatibility the first law black hole with θ = 0.1, l = 10 and J = 0.03 as well as the of thermodynamics with Bekenstein-Hawking entropy for free energy of the rotating BTZ black hole with l = 10 the noncommutative inspired black hole. Because the and J = 0.03 in Fig. 4. Clearly, the free energy of the heat capacity is positive, this black hole is stable. In NCBTZ black hole is higher than that of the rotating addition, our numerical calculation shows that there ex-

BTZ black hole at any value of rH , therefore, there exists a possible decay of the NCBTZ black hole into the ists a possible decay of the noncommutative black hole rotating commutative BTZ black hole.

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