Solutions and Thermodynamics of Charged Rotating Black Holes in a Fuzzy Space

Abstract

We investigate the effect of fuzzy space on black hole and its thermodynamic Properties; we also present our results in different types of black holes according to the parameters 𝑄 and 𝐽 representing electric and magnetic charges.

1. Introduction

In general relativity, there exist three classes of black holes, namely, Schwarzschild black hole, Kerr black hole and Reissner-Nordström black hole, or Kerr-Newman black hole [1]. In fact, the solutions of Einstein’s field equation for the gravitational field of an electrically charged point mass were obtained in 1918 by Nordström and Reissner [2, 3], not long after Karl Schwarzschild had found the Schwarzschild metric as a solution for a point mass without electric charge and angular momentum [4].

However, the cosmic behavior of black holes has been studied in different points such as the cosmic history of black hole [5], the dynamical evolution and observational signatures of massive black hole [6].

In this paper we will examine the effect of fuzzy space [7] on a static, charged, and rotating black hole. We will investigate thermodynamic properties such as Hawking temperature and entropy.

This paper is organized as follows. In Section 2, we present Schwarzschild black hole in fuzzy space. In Section 3, we present the electric charged black hole in fuzzy space. We study a magnetic fuzzy black hole in Section 4. In Section 5, we give the combined case where 𝐽>0 and 𝑄>0. We give some thermodynamic properties of fuzzy black hole in diverse cases in Section 6. Finally, we discuss our results in Section 7.

2. Fuzzy Schwarzschild Black Hole

We consider Schwarzschild black hole in a fuzzy space with the mass density of point particle described by [8] 𝜌_ℎ(𝑟)=12𝜋ℎ²−𝑟expℎ. (1) The covariant conservation law ∇_𝜇𝑇^𝜇𝜈 and the condition of the metric coefficient 𝑔₀₀=−𝑔⁻¹_𝑟𝑟 for the noncommutative Schwarzschild-like metric and the energy momentum tensor are given by 𝑇^𝜈_𝜇=Diag−𝜌_ℎ(𝑟),𝑝_𝑟(𝑟),𝑝_⟂(𝑟),𝑝_⟂(𝑟), (2) where 𝑝_𝑟=−𝜌_ℎ,𝑝_⟂=𝑝_𝑟−𝑟2𝜕_𝑟𝜌_ℎ. (3) The asymptotic solution of Einstein’s equation (2), using (1) as the matter source, is the same as replacing the mass of Dirac delta function source in Schwarzschild spacetime by the effective mass of smeared source [9] 𝑀_ℎ(𝑟)=𝑀𝑑³𝑥𝜌_ℎ(𝑟)=4𝜋^𝑟₀𝑟^′2𝜌_ℎ𝑟^′𝑑𝑟^′1=𝑀1−2𝑟ℎ²+𝑟ℎ−𝑟+1expℎ. (4) The geometry of fuzzy black hole is described by the line element 𝑑𝑆²=−1−2𝑀_ℎ(𝑟)𝑟𝑑𝑡²+1−2𝑀_ℎ(𝑟)𝑟𝑑𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜙², (5) where 𝑔₀₀(𝑟)=𝑔^𝑟𝑟(𝑟)=𝑓(𝑟)=1−2𝑀_ℎ(𝑟)𝑟. (6) The event horizon of the black hole is given by setting 𝑓(𝑟_ℎ)=0, and the nonzero solution of this equation satisfies 𝑟_ℎ=2𝑀_ℎ(𝑟)1=2𝑀1−2𝑟_ℎℎ²+𝑟_ℎℎ−𝑟+1exp^ℎℎ. (7) Instead of a single-event horizon, there are different possibilities: (i) we have two distinct horizons for 𝑀>𝑀₀, (ii) in this case, we have one degenerate horizon (external black hole) for 𝑀=𝑀₀, and (iii) no horizon for 𝑀<𝑀₀.

3. Charged Black Hole in a Fuzzy Space

In this case, we have a black hole with 𝐽=0 and 𝑄>0. As mentioned in [10], the geometry of a fuzzy black hole is described by the line element 𝑑𝑆²=−𝑓(𝑟)𝑑𝑡²+1𝑓(𝑟)𝑑𝑟²+𝑟²𝑑𝜙². (8) Its Einstein’s field equations are written accordingly as 1𝑟𝑑𝑓(𝑟)𝑑𝑟=−32𝜋𝑀𝜌_ℎ(𝑟)−12𝐸²,𝑑²𝑓(𝑟)𝑑𝑟²=32𝜋𝑝_⟂(𝑟)+12𝐸². (9) Solving the previous equations, we find 𝑓(𝑟)1=−8𝑀1−2𝑟ℎ²+𝑟ℎ−𝑟+1expℎ+𝑟²𝑙²−𝑄²8𝜋²ln𝑟+2𝑒^{−(𝑟/ℎ)}−2𝑟ℎ+54𝑒^{−2(𝑟/ℎ)}𝑒−2^{−(𝑟/ℎ)}𝑟𝑒𝑑𝑟+^{−2(𝑟/ℎ)}𝑟𝑑𝑟, (10) where Ei(𝑧) represents the exponential integral function defined as Ei(𝑧)=−^∞_−𝑧𝑒^−𝑡𝑡𝑑𝑡, (11) and a smeared distribution function is described by 𝜌_ℎ(𝑟)=12𝜋ℎ²−𝑟expℎ,2Dfuzzyspace. (12) In this case, the metric 𝑓(𝑟) takes the following form: 𝑓(𝑟)1=−8𝑀1−2𝑟ℎ²+𝑟ℎ−𝑟+1expℎ+𝑟²𝑙²−𝑄²8𝜋²−𝑟ln𝑟−2ℎEiℎ+ℎ2𝑟Ei−2ℎ+2𝑒^{−(𝑟/ℎ)}𝑟−2ℎ𝑒^{−(2𝑟/ℎ)}−54𝑒^{−(2𝑟/ℎ)}. (13) Note that when 𝑟/ℎ→∞, either when considering a large black hole (𝑟→∞) or the commutative limit (ℎ→0), we obtain the following metric which defines charged BTZ solution [11]: 𝑓(𝑟)𝑟=−8𝑀+²𝑙²−𝑄²8𝜋²ln𝑟. (14) The metric (13) describes the geometry of a fuzzy black hole with the corresponding event horizon given by the following condition imposed on 𝑓(𝑟). Equation (13) cannot be solved in closed form. However, by plotting 𝑓(𝑟), one can see obvious intersections with the 𝑟-axis and determine numerically the existence of horizons and their radii.

In view of these results, there can be no black hole if the original mass is less than the lower limit mass 𝑀₀. The horizon of the external black hole is determined by the conditions 𝑓=0 and 𝜕_𝑟𝑓=0, and the mass of the external black hole can be written as 𝑀₀=ℎ³2𝑙²𝑒^𝑟0/ℎ−𝑄²ℎ³32𝜋²𝑟₀×1𝑟₀𝑒^𝑟0/ℎℎ+2𝑟₀−1ℎ+4𝑟₀ℎ²−ℎ²4𝑟₀+1𝑒2ℎ^{−(𝑟0/ℎ)}. (15)

4. Rotating Fuzzy Black Hole

The metric of a fuzzy black hole (with 𝐽>0 and 𝑄=0) is given in [12, 13] by 𝑑𝑆²=−𝑓(𝑟)𝑑𝑡²+1𝑓(𝑟)𝑑𝑟²+𝑟²𝑑𝜙², (16) where 𝑓(𝑟)1=−8𝑀1−2𝑟ℎ²+𝑟ℎ+1𝑒^{−(𝑟/ℎ)}+𝑟²𝑙²+𝐽²4𝑟². (17) Note that when ℎ→0, we obtain the well-known BTZ rotating solution with angular momentum 𝐽 and total mass 𝑀, 𝑓(𝑟)=−8𝑀+(𝑟²/𝑙²)+(𝐽²/4𝑟²). The metric (16) describes the geometry of a fuzzy black hole with the corresponding event horizon given by the following condition imposed on 𝑓(𝑟): 𝑓(𝑟)1=−8𝑀1−2𝑟_±ℎ²+𝑟_±ℎ+1𝑒^𝑟±/ℎ+𝑟²_±𝑙²+𝐽²4𝑟²_±=0. (18) This equation cannot be solved in closed form. However, by plotting 𝑓(𝑟), one can read intersections with the r-axis and determine numerically the existence of horizon(s) and their radii.

Figure 1 shows that the existence of angular momentum introduces new behavior with respect to the noncommutative black hole studied by Myung and Yoon [14]. Instead of a single-event horizon, there are different possibilities: we have two distinct horizons for 𝑀>𝑀₀, we have one degenerate horizon (external black hole) for 𝑀=𝑀₀, and no horizon for 𝑀<𝑀₀.

Figure 1 

Metric function 𝑓 as a function of 𝑟. We have taken the values (ℎ=0.1, 𝑙=10,𝐽=1, and 𝑀0≈0.001).

In view of these results, there can be no black hole if the original mass is less than the lower limit mass 𝑀₀. The horizon of the external black hole is determined by the conditions 𝑓=0 and 𝜕_𝑟𝑓=0, which gives 4𝑟⁴−4𝑟⁴12𝑟ℎ²+𝑟ℎ+1𝑒^−𝑟/ℎ−𝑟𝑒^{−(𝑟/ℎ)}=𝐽²𝑙²4𝑟𝑒^{−(𝑟/ℎ)}2𝑟+4−ℎ²𝑟+4ℎ+4𝑒^{−(𝑟/ℎ)}, (19) and then, the mass of the extremal black hole can be written as 𝑀₀=𝑟²₀8𝑙²+𝐽²32𝑟²₀×11−2𝑟²₀ℎ+𝑟₀ℎ−𝑟+1exp⁰ℎ⁻¹. (20) In the commutative limit ℎ→0, the extreme black hole has the horizon at 𝑟^BTZ₀=𝐽𝑙2, (21) and its mass is 𝑀^BTZ₀=18𝐽𝑙. (22)

5. Charged Rotating Fuzzy Black Hole

In this section, we will combine our results for the case of an electric and magnetic charged black hole; such type is known as Kerr-Newman black hole (𝐽>0 and 𝑄>0). Its metric is given by 𝑑𝑆²=−𝑓(𝑟)𝑑𝑡²+1𝑓(𝑟)𝑑𝑟²+𝑟²𝑑𝜙²,𝑓(𝑟)𝑟=−8𝑀+²𝑙²−𝑄²8𝜋²𝐽ln𝑟+²4𝑟². (23)(24) The metric (23) describes the geometry of a fuzzy black hole with the corresponding event horizon given by the following condition imposed on 𝑓(𝑟): 𝑓(𝑟)1=−8𝑀1−2𝑟ℎ²+𝑟ℎ−𝑟+1expℎ+𝑟²𝑙²+𝐽²4𝑟²−𝑄²8𝜋²−𝑟ln𝑟−2ℎEiℎ+ℎ2𝑟Ei−2ℎ−𝑟+expℎ𝑟−2ℎexp−2𝑟ℎ−54exp−2𝑟ℎ. (25) The derivative of the metric function is given by 𝜕_𝑟𝑓=−8𝑀ℎ³−𝑟expℎ+2𝑟𝑙²−𝐽²2𝑟³−𝑄²8𝜋²𝑟−𝑟1+2expℎ𝑟−exp−2ℎ𝑟−2ℎ−𝑟expℎ𝑟−2ℎ𝑟exp−2ℎ×4𝑟²ℎ𝑟exp−2ℎ−5𝑟𝑟2ℎexp−2ℎ. (26) In view of these results, there can be no black hole if the original mass is less than the lower limit mass 𝑀₀. The horizon of the external black hole is determined by the conditions 𝑓=0 and 𝜕_𝑟𝑓=0, which gives 𝑟²−𝑟expℎ−4𝑟ℎ³()−𝑟1−𝜆𝑟,ℎexpℎ𝐽=−²𝑙²4𝑟²−𝑟expℎ×𝑒^{−(𝑟/ℎ)}+4ℎ³𝑟()−𝑟1−𝜆𝑟,ℎexpℎ−𝑄²𝑙²8𝜋²−𝑟𝑟expℎ−𝑟ln𝑟+2Eiℎ𝑟−Ei−2ℎ−𝑟+2expℎ𝑟−2ℎexp−2𝑟ℎ−54exp−2𝑟ℎ+2ℎ³()−𝑟1−𝜆𝑟,ℎexpℎ×−𝑟1+2expℎ−exp−2𝑟ℎ−2ℎ−𝑟𝑟expℎ−ℎ2+5−2ℎ𝑟exp2𝑟ℎ+𝑟²exp−2𝑟ℎ, (27) where 𝜆(𝑟,ℎ)=(1/2)(𝑟/ℎ)²+(𝑟/ℎ)+1, and then, the mass of the external black hole can be written as 𝑀₀=4ℎ³−𝑟exp⁰ℎ×2𝑟₀ℎ−𝐽²2𝑟³₀−𝑄²8𝜋²𝑟₀×−𝑟1+2exp⁰ℎ𝑟−exp−2⁰ℎ−2ℎ𝑟₀−𝑟exp⁰ℎ−ℎ2+5𝑟2ℎ⁰𝑟exp−2⁰ℎ+𝑟²₀𝑟exp−2⁰ℎ. (28)

6. Thermodynamic Properties

6.1. Static Black Hole

The Hawking temperature is calculated by [15, 16] 𝑇_𝐻=14𝜋𝑑𝑔₀₀𝑑𝑟_𝑟=𝑟ℎ=14𝜋𝑟_ℎ×()𝑟1−1/2_ℎ/ℎ³exp−𝑟_ℎ/ℎ1−(1/2)𝑟_ℎ/ℎ²+𝑟_ℎ/ℎ+1exp−𝑟_ℎ./ℎ (29)

6.2. Charged Black Hole

Hawking temperature [17] is as follows: 𝑇_𝐻=1(𝑟)4𝜋𝜕𝑓𝜕𝑟_𝑟=𝑟+=1−4𝜋4𝑀_𝐻𝑟²₊ℎ³𝑒^{−(𝑟+/ℎ)}𝑟+2⁺𝑙²−𝑄²8𝜋²1𝑟₊ℎ+2²𝑟₊−1ℎ𝑒^{−(𝑟+/ℎ)}+4𝑟₊ℎ²+1−ℎ2ℎ4𝑟²₊𝑒^{−(2𝑟+/ℎ)}, (30) where 𝑀_𝐻=8𝜋²𝑟²₊−𝑄²𝑙²×ln𝑟₊−𝑟−2ℎEi⁺ℎ+ℎ2𝑟Ei−2⁺ℎ+2𝑒^{−(𝑟+/ℎ)}𝑟−2⁺ℎ𝑒^{−(𝑟+/ℎ)}−54𝑒^{−2(𝑟+/ℎ)}×64𝜋²𝑙²11−2𝑟₊ℎ²+𝑟₊ℎ−𝑟+1exp⁺ℎ⁻¹. (31) The first law of thermodynamics for a charged black hole form is 𝑑𝑀=𝑇_𝐻𝑑𝑆+Φ𝑑𝑄, (32) where the electrostatic potential is given by Φ=𝜕𝑀𝜕𝑄_𝑟+=−𝑄32𝜋²ln𝑟₊−𝑟−2ℎEi⁺ℎ+ℎ2Ei−2𝑟₊ℎ+2𝑒^{−(𝑟+/ℎ)}𝑟−2⁺ℎ𝑒^{−2(𝑟+/ℎ)}−54𝑒^{−(2𝑟+/ℎ)}×11−2𝑟²₊ℎ+𝑟₊ℎ−𝑟+1exp⁺ℎ⁻¹. (33) We calculate the entropy as 𝑆=^𝑟+_𝑟01𝑇_𝐻𝑑𝑀, (34) which gives 𝜋𝑆=2^𝑟+_𝑟01(−(=1−exp𝜉/ℎ))𝑑𝜉𝜋ℎ2−𝑟lnexp₊/ℎ−1exp𝑟₀./ℎ−1 (35)

6.3. Rotating Black Hole

For rotating fuzzy black hole, the hawking temperature of the geometry of the fuzzy black hole is given by [18] 𝑇_𝐻=1(𝑟)4𝜋𝜕𝑓𝜕𝑟_𝑟=𝑟+=1𝑟2𝜋𝑙⁺²×𝐽1−²𝑙²4𝑟²₊+𝑟³₊+𝐽²𝑙²/4𝑟₊4ℎ³1+𝑟₊/ℎ+(1/2)𝑟₊/ℎ²−𝑒^{−(𝑟+/ℎ)}, (36) where 𝑀_𝐻=𝑟²₊8𝑙²+𝐽²32𝑟²₊×11−2𝑟₊ℎ²+𝑟₊ℎ+1𝑒^{−(𝑟+/ℎ)−1}. (37) The temperature is a monotonically increasing function towards the horizon radius for large black holes, and the temperature of the extreme black hole is zero.

For large black holes, that is, (𝑟₊/ℎ)0𝑥02𝑎𝑏𝑐0, one recovers the temperature of the rotating BTZ black hole 𝑇^BTZ_𝐻=1𝑟2𝜋𝑙⁺²𝐽1−²𝑙²4𝑟⁴₊. (38) The first law of thermodynamics for a charged black hole form is 𝑑𝑀=𝑇_𝐻𝑑𝑆+Φ𝑑𝑄, (39) where the angular velocity of the black hole is given by Φ=𝜕𝑀𝜕𝐽_𝑟+=𝐽²2𝑟²₊. (40) It is the same rotating BTZ solution. Equation (37) can be written in the Gibbs form as [10, 12] 1𝑑𝑆=𝑇Φ𝑑𝑀−𝑇𝑑𝐽. (41) At the outer horizon, 𝑟=𝑟₊, we have 𝑀=𝑀(𝑟₊,𝐽) and 𝑑𝑀=𝜕𝑀𝜕𝑟₊𝑑𝑟₊+Φ𝑑𝐽. (42) Using (39) we get 1𝑑𝑀=𝑇𝜕𝑀𝜕𝑟₊𝑑𝑟₊, (43) where 𝑆 is the entropy of the noncommutative black hole. From (20) we finally obtain 𝑆=^𝑟+_𝑟011−𝑒^{−(𝜉/ℎ)}𝑑𝜉, (44) where the entropy is a function of 𝑟₊. Note that, in the large black hole limit, the entropy function corresponds to the Bekenstein-Hawking entropy (area law), 𝑆_BH=𝜋𝑟₊/2, for the rotating BTZ geometry.

7. Conclusion

In this paper, we have constructed a static, electric, and magnetic (rotating) black hole in a fuzzy space by using an anisotropic perfect fluid fuzzy black hole and a smeared distribution function as an electric and magnetic charge. We have investigated the Hawking temperature and entropy.

We have also studied the combined case where both magnetic and electric charges of the black holes in a fuzzy space exist.

Note that when 𝑟/ℎ→∞, either when considering a large black hole (𝑟→∞) or the commutative limit (ℎ→0), we obtain the metric of charged BTZ solutions with the electric charge 𝑄 and total mass 𝑀𝑓(𝑟)𝑟=−8𝑀+²𝑙²−𝑄²8𝜋²ln𝑟, (45) and we obtain the well-known BTZ rotating solution with angular momentum 𝐽 and the total mass 𝑀𝑓(𝑟)𝑟=−8𝑀+²𝑙²+𝐽²4𝑟². (46) We have obtained similar results in the case of a black hole in Moyal space, namely, the existence of the event horizon and the Hawking temperature and entropy [10, 12].