Accretion and Evaporation of Modified Hayward Black Hole

Eur. Phys. J. C (2015) 75:129 DOI 10.1140/epjc/s10052-015-3349-1

Regular Article - Theoretical Physics

Accretion and evaporation of modiﬁed Hayward black hole

Ujjal Debnatha,b,c Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711 103, India

Received: 18 February 2015 / Accepted: 3 March 2015 / Published online: 20 March 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We assume the most general static spherically isfies w<−1) [14]. Till now a lot of dark-energy models symmetric black hole metric. The accretion of any general have been considered. A brief review of dark-energy models kind of fluid flow around the black hole is investigated. The is found in Ref. [15]. accretion of the fluid flow around the modified Hayward A condensed object (e.g. a neutron star, a black hole, etc.) black hole is analyzed, and we then calculate the critical surrounded by a fluid can capture particles of the fluid that point, the fluid’s four-velocity, and the velocity of sound dur- pass within a certain distance from the condensed object. This ing the accretion process. Also the nature of the dynamical phenomenon is termed accretion of the fluid by condensed mass of the black hole during accretion of the fluid flow, objects. In Newtonian theory of gravity, the problem of accre- taking into consideration Hawking radiation from the black tion of matter onto the compact object was first formulated by hole, i.e., evaporation of the black hole, is analyzed. Bondi [16]. Michel [17] first obtained an analytic relativis- tic accretion (of gas) solution onto the static Schwarzschild black hole. Such accretion processes are candidates to the 1 Introduction mechanisms of the formation of supermassive black holes (SMBH) in the center of most active galaxies [18]. In partic- At present we live in a Universe which is expanding and ular, it should show some analogies with the process proposed the expansion rate is increasing, i.e., the Universe is accel- by Salpeter et al. [19] where galaxies and quasars could get erating, which was confirmed by recent Supernova type Ia some of their energy from processes of accretion. Using this observations [1,2]. The large scale structure [3Ð5] and cos- accretion procedure, Babichev et al. [20,21] formulated the mic microwave background radiation [6] WMAP observa- accretion of phantom dark energy onto a static Schwarzschild tions [7Ð9] also support this acceleration of the Universe. black hole and showed that static Schwarzschild black hole This acceleration is caused by some unknown matter which mass will gradually decrease due to the strong negative pres- produces a sufficiently strong negative pressure (with posi- sure of the phantom energy and finally all the masses tend tive energy density), known as dark energy. The present Uni- to zero near the big rip singularity. Sun [22] discussed phan- verse occupies ∼4 % ordinary matter, ∼74 % dark energy tom energy accretion onto a black hole in the cyclic universe. and ∼22 % dark matter. Dark energy and dark matter are Jamil [23] has investigated accretion of a phantom like mod- the two main components in our universe; the present dark- ified variable Chaplygin gas onto the Schwarzschild black energy and dark-matter densities are 7.01 × 10−27 and hole. Phantom energy accretion by a stringy charged black 2.18 × 10−27 kg/m3, respectively. The simplest candidate hole has been discussed by Sharif et al. [24]. Dark matter of the dark energy is the cosmological constant , which and dark energy accretion onto a static black hole has been obeys the equation of state EoS p = wρ with EoS parameter discussed by Kim et al. [25]. Also the accretion of the dark w =−1[10,11]. Other candidates for the dark energy are energy onto the more general KerrÐNewman black hole was quintessence (where the EoS parameter satisfies −1

123 129 Page 2 of 5 Eur. Phys. J. C (2015) 75 :129 √ d ( 1 − ) = In the present work, first we assume the most general static obtain dr T0√ g 0, which provides the first integral, 1 spherically symmetric black hole metric in Sect. 2. The accre- (ρ + p)u0u −g = C1. This simplifies to tion of any general kind of the fluid flow around the black − A hole will be investigated. The accretion of the fluid flow ur2 M 2(ρ + p) u2 + B = C (3) B 1 around the modified Hayward black hole will be analyzed in Sect. 3 and we then calculate the critical point, the fluid’s where C1 is an integration constant, which has the dimension four-velocity, and the velocity of sound during the accre- of the energy density. Moreover, the energy flux equation can tion process. Also the nature of the dynamical mass of the be derived by the projection of the conservation law for the black hole during accretion of the fluid flow and taking into energyÐmomentum tensor onto the fluid four-velocity, i.e., μν μ μ consideration Hawking radiation from the black hole, i.e., uμT;ν = 0, which gives u ρ,μ + (ρ + p)u;μ = 0. From evaporation of the black hole, will be analyzed in Sect. 4. this, we obtain Finally, we shall present fruitful discussions of the accre- ρ − A h dρ tion of the fluids upon the modified Hayward black hole in ur2 M 2 exp =−C (4) B ρ ρ + p(ρ) Sect. 5. ∞ where C is an integration constant (energy flux onto the black hole) and the associated minus sign is taken for convenience. 2 Accretion phenomena of general static spherically Also ρh and ρ∞ represent the energy densities at the black symmetric black hole hole horizon and at infinity, respectively. Combining Eqs. (3) and (4), we obtain First we consider general static spherically symmetric metric ρ h ρ given by 2 A d (ρ + p) u + B exp − = C2 (5) B ρ ρ + p(ρ) 1 ∞ ds2 =−A(r)dt2 + dr 2 + r 2(dθ 2 + sin θdφ2) (1) B(r) where C2 =−C1/C = ρ∞ + p(ρ∞). The equation of mass μ √ flux J = 0 is given by d (J 1 −g) = 0, which integrates where A(r)>0 and B(r)>0 are functions of r only. We ;μ√ dr 1 can choose A(r) and B(r) in such a way that the above metric to ρu −g = A1 and yields represents a black hole metric. Let us assume M is the mass ( ) = ( ) = − 2M ρ 2 −2 A = of the black hole. For instance, if A r B r 1 r , ur M C3 (6) the above metric represents a Schwarzschild black hole. B The energyÐmomentum tensor for the fluid is given by where C3 is an integration constant. From (3) and (6), we obtain Tμν = (ρ + p)uμuν + pgμν (2) ρ + p A C1 where ρ and p are the energy density and pressure of the u2 + B = = C = constant. (7) ρ B C 4 fluid. The four-velocity vector of the fluid flow is given by 3 μ uμ = dx = (u0, u1, 0, 0) where u0 and u1 are the non- Now let us assume ds μ zero components of velocity vector satisfying uμu =−1. dln(ρ + p) 0 0 1 1 V 2 = − 1. (8) This implies g00u u + g11u u =−1. So we can obtain ρ ( 1)2+ dln (u0)2 = u B and let the radial velocity of the flow u1 = AB √ √ Thus, from Eqs. (6), (7), and (8), we obtain 0 A 2 u, thus we have u0 = g00u = u + B.Here −g = B 2 2 u du 2 1 A B A r 2sinθ. From the above Eq. (2), we obtain T 1 = (ρ + V − + −2V + − B 0 u2 + B u 2 A B p)u u. It is assumed that u < 0 for inward flow of the fluid 0 rB dr toward the black hole. ×(V 2 + 1)r + = 0. (9) 2(u2 + B) r In the fluid flow, we may assume that the fluid is dark matter or any kind of dark energy. A proper dark-energy Now if one or the other of the bracketed terms in (9) vanishes, accretion model for a static spherically symmetric black hole we get a turn-around point, and in this case, the solutions will should be obtained by generalizing Michel’s theory [17]. In be the double-valued in either r or u. There are only solutions the dark-energy accretion onto Schwarzschild black hole, which pass through a critical point that correspond to material Babichev et al. [20,21] have performed the above gener- falling into (or flowing out of) the object with monotonically alization. We shall follow now the above procedure in the increasing velocity along with the particle trajectory. A point case of static spherically symmetric black hole. The rel- where the speed of the flow is equal to the speed of sound is ativistic Bernoulli equation (the time component) of the called a critical point. It is assumed that the critical point of μν energy-momentum conservation law is T;ν = 0, and we accretion is located at r = rc, which is obtained by taking 123 Eur. Phys. J. C (2015) 75 :129 Page 3 of 5 129 the two bracketed terms (coefficients of du and dr)inEq.(9) Schwarzschild black hole, but it becomes a de Sitter black 2 B(r) ≈ − r r ≈ to be zero. So at the critical point, we obtain hole as 1 l2 near the center ( 0), so it is a regular space-time without singularity. Thus a Hayward black u2 V 2 = c (10) hole is the simplest regular black hole. Some physical con- c u2 + B(r ) c c sequences of Hayward black holes have been discussed by and several authors [41Ð43]. After that the Hayward metric was ( ) = ( ) ( ) 4V 2 A (r ) B (r ) B (r ) modified [44] by choosing A r f r B r , satisfying the c = c − c (V 2 + ) + c . c 1 2 (11) following conditions: it rc A(rc) B(rc) u + B(rc) c (i) preserves the Schwarzschild behavior at large r, Here, the subscript c denotes the critical value and uc is the (ii) includes the one-loop quantum corrections, and critical speed of the flow at the critical point rc.Fromthe (iii) allows for a finite time dilation between the center above two expressions, we have and infinity. ( ) ( ) ( ) ( ) −1 So the modified Hayward black hole metric is given by 2 = B rc A rc 2 − A rc + B rc uc (12) [44] 2 A(rc) rc A(rc) B(rc) 1 and s2 =−f (r)B(r) t2 + r 2 + r 2( θ 2 + θ φ2) d d ( ) d d sin d − B r A(r ) B(r) 2 A(r ) B(r ) 1 V 2 = 1 + 2 c − c + c . (17) c A(r ) B(r ) r A(r ) B(r ) c c c c c where (13) 2Mr2 αβ M B(r) = 1 − , f (r) = 1 − (18) At the critical point rc, the sound speed can be determined r 3 + 2Ml2 αr 3 + β M by α, β ( ) = with positive constants. Now from the relation A r d p C V (V 2 + 1) B(r ) f (r)B(r), we may obtain c2 = = 4 c c c − 1. (14) s ρ ( ) d r=r uc A rc A (r) f (r) B (r) c = + . (19) The physically acceptable solutions of the above equations A(r) f (r) B(r) 2 > 2 > may be obtained if uc 0 and Vc 0, which leads to Also from the expressions of B(r) and f (r) [Eq. (18)], we get 2 A (r ) B (r ) A (r )B (r )>0 and > c − c . (15) c c r A(r ) B(r ) B(r) 2Mr(r 3 − 4Ml2) c c c = , (20) ( ) ( 3 + 2)[ 3 + ( 2 − 2)] From the above equation we can obtain the bound of rc if A B r r 2Ml r 2M l r and B are known for several kinds of static black holes. f (r) 3α2β Mr2 = . (21) f (r) (αr 3 + β M)[αr 3 + (1 − α)βM] Since outside the horizon, 3 Accretion phenomena of modified Hayward black (i) B(r)>0, which implies r 3 > 2M(r 2 − l2) and (ii) hole f (r)>0, β(α−1)M 1 we get r > [ α ] 3 with α>1. So from Eq. (21), we The static spherically symmetric space-time is described by have f (r)>0. the Hayward metric which is obtained by A(r) = B(r) in If we assume that the fluid flow accretes upon the mod- Eq. (1) and is given by [36] ified Hayward black hole, we can calculate the expressions 1 of u2, V 2 and c2 at the critical point r . The expressions are ds2 =−B(r)dt2 + dr 2 + r 2(dθ 2 + sin θ dφ2). (16) c c s c B(r) given below (using Eqs. (12), (13), and (14)): Here, M is the mass of the Hayward black hole and B(r) = ( ) ( ) ( ) ( ) −1 2 B rc B rc f rc 2 f rc 2Mr2 u = + − , 1− , where l is a parameter with dimensions of length c ( ) ( ) ( ) (22) r3+2Ml2 2 B rc f rc rc f rc (Hubble length) with a small scale related to the inverse − B(r ) B(r ) f (r ) 1 cosmological constant (l is a convenient encoding of the V 2 = + c c + c c 1 2 ( ) ( ) ( ) central energy density 3 ∼ , assumed positive). Such B rc B rc f rc 8πl2 a behavior has been proposed by Sakharov [37,38]asthe −1 2 f (r ) equation of state of matter at high density and by Markov × − c , ( ) (23) [39,40] based on an upper limit on the density or curvature, rc f rc to be ultimately justified by a quantum theory of gravity. In C V (V 2 + ) 2 = 4 c√ c 1 − the limit r →∞, B(r) ≈ 1 − 2M , which represents the cs 1(24) r uc f (rc) 123 129 Page 4 of 5 Eur. Phys. J. C (2015) 75 :129 where B(r), f (r), and their derivatives are given in (18), We may also assume that the black hole evaporates by the (20), and (21) at the point r = rc. The physically acceptable Hawking radiation process. The rate of change of mass for 2 > solutions of the above equations may be obtained if uc 0 the evaporation is given by and V 2 > 0, which leads to c ˙ D Meva =− (29) f (r ) 2 M2 B (r )> < c < . c 0 and 0 ( ) (25) f rc rc where D > 0 is a constant whose value depends on the model

From the above restrictions, we may get the bounds of rc, [45]. Now due to accretion of the fluid flow and evaporation that is (α>1): of the mass of the black hole, we get the rate of change of √ the mass of the black hole as β(−4 + 5α + α(25α − 24))M r 3 > Max 4Ml2, . D c 4α M˙ = M˙ + M˙ = 4πCM2(ρ + p) − . (30) acc eva M2 (26) For the accretion scenario, the change of the mass of the For example, we assume the fluid flow obeys a linear equation black hole completely depends on the nature of the fluid = wρ w = of state p ( constant) as it accretes upon the that accretes. But for an evaporation process, the change of 2 = w modified Hayward black hole. Then we obtain cs and the mass of the black hole is independent of the nature of 2 = = Vc 0 and from (10), we obtain uc 0. From Eqs. (22) and the fluid, because this is internal process. In fact, when the = (24), we see that the critical point occurs at the point rc accretion fluid is only the cosmological constant (p =−ρ), 1 (4Ml2) 3 . For the general equation of state where w = w(t), the mass of the black hole for only the accretion scenario 2 = 2 = 2 = we obtain cs constant, Vc 0, and uc 0. In this case, is always the same throughout the time evolution. Only in it is very difficult to obtain the critical point rc. the accretion process, the mass of the black hole increases for a normal fluid and quintessence type dark energy fluid 4 Changes of black hole mass during accretion and decreases for phantom dark energy. But due to accre- M˙ > M4 > D and evaporation tion as well as evaporation, 0for 4πC(ρ+p) ˙ < 4 < D and M 0forM 4πC(ρ+p) for a normal fluid and a The rate of change of mass M˙ of the black hole is computed quintessence type dark energy, but for phantom energy, the ˙ < by integrating the flux of the fluid over the 2-dimensional black hole mass always decreases (M 0). Thus evapora- ˙ =− 1 tion supports the decreasing of the mass of the black hole surface√ of the black hole and given by M T0 dS where dS = −gdθdφ.UsingEq.(3), we obtain the rate of change with some restrictions of the minimum values of the mass of of the mass of the black hole in the following form: the black hole. ˙ 2 M = 4πCM (ρ∞ + p(ρ∞)). (27) The above result is also valid for any equation of state p = 5 Discussions and concluding remarks p(ρ). So the rate of change of mass for the accreting fluid around the black hole will be First we have assumed the most general static spherically symmetric black hole metric. The accretion of any general ˙ = π 2(ρ + ). Macc 4 CM p (28) kind of the fluid flow around the black hole has been inves- We see that the rate of change of mass for the general spheri- tigated. For this general kind of static black hole, the critical cally symmetric static black hole due to accretion of the fluid point, velocity of sound, and the fluid’s four-velocity have flow becomes exactly the rate of the case of a Schwarzschild been calculated and it was shown that these values depend black hole. From the expression (28) it is to be noted that the completely on the metric coefficients. Next, the accretion of rate of change of the mass of any static spherically symmet- the fluid flow around the modified Hayward black hole has ric black hole is completely independent of A(r) and B(r). been analyzed and we then calculated the critical point, the When some fluid accretes outside the black hole, the mass fluid’s four-velocity, and the velocity of sound during the function M of the black hole is considered as a dynamical accretion process. We can mention that outside the horizon, ( )> 3 > ( 2 − 2) mass function and hence it should be a function of time also. (i) B r 0, which implies r 2M r l , and (ii) ˙ β(α−1)M 1 So M is time dependent and the increasing or decreasing of f (r)>0, so we get r > [ α ] 3 with α>1 and also the black hole mass M sensitively depends on the nature of f (r)>0. For the physical region of accretion the bounds the fluid which accretes upon the black hole. If ρ + p < 0 of the critical point have been√ generated and were found to 3 > { 2, β(−4+5α+ α(25α−24))M } i.e., for phantom dark-energy accretion, the mass of the black be rc Max 4Ml 4α . When the per- hole decreases but if ρ + p > 0, i.e., for quintessence dark- fect fluid satisfies a linear equation of state, p = wρ (w = energy accretion, the mass of the black hole increases. constant), it accretes upon the modified Hayward black hole, 123 Eur. Phys. J. C (2015) 75 :129 Page 5 of 5 129

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