Accretion and Evaporation of Modified Hayward Black Hole

First we have assumed the most general static spherically symmetric black hole metric. The accretion of any general kind of fluid flow around the black hole have been investigated. The accretion of fluid flow around the modified Hayward black hole have been analyzed and we then calculated the critical point, fluid 4 velocity and velocity of sound during accretion process. Also the nature of the dynamical mass of black hole during accretion of fluid flow and taking into consideration of Hawking radiation from black hole i.e., evaporation of black hole have been analyzed.


I. INTRODUCTION
At present we live in a Universe which is expanding and the expansion rate is increasing i.e, the Universe is accelerating which was confirmed by recent Supernova type Ia observations [1,2]. The large scale structure [3,4], cosmic microwave background radiation [5], WMAP observations [6][7][8] also support this acceleration of the Universe. This acceleration is caused by some unknown matter which produce strong sufficient negative pressure (with positive energy density), known as dark energy. The present Universe occupies ∼ 4% ordinary matter, ∼ 74% dark energy and ∼ 22% dark matter. Since dark energy and dark matter are main two components in our universe, such that the present dark energy and dark matter densities are 7.01 × 10 −27 kg/m 3 and 2.18 × 10 −27 kg/m 3 , respectively. Most simplest candidate of dark energy is the cosmological constant Λ which obeys the equation of state EoS p = wρ with EoS parameter w = −1 [9,10]. Another candidates of dark energy are quintessence (where EoS parameter satisfies −1 < w < −1/3) [11,12] and phantom (where EoS parameter satisfies w < −1) [13]. Till now there are lot of dark energy models have been considered. A brief review of dark energy models is found in the ref. [14].
A condensed object (e.g. neutron stars, black holes, etc.) surrounded by a fluid can capture particles of fluid that pass within a certain distance from the condensed object. This phenomena is termed as accretion of fluid by condensed objects. In Newtonian theory of gravity, the problem of accretion of matter onto the compact object was first formulated by Bondi [15]. Michel [16] first obtained analytic relativistic accretion (of gas) solution onto * ujjaldebnath@gmail.com , ujjal@iucaa.ernet.in the static Schwarzschild black hole. Such accretion processes are candidates to mechanisms of formation of supermassive black holes (SMBH) in the center of most active galaxies [17]. In particular, it should follow some analogies with the process proposed by Salpeter et al [18] where galaxies and quasars could get some of their energy from processes of accretion. Using this accretion procedure, Babichev et al [19,20] formulated the accretion of phantom dark energy onto a static Schwarzschild black hole and shown that static Schwarzschild black hole mass will gradually decrease due to strong negative pressure of phantom energy and finally all the masses tend to zero near the big rip singularity. Sun [21] discussed phantom energy accretion onto a Black hole in the cyclic universe. Jamil [22] has investigated accretion of phantom like modified variable Chaplygin gas onto Schwarzschild black hole. Phantom energy accretion by stringly charged black hole has been discussed by Sharif et al [23]. Dark matter and dark energy accretion onto static black hole has been discussed by Kim et al [24]. Also the accretion of dark energy onto the more general Kerr-Newman black hole was studied by Madrid et al [25]. The new variable modified Chaplygin gas and generalized cosmic Chaplygin gas dark energy accretions and onto Kerr-Newman black hole and their features were studied Bhadra et al [26]. Several authors [27][28][29][30][31][32][33][34] have discussed the accretions of various components of dark energy onto several types of black holes.
In the present work, first we assume the most general static spherically symmetric black hole metric in section II. The accretion of any general kind of fluid flow around the black hole will be investigated. The accretion of fluid flow around the modified Hayward black hole will be analyzed in section III and we then calculate the critical point, fluid 4 velocity and velocity of sound during accretion process. Also the nature of the dynamical mass of black hole during accretion of fluid flow and taking into consideration of Hawking radiation from black hole i.e., evaporation of black hole will be analyzed in section IV. Finally we shall draw fruitful discussions about accretion of fluids upon modified Hayward black hole in section V.

II. ACCRETION PHENOMENA OF GENERAL STATIC SPHERICALLY SYMMETRIC BLACK HOLE
First we consider general static spherically symmetric metric given by where A(r) > 0 and B(r) > 0 are functions of r only. We can choose A(r) and B(r) in such a way that the above metric represents a black hole metric. Let us assume M is the mass of the black hole. For instance, if A(r) = B(r) = 1 − 2M r , the above metric represents Schwarzschild black hole.
The energy-momentum tensor for the fluid is given by where ρ and p are the energy density and pressure of the fluid. The four velocity vector of the fluid flow is given by u µ = dx µ ds = (u 0 , u 1 , 0, 0) where u 0 and u 1 are the non-zero components of velocity vector satisfying u µ u µ = −1. This implies g 00 u 0 u 0 + g 11 u 1 u 1 = −1. So we can obtain (u 0 ) 2 = (u 1 ) 2 +B AB and let the radial velocity of the flow From above equation (2), we obtain T 1 0 = (ρ + p)u 0 u. It is assumed that u < 0 for inward flow of the fluid towards the black hole.
In the fluid flow, we may assume that the fluid is dark matter or any kind of dark energy. A proper dark-energy accretion model for static spherically symmetric black hole should be obtained by generalizing the Michel's theory [16]. In the dark energy accretion onto Schwarzschild black hole, Babichev et al [19,20] have performed the above generalization. We shall follow now the above procedure in the case of static spherically symmetric black hole. The relativistic Bernoulli's equation (the time component) of the energy-momentum conservation law T µν ;ν = 0, we obtain d dr (T 1 0 √ −g) = 0 which provides the first integral, (ρ + p)u 0 u 1 √ −g = C 1 , that simplifies to where C 1 is the integration constant which has the dimension of the energy density. Moreover, the energy flux equation can be derived by the projection of the conservation law for energy-momentum tensor onto the fluid four-velocity, i.e., u µ T µν ;ν = 0, which gives u µ ρ ,µ + (ρ + p)u µ ;µ = 0. From this, we obtain where C is integration constant (energy flux onto the black hole) and the associated minus sign is taken for convenience. Also ρ h and ρ ∞ represent the energy densities at the black hole horizon and at infinity respectively. Combining equations (3) and (4), we obtain, where, C 3 is the integration constant. From (3) and (6), we obtain, (7) Now let us assume, So from equations (6), (7) and (8), we obtain Now if one or the other of the bracketed terms in (9) vanishes, we get a turn-around point and in this case, the solutions will be the double-valued in either r or u. There are only solutions which pass through a critical point that correspond to material falling into (or flowing out of) the object with monotonically increasing velocity along with the particle trajectory. A point where speed of flow is equal to the speed of sound, such a point is called a critical point. It is assumed that the critical point of accretion is located at r = r c which is obtained by taking the both bracketed terms (coefficients of du and dr) in Eq. (9) to be zero. So at the critical point, we obtain and Here, subscript c denotes the critical value and u c is the critical speed of flow at the critical point r c . From above two expressions, we have and At the critical point r c , the sound speed can be determined by The physically acceptable solutions of the above equations may be obtained if u 2 c > 0 and V 2 c > 0 which leads to From the above equation we can obtain the bound of r c if A and B are known for several kinds of static black holes.

III. ACCRETION PHENOMENA OF MODIFIED HAYWARD BLACK HOLE
The static spherically symmetric space-time is described by the Hayward metric which is obtained by A(r) = B(r) in equation (1) and is given by [35] Here, M is the mass of Hayward black hole and B(r) = 1 − 2Mr 2 r 3 +2Ml 2 , where l is a parameter with dimensions of length (Hubble length) with small scale related to the inverse cosmological constant Λ (l is a convenient encoding of the central energy density 3 8πl 2 ∼ Λ, assumed positive). Such behavior has been proposed by Sakharov [36,37] as the equation of state of matter at high density and by Markov [38,39] based on an upper limit on density or curvature, to be ultimately justified by a quantum theory of gravity. In the limit r → ∞, B(r) ≈ 1 − 2M r which represents Schwarzschild black hole, but it becomes de-Sitter black hole as B(r) ≈ 1 − r 2 l 2 near the center (r ≈ 0), so it is a regular space-time without singularity. Thus Hayward black hole is the simplest regular black hole. Some physical consequences of Hayward black hole have been discussed by several authors [40][41][42]. After that the Hayward metric was modifies [43] by choosing A(r) = f (r)B(r), satisfying the conditions: it (i) preserves the Schwarzschild behaviour at large r, (ii) includes the 1-loop quantum corrections and (iii) allows for a finite time dilation between the center and infinity. So the modified Hayward black hole metric is given by [43], where with α, β are positive constants. Now from the relation A(r) = f (r)B(r), we may obtain Also from the expressions of B(r) and f (r) (equation (18)), we get , (20) Since for outside the horizon, (i) B(r) > 0 which implies r 3 > 2M (r 2 − l 2 ) and (ii) f (r) > 0, so we get, r > β(α−1)M α 1 3 with α > 1. So from equation (21), If we assume that the fluid flow accretes upon modified Hayward black hole, we can calculate the expressions of u 2 c , V 2 c and c 2 s at the critical point r c . The expressions are given below (using equations (12), (13) and (14)): where B(r), f (r) and their derivatives are given in (18), (20) and (21) at the point r = r c . The physically acceptable solutions of the above equations may be obtained if u 2 c > 0 and V 2 c > 0 which leads to From above restrictions, we may get the bounds of r c and that is (α > 1): For example, we assume a fluid flow obeys linear equation of state p = wρ (w = constant) accretes upon modified Hayward black hole. Then we obtain c 2 s = w and V 2 c = 0 and from (10), we obtain u c = 0. From equations (22) and (24), we see that the critical point occurs at the point r c = (4M l 2 ) The above result is also valid for any equation of state p = p(ρ). So the rate of change of mass for the accreting fluid around the black hole will bė We see that the rate of change of mass for the general spherically symmetric static black hole due to accretion of fluid flow becomes exactly similar rate in the case of a Schwarzschild black hole. From the expression (28) it is to be noted that the rate of change of mass of any static spherically symmetric black hole is completely independent of A(r) and B(r). When some fluid accretes outside the black hole, the mass function M of the black hole is considered as a dynamical mass function and hence it should be a function of time also. SoṀ is time dependent and the increasing or decreasing of the black hole mass M sensitively depends on the nature of the fluid which accretes upon the black hole. If ρ + p < 0 i.e., for phantom dark energy accretion, the mass of the black hole decreases but if ρ + p > 0 i.e., for quintessence dark energy accretion, the mass of the black hole increases.
We may also assume that the black hole evaporates by Hawking radiation process. The rate of change of mass for the evaporation is given bẏ where D > 0 is a constant whose value depends on the model [44]. Now due to accretion of fluid flow and evaporation of mass of black hole, we get the rate of change of mass of black hole aṡ For accretion scenario, the changes of mass of black hole completely depends on the nature of the fluid accretes. But for evaporation process, the change of mass of black hole is independent of the nature of fluid, because this is internal process. In fact, when the accretion fluid is only the cosmological constant (p = −ρ), the mass of black hole for only accretion scenario is always same throughout time evolution. Only in accretion process, the mass of black hole increases for normal fluid and quintessence type dark energy fluid and decreases for phantom dark energy. But due to accretion as well as evaporation,Ṁ > 0 for M 4 > D 4πC(ρ+p) andṀ < 0 for M 4 < D 4πC(ρ+p) for normal fluid and quintessence type dark energy, but for phantom energy, black hole mass always decreases (Ṁ < 0). Thus evaporation supports the decreasing of the mass of black hole with some restrictions of minimum values of mass of black hole.

V. DISCUSSIONS AND CONCLUDING REMARKS
First we have assumed the most general static spherically symmetric black hole metric. The accretion of any general kind of fluid flow around the black hole have been investigated. For this general kind of static black hole, the critical point, velocity of sound, fluid 4 velocity have been calculated and shown that these value depend completely on the metric coefficients. Next, the accretion of fluid flow around the modified Hayward black hole have been analyzed and we then calculated the critical point, fluid 4 velocity and velocity of sound during accretion process. We can mention that for outside the horizon, . When the perfect fluid satisfying linear equation of state p = wρ (w = constant) accretes upon modified Hayward black hole we have obtained c 2 s = w, u c = 0 and V 2 c = 0. In this accretion process, we have seen that the critical point occurs at the point r c = (4M l 2 ) 1 3 . Also the nature of the dynamical mass of black hole during accretion of fluid flow and taking into consideration of Hawking radiation from black hole i.e., evaporation of black hole have been analyzed. Only in accretion process, the mass of black hole increases for normal fluid and quintessence type dark energy fluid and decreases for phantom dark energy.
But due to accretion as well as evaporation,Ṁ > 0 for M 4 > D 4πC(ρ+p) andṀ < 0 for M 4 < D 4πC(ρ+p) for normal fluid and quintessence type dark energy, but for phantom energy, black hole mass always decreases (Ṁ < 0). Thus evaporation supports the decreasing of the mass of black hole with some restrictions of minimum values of mass of black hole.