Adiabatic evolution of Hayward black hole
aa r X i v : . [ g r- q c ] J a n Adiabatic evolution of Hayward black hole
Mohsen Fathi, ∗ Martín Molina, † and J.R. Villanueva ‡ Instituto de Física y Astronomía, Universidad de Valparaíso, Avenida Gran Bretaña 1111, Valparaíso, Chile (Dated: February 1, 2021)In this paper, we continue studying the thermodynamics of Hayward black hole, which has beenrecently approached by Molina & Villanueva regarding the laws of black hole thermodynamics,by introducing the Hayward’s parameter as being responsible for a possible regularization of theSchwarzschild black hole. Here, we show that the adiabatic foliations of the thermodynamic manifoldare confined by the extremal subspace, and therefore, the latter cannot be reached adiabatically.A direct consequence of this features, is the impossibility of the merger of two extremal Haywardblack holes. keywords : Black Holes, Thermodynamic, Hayward black holePACS numbers: 04.20.Fy, 04.20.Jb, 04.25.-g
CONTENTS
I. Introduction 1II. The Hayward black hole spacetime 2III. Review of the ( r s , l ) thermodynamics 5A. The basis 5B. The T - S diagram revisited 6IV. The adiabatic processes and the extremal limit 7A. The y branch 8B. The y branch 9C. The physical parts of the solutions 11V. Classical scattering of two EHBHs and the second law 12VI. Summary 12Acknowledgments 12A. Derivation of the solutions to the Cauchy equation 13References 14 I. INTRODUCTION
The reconciliation of the laws of thermodynamics with black hole mechanics [1], beside its interesting featuresin the study of the black hole event horizon, has provided a criterion, in which, the celestial phenomena such asastrophysical mergers, can be assessed for theoretical black holes deduced from diversified conditions. This way, theentropy assigned by Bekenstein to the black holes [2–5], and the so-called no-hair theorem [6, 7], have brought upseveral consequent theories, that are supposed to dominate the evolutionary phases of these astrophysical objects.These theories, together with the possibility of black hole evaporation through the Hawking radiation [8] (also seeRef. [9] for a review), have been being tested constantly by experiments and observations [10, 11], and also, constitute ∗ [email protected] † [email protected] ‡ [email protected] the foundations of the holographic principle [12]. However, although the famous Bekenstein-Hawking (B-H) entropyformula has been applied widely for regular black holes, nevertheless, its direct application to the extremal black holes(EBHs) is not that simple. In fact, EBHs are supposed to consist of the minimal possible mass with given physicalcharacteristics, and are therefore, as smallest as possible. They are sometimes assumed to be super-symmetric andfor this reason, the entropy of EBHs in such cases is commonly calculated in string theory [13]. The results arestill in agreement with the B-H formula, although the case of Schwarzschild black hole is still under debate. In theforthcoming sections of this paper, however, we use this special suggestion that the EBHs have zero entropy [14], andtherefore, the direct relationship between the entropy and the event horizon’s area, as demanded by the B-H formula,is overlooked for the EBHs.Along with the strong interest of the community in the study of the thermodynamics of regular black holes, inthis paper, we focus on a particular, non-singular minimal black hole model proposed by Hayward in Ref. [15],which constructs a static spherically symmetric and asymptotically flat spacetime, and has been investigated fromseveral points of view since its advent (see for example Refs. [16–18]). Recently, in Ref. [19] this solution has beengeneralized to certain scalar-tensor theories (i.e. DHOST theories). In addition, new regular black holes in the contextof quasitopological electromagnetic theories have been also reported by in Refs. [20–22]. Moreover, the non-singularnature of this black hole has made it an interesting topic studying its thermodynamics, and in particular, its formationand evaporation. Accordingly, and in Ref. [23], the laws of black hole thermodynamics have been investigated forthe Hayward black hole (HBH) through studying the relations between its dynamical parameters. In this paper, wecontinue the mentioned study, by taking into account the possible adiabatic processes inside the system. In particular,we solve the Cauchy problem for near horizon regions outside the black hole, and confine the obtained values, to thedomains that respect the physical laws. Furthermore, we ramify the possible adiabatic transitions for the cases ofextremal and regular HBHs and calculate the relevant physical limits.The paper is organized as follows: In Sec. II, we introduce the HBH solution and the variables, by mean of which,the thermodynamics of the black hole is investigated. In Sec. III, the mentioned parameters are dealt with in moredetail, in the way that the thermodynamic quantities are expressed in terms of the physical characteristics of the HBH.In Sec. IV, we consider the Pfaffian form of the first law for the HBH, and obtain solutions to the Cauchy problem,regarding the adiabatic processes that can be feasible for the system. These solutions are then scrutinized regardingtheir physically accepted segments, as well as being confronted with the extremal case, as a limit that cannot beachieved by means of adiabatic transitions. In Sec. V, the property extremality is inspected deeper, by consideringthe merger of two extremal Hayward black holes (EHBHs). We summarize in Sec. VI. II. THE HAYWARD BLACK HOLE SPACETIME
It is well-known that, the general relativistic non-vacuum solutions with electromagnetic components, are commonlygenerated by coupling the Einstein-Hilbert action with a non-linear electromagnetic field, in the form [24–27] I = 12 K Z d x √− g (cid:16) R − KL ( F ) (cid:17) , (1)where K = 8 πG/c with G as the Newton’s constant and c as the speed of light, g ≡ det( g µν ) , R = R µµ is the Ricciscalar, F ≡ F µν F µν with F = d A as the field strength tensor defined in terms of the vector potential A , and L ( F ) isthe Lagrangian density which is conceived as a function of F [28–33]. The resultant Einstein-Maxwell covariant fieldequations G µν = K T µν , (2a) ∇ µ ( L F F µν ) = , (2b)are then derived by varying the action (1) with respect to the metric g µν . where G µν = R µν − R g µν is the Einsteintensor, L F ≡ ∂ L ( F ) ∂ F , and T µν = 2 L F F µν − g µν L ( F ) , (3)is the energy-momentum tensor. In Ref. [15], Hayward obtained a regular, non-singular, static spherically symmetricsolution to the Einstein field equations, in the form d s = − f ( r ) d( ct ) + d r f ( r ) + r d θ + r sin θ d φ , (4)in which the lapse function f ( r ) is given by f ( r ) = 1 − r s r r + r s l = P ( r ; r s , l ) r + r s l , (5)where r s = 2 GM/c is the classical radius of a Schwarzschild black hole (SBH) of mass M , l is the Hayward’sparameter ( ≤ l < ∞ ), and we have assigned P ( r ; r s , l ) = r − r s r + r s l . (6)Although the Hayward’s solution does not rely on any electromagnetic couplings, it has however shown in Refs. [28–33]that, the same solution can be regenerated from the action (1), by applying the Lagrangian density L ( F ) = 6 l (cid:0) l F (cid:1) h l F ) i , (7)which associates the lapse function (5) with the vector potential A φ = Q m cos θ , and hence, the magnetic charge Q m = 12 (cid:0) r s l (cid:1) , (8)and is rewritten as f ( r ) = 1 − (2 lQ m ) r l ( r + 2 lQ m ) . (9)One can therefore deduce the field strength scalar F = (cid:0) r s l (cid:1) r . (10)It can be checked from expressions in Eqs. (5) and (9) that, for r = 0 , the solution remains regular and for l = 0 , theSBH is regenerated. Note that, the reduction to the SBH spacetime can be done, by approximation, in the case of < l ≪ r . In this case, the lapse function (5) can be recast as f ( r ) ≃ − r s r + r s l r − . . . , (11)which is asymptotically flat and returns the SBH at large distances, even in the case of l = 0 , and is also asymptoticallyflat. On the other hand, when l ≫ r , the solution is approximated to f ( r ) ≃ − r l + r r s l − . . . , (12)which is flat and regular at r = 0 , and resembles the de Sitter spacetime.The horizons of the HBH are roots of the cubic equation f ( r ) = 0 . In general, some standard methods are usedto solve cubic equations (see for example, the appendices in Refs. [34, 35]). Applying these methods, the solutions to f ( r ) = 0 are [23, 36] r + = r s (cid:18) α (cid:19) ≡ r s R + , (13) r − = r s − cos α √
33 sin α ! ≡ r s R − , (14) r n = r s − cos α − √
33 sin α ! ≡ r s R n , (15)given in terms of trigonometric functions, where α = 13 arccos (cid:18) − l l e (cid:19) , (16)and l e = 2 r s / √ . In fact, if the condition ≤ l < l e is satisfied, there will be three different real solutions to f ( r ) = 0 , which are located in the domain R + > R − ≥ ≥ R n . In the case that l = l e , the roots reduce to the twodegenerate positive values R + = R − = 2 / , and a negative value R n = − / (which is ruled out), and the EHBHis obtained that corresponds to the thermodynamic limit of the black hole. Hence, l e is the extremal limit of theHayward’s parameter. For l > l e , the above roots reshape to r = r ∗ = r s β − ı √
33 sinh β ! ≡ r s R = r s R ∗ , (17) r n = r s − β ) ≡ r s R n , (18)which are given in terms of the hyperbolic functions, and β = 13 arccosh (cid:18) l l e − (cid:19) . (19)As it is observed, the above solutions provide the complex conjugate pair R = R ∗ . In Fig. 1, the complete setof roots have been plotted together, as functions of l/l e . Now, before proceeding any further, let us introduce the R oo t s o f f ( r ) R + R - R n - R e (R ) I m (R ) I m (R ) l l e / FIG. 1. The behavior of the roots of the Hayward’s lapse function in terms of l/l e . The corresponding domains are therefore,divided by the line l = l e , as demanded by Eqs. (13–15) and Eqs. (17–18). following functions: (cid:18) d R + d α (cid:19) = f ∗ ( α ) , (20) r s (cid:18) ∂α∂r s (cid:19) l = − l (cid:18) ∂α∂l (cid:19) r s = h ∗ ( α ) , (21) r s (cid:18) ∂R + ∂r s (cid:19) l = − l (cid:18) ∂R + ∂l (cid:19) r s = g ∗ ( α ) , (22)in which f ∗ ( α ) = −
23 sin α, (23) h ∗ ( α ) = −
23 (csc 3 α − cot 3 α ) , (24) g ∗ ( α ) = f ∗ ( α ) h ∗ ( α ) = 49 sin α (csc 3 α − cot 3 α ) . (25)In the next section, we relate the concepts discussed above, to thermodynamics of the HBH. III. REVIEW OF THE ( r s , l ) THERMODYNAMICS Following the discussion given in Ref. [23], we consider the pair ( r s , l ) as the thermodynamic variables of the HBH,and establish our study on this postulate [1–5, 8]: The area of the black hole event horizon cannot decrease; it increases during most of the physical processes of theblack hole.
This means that, the thermodynamics of the system can be approached, geometrically, in the context of theCarathéodory’s theorem, and this way, as highlighted in Ref. [37], it is connected with the Gibbs’s thermodynamics.Accordingly, and considering the black hole event horizon r + , the above postulate allows for establishing the well-known Hawking-Bekenstein/area-entropy, that is formulated as [1–5] S = k B πr ℓ p , (26)in which, k B and ℓ p are, respectively, the Boltzmann constant and the Planck length. It is then useful to define theentropy function S ( r s , l ) ≡ ℓ p Sπk B = r s R ( r s , l ) . (27)which will be exploited in the forthcoming subsections. A. The basis
To apply the approach of Carathéodory to the black hole thermodynamics, we first define the infinitesimal heatexchanged reversibly δQ rev with the Pfaffian form δQ rev = T d S , (28)that introduces the so-called metric entropy , S , and metric temperature , T [38–42]. If r s and l are seen as independentthermodynamic variables, the homogeneity of the system is then reflected in the homogeneity of the integrable Pfaffianform δQ rev = d r s − F H d l ≡ d U + d W, (29)where U and W represent respectively, the system’s internal energy work. Furthermore, F H is an intensive variable(i.e. homogeneous of degree zero), which is termed as the generalized Hayward’s force , and is defined as [23] F H = g ∗ ( α ) R + ( α ) + g ∗ ( α ) r s l . (30)It is important to note that, the HBH is regarded as an equilibrium geometrical state. According to the discussionsgiven in Sec. II, one can consider a non-extremal thermodynamic manifold, corresponding to the case of l < r s / √ ,that is encompassed by the extremal sub-manifold (thermodynamic limit), formed by l = 2 r s / √ .Now, following the standard theory of thermodynamics, we define the metric temperature as T = (cid:18) ∂ S ∂r s (cid:19) l > , (31)which, by applying Eq. (27) together with Eqs. (20–22) yields T = T s R + ( α ) [ R + ( α ) + g ∗ ( α )] , (32)where T s ≡ (2 r s ) − ≡ (4 S s ) − is the SBH temperature.Furthermore, to concern with the homogeneity of the system, let us consider the Euler field D (The Liouvilleoperator) D = r s ∂∂r s + l ∂∂l . (33)It is immediately seen from Eqs. (21) and (22) that D α = D R + = 0 , which means that both α and R + are homogeneousfunctions of degree zero. Same holds for F H (i.e. DF H = 0 ), as observed from Eq. (30). Similarly, one finds fromEq. (32) that DT = −T . Therefore, this parameter is homogeneous of degree − , and hence, is not intensive.Note that, if we double each of r s and l , then T becomes one-half of its value at the initial state of the black hole.Additionally, from Eq. (27) we have DS = 2 S , which indicates that the entropy is homogeneous of degree two.Accordingly, we can infer from Eqs. (28) and (29), that T S = r s − F H l, (34)which is known as the fundamental equation or the Gibbs-Duhem relation for the HBH. Moreover, the mentionedequations indicate that the Pfaffian form of infinitesimal heat exchanged reversibly is homogeneous of degree one, andtherefore, we are allowed to construct the thermodynamics. To proceed with this, we study the mutual behavior ofthe temperature and entropy for the HBH, in the next subsection.
B. The T - S diagram revisited Here we review some important features of the and T - S diagram which have been formerly reported in Ref. [23].Let us consider Fig. 2, where we have plotted the thermodynamics parameters in Eqs. (27) and (32), as functionsof the variables ( r s , l ). The physically accepted segment, lies within the domain ≤ T ≤ T s (the blue curve). Note ∞ TS l=0r s r s EBH
00 (2)(1)
FIG. 2. The T - S diagram of the HBH, indicating both the EHBH and the SBH limits. The thermodynamic processes (redcurves) will finally reach the SBH, through which, the entropy and the temperature increase. This corresponds to fixed valuesof r s and decrease in l . In the adiabatic processes (the green arrow), on the other hand, both of the parameters ( r s , l ) decrease. that, if r s is fixed, then ∆ S > implies ∆ l < . Therefore, by varying l while keeping r s fixed (the red curves), theHBH transits to the SBH. It is also straightforward to see that, going from state (1) to state (2) (for which T < T ),the variable r s decreases. We can therefore infer that, in an adiabatic process (the green arrow), both of the variables ( r s , l ) decrease simultaneously.In this section, we characterized analytically the thermodynamic parameters of the HBH, in terms of its mechanicalcharacteristics and figured out the way they behave during the evolution of the black hole. If the HBH is consideredas an isolated system, these processes can be regarded as being adiabatic, and as indicated above, this takes placeunder certain conditions. In the next section, we deal with this kind of evolution in more detail, and highlight theextent the black hole can evolve, without violating the physical limits. IV. THE ADIABATIC PROCESSES AND THE EXTREMAL LIMIT
Recalling the relation r s = √ l e / , one can consider the extremal limit of r s , as r s ≡ r es = √ l. (35)Clearly, r s is homogeneous of degree one, and the above equation yields d r s = F eH d l, (36)where F eH = √ / is the Hayward’s force for the EHBH. Note that, similar to the case of the extremal Kerr-Newmanblack hole (EKNBH) discussed in Ref. [40], the EHBH can be regarded as an extremal submanifold that resides in theintegral manifold of δQ rev in Eq. (29). Accordingly, the EHBH corresponds to an adiabatic submanifold. Furthermore,considering A e = 4 π (cid:0) r e + (cid:1) = 4 π (cid:0) r es R e + (cid:1) = 169 π r s , (37)as the area of the EHBH, it is inferred from Eq. (37) that d A e = 24 πl d l. (38)It is therefore observed that the extremality condition does not imply d A e = 0 , and this latter, is satisfied simply forall constant values of l .Now, getting back to the Carathéodory approach, we note that, Eq. (29) allows for foliating the thermodynamicmanifold by means of solutions to the Pfaffian equation δQ rev = 0 . Therefore, these solutions correspond to a smoothand continuous 1-form field that resides in the non-extremal sub-manifold. Accordingly, the integral manifolds of δQ rev are surfaces with constant S , which together with the paths that solve the Pfaffian equation, construct an isentropic surface (i.e. adiabatic and reversible) [40]. On the other hand, the extremal submanifold corresponding tothe line T = 0 , also solves δQ rev = 0 and therefore, it is an integral manifold of the Pfaffian form (see Fig. 3). Toelaborate on this point, let us apply the changes of variables x . = r s and y . = ( F eH ) l , so that Eq. (29) can be recastas δQ rev = d x √ x − F H ( x, y ) F eH d y √ y , (39)which holds as long as y ≤ x . Accordingly, the states which are connected adiabatically with the initial black holestate ( x , y ) , are solutions to the Cauchy problem d y d x = r yx F eH F H ( x, y ) , (40a) y ( x ) = y , (40b)where x = y . Applying Eq. (30), one can rewrite Eq. (40a) as d y d x = yx (cid:18) R + ( x, y ) g ∗ ( x, y ) (cid:19) , (41)which is reshaped to d y d x = 18 (cid:26) (cid:20)
13 arccos (cid:18) − yx (cid:19)(cid:21)(cid:27) , (42)by means of Eqs. (13) and (25). The above problem allows for two solutions, say y , , which are given by (seeappendix A): y ( x ) = x − y ( x ) , (43)in which y ( x ) = 27 ρ (cid:18) − ρ √ x (cid:19) , (44) E x t r e m e B l a c k H o l e ( x ; x ) y ( x ) = x ( x ; x ) y ( x ) ( x ; y ) A d i a b a t i c Su r f a ce ( T > ) ( x ; y ) y ( x ) ( x ; x ) ( x ; y ) ( T = ) ( a ) (b) (c) y ( x ) = x = FIG. 3. The adiabatic surface with the virtual intersection with the extremal black hole limit. Among the three conceivedadiabatic processes fulfilled by y ( x ) , only (a) and (b) are physically possible. The process (c) must be then excluded fromconsideration. The curve y ( x ) = x/ , is shown to divide the adiabatic surface into two equal halves. where ρ is a constant determined by the initial condition.Before continuing, it is important to note that, obviously, the straight line representing the set of all extremal states y ( x ) = x , is a solution to the problem with the condition y ( x ) = x . This indicates that, the extremal states areadiabatically connected to each other. In the equation of state presented in Ref. [23], this fact is evident for theextremal submanifold (a plane) of T = 0 that borders the thermodynamic submanifold T > (see Fig. 3). A. The y branch If we consider a non-extremal state ( x , y ( x )) = ( x , y ) as the initial point, the constant ρ satisfies the followingcubic equation: ρ − √ x ρ + 3227 √ x ( x − y ) = 0 , (45)so that performing the change of variable ξ . = ρ − √ x / , one gets ξ − x ξ − √ x (2 y − x ) = 0 , (46)which yields [34, 35] ρ n ( x , y ) = 2 √ x (cid:20) (cid:18) ω + 2 nπ (cid:19)(cid:21) , (47)where n = 0 , , and ω ≡ ω ( x , y ) = 13 arccos (cid:12)(cid:12)(cid:12)(cid:12) − y x (cid:12)(cid:12)(cid:12)(cid:12) . (48)Without loss of generality, we choose the positive root (corresponding to n = 0 ), i.e., ρ ≡ ρ ( x , y ) = 2 √ x ω ] . (49) x y ( x ) = . = . = . y = x SBH
FIG. 4. The behavior of the solution y ( x ) plotted for ρ = 0 . , . and 0.3, and the line y = 0 corresponds to the SBH.The physically accepted segments are demonstrated by solid curves, for which, the flow of the lines towards the SBH is basedon decreasing both y (Hayward’s parameter) and x (mass). The other segments violate these evolutionary thermodynamicconditions and are therefore indicated by dotted and dashed curves. The system can approach adiabatically to the extremalblack hole (for which y = x ). It is however, physically impossible for the system to coincide with the extremality. Hence, thepoints of intersection with the line y = x have been excluded. The other points indicated on the curves correspond to theintersections with y = 0 . One consequence of the solution (43) is that extremal states are also adiabatically connected to non-extremal ones.In fact, if y = x , then ρ = 2 √ x and the solution becomes y ( x ) = x − x (cid:18) − r x x (cid:19) . (50)Another interesting feature of the solution is the nature of the points ( x s , , since they correspond to the zeros aswell as to the minimum of the function (see Fig. 4). Obviously, their values depend on the initial states through x s = 9 ρ , (51)and correspond to intersections with the SBH limit. Therefore, the thermodynamic evolution of any non-extremalinitial state corresponds to an isoareal solution, which is here of the form y ( x ) = y + ( x − x ) − s x s x (cid:18) − r x x (cid:19) , (52)that satisfies y ( x ) = y < x . B. The y branch In this case, the initial state ( x , y ) provides the equation ρ − √ x ρ + 3227 √ x y = 0 , (53)or ξ − x ξ + 1627 √ x (2 y − x ) = 0 . (54)0 x y ( x ) ρ = . ρ = . ρ = . y = x SBH
FIG. 5. The behavior of the solution y ( x ) plotted for ρ = 0 . , . and 0.3. As explained for the case of y ( x ) , the physicallyaccepted segments are indicated by solid curves, which lie between the points of exclusion and intersections with the SBH line. Let us denote x i and x s , respectively, for the points of intersection of y ( x ) with the lines y ( x ) = x and y ( x ) = 0 .The same holds for the points x i and x s for the case of y ( x ) . It is then straightforward to see from Eqs. (43) and(44), that x s = x i ≡ x and x i = x s ≡ x . We can therefore infer that the points of exclusion in the y branch,correspond to the points of intersection with the SBH limit in that of y (see Figs. 4 and 5).In fact, the points x and x depend on the initial condition ( x , y ) , through the constant ρ ≡ ρ ( x , y ) , whichrespects the condition x > y . This way, and introducing the parameter ϕ = 1 − y x / , (55)the two following conditions are obtained from Eqs. (46) and (54): ξ − x ξ − q x ϕ = 0 , (56a) ξ − x ξ + 1627 q x ϕ = 0 , (56b)for ϕ > in the domain < y < x / , and ξ − x ξ + 1627 q x | ϕ | = 0 , (57a) ξ − x ξ − q x | ϕ | = 0 , (57b)for ϕ < in the domain x / < y < x . As it is observed from Eqs. (56) and (57), these conditions, in fact, havethe same essence and the only difference is the domain, to which, they are applied. These domains will be elaboratedmore in the next subsection, regarding the physically accepted segments of the solutions.However, for now, let us comment on the way that the correct construction of the thermodynamic foliation isensured. We assume that the surface T = 0 is, itself, a leaf and therefore, we exclude the solution branch given byEq. (50) to the Cauchy problem. In this sense, as remarked in Ref. [40], the extremal states with the same area asthat for the non-extremal ones, are allowed. However, the relation S ∝ A is not obviated in the case of extremal blackholes. Strictly speaking, S ( r s , l ) = S B − H = A / , non-extremal states, , extremal states, (58)1 x y ( x ) y y y = x y = x / SBH
FIG. 6. The y , branches plotted for ρ = 0 . . The physically accepted segments of the solutions have been indicated by strongblack curves. where S B − H denotes the B-H entropy. In Sec. V, this will be demonstrated more clearly in the context of the secondlaw of black hole thermodynamics, by considering an isolated merger of two extremal black holes. However, beforeproceeding to that, let us make some clarifications regarding the physically accepted parts of the solutions. C. The physical parts of the solutions
So far, we have been able to express our solutions in terms of the general initial conditions, under the strongconstraint x > y (and obviously, x > y ). The Cauchy problem is therefore solved mathematically. However, we arestill in need of a correct physical description of these solutions. To proceed with this, it is of importance to recall fromSubsec. III B, that because of the equal homogeneity of the variables ( r s , l ) , both grow or decrease, simultaneously,in an adiabatic path. This can be summarized in the following conditions:• Condition (i): The slope of the x - y curves must be positive, or d y d x > . (59)• Condition (ii): As the variables decrease, the system evolves towards the SBH, while when they grow, the EHBHis approached.In Fig. 6, where the y , branches have been shown together, the physically accepted segments of the solutions havebeen demonstrated. As indicated in the figure, despite the fact that, mathematically, the solutions are continuous,however, the physically accepted segments indicate the piecewise nature of the complete solution to the Cauchyproblem. According to the notations introduced in the previous subsection, y ( x ) satisfies the conditions (i) and (ii)within x i ≤ x < x s , while this is done by y ( x ) within x s ≤ x < ∞ (see Fig. 6). Therefore, the adiabatic pathsthat satisfy the appropriate initial conditions of the Cauchy problem, are given by y ( x ) = y ( x ) , for x i ≤ x < x s y ( x ) , for x s ≤ x < ∞ . (60)Note that, asymptotically, the above piecewise function behaves as y ( x ) ≃ x − b, (61)2in which x ≫ x s , and b ≡ ( x − y ) + 2 r x s x > , (62)that ensures the condition x > y . V. CLASSICAL SCATTERING OF TWO EHBHS AND THE SECOND LAW
In this section, we argue that merging two EHBHs violates the second law of black hole thermodynamics, and itis therefore impossible. To elaborate this, let us consider two HBHs characterized by the initial states ( x a , y a ) and ( x b , y b ) . We assume that these black holes merge through a process, in which no exchange of energy is done with therest of the universe, and the final state ( x a + x b , y a + y b ) is produced. As it is shown in what follows, if the initialblack holes are extremal, this final state, as well, will correspond to an extremal black hole. Similar to Refs. [40, 43],we define the quantity ζ ( x, y ) = x − y. (63)Accordingly, the initial state of the process is characterized by ζ in = ζ a + ζ b , with ζ a ≡ ζ ( x a , y a ) ≥ and ζ b ≡ ζ ( x b , y b ) ≥ . In the same manner, the final state becomes ζ fin = ζ ab , where ζ ab ≡ ζ ( x a + x b , y a + y b ) , and Eq. (63)yields ζ ab = ζ a + ζ b + 2 ( x a x b − y a y b ) . (64)Exploiting Eq. (64) together with the given definitions, we obtain ζ ab − ( ζ a + ζ b ) = 2 h x a x b − y a y b − q ( x a − y a )( x b − y b ) i . (65)In the case that the initial states are constituted by extremal black holes (with zero entropy according to Eq. (58)),we have x a = y a and x b = y b , giving ζ a = 0 = ζ b and hence, ζ ab = 0 , which implies that the final black hole isas well, extremal and therefore, has zero entropy. Since black hole merging is irreversible, the process of merging oftwo EHBHs violates the second law of thermodynamics, that demands the increase of entropy through irreversibleprocesses. We can therefore infer that two EHBHs cannot merge, because they never produce a regular black hole toincrease the total entropy of the system. VI. SUMMARY
In the study of the evolutionary states of black hole thermodynamics, the extremal limit can put a concreteconceivable boundary. In this work, we demonstrated this by following the adiabatic, reversible (isentropic) paths on athermodynamic manifold constructed by the Pfaffian 1-form states. Following the work done in Ref. [23], we consideredthe general variables of the HBH, namely r s and l , as its thermodynamic parameters, and built our discussion on thesecond law of black hole thermodynamics. To proceed with this, we considered the Pfaffian form of the infinitesimalheat exchange reversibly, which construct a non-extremal 1-form manifold, bounded by the thermodynamic limit,corresponding to the EHBH. We showed that, although the changes in the variables can evolve the system to eitherthe SBH or the EHBH, this evolution, however, can be adiabatic if and only if the variables increase or decreasesimultaneously. This fact was later elaborated through the Cauchy problem, where we obtained solutions to thePfaffian form and ramified their physically valid parts. In this regard, the general physical solution to the Pfaffian hasa piecewise nature. Furthermore, as it was elaborated, the adiabatic paths that foliate the non-extremal sub-manifoldcan approach the thermodynamic limit, without intersecting the T = 0 line. Same holds for the EHBH, for which,the adiabatic paths are available, but are confined inside the extremal sub-manifold. The extremal limit appears tohave also other features for the case of HBH. In fact, according to the small number of degrees of freedom, it turnsout that the EHBHs should not merge, otherwise the second law is violated. This is, for example, in contrast withthe case of EKNBHs, investigated in Ref. [40]. For future studies, the thermodynamics of a rotating version of theHBH is in our perspective. ACKNOWLEDGMENTS
M. Fathi has been supported by the Agencia Nacional de Investigación y Desarrollo (ANID) through DOCTORADOGrant No. 2019-21190382. J.R.V. was partially supported by Centro de Astrofísica de Valparaíso.3
Appendix A: Derivation of the solutions to the Cauchy equation
Applying the change of variable z . = x − y Eq. (42) can be rewritten as d z d x = − (cid:20) z cos (cid:18)
13 arccos (cid:16) z x (cid:17)(cid:19)(cid:21) − z x + 32 . (A1)Performing another change of variable u . = z /x , then changes the above equation to d u d x = 34 x (cid:20) − u − (cid:18)
13 arccos u (cid:19) − (cid:18)
13 arccos u (cid:19)(cid:21) . (A2)Defining ω . = arccos u , now gives d ω d x = − P ( ω )4 x sin(3 ω ) , (A3)with P ( ω ) = 1 − ω ) − ω − ω. (A4)The elements in the numerator and denominator of Eq. (A4) can then be recast by means of the identities cos(3 ω ) = 4 cos ω − ω, (A5) sin(3 ω ) = 3 sin ω − ω. (A6)Accordingly, we have P ( ω ) = (1 + 2 cos ω ) (1 − ω ) , (A7a) sin(3 ω ) = − sin ω (cid:0) − ω (cid:1) . (A7b)Substitution of Eqs. (A7) in Eq. (A3) results in d ω d x = 1 + 2 cos ω x sin ω , (A8)which by applying the change of variable γ . = 1 + 2 cos ω , changes to d γ d x = − γ x . (A9)Direct integration of the above equation yields γ = ± b √ x , (A10)where b is an integration constant. Taking into account the changes of variables applied above, we get to the solutions y , ( x ) = x (cid:20) ± cos (cid:18) (cid:18) (cid:20) b √ x − (cid:21)(cid:19)(cid:19)(cid:21) . (A11)Defining ϕ ( x ) = arccos (cid:18) (cid:20) b √ x − (cid:21)(cid:19) , (A12)the above solutions change their form to y ( x ) = x − b (cid:18) − b √ x (cid:19) , (A13a) y ( x ) = b (cid:18) − b √ x (cid:19) . (A13b)4Finally, applying the identity (A5) and defining ρ = 2 b / , we get y ( x ) = x − ρ (cid:18) − ρ √ x (cid:19) , (A14a) y ( x ) = 27 ρ (cid:18) − ρ √ x (cid:19) . (A14b) [1] J. M. Bardeen, B. Carter, and S. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. , vol. 31,pp. 161–170, 1973.[2] J. D. Bekenstein, “Black holes and the second law,”
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