Constant curvature f(R) gravity minimally coupled with Yang-Mills field
aa r X i v : . [ g r- q c ] M a r Constant curvature f(R) gravity minimally coupled with Yang-Mills field
S. Habib Mazharimousavi, ∗ M. Halilsoy, † and T. Tahamtan ‡ Department of Physics, Eastern Mediterranean University,G. Magusa, north Cyprus, Mersin 10, Turkey.
We consider the particular class of f ( R ) gravities minimally coupled with Yang - Mills (YM) fieldin which the Ricci scalar = R = constant in all dimensions d ≥
4. Even in this restricted classthe spacetime has unlimited scopes determined by an equation of state of the form P eff = ωρ .Depending on the distance from the origin (or horizon of a black hole) the state function ω ( r ) takesdifferent values. It is observed that ω → (the ultra relativistic case in 4 - dimensions) and ω → − ω ( r ) in a spacetime centeredby a black hole. This suggests that having a constant ω throughout spacetime around a chargedblack hole in f ( R ) gravity with constant scalar curvature is a myth. I. INTRODUCTION
For a number of reasons, ranging from dark energy and accelerated expansion of the universe to astronomicaltests, modified version of general relativity gained considerable interest in recent times. f ( R ) gravity, in particular,attracted much attention in this context (see [1] for comprehensive reviews of the subject). The reason for this trendmay be attributed to the dependence of its Lagrangian on the Ricci scalar alone, so that it can be handled relativelysimpler in comparison with the higher order curvature invariants. Depending on the structure of the function f ( R ) thenonlinearity creates curvature sources which may be interpreted as ’sources without sources’, manifesting themselvesin the Einstein equations. Beside these curvature (or geometrical) sources there may be true physical sources thatcontribute together with the former to determine the total source in the problem. It should be added that owing tohighly nonlinear structure of the underlying field equations attaining exact solutions is not an easy task at all. In spiteof all odds many exact solutions have been obtained from ab initio assumed f ( R ) functions. To recall an examplewe refer to the choice f ( R ) = R N , ( N = an arbitrary number) which attains an electromagnetic - like curvaturesource, so that N = 1 can be interpreted as an ’electric charge without charge’ [2]. That is, the resulting geometrybecomes equivalent to the Reissner-Nordstrom (RN) geometry in a spherically symmetry metric ansatz of Einstein’sgravity. This particular example reveals that the failure of certain tests related to Solar System / Cosmology in f ( R )gravity is accountable by the curvature sources in the Einstein Hilbert action. Equivalence with f ( R ) = R +(scalarfields) provides another such example beside the electromagnetic one. More recently we obtained a large class ofnon-analytical f ( R ) gravity solutions minimally coupled with Yang-Mills (YM) field [3]. Even more to this the YMfield was allowed to be a nonlinear theory in which the power-YM constitutes a particular example in all higherdimensions. In particular, in d = 6, f ( R ) = √ R solves the Einstein-Yang-Mills (EYM) system exactly. For d = 4our solution for nonabelian gauge reduces to an abelian one which may be considered as an Einstein-Maxwell (EM)solution [4].Previously f ( R ) gravity coupled non-minimally with Yang-Mills and Maxwell matter sources have been considered[5]. In this paper we consider a particular class within minimally coupled YM field in f ( R ) gravity with the conditionsthat the scalar curvature R = R = constant and the trace of the YM energy-momentum tensor is zero. (To seeother black hole solutions with matter in f ( R ) gravity we refer to Ref. [6]). Contrary to our expectations thisturns out to be a non-trivial class with far-reaching consequences. Our spacetime is chosen spherically symmetricto be in accord with the spherically symmetric Wu-Yang ansatz for the YM field. The field equations admit exactsolutions in all dimensions d ≥ m ) of the black hole, YM charge ( Q ) and thescalar curvature ( R ) of the space time. In this picture we note that cosmological constant arises automatically asproportional to R . From physics stand point, considering equation of state in the form P eff = ωρ , with effectivepressure ( P eff ) and density ( ρ ), important results are obtained as follow: For a critical value of r = r c we have − < ω ( r ) < r > r c and 0 < ω ( r ) < for r < r c . Remarkably this amounts to a sign shift in the effectivepressure to account for the accelerated expansion of a universe centered by a charged black hole. In general the criticaldistance is thermodynamically unstable so that the universe undergoes the phase of accelerated expansion beyond ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] that particular distance. Absence of the phantom era (i.e. ω < −
1) is also manifest. Alternatively, in the limit r → ω → (i.e. 4 − dimensional ultra relativistic case), while for r → ∞ we have ω → −
1, the case of a purecosmological constant. Let us note that the latter case corresponds to vanishing of the YM field. Stated otherwise, inthe overall space time we do not have a fixed value for ω . Depending on the distance from the center (or horizon) of ablack hole we have a varying state parameter ω ( r ). The same argument in the Friedmann-Robertson-Walker (FRW)version of the theory implies that beyond a critical time t = t c , ω ( t ) changes its role and a different type of matterbecomes active. It is known that for the normal and dark matters which provide clustering both the weak energycondition (WEC) and the strong energy condition (SEC) must be satisfied. In the case of dark energy on the otherhand WEC is satisfied while SEC is violated. In Appendix A we analyze the energy conditions thoroughly covering alldimensions. Although our metric ansatz is chosen to be spherically symmetric so that the constant scalar curvature R >
0, in order to prepare ground for the topological black holes we consider the case of R < II. THE FORMALISM AND SOLUTION FOR R = CONSTANT.
We choose the action as (Our unit convention is chosen such that c = G = 1 so that κ = 8 π ) S = Z d d x √− g (cid:20) f ( R )2 κ + L ( F ) (cid:21) (1)in which f ( R ) is a real function of Ricci scalar R and L ( F ) is the nonlinear YM Lagrangian with F = tr (cid:16) F ( a ) µν F ( a ) µν (cid:17) . Obviously the particular choice L ( F ) = − π F will reduce to the case of standard YM theory. TheYM field 2 − form components are given by F ( a ) = 12 F ( a ) µν dx µ ∧ dx ν (2)with the internal index ( a ) running over the degrees of freedom of the nonabelian YM gauge field. Variation of theaction with respect to the metric g µν gives the EYM field equations as f R R νµ + (cid:18) (cid:3) f R − f (cid:19) δ νµ − ∇ ν ∇ µ f R = κT νµ (3)in which T νµ = L ( F ) δ νµ − tr (cid:16) F ( a ) µα F ( a ) να (cid:17) L F ( F ) , (4) L F ( F ) = d L ( F ) dF . Our notation here is as follows: f R = df ( R ) dR , (cid:3) f R = ∇ µ ∇ µ f R = √− g ∂ µ ( √− g∂ µ ) f R , R νµ is the Ricci tensor and ∇ ν ∇ µ f R = g αν ( f R ) ,µ ; α = g αν h ( f R ) ,µ,α − Γ mµα ( f R ) ,m i . (5)The trace of the EYM equation (3) yields f R R + ( d − (cid:3) f R − d f = κT (6)in which T = T µµ . The SO ( d −
1) gauge group YM potentials are given by A ( a ) = Qr C ( a )( i )( j ) x i dx j , Q = YM magnetic charge, r = d − X i =1 x i , (7)2 ≤ j + 1 ≤ i ≤ d − , and 1 ≤ a ≤ ( d −
2) ( d − / ,x = r cos θ d − sin θ d − ... sin θ , x = r sin θ d − sin θ d − ... sin θ ,x = r cos θ d − sin θ d − ... sin θ , x = r sin θ d − sin θ d − ... sin θ ,...x d − = r cos θ , in which C ( a )( b )( c ) are the non-zero structure constants of ( d − d − − parameter Lie group G [7, 8]. The metric ansatzis spherically symmetric which reads ds = − A ( r ) dt + dr A ( r ) + r d Ω d − , (8)with the only unknown function A ( r ) and the solid angle element d Ω d − = dθ + d − P i =2 i − Q j =1 sin θ j dθ i , (9)with 0 ≤ θ d − ≤ π, ≤ θ i ≤ π, ≤ i ≤ d − . Variation of the action with respect to A ( a ) implies the YM equations d h ⋆ F ( a ) L F ( F ) i + 1 σ C ( a )( b )( c ) L F ( F ) A ( b ) ∧ ⋆ F ( c ) = 0 , (10)in which σ is a coupling constant and ⋆ means duality. One may show that the YM invariant satisfies F = 14 tr (cid:16) F ( a ) µν F ( a ) µν (cid:17) = ( d −
2) ( d − Q r (11)and tr (cid:16) F ( a ) tα F ( a ) tα (cid:17) = tr (cid:16) F ( a ) rα F ( a ) rα (cid:17) = 0 , (12)while tr (cid:16) F ( a ) θ i α F ( a ) θ i α (cid:17) = ( d − Q r , (13)which leads us to the exact form of the energy momentum tensor T νµ = diag (cid:20) L , L , L − ( d − Q r L F , L − ( d − Q r L F , ..., L − ( d − Q r L F (cid:21) . (14)Here the trace of T νµ becomes T = T µµ = d L − F L F , (15)and therefore with Eq. (3) we find f = 2 d [ f R R + ( d − (cid:3) f R − κ ( d L − F L F )] . (16)To proceed further we set the trace of energy momentum tensor to be zero i.e., d L − F L F = 0 (17)which leads to a power Maxwell Lagrangian [9] L = − π F d . (18)Here for our convenience the integration constant is set to be − π . On the other hand, the constant curvature R = R ,and the zero trace condition together imply f ′ ( R ) R − d f ( R ) = 0 . (19)This equation admits f ( R ) = R d , (20)where the integration constant is set to be one. One can easily write the Einstein equations as G νµ = κ ˜ T νµ (21)where ˜ T νµ = 2 R f ( R ) d T νµ − Λ eff κ δ νµ , (22)Λ eff = ( d − R d , (23)and in which T νµ is given by (4). The constancy of the Ricci scalar amounts to − r A ′′ + 2 ( d − rA ′ + ( d −
2) ( d −
3) ( A − r = R (24)which yields A = 1 − R d ( d − r − mr d − + σr d − , (25)where σ and m are two integration constants. From the Einstein equations one identifies the constant σ as σ = 8 d ( d − R d − (cid:18) ( d −
3) ( d − Q (cid:19) d . (26)In the next section we investigate physical properties of our solution in all dimensions. III. ANALYSIS OF THE SOLUTIONA. − dimensions
1. Thermodynamics
In 4 − dimensions, we know that the nonabelian SO (3) gauge field coincides with the abelian U (1) Maxwell field [4].Due to its importance we shall study the 4 − dimensional case separately and give the results explicitly. First of all,in 4 − dimensions the metric function becomes A = 1 − R r − mr + Q R r , < | R | < ∞ (27)and the form of action reads as S = Z d x √− g (cid:20) f ( R )2 κ + L ( F ) (cid:21) (28)in which f ( R ) = R , (29)with R = R and L ( F ) = − π F. (30)By assumption, R gets positive / negative values and the resulting spacetime becomes de-Sitter / anti de-Sitter,type in f ( R ) = R theory respectively, with effective cosmological constant Λ eff = R . Let us add that in orderto preserve the sign of the charge term in (27) we must abide by the choice R >
0. However, simultaneous limits Q → R →
0, so that Q R = λ =constant, leads also to an acceptable solution within f ( R ) gravity [2]. It isnot difficult to see here that m is the ADM mass of the resulting black hole. Viability of the pure f ( R ) = R modelwhich has recently been considered critically [10] is known to avoid the Dolgov-Kawasaki instability [11]. Further, inthe late time behaviour of the expanding universe (i.e. for r → ∞ ) it asymptotes to the de Sitter / anti de Sitterform. With reference to [10] we admit that sourceless f ( R ) = R model doesn’t possess a good record as far as theSolar System tests are concerned. Herein we have sources and wish to address the universe at large. Now, we follow[12] to give the form of the entropy akin to the possible black hole solution. From the area relation the entropy of themodified gravity with constant curvature is given by S = A h G f ′ ( R ) (31)which upon insertion from (19) becomes S = A h GR f ( R ) = 2 πR r h (32)where r h indicates the event horizon. The Hawking temperature and heat capacity are given respectively by T H = A ′ ( r h )4 π = 4 R r h − Q − R r h πR r h , (33)and C Q = T H ∂S∂T H = 4 πR r h (cid:0) R r h − R r h + 2 Q (cid:1) ( R r h + 4 R r h − Q ) . (34)Here we note that for the case of zero YM charge ( Q = 0) one finds C Q = T H ∂S∂T H = 4 πR r h (cid:0) R r h − (cid:1) (4 + R r h ) (35)which clearly shows from (34) that for R > , the YM source brings in the possibility of having a phase change. Thisis depicted in Fig. 1. In Fig.s 1A and 1C we plot the horizon radius versus mass for Q = 0 and Q = 1. Similarly inFig.s 1B and 1D we plot the heat capacity C for Q = 0 and C Q for Q = 1 to see the drastic difference. It is observedthat for Q = 0 (Fig. 1B) the heat capacity is regular whereas for Q = 1 (Fig. 1D), C Q is a discontinuous functionsignalling a phase change.
2. Energy Conditions
From the energy conditions (see Appendix, A2) the density and principal pressures are given as ρ = − ˜ T = 18 πR (cid:18) F + 14 R (cid:19) , p = ˜ T = − πR (cid:18) F + 14 R (cid:19) , p i = ˜ T ii = 18 πR (cid:18) F − R (cid:19) , i = 2 , . These conditions imply that for R ≥
0, both the WEC and SEC are satisfied. DEC implies, on the other hand, from(A7) that P eff = 13 P i =1 ˜ T ii = 124 πR (cid:18) F − R (cid:19) ≥ , (36)which yields R ≥ F ≥ R → r ≤ s Q R . (37)In addition to the energy conditions one can impose the causality condition (CC) from (A9)0 ≤ P eff ρ = (cid:0) F − R (cid:1) (cid:0) F + R (cid:1) < , (38)which is satisfied if F ≥ R or r ≤ q Q R . Finally, if one introduces a new parameter (as the equation of state function ω ) by ω = P eff ρ , one observes that inthe range for 0 < r < ∞ we have − ≤ ω < . (39)In terms of the physical parameters, if s Q R ≤ r (40)then − ≤ ω ≤
0, and if s Q R > r (41)we have 0 < ω < . It is clearly seen that the foregoing bounds serve to define possible critical distances where thesign of the effective pressure changes sign. This may be interpreted as changing phase for example, from contractionto expansion or vice versa in a universe centered by a black hole. We note that scaling the mass and distance by R the results will not be affected. For this reason we set R = 1 . From Eq. (34) we plot in Fig.2, C Q , (with R = 1)versus r h and Q . The shaded region for r h < r c and C Q > , which lies below the curve r h = r c is the stable regionoutside the black hole. All the rest with C Q < r c at which the effective pressure turnssign and continues into opposite pressure, i.e. expansion reverses into contraction or vice versa. B. d − dimensions
1. Thermodynamics
In higher dimensions one obtains for the entropy and Hawking temperature the following expressions (for n ≥ S = (2 n − nπ n − r n − h R n − ( n + ) , d = 2 n (2 n +1) nπ n r n − h R (2 n − / n +1) , d = 2 n + 1 , (42) T H = − nπr n − h R n − (cid:20) Q n ( n − (cid:16) (2 n − n − (cid:17) n + (cid:0) n − n + R r h (cid:1) r n − h R n − (cid:21) , d = 2 n − π (2 n +1) r nh R n − (cid:20) Q n +12 n − (cid:16) ( n − n − (cid:17) n +14 + (cid:2) R r h − n + 1) ( n − (cid:3) r n − h R n − (cid:21) , d = 2 n + 1 . (43)The specific heat also fallows as C Q = π n − r n − h nR n − ( n − n − ( n + ) Φ , d = 2 n π n r n − h R n − n ( n − ) Ψ n +1)Φ , d = 2 n + 1 (44)in which we have used the following abbreviationsΨ = 4 Q n (cid:18) (2 n −
3) ( n − (cid:19) n + (cid:0) n − n + R r h (cid:1) r n − h R n − (45)Φ = − Q n (cid:18) (2 n −
3) ( n − (cid:19) n (2 n −
1) + (cid:0) − n + 4 n + R r h (cid:1) r n − h R n − Ψ = 4 Q n +12 (cid:18) ( n −
1) (2 n − (cid:19) n +14 + (2 n − (cid:2) R r h − n + 1) ( n − (cid:3) r n − h R n − Φ = − nQ n +12 (cid:18) ( n + 1) (2 n − (cid:19) n +14 + (2 n − (cid:2) R r h + 2 (2 n + 1) ( n − (cid:3) r n − h R n − . We notice that in odd dimensions from f ( R ) = R d , R can not get negative values for d =odd integer. The detailcan be seen in Appendix.
2. The First Law of Thermodynamics
As it was shown in Ref. [3] the first law of thermodynamics in f ( R ) gravity can be expressed as T dS − dE = P dV (46)in which E is the Misner-Sharp [13] energy stored inside the horizon such that dE = 12 κ (cid:20) ( d −
2) ( d − r h f R + ( f − Rf R ) (cid:21) A h dr h , (47) T = A ′ π is the Hawking temperature, S = π A h κ f R , is the entropy of the black hole P = T rr = T is the radial pressureof matter fields at the horizon and dV = A h dr h is the change of volume of the black hole at the horizon. In the caseof constant curvature i.e., R = R one gets dE = 12 κ (cid:20) ( d −
2) ( d − r h d R + (cid:18) − d (cid:19)(cid:21) R d A h dr h (48)which implies E = ( d − κ (cid:20) d ( d − r h ( d + 1) R − r h d − (cid:21) R d A h . (49)Here we show that the first law of thermodynamics for the metric function (25) is satisfied. Herein P = − π (cid:16) ( d − d − Q r h (cid:17) d and therefore the right hand side reads P dV = − π (cid:18) ( d −
2) ( d − Q r h (cid:19) d A h dr h . (50)On the other side we have T dS − dE = A ′ A h κ d ( d − r h R d − dr h − κ (cid:20) ( d −
2) ( d − r h d R + (cid:18) − d (cid:19)(cid:21) R d A h dr h . (51)We combine the latter with (46) and (50) to rewrite the first law as A ′ κ d ( d − r h R d − − κ (cid:20) ( d −
2) ( d − r h d R + (cid:18) − d (cid:19)(cid:21) R d = − π (cid:18) ( d −
2) ( d − Q r h (cid:19) d (52)or equivalently A ′ = ( d − r h − r h d R − d ( d − r d − h R d − (cid:18) ( d − d − Q (cid:19) d , (53)which is the derivative of the metric function at r = r h . This shows that the first law of thermodynamics by using thegeneralized form of the entropy for the Misner-Sharp energy is satisfied. To conclude this section of thermodynamicswe must admit that we don’t feel the necessity of addressing the second law. This originates from the fact that weare entirely in the static gauge so that the entropy change is assumed trivially satisfied i.e. ∆ S = 0 . IV. CONCLUSION
A relatively simpler class of solutions within f ( R ) gravity is the one in which the scalar curvature R is a constant R (both R > R < d ≥ L ( F ) ∼ ( F a .F a ) d . Since exact solutions in f ( R ) gravity with externalmatter sources, are rare, such solutions must be interesting. The equation of state for effective matter is consideredin the form P eff = ωρ , which is analyzed in Appendix A. The general forms of ω ( r ) given in (A21) determine ω within the ranges of − < ω < d − and 0 < ω < d − respectively. The fact that ω < − R . In case that the YMfield vanishes ( Q →
0) the only source to remain is the effective cosmological constant Λ eff = ( d − R d , which arisesnaturally in f ( R ) gravity. Another interesting result to be drawn from this study is that the effective pressure P eff changes sign before / after a critical distance. Thus, it is not possible to introduce a simple ω =constant, so that thepressure preserves its sign in the presence of a physical field (here YM) in the entire spacetime. From cosmologicalconsiderations the interesting case is when the critical distance lies outside the event horizon. This is depicted inthe projective plot (Fig. 2) of the heat capacity versus horizon and the charge. Finally it should be added thatalthough f ( R ) = R d/ gravities face viability problems in experimental tests the occurrence of sources may renderthem acceptable in this regard. [1] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. , 451 (2010);S. Nojiri and S. D. Odintsov, Physics Reports,
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Energy conditions
When a matter field couples to any system, energy conditions must be satisfied for physically acceptable solutions.We follow the steps as given in [14]. A. R > Weak Energy Condition (WEC):
The WEC states that ρ ≥ , (A1) ρ + p i ≥ . In which ρ is the energy density and p i are the principal pressure components given by ρ = − ˜ T = R πd F d R d + ( d − ! , (A2) p i = ˜ T ii = R πd d − F d R d − ( d − ! , i = 2 , · · · , ( d − ,p = ˜ T = − R πd F d R d + ( d − ! . Both conditions are satisfied. So WEC is held.
Strong Energy Condition (SEC):
This condition states that ρ + d − P i =1 p i ≥ , (A3) ρ + p i ≥ . The second condition is satisfied but first condition implies that ρ + d − P i =1 p i = R πd F d R d − ( d − ! ≥ (cid:18) FR (cid:19) d − ( d − ! ≥ . (A5)By a substitution from (11) for F one finds that for r < r c the condition is satisfied in which r c = d s d − s ( d −
2) ( d − Q R . (A6)0 Dominant Energy Condition (DEC):
In accordance with DEC, the effective pressure must not be negative. This amounts to P eff = 1 d − d − P i =1 T ii = 1( d − R πd F d R d − ( d −
2) ( d − ! ≥ , (A7)which for r < ˜ r c it is fulfilled in which˜ r c = d s d −
2) ( d − s ( d −
2) ( d − Q R . (A8) Causality Condition (CC):
In addition to the energy conditions one can impose the causality condition0 ≤ P eff ρ = (cid:16) F d R − d − ( d − d − (cid:17) ( d − (cid:16) F d R − d + ( d − (cid:17) < . (A9)This is equivalent to F d R − d − ( d −
2) ( d − > r < ˜ r c is satisfied.Finally we introduce ω = P eff ρ , given by ω = (cid:18)(cid:16) FR (cid:17) d − ( d − d − (cid:19) ( d − (cid:18)(cid:16) FR (cid:17) d + ( d − (cid:19) , (A11)which is bounded as − ≤ ω < d − . (A12)It is observed that (cid:26) ≤ ω < d − if r < ˜ r c − ≤ ω < r c < r . (A13) B. R < As one may see, presence of R d in the definition of ρ and p i imposes that d = 2 n + 1 where for n = 2 , , , ... . For d = 4 n we get ρ = − ˜ T = − | R | πn (cid:18) F n R n + 2 n − (cid:19) ,p i = ˜ T ii = − | R | πn (cid:18) n − F n R n − n − (cid:19) , (A14) p = ˜ T = | R | πn (cid:18) F n R n + 2 n − (cid:19) , WEC:
These expressions reveal that the condition ρ ≥ ρ + p i ≥ R < d = 4 n + 2 for n = 1 , , , ... in which ρ = − ˜ T = | R | π (2 n + 1) F n +12 | R | n +1 − n ! , (A15) p i = ˜ T ii = | R | π (2 n + 1) n F n +12 | R | n +1 + n ! , i = 2 , · · · , ( d − ,p = ˜ T = − | R | π (2 n + 1) F n +12 | R | n +1 − n ! . WEC: ρ ≥ F n +12 | R | n +1 − n ≥ r < ¯ r c (A17)where ¯ r c = n +2 r n s n (4 n − Q | R | . (A18) SEC:
The conditions are simply satisfied.
DEC:
This amounts to P eff = 14 n + 1 | R | π (2 n + 1) F n +12 | R | n +1 + n n ! ≥ , (A19)which is also satisfied. CC:
The causality condition implies0 ≤ P eff ρ = (cid:18) F n +12 | R | n +1 + n + 2 n (cid:19) (4 n + 1) (cid:18) F n +12 | R | n +1 − n (cid:19) < , (A20)or equivalently | R | n +1 n < F n +12 (A21)which is satisfied for r < ˘ r c (A22)where ˘ r c = n +2 r
41 + 4 n s n (4 n − Q | R | . (A23)Here the state function ω = P eff ρ becomes2 ω = (cid:18) F n +12 | R | n +1 + n + 2 n (cid:19) (4 n + 1) (cid:18) F n +12 | R | n +1 − n (cid:19) , (A24)which is bounded as − ≤ ω < n + 1 . (A25)One can show that (cid:26) ≤ ω < n +1 if r < ¯ r c − ≤ ω < r c < r . (A26) Figure Captions:Fig. 1 : The plot of horizon radius r h in 4 − dimensions versus mass m for different charges, Q = 0 (Fig. 1A) and Q = 1 (Fig. 1C). We also plot the heat capacity C Q versus the horizon radius for Q = 0 (Fig. 1B) and Q = 1 (Fig.1D). Fig. 1D, displays in particular the instability caused by the nonzero charge. Fig. 2:
The 3-dimensional picture of C Q versus r h and Q as projected into the ( r h , Q ) plane. The shaded regionwith C Q > r c is outside the event horizon. As shown, below the curve r h = r c we obtain stability (dark) regions.Above the curve r h = r c , the region is already inside the black hole and no stability is expected., the region is already inside the black hole and no stability is expected.