Distinct thermodynamic and dynamic effects produced by scale factors in conformally related Einstein-power-Yang-Mills black holes
DDistinct thermodynamic and dynamic effects produced by scalefactors in conformally related Einstein-power-Yang-Mills blackholes
Yang Li * and Yan-Gang Miao † School of Physics, Nankai University, Tianjin 300071, China
Abstract
We study the thermodynamics and dynamics of high-dimensional Einstein-power-Yang-Mills blackholes in conformal gravity. Specifically, we investigate a class of conformally related black holes whosemetrics differ by a scale factor. We show that a suitable scale factor cures the geodesic incompleteness andthe divergence of Kretschmann scalars at the origin of spacetimes. In the aspect of thermodynamics, weanalyse the Hawking temperature, the entropy, and the specific heat, and verify the existence of second-order phase transitions. We find that the thermodynamics of this class of conformally related black holesis independent of scale factors. In the aspect of dynamics, we find that the quasinormal modes of scalarfield perturbations are dependent on scale factors. Quite interesting is that the behavior of quasinormalmode frequencies also supports the independence of scale factors for the second-order phase transitions.Our results show that the scale factors produce distinct thermodynamic and dynamic effects in the confor-mally related Einstein-power-Yang-Mills black holes, which provides an interesting connection betweenthermodynamics and dynamics of black holes in conformal gravity.
General Relativity (GR) is the most widely accepted theory of gravitation because it has been testedby numerous observations. In particular, the direct observation of gravitational waves from a binary blackhole merger reported by LIGO Scientific and Virgo Collaborations [1] opens a new era of gravitationalwave astrophysics. Despite the convincible experimental supports, GR encounters multiple severe prob-lems. First of all, the singularity at the origin of spacetimes is inevitable, where the curvature is divergentand the current theory comes into failure. It is hard to believe that a physically complete theory wouldallow the existence of divergence. Secondly, GR was modified [2] to include a cosmological constant toexplain the cosmic inflation. From the perspective of modern views, the introduction of the cosmolog-ical constant term is merely a compromised disposal in order to explain the cosmic inflation, however,it does not provide an ultimate answer about the physical origin of this term. Although many fancy hy-potheses have been proposed to explain this term, such as dark matter and dark energy [3], a persuasiveexperimental confirmation of these hypotheses is still beyond our reach. Thirdly, a satisfactory theory * E-mail address: [email protected] † Corresponding author. E-mail address: [email protected] a r X i v : . [ g r- q c ] F e b f quantum gravity has not been established. The essential problem is that GR is non-renormalizable inthe perspective of quantum field theory (QFT), and many theories in which gravity is consistent with theframe of quantum theory are very complicated in formalism, e.g. , string theory [4], thus they fail to makeapplicable predictions for physical observations.The three problems mentioned above imply that GR is probably not the ultimate theory of gravity,which prompts us to consider other possible schemes, such as conformal gravity. As early as 1918, Weylfound [5] a theory of gravity different from Einstein’s, in which not only the diffeomorphic invariancebut also the conformal invariance were considered. In a system already endowed with the diffeomorphicinvariance, such as GR, the conformal transformation is equivalent to the Weyl transformation. Thus, weidentify the conformal transformation with the Weyl transformation, and sometimes refer to the Weyl fac-tor as a scale factor. The conformal invariance is a strong constraint, under which the freedom of choiceof actions is very limited. The original conformal action of gravity was constructed [5] to be square of theWeyl tensor in the 4-dimensional spacetimes. However, such an action has no conformal invariance in thedimensions other than 4 due to the specific scaling behavior of the Weyl tensor. Therefore, an alternativeversion of conformal gravity was given [6] in which a massless scalar field was introduced. The invari-ance of all physical quantities under the conformal transformation implies that the conformal invariancecan be regarded as a gauge symmetry of spacetimes. However, it is obvious that the universe we live indoes not possess the conformal invariance, otherwise all the fundamental particles would be massless. Itwas suggested [7, 8] that the conformal symmetry was simultaneously broken in the early stage of theuniverse, similar to the simultaneous breaking of electroweak gauge symmetry in some sense, and all theequivalent solutions of conformal gravity became solutions of inequivalent gravitational theories, e.g. ,the conformal action could reduce to the Einstein-Hilbert action. Since there exists an extra degree offreedom for the choice of conformal transformations in conformal gravity, one can remove the singular-ity of conformal gravity at the spacetimes origin. For instance, the initial Big Bang singularity of theFriedman-Robertson-Walker (FRW) model is actually non-singular if measured [7] by the Weyl tensor.As a result, the FRW spacetimes are conformally equivalent to the flat spacetimes, i.e. , the appearance ofsingularity is simply caused by an improper choice of scale factors. Moreover, the inflation of universecould be well explained [9] in conformal gravity without the hypothesis of dark matter and dark energy.It was also suggested [10] in string theory that the conformal symmetry is an important property of theultimate quantum gravity theory since non-renormalizability can be fixed in conformal gravity. Generallyspeaking, the conformal symmetry is important even necessary if one wants to overcome the defects ofGR.As is known, Yang-Mills (YM) fields have been proved to be an exception of the black hole (BH) no-hair theorem in both GR [11] and conformal gravity [12, 13]. Furthermore, the achievement of Yang-Millstheory in particle physics is so remarkable that the standard model has been established, see, for example,the monograph [14]. On the other hand, the theory of Yang-Mills fields coupled to Einstein’s gravity, i.e. , the Einstein-Yang-Mills (EYM) theory has been developed [15, 16], where numerical and analyticBH solutions have been found in various dimensions. The dynamical properties of high-dimensionalEYM black holes have been investigated [17] recently through the resonance behaviors of EYM BHs un-der scalar field perturbations, characterized by quasinormal modes (QNMs) [18, 19, 20, 21]. QNMs are2igenmodes of oscillation of a dissipative system and thus they consist of a real part (Re ω ) denoting thepure oscillation and an imaginary part (Im ω ) denoting the time scale of damping. The behaviors of Re ω and Im ω represent BHs’ dynamic responses to perturbations. Based on the significance of conformalgravity mentioned in the above paragraph, it is warrant for us to combine the Einstein-power-Yang-Mills(EPYM) theory ‡ with conformal gravity, i.e. , to find the BH solutions of the conformally related EPYMtheory, to analyze the thermodynamics and dynamics of these BHs, and investigate the relationship be-tween thermodynamics and dynamics.The outline of this paper is as follows. In Sec. 2, we briefly review the conformal gravity and EPYMtheory, and then generalize the static spherically symmetric EPYM BH solution to its conformally relatedone, i.e. , the conformal-Einstein-power-Yang-Mills (CEPYM) BH solution. We calculate the Ricci scalarsand Kretschmann scalars of the CEPYM BHs and find that their divergence at the origin of spacetimes canbe removed by choosing a suitable conformal transformation. We also verify the geodesic completenessin the CEPYM BH spacetimes. In Sec. 3, we focus on the thermodynamics of the CEPYM BHs byanalyzing the Hawking temperature, the entropy, and the second-order phase transition. We then turn tothe dynamics of the CEPYM BHs by computing the QNMs of scalar field perturbations in terms of thesixth-order WKB method in Sec. 4. Finally, we present our conclusions in Sec. 5. In a gravitational system with a diffeomorphic invariance, the essence of conformal transformation isthe Weyl transformation, which is simply the metric g µν being multiplied by a scale factor Ω ( x ) ,˜ g µν = Ω ( x ) g µν , (1)and the transformation of the determinant √− g depends on the spacetime dimension D , (cid:112) − ˜ g = Ω D ( x ) √− g . (2)The action of conformal gravity proposed by Weyl [5] takes the form, I W = (cid:90) d D x √− gC µνσρ C µνσρ , (3)where two Weyl tensors are fully contracted with each other. Note that the original Weyl tensor is C ρµνσ ,which is invariant under the conformal transformation, Eq. (1). However, when index ρ is lowered, thecovariant tensor, C µνσρ , is no longer invariant,˜ C µνσρ = Ω ( x ) C µνσρ , (4)and neither is its contravariant tensor, ˜ C µνσρ = Ω − ( x ) C µνσρ . (5) ‡ It is an extension of EYM theory. For the details, see Sec. 2.
3n the 4-dimensional spacetimes, √− g gives the scale factor Ω ( x ) , see Eq. (2). Thus, the Weyl action,Eq. (3), is invariant under the conformal transformation. However, in dimensions other than 4, this actionis not conformally invariant.Here we focus on an alternative action of conformal gravity [6], I C = (cid:90) d D x √− g φ (cid:18) D − D − R φ − (cid:3) φ (cid:19) , (6)where φ is a massless scalar field, R the Ricci scalar, and (cid:3) ≡ g µν ∇ µ ∇ ν the covariant d’Alembertian. Asshown in Ref. [6], Eq. (6) is invariant under the conformal transformations of g µν and φ as follows,˜ g µν = Ω ( x ) g µν , (7)˜ φ = Ω − D ( x ) φ . (8)In our following contexts, g µν will describe a EPYM black hole, the scale factor Ω ( x ) will properly bechosen due to the conformal invariance, and thus the metric ˜ g µν which describes a family of conformallyrelated EPYM black holes will be determined by Eq. (7). We focus on SO ( D − ) Yang-Mills fields coupled with Einstein’s gravity, i.e. , the analytic solutionsof Einstein-Yang-Mills (EYM) black holes [16, 22, 23, 24] and of Einstein-power-Yang-Mills (EPYM)black holes [25].In SO ( D − ) gauge theory, the Yang-Mills invariant has the form, F YM = ( D − )( D − ) / ∑ a = F a µν F a µν , (9)where F a µν = ∂ µ A a ν − ∂ ν A a µ + σ C abc A b µ A c ν are the field strengths of SO ( D − ) Yang-Mills fields withthe structure constants C abc and coupling constant σ , and A a µ are the gauge potentials, the Latin indices, a , b , c , · · · = , , . . . , ( D − )( D − ) /
2, represent the internal space of the gauge group, and the Greekindices, µ , ν , α , β , · · · = , , . . . , D −
1, describe D -dimensional spacetimes. The action of SO ( D − ) gauge theory, I YM = − (cid:82) d D x √− g F YM , is conformally invariant only in the 4-dimensional spacetimes.In order to have a conformally invariant version in D dimensions, a power-Yang-Mills (PYM) invari-ant, F q YM , should be introduced, whose action, I PYM = − (cid:82) d D x √− g F q YM , has the conformal invariancewhen the power exponent q satisfies 4 q = D . Correspondingly, the EPYM action takes [25] the form, I EPYM = (cid:90) d D x √− g ( R − F q YM ) , (10)which reduces to the EYM action if q =
1, and its static spherically symmetric black hole solution underWu-Yang ansatz [26, 27] reads d s = − f ( r ) d t + f ( r ) d r + r d Ω D − , (11)4 ( r ) = − M ( D − ) r D − − Q r D − , (12) Q ≡ (( D − )( D − ) Q ) q ( D − )( D − − q ) , (13)if D − (cid:54) = q , § where Q is the only non-zero gauge charge of SO ( D − ) gauge group and M the BH mass.Note that the EPYM action has no conformal invariance although its second part, see Eq. (10), the PYMaction has such an invariance when 4 q = D .By combining Eq. (6) and Eq. (10), we generalize the EPYM theory to conformal gravity and writethe CEPYM action, I CEPYM = (cid:90) d D x √− g (cid:18) D − D − R φ − φ (cid:3) φ − F q YM (cid:19) , (14)which is invariant under the conformal transformations, Eqs. (7) and (8), when D = q . Note that theCEPYM action reduces to the EPYM action when the conformal (Weyl) symmetry is simultaneouslybroken with the specific choices of φ = (cid:113) D − D − and Ω ( x ) = S ( r ) ≡ Ω ( r ) = (cid:18) + L r (cid:19) N , (15)where N is a positive integer called the scale exponent of conformal transformations and L a length scale.We can see that the scale factor reduces to unity when L (cid:28) r . In this limit, the EPYM theory is recovered.Moreover, this scale factor was also chosen in Refs. [28, 29] where a conformally related SchwarzschildBH was obtained and its QNMs of scalar, electromagnetic, and axial gravitational perturbations werecomputed. As a result, we obtain from Eqs. (7) and (11)-(13) the metric of a conformally related EPYMblack hole, in short, the metric of a CEPYM black hole,d ˜ s = S ( r ) (cid:20) − f ( r ) d t + f ( r ) d r + r d Ω D − (cid:21) . (16)It is worth mentioning that Eq. (16) actually represents a family of CEPYM BH spacetimes which arerelated by the scale factor Eq. (15). Next, we shall show that the Ricci scalars and Kretschmann scalars ofthe CEPYM BH spacetimes have no singularities everywhere by choosing a suitable scale exponent andverify that the geodesic completeness is guaranteed in the CEPYM BH spacetimes.For the sake of convenience in the following discussions, we introduce the dimensionless rescaling of r , L , and Q , rM / ( D − ) → r , LM / ( D − ) → L , QM ( D − ) / ( D ( D − )) → ˆ Q . (17) § When D − = q , there exists the other branch of solutions in which f ( r ) contains a logarithmic term, see Ref. [25] forthe details. As we shall need a conformally invariant EPYM model, that is, D = q is required, we thus consider the solutionEq. (12) for the case of D − (cid:54) = q . Q after rescaling, which will not cause confusion in the contexts below. The lapse function Eq. (12)can be recast into the form consisting of dimensionless quantities, f ( r ) = − ( D − ) r D − − ˆ Q r D − , (18)ˆ Q ≡ (( D − )( D − ) ˆ Q ) q ( D − )( D − − q ) , (19)but the form of S ( r ) , see Eq. (15), maintains unchanged under such a dimensionless rescaling.We plot the image of Eq. (18) in Fig. 1, where the conditions that horizons exist can be seen clearly, i.e. ,ˆ Q (cid:46) .
00 in 4 dimensions, ˆ Q (cid:46) .
514 in 5 dimensions, and ˆ Q (cid:46) .
357 in 6 dimensions. In particular, theseconditions also give the horizons of the extreme BHs when the dimensionless charge ˆ Q approximatelytakes 1 .
00, 0 . . Q = . r + = . , . , .
00, respectively, and such a case will typically be chosenfor analyzing the Ricci scalars and Kretschmann scalars in Fig. 2 and Fig. 3, respectively. Q = Q = Q = Q = - - rf D = Q = Q = Q = Q = - - rf D = Q = Q = Q = Q = - - rf D = Figure 1: The lapse functions of 4, 5 and 6 dimensions with respect to the dimensionless radial coordinate.6 .2 Ricci and Kretschmann scalars of CEPYM black hole spacetimes
The Ricci scalar is defined as the trace of Ricci tensors, R ≡ g µν R µν . (20)For the EPYM black holes, the energy momentum tensor has a vanishing trace when 4 q = D , as shown inRef. [25]. Therefore, the Ricci scalars of the EPYM black holes vanish, namely, R =
0. For the CEPYMblack holes, when substituting Eqs. (15), (16), (18) and (19) into Eq. (20), ¶ we obtain the correspondingRicci scalars to their first orders in the vicinity of the origin in the 4-, 5-, and 6-dimensional spacetimesas follows, ˜ R ≈ − ( N + N ) ˆ QL N r N − , ˜ R ≈ − (cid:16) √ N + √ N ) (cid:17) ˆ Q / L N r N − , ˜ R ≈ − (cid:0) √ N + √ N (cid:1) ˆ Q / L N r N − . (21)It is obvious that they are convergent in the limit of r → N ≥ D .We turn to the Kretschmann scalar defined as two fully contracted Riemann tensors, K ≡ R µνσρ R µνσρ . (22)For the EPYM black holes, we obtain the Kretschmann scalars to their first orders in the vicinity of theorigin in 4, 5, and 6 dimensions by substituting Eqs. (11), (18) and (19) and into Eq. (22), K ≈
28 ˆ Q r , K ≈ √ Q / r , K ≈ Q r , (23)which are singular at the origin. For the CEPYM black holes, when substituting Eqs. (15), (16), (18) and(19) into Eq. (22), we obtain the corresponding Kretschmann scalars as follows,˜ K ≈ ( + N + N − N + N ) ˆ Q L N r ( N − ) , ˜ K ≈ ( √ + √ N + √ N − √ N + √ N ) ˆ Q / L N r ( N − ) , ˜ K ≈ ( + N + N − N + N ) ˆ Q L N r ( N − ) . (24) ¶ In the contexts below, the condition q = D / r → N ≥ D . We notice that the constraintof N for the CEPYM BHs is stronger than that given in Ref. [29] for conformally related SchwarzschildBHs, which implies that the Kretschmann scalars in the CEPYM BH spacetimes will be divergent at theorigin if N is not large enough in higher (than four) dimensions. Therefore, we conclude that 4 N ≥ D is the condition to remove the singularity of the Ricci and the Kretschmann scalars in the CEPYM BHspacetimes.In order to have a more intuitive description, we present the relationship between the Ricci scalarand the dimensionless radial coordinate r in 4, 5, and 6 dimensions in Fig. 2, where the Ricci scalars inthe EPYM BH spacetimes ( N =
0) are not shown because they apparently vanish in any dimensions asdiscussed above. D = = = - - rR N = D = = = - - rR N = D = = = - - - - - rR N = Figure 2: The Ricci scalar with respect to the dimensionless radial coordinate r , where ˆ Q = . L = N = , ,
10. The vertical lines mark the horizon positions in 4, 5, and 6 dimensions .From Fig. 2, we can see that the Ricci scalars in the CEPYM BH spacetimes (4 N ≥ D ) vanish atboth r = r → ∞ , which implies that the EPYM BHs and the CEPYM BHs are alike asymptotically.However, there is a major distinction between these two kinds of BHs in the near-horizon region. Aswe know, the Ricci scalar actually equals the trace of energy-momentum tensors. It is zero [25] for theEPYM BHs since the Yang-Mills field is conformally coupled to gravity and the condition 4 q = D ismatched, but the situation is different for the CEPYM BHs, i.e. , the Ricci scalar can be positive and/or8egative outside an event horizon. The reason is that the trace of energy-momentum tensors includes anextra contribution from the scalar field φ which is related to the scale factor via Eq. (8). Although theYang-Mills field still does not contribute to the trace, the scale factor is responsible for the non-vanishingtrace of energy momentum tensors, and hence the non-vanishing Ricci scalar. Note that the null energycondition requires a positive trace of energy momentum tensors. In the regions where the Ricci scalar isnegative, the null energy condition is violated and the gravitation becomes repulsive rather than attractive.Now we turn to the relationship between the Kretschmann scalar and the dimensionless radial coordi-nate r in 4, 5, and 6 dimensions in Fig. 3. D = = = rK N = D = = = rK N = D = = = rK N = D = = = rK N = Figure 3: The Kretschmann scalar with respect to the dimensionless radial coordinate r , where ˆ Q = . L =
1, and N = , , ,
10. Note that the case of N = N =
0) are indeedsingular at r =
0, but they have no singularity in the CEPYM BH spacetimes (4 N ≥ D ). In addition, thegeometries of the EPYM BHs and CEPYM BHs are similar when r (cid:29) r increases in a large r region.9 .3 Radial geodesic Now we prove the geodesic completeness in the CEPYM BH spacetimes by following Ref. [7]. Theequation that describes the radial geodesic motion takes the form, g tt ˙ t + g rr ˙ r = − δ , (25)where the dot represents the derivative of coordinates with respect to the affine parameter τ , e.g. , ˙ t = d t / d τ , and δ equals zero for a null particle and one for a time-like particle. For a free radially infallingparticle, energy E is a constant due to the conservation law of energy in static spacetimes and is related tothe time component of momenta as follows, P t = g tt ˙ t = − E . (26)Combining Eq. (25) with Eq. (26), we obtain the affine parameter in terms of integration with respect tothe radial coordinate r , ∆ τ = (cid:90) r i r f d r (cid:114) − g tt g rr δ g tt + E , (27)where r i represents the initial position of motion and r f the final position. When substituting Eq. (16) intoEq. (27), we derive the affine parameter in its convenient form in the CEPYM BH spacetimes, ∆ τ = (cid:90) r i r f d r (cid:115) [ S ( r )] E − δ S ( r ) f ( r ) . (28)In the following analyses, we shall adopt the dimensionless parameters introduced in Eq. (17) for conve-nience.For null geodesics with δ =
0, the proper time is independent of the lapse function and spacetimedimension D . We plot the graph of the affine parameter with respect to r f in Fig. 4, where we fix thevalue of r i and vary the value of r f . In the EPYM BH spacetimes with N = δ =
0, the integrand N = = = = r f Δτ Figure 4: The affine parameter with respect to r f , where E = . L = .
2, and r i = r f approaches the origin, see theblue line in Fig. 4. In the CEPYM BH spacetimes, a particle moving towards the origin along the radialnull geodesics actually never reaches the origin in a finite affine parameter, because the affine parameter isdivergent when r f goes to the origin, see, for instance, the cases of N = , ,
3. Here, the constraint 4 N ≥ D is not needed, since Eq. (28) with δ = δ =
0, a more general conclusion can be obtained: Null geodesics in the spacetimesdescribed by Eq. (15) and Eq. (16) with an arbitrary lapse function share the same behavior shown inFig. 4.For time-like geodesics with δ =
1, the result of integration Eq. (28) cannot be expressed analyticallydue to the complicacy of the integrand. Instead, we expand Eq. (28) in the vicinity of the origin andanalyze its approximate formulation. Starting from Eqs. (25) and (26) and considering the dimensionlessrescaling of Eq. (17), we obtain the approximate formula assuming r (cid:28) r = ˆ Q L N r N + − D + E L N r N . (29)Now we discuss the competition of the two terms on the right-hand side of Eq. (29). There are threesituations.(a) When 4 N + − D > N , the first term is infinitesimal compared to the second term and can beomitted. Note that 4 N + − D > N is equivalent to D < − N , and that the scale exponent N is apositive integer as well as the spacetime dimension D . In conclusion, the only allowed value of N is zero,which means that the CEPYM BH spacetimes reduce to the EPYM ones. The spacetime dimension nowsatisfies D < i.e. , D = ,
1. In this case, the spacetime geometry is trivial since there is no gravitationin the dimensions lower than 2.(b) When 4 N + − D < N , the second term on the right-hand side of Eq. (29) is infinitesimal com-pared to the first term and can be omitted. Now Eq. (29) leads to˙ r = (cid:112) ˆ Q L N r N + − D / . (30)The constraint 4 N ≥ D obtained in Sec. 2.2 can be written as 4 N + − D ≥
2. Combining with 4 N + − D < N , we have 8 N > N + − D ≥
2. For simplicity, we introduce a parameter defined as a ≡ N + − D /
2. Equivalently, we have 4 N > a ≥
1. The integration of Eq. (30) with respect to r can beexpressed as ∆ τ = L N (cid:112) ˆ Q (cid:90) r i r f r − a d r . (31)When the lower bound of a is reached, i.e. , a = N = D , Eq. (31) leads to ∆ τ = L N (cid:112) ˆ Q (cid:0) ln r i − ln r f (cid:1) , (32)11hich implies that the proper time is logarithmically divergent when r f →
0. When a >
1, Eq. (31) leadsto ∆ τ = L N (cid:112) ˆ Q ( a − ) (cid:32) r a − f − r a − i (cid:33) , (33)which implies that the proper time is divergent in power law when r f → N + − D = N , the two terms on the right-hand side of Eq.(29) have the same order andnone of them can be omitted. The proper time becomes ∆ τ = L N ( N − ) (cid:18) ˆ Q + E L N (cid:19) − / (cid:32) r N − f − r N − i (cid:33) . (34)The condition under which Eq. (34) diverges is 4 N >
1. Because spacetime dimension D is of courselarger than 1, we have 4 N ≥ D > N ≥ D , which means that such CEPYM BH spacetimes have no singularity. In this section, we study the Hawking temperature, the entropy, and the second-order phase transitionof the CEPYM BHs.
The Hawking temperature is T H = κ H π , (35)where κ H is the surface gravity at an event horizon which is defined as κ ≡ − ∇ µ χ ν ∇ µ χ ν (cid:12)(cid:12)(cid:12)(cid:12) r = r + . (36)In a D -dimensional static spacetime, χ ν is the time-like Killing vector, χ ν = ( , , . . . , ) , and r + is horizonradius. For a CEPYM BH, the Hawking temperature takes the form, T H = [ S ( r ) f ( r )] (cid:48) π S ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = r + = f (cid:48) ( r ) π (cid:12)(cid:12)(cid:12)(cid:12) r = r + , (37)where the prime represents the derivative with respect to the radial coordinate r and the definition of r + , i.e. , f ( r + ) = T H = π (cid:20) ( D − ) D − r − D + − (cid:0) ( D − )( D − ) ˆ Q (cid:1) D / r − D + (cid:21) , (38)12here the horizon radii in the 4-, 5-, and 6-dimensional spacetimes have the following forms, r + = (cid:113) − ˆ Q + , ( D = ) r + = (cid:18) A ( ˆ Q ) + A ( ˆ Q ) (cid:19) , ( D = ) r + = (cid:113) B ( ˆ Q ) + (cid:118)(cid:117)(cid:117)(cid:116) − B ( ˆ Q ) + (cid:113) B ( ˆ Q ) , ( D = ) (39)with A ( ˆ Q ) and B ( ˆ Q ) defined by A ( ˆ Q ) ≡ (cid:114)(cid:113) √ Q − − √ Q / , B ( ˆ Q ) ≡ √ √ Q (cid:113)(cid:112) − √ Q + + (cid:113)(cid:112) − √ Q + √ . (40)For the extreme BHs in 4, 5, and 6 dimensions, we have obtained the values of ˆ Q in Sec. 2.1, i.e. ,ˆ Q ext ≈ .
00, 0 . . r ext + ≈ .
00, 0 . . D = = = Q T H Figure 5: The Hawking temperature with respect to the dimensionless charge ˆ Q , where ˆ Q ext approxi-mately equals 1 .
00, 0 . .
357 for the 4-, 5-, and 6-dimensional extreme CEPYM BHs, respec-tively. 13 .2 Entropy
We investigate the Bekenstein-Hawking entropy for the CEPYM BHs, which is different from that ofthe EPYM BHs because the scale factor imposes a non-trivial correction to the area of event horizons, A D − = (cid:90) d θ . . . d θ D − √ h = [ S ( r + )] ( D − ) / π ( D − ) / r D − + Γ (cid:0) D − (cid:1) , (41)where A D − denotes the area of a ( D − ) -dimensional surface, θ i , i = , . . . , D −
2, the D − h the determinant of the induced metric on the horizon r + , and Γ ( x ) the Gamma function.Using the relation between entropy and horizon area for a static and spherically symmetric black hole [30],we write the Bekenstein-Hawking entropy of the D -dimensional CEPYM BHs, S D = A D − /
4, whichyields in 4, 5 and 6 dimensions as follows, S = A = S ( r + ) π r + , S = A = [ S ( r + )] / π r + , S = A = [ S ( r + )] π r + . (42)Substituting Eqs. (15), (39) and (40) into Eq. (42), we plot the graph presenting the Bekenstein-Hawking entropy with respect to the dimensionless charge in 4, 5, and 6 dimensions in Fig. 6, wherethe cases of N = , ,
10 are taken for the CEPYM BHs and the case of N = L is set to be unity for convenience.From this figure, we find that the Bekenstein-Hawking entropy of a CEPYM BH increases when theBH is approaching its extreme configuration but that of a EPYM BH decreases. The reason is that thescale factor has a non-trivial impact on the horizon area in the CEPYM BH spacetimes. In particular,the Bekenstein-Hawking entropy of a CEPYM BH is larger than that of a EPYM BH by several ordersin magnitude. This blow-up of Bekenstein-Hawking entropy is caused by the scale factor S ( r ) whichequals a factor (larger than one) to the ( N ) -th power, see Eq. (15). As a result, the larger N is, the higherorders in magnitude the entropy becomes.The behavior of the Bekenstein-Hawking entropy with respect to the spacetime dimension in theCEPYM BH spacetimes is also different from that in the EPYM BH spacetimes. In the former spacetimes,the entropy in higher dimensions is larger, however, it is just the opposite in the latter. Moreover, thedisparities between the entropies in two different dimensions increase with the increasing of the scaleexponent in the CEPYM BH spacetimes. We thus conclude that the influence of S ( r ) on Bekenstein-Hawking entropy is more obvious in higher dimensions.Here a comment to L is necessary. The choice of L seems to be artificial. Theoretically, L can bearbitrarily small such that S ( r ) approximates unity and the CEPYM BHs become arbitrarily close to theEPYM BHs. Indeed, a small L would be more reasonable since Einstein’s theory has been well verifiedby numerous experiments up to a decent precision. However, as mentioned above, N is an arbitrarypositive integer larger than D /
4. Thus, the blow-up of Bekenstein-Hawking entropy still happens even14 = = = Q S D N = D = = = Q S D N = D = = = Q S D N = D = = = Q S D N = Figure 6: The Bekenstein-Hawking entropy of the CEPYM BHs with respect to the dimensionless chargeˆ Q in 4, 5, and 6 dimensions for the cases of N = , ,
10, where the case of the EPYM BHs with N = L , as long as N inflates to a very large value, e.g. , N = L is sophisticatedly tuned. Nonetheless, we emphasize that such a blow-upof Bekenstein-Hawking entropy does not represent the physical entropy of the CEPYM BHs since themicroscopic degrees of freedom of a thermodynamic system do not blow up under a spacetime dilation,which will be seen clearly in the First Law of BH thermodynamics below.Next, we turn to discussions about the First Law of BH thermodynamics. Note that the Bekenstein-Hawking entropy of the CEPYM BHs bears a non-trivial correction compared to that of the EPYM BHsdue to the scale factor S ( r ) , while the Hawking temperature does not, see Eq. (37), and the ADMmass M maintains unchanged since S ( r ) does not influence the asymptotic structure of the CEPYM BHspacetimes. Therefore, we deduce the violation of the First Law of BH thermodynamics, ∂ S D ∂ M (cid:54) = T H . (43)15f we want to maintain the First Law of BH thermodynamics, we have to modify the unphysical entropy.We derive the expression of the derivative of S D with respect to M . Considering Eq. (41), we re-cast [30] S D as follows, S D = A D − = [ S ( r + )] ( D − ) / ¯ S D , (44)where ¯ S D ≡ π ( D − ) / r D − + Γ ( D − ) denotes the entropy of the D -dimensional EPYM BHs. Because the First Law ofthermodynamics in the EPYM BH spacetimes is valid, c.f. Ref. [25], ∂ ¯ S D ∂ M = T H , (45)we then obtain the derivative of S D with respect to the mass M , ∂ S D ∂ M = η T H , (46)where η is defined as η ≡ [ S ( r + )] ( D − ) / + D − T H S D ∂ ln S ( r + ) ∂ M . (47)The appearance of η implies the invalidity of the First Law of thermodynamics in the CEPYM BHs. Weshould redefine the entropy of the CEPYM BHs in such a way that it recovers its First Law, that is, themodified entropy should absorb η , d S Mod D = d S D η , (48)which leads to ∂ S Mod D ∂ M = T H . (49)Comparing Eq. (45) with Eq. (49), we can see that the modified entropy is nothing else but ¯ S D .This result implies that the quantity which is thermodynamically conjugated to the Hawking temperatureEq. (37) remains ¯ S D although the metric is modified by the scale factor.The essence of the entropy modification is that the relation S D = A D − / The Davies point at which a phase transition happens corresponds to the solution of the algebraicequation, 1 / C Q =
0, where the specific heat is defined as C Q ≡ (cid:18) ∂ M ∂ T (cid:19) Q = (cid:30)(cid:18) ∂ T ∂ M (cid:19) Q . (50)16s analyzed in the above subsection, the specific heat is independent of the scale factor S ( r ) . Note thatthe specific heat cannot be rescaled in terms of Eq. (17) because M should be regarded as a variable.We adopt an alternative dimensionless rescaling here. At first, we derive the specific heat by using theoriginal lapse function Eq. (12) and the definition of the Hawking temperature Eq. (37). Then, we rescalethe specific heat in the dimensionless formalism in the D -dimensional spacetimes, C Q M ( D − ) / ( D − ) −→ C Q . (51)After rescaling, the dimensionless C Q becomes a function of the dimensionless mass m defined as m ≡ M / Q D ( D − ) / ( D − ) . (52)Specifically, in 4 dimensions, Eq. (51) yields, C Q M −→ C Q = − π √ − m − (cid:16) √ − m − + (cid:17) √ − m − − m − + , (53)where m = M / Q . In 5 and 6 dimensions, the rescaling of the specific heat takes the forms, C Q M / −→ C Q , ( D = ) C Q M / −→ C Q , ( D = ) (54)where the corresponding m equals M / Q / and M / Q / , respectively. The derivation of the explicit formsof the specific heat in Eq. (54) is tedious but straightforward, so we do not present it here. Instead, weplot the dimensionless C Q in 4, 5, and 6 dimensions with respect to the dimensionless mass in Fig. 7. D = = = - - m C Q Figure 7: The dimensionless specific heat with respect to the dimensionless mass for the CEPYM BHs in4, 5, and 6 dimensions.From Fig. 7, we can see that the second-order phase transition exists in 4, 5, or 6 dimensions. Thecurves of specific heat for each spacetime are divided into two branches by a Davies point. By substituting17q. (53) and Eq. (54) into 1 / C Q = m Davies = / √ ≈ . m Davies ≈ .
82, and m Davies ≈ . m in a higher dimension. At the left branch in4, 5, or 6 dimensions, the specific heat is positive before the mass reaches the Davies point at which thesecond-order phase transition occurs, and the Hawking temperature increases with the increasing of m .After the mass exceeds the Davies point, i.e. at the right branch, the specific heat becomes negative, whichimplies that the Hawking temperature decreases with the increasing of m , and the lost energy convertsinto the Hawking radiation. Moreover, C Q has a zero point at its left branch located at m ext = . m ext ≈ .
00 in 5 dimensions, and m ext ≈ . m Davies with m ext , which means that thesecond-order phase transition occurs near the extreme configurations.In summary, we conclude that the thermodynamics the CEPYM BHs is independent of the scale factor. The Klein-Gordon equation for a massless scalar field in a curved spacetime is ∇ µ ∇ µ Φ = , (55)or it can be written as 1 √− g ∂ µ √− gg µν ∂ ν Φ = . (56)In Ref. [28], the decomposition of scalar field Φ in the background of a 4-dimensional conformal non-singular BH was introduced. Here we generalize it to the D -dimensional spacetimes as follows, Φ = ∑ l , m r ( D − ) / [ S ( r )] ( D − ) / e − i ω t ψ l ( r ) Y lm ( θ , . . . , θ D − ) , (57)where Y lm ( θ , . . . , θ D − ) stands for the spherical harmonics of D − ≤ ( θ , θ , . . . , θ D − ) ≤ π , and 0 ≤ θ D − ≤ π . Substituting Eq. (57) into Eq. (56), we obtain the radial equation, f ψ (cid:48)(cid:48) l + f f (cid:48) ψ (cid:48) l + (cid:0) ω − V eff (cid:1) ψ l = , (58)where V eff is the effective potential, V eff = f ( r ) l ( l + D − ) r + (cid:18) f ( r ) (cid:16) r ( D − ) / [ S ( r )] ( D − ) / (cid:17) (cid:48) (cid:19) (cid:48) [ S ( r )] ( D − ) / r ( D − ) / . (59)18he prime represents the derivative with respect to r . For the special case of 4 dimensions, Eq. (59) turnsback to Eq. (12) of Ref. [28]. Using Eqs. (15), (18), (19), and (59), we write the effective potential in itsexplicit form, V eff = ( D − ) r ( D − ) (cid:2) ( D + l − )( D + l − ) r D − + D ( D − ) ˆ Q + ( D − ) r (cid:3) × (cid:2) − ( D − ) ˆ Q + ( D − ) r D − − r (cid:3) . (60)Again using the “tortoise” coordinate r ∗ defined as d r ∗ ≡ d r / f ( r ) , we finally derive the Schr ¨odinger-likeequation of ψ l ( r ) , ∂ r ∗ ψ l + (cid:0) ω − V eff (cid:1) ψ l = . (61)We use the WKB method to compute the quasinormal mode frequencies of Eq. (61). The WKBmethod is an approximation method for us to find the eigenvalues of Schr¨odinger-like equations, espe-cially when the effective potential is too complicated to allow any analytic solutions. The first-order WKBmethod was introduced into BH perturbation theory by Schutz and Will [31], the third-order WKB by Iyerand Will [32], and the sixth-order WKB by Konoplya [33]. In the perspective of scattering theory, pertur-bations in a BH background can be regarded as waves scattered by effective potential V eff . The scatteringis restricted by the boundary condition, which requires that the waves are purely ingoing at horizon andoutgoing at infinity. For the scalar perturbation above, the boundary condition is ψ l ∼ e ± i ω r ∗ , r ∗ → ± ∞ . (62)Impacted by the boundary condition, the frequency ω is discretized. Because the perturbative scalar fieldis dissipated due to the existence of event horizons, ω is complex. These discrete frequencies are thequasinormal mode frequencies of perturbation, in short, quasinormal modes (QNMs).Using the sixth-order WKB method, we calculate the QNMs with the overtone number n = l =
2. Specifically, we plot QNMs with respect to ˆ Q in Fig. 8. Again, the dimensionlessrescaling introduced by Eqs. (17), (18), and (19) is adopted here.Re ω differs on a varying scale exponent. In general, it increases monotonically with the increasing ofˆ Q for a fixed N . For a large N , the value of Re ω is apparently large for a fixed ˆ Q , e.g. , Re ω with N = N = , , ,
4. However, for a small N , such as N = , ,
4, Re ω does notincrease with an increasing N monotonically when compared with Re ω in the case of N =
0. This resultshows that the oscillation frequency of the CEPYM BHs with a small N may be higher or lower than thatof the EPYM BHs.Im ω also differs on a varying scale exponent, and its absolute value simply increases monotonicallywith an increasing N for a fixed ˆ Q . More significantly, the behavior of Im ω provides the information aboutthe second-order phase transition mentioned in Sec. 3.3. Note that the relation between the dimensionlessmass m and dimensionless charge ˆ Q is ˆ Q = m − ( D − ) / ( D ( D − )) . Substituting m Davies into the relation, weobtain the corresponding values of ˆ Q Davies ≈ . Q Davies ≈ . Q Davies ≈ . .0 0.2 0.4 0.6 0.8 1.00.240.260.280.300.320.34 Q Re ω N = = = = = D = - - - - Q Im ω N = = = = = D = Q Re ω N = = = = = D = - - - - - - Q Im ω N = = = = = D = Q Re ω N = = = = = D = - - - - - Q Im ω = = = = = D = Figure 8: The real and imaginary parts of fundamental QNMs with respect to dimensionless charge ˆ Q fordifferent values of the scale exponent, N = , , , ,
10. For N = , , ,
4, the differences of Re ω ’s aresmall. The dimensionless parameter L is set to be unity for convenience.20he second-order phase transition is irrelevant of the scale exponent as shown in Sec. 3.3. Consequently,the behavior of QNMs of the scalar perturbation proves from the point of view of BH dynamics that thesecond-order phase transition of the CEPYM BHs is independent of N .In summary, we conclude that the scale factor has a significant influence on the dynamic properties ofthe CEPYM BH spacetimes. In the present paper, we investigate the non-singularity of the CEPYM BH spacetimes, the thermody-namics and dynamics of the CEPYM BHs, and their relationship.We prove that the CEPYM BH spacetimes are non-singular in two aspects. At first, the geometricquantities (the Kretschmann scalar and the Ricci scalar) are non-singular at the origin. Secondly, theCEPYM BH spacetimes are geodesically complete.We discuss the thermodynamics of the CEPYM BHs in the following three aspects.(a) The Hawking temperature does not depend on the scale factor S ( r ) , which is consistent withthe result in Ref. [7] where the Schwarzschild BH and its conformally related counterparts were ana-lyzed. Meanwhile, the extreme configurations of the CEPYM BHs have a vanishing Hawking tempera-ture, which implies no Hawking radiation.(b) In the CEPYM BH spacetimes, the relation S D = A D − / S ( r ) .(c) The specific heat of the CEPYM BHs is independent of the scale factor S ( r ) and the Davies pointsof the CEPYM BHs associated with second-order phase transitions are same as those of the EPYM BHs.A CEPYM BH undergoes one second-order phase transition near its extreme configuration.All the discussions about the Hawking temperature, the entropy, and the second-order phase transitionlead to the conclusion that the thermodynamics of the CEPYM BHs is independent of the scale factor,which can be understood in the perspective of statistical physics and thermodynamics: A pure dilationof spacetime volume by S ( r ) should have no influence on the microscopic states of a BH, and thus noinfluence on its macroscopic thermodynamic properties.For the dynamics of the CEPYM BHs, we obtain the real and imaginary parts of QNMs by using thesixth-order WKB method. The dependence of QNMs on the scale exponent is obvious. Re ω increasesmonotonically with the increasing of ˆ Q for a fixed N . For a large N , for instance, N ≥
10, the value of Re ω is apparently large. Quite interesting is that Re ω with a small N , for instance, N = , ,
4, is not certainlygreater than that with N = Q , which means that the oscillation frequency of the CEPYMBHs with a small N may be higher or lower than that of the EPYM BHs. Moreover, | Im ω | increasesmonotonically with the increasing of the scale exponent for a fixrd ˆ Q . The inflection points of Im ω correspond to the Davies points which are related to the second-order phase transition. In other words,both the QNMs and the specific heat confirm the existence of second-order phase transitions. Therefore,21he Hawking temperature Eq. (35) and specific heat Eq. (50) are well-defined and indeed manifest thethermodynamic properties of the CEPYM BHs.Finally, we conclude that the scale factor S ( r ) produces the distinct thermodynamic and dynamiceffects in the CEPYM BH spacetimes: The thermodynamics does not depend on the scale factor S ( r ) ,while the dynamics does. This distinction reveals a new relationship between the thermodynamics and dy-namics of the CEPYM BHs. As the relation between the thermodynamics and dynamics of BHs plays [34]an important role in BH quantization, our result may provide helpful information in the study of BH quan-tization in conformal gravity. Acknowlegement
The authors would like to thank Y. Guo, C. Lan, and H. Yang for helpful discussions. This work wassupported in part by the National Natural Science Foundation of China under Grant No. 11675081.
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