Phase transition and Quasinormal modes for Charged black holes in 4D Einstein-Gauss-Bonnet gravity
PPhase transition and Quasinormal modes for Charged black holes in 4DEinstein-Gauss-Bonnet gravity
Ming Zhang ∗ , Chao-Ming Zhang † , De-Cheng Zou ‡ and Rui-Hong Yue § Faculty of Science, Xi’an Aeronautical University, Xi’an 710077 China Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China (Dated: September 8, 2020)In this paper, we study the quasinormal modes(QNMs) of massless scalar perturbationsto probe the Van der Waals like small and large black holes(SBH/LBH) phase transitionof (charged) AdS black holes in the 4-dimensional Einstein Gauss-Bonnet gravity. We findthat the signature of this SBH/LBH phase transition in the isobaric process can be detectedsince the slopes of the QNMs frequencies change drastically different in small and large blackholes near the critical point. The obtained results further support that the QNMs can be adynamic probe of the thermodynamic properties in black holes.
I. INTRODUCTION
During the past decades, higher order derivative curvature gravities, as the effective models ofgravity in their low-energy limit string theories, have attracted considerable interest. Among thesehigher order derivative curvature gravities, the most extensively studied theory is the so-calledGauss-Bonnet gravity ([1]-[18]), which naturally emerges when we want to generalize Einstein’stheory in higher dimensions by keeping all characteristics of usual general relativity excepting thelinear dependence of the Riemann tensor. However, it is well known that the Gauss-Bonnet(GB)term’s variation is a total derivative in 4 dimensional spacetime, which has no contribution tothe gravitational dynamics. Therefore, one requires D ≥ α → α/ ( D − D the number of spacetime dimensions, and defining the 4-dimensional theory as the limit D →
4, the GB term gives rise to non-trivial dynamics. Furthermore, the spherically symmetric ∗ e-mail: [email protected]; [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] a r X i v : . [ h e p - t h ] S e p black hole solutions have been also constructed in this paper. The generalization to other blackholes has also appeared in refs. [3]-[11].In the black hole physics, the thermodynamical phase transition of black hole is always a hottopic. Due to the AdS/CFT correspondence[19–21], lots of attentions have been attracted to studythe phase transition of black holes in anti-de Sitter(AdS) space mainly. Recently thermodynamicsof AdS black holes has been studied in the extended phase space where the cosmological constant istreated as the pressure of the system [22], where it was found that a first order small and large blackholes phase transition is allowed and the P − V isotherms are analogous to the Van der Waals fluid.More discussions in this direction can be found as well in, including reentrant phase transitions andmore general Van der Waals behavior [23–40]. On the other hand, the quasinormal modes (QNMs)of dynamical perturbations are considered as the characteristic sounds of black holes. The QNMsof the dynamical perturbations are expected to reflect the black hole phase transitions in theirsurrounding geometries through frequencies and damping times of the oscillations. In fact, thethermodynamic phase transition of the AdS black hole in the dual field theory corresponds to theonset of instability of a black hole. With lots of researches on this issue, more and more evidencesof the connections between the QNMs of black holes and the thermodynamic phase transitionswere found[41–52].Until now, this Van der Waals-like (SBH/LBH) phase transition for charged and neutral blackhole was also recovered in Gauss-Bonnet gravity[53–57]. Motivated by these results and the exten-sive importance of AdS/CFT correspondence, the aim of this paper is to study whether signatureof Van der Waals like SBH/LBH phase transition of AdS black holes in 4D EGB gravity can bereflected by the dynamical QNMs behavior with the massless scalar perturbation.The paper is organized as follows: in Sect.II, we firstly review the Van der Waals like phasetransition of (charged) AdS black holes in 4-dimensional EGB gravity in the extended phase space.Then, we give discussions for QNM frequencies under test scalar field perturbations in Sect.III, anddisclose the phase transition can be reflected by the QNM frequencies of dynamical perturbations.We end the paper with closing remakes in the last section. II. THERMODYNAMICS AND PHASE TRANSITION OF CHARGED ADS BLACKHOLES
The action of D -dimensional charged EGB gravity in the presence of a negative cosmologicalconstant Λ ≡ − ( D − D − l is given by S = 116 π (cid:90) d D x √− g (cid:20) R + ( D − D − l + αD − G − F µν F µν (cid:21) , (1)where the Gauss-Bonnet term is G = R − R µν R µν + R µνρσ R µνρσ , the Gauss-Bonnet coefficient α with dimension ( length ) is positive in the heterotic string theory. F µν = ∂ µ A ν − ∂ ν A µ is theusual Maxwell tensor.Taking the limit D → ds = − f ( r ) dt + 1 f ( r ) dr + r d Ω D − , (2) f ( r ) = 1 + r α (cid:32) − (cid:115) α (cid:18) Mr − Q r − l (cid:19)(cid:33) , (3)with the nonvanishing electrostatic vector potential A t = Qr . Here M and Q are the mass andcharge of black hole. In the extended phase space, the cosmological constant Λ is regarded asa variable and also identified with the thermodynamic pressure P = − Λ8 π in the geometric units G N = (cid:126) = c = k = 1. We will only consider the positive GB coefficient α in the followingdiscussion. If we take α →
0, the solution f ( r ) reduces to RN AdS case in general relativity.In terms of the horizon radius r + , mass M , Hawking temperature T and entropy S of 4Dcharged EGB-AdS black holes can be written as [5] M = Q r + + α r + + r + πP r , (4) T = 2 r r + α P − Q − r + α πr + ( r + 2 α ) , (5) S = πr + 4 πα ln r + . (6)In the extended phase space, the black hole mass M is considered as the enthalpy H rather thanthe internal energy of the gravitational system.From the Hawking temperature (6), we can obtain the equation of state P = ( αr + 12 r + ) T + Q + α − r πr (7)As usual, a critical point occurs when P has an inflection point, ∂P∂r + (cid:12)(cid:12)(cid:12) T = T c ,r + = r c = ∂ P∂r (cid:12)(cid:12)(cid:12) T = T c ,r + = r c = 0 . (8)Then we can obtain corresponding critical values T c = r c − Q − α πr c ( r c + 6 α ) , (9) P c = − Q (3 r c + 2 α )8 πr c ( r c + 6 α ) + r c − αr c − α πr c ( r c + 6 α ) , (10)where r c = (cid:113) Q + 2 α ) + (cid:112) Q + 4 α )( Q + 4 α ) . (11)Here the subscript “c” represents the critical values of the physical quantities. For instance, we canobtain a critical point with r c = 0 . T c = 0 .
219 and P c = 0 . α = 0 .
01 and Q = 0 . P − r + isotherm diagram around the critical temperature T c for this charged AdS blackhole, see Fig.1. The dotted line with T > T c corresponds to the “idea gas” phase behavior, and theVan der Waals like small/large black hole phase transition appears in the system when T < T c . T = T c T = T c T = T c T > T c T = T c T < T c r + P (a) Q = 0 . T c = 0 . T = T c T = T c T = T c T > T c T = T c T < T c r + P (b) Q = 0 and T c = 0 . FIG. 1: The P − r + diagram of charged and uncharged AdS black holes. The behavior of Gibbs free energy G is important to determine the thermodynamic phasetransition. The free energy G obeys the following thermodynamic relation G = H − T S with G = 2 πr P (cid:18) − r + 4 α ln r + r + 2 α (cid:19) + ( Q − r + α )( r + 4 α ln r + )4 r + ( r + 2 α ) + Q + r + α r + . (12)Here r + is understood as a function of pressure and temperature, r + = r + ( P, T ), via equation ofstate (7).In left panel of Fig.2, we see that the G surface demonstrates the characteristic “swallow tail”behavior, which shows that there is a Van der Waals like first order phase transition in the system. T G P = P c P = P c P = P c P < P c P = P c P > P c SBH LBHCritical point T P FIG. 2: The Gibbs free energy G (left panel) and coexistence line of small/large black holes phase transition(right panel) with α = 0 .
01 and Q = 0 . The right panel of Fig.2 shows the coexistence line in the (
P, T ) plane by finding a curve wherethe Gibbs free energy and temperature coincide for small and large black holes. The coexistenceline is very similar to that in the Van der Waals fluid. The critical point is shown by a small circleat the end of the coexistence line. The small-large black hole phase transition occurs for
T < T c .Moreover, in the uncharged case, Eq.(9)-Eq.(11) can be written as T c = 1 + √ √ (cid:112) √ π √ α , (13) P c = 13 + 7 √ πα + 1824 √ πα , (14) r c = √ (cid:113) α + 2 √ α. (15)For instance, we can get a critical point with r c = 0 . T c = 0 . P c = 0 . α = 0 .
01. The corresponding “ P − r + ” and “ G − T ” diagrams of black hole are also qualitativelysimilar to the charged case.For Van der Waals liquid-gas system, the liquid-gas structure does not change suddenly butundergoes the second order phase transition at the critical point ( V = V c , T = T c , P = P c ). Thisis described by the Ehrenfest’s description [58, 59]. In conventional thermodynamics, Ehrenfest’sdescription consists of the first and second Ehrenfest’s equations [60, 61] ∂P∂T (cid:12)(cid:12)(cid:12) S = C P − C P T V ( ζ − ζ ) = ∆ C P T V ∆ ζ , (16) ∂P∂T (cid:12)(cid:12)(cid:12) V = ζ − ζ κ T − κ T = ∆ ζ ∆ κ T . (17)For a genuine second order phase transition, both of these equations have to be satisfied simulta-neously. Here ζ and κ T denote the volume expansion and isothermal compressibility coefficients ofthe system respectively ζ = 1 V ∂V∂T (cid:12)(cid:12)(cid:12) P , κ T = − V ∂V∂P (cid:12)(cid:12)(cid:12) T . (18)Using the similar method as Ref.[54], we can find that this phase transition at the critical point inthe 4-dimensional EGB-AdS black hole is of the second order (the Ehrenfests equations is satisfiedin this case). This result is consistent with the nature of the liquid-gas phase transition at thecritical point. III. PERTURBATION OF ADS BLACK HOLE IN 4D EINSTEIN-GAUSS-BONNETGRAVITY
In order to reflect the thermodynamical stabilities in dynamical perturbations, we can study theevolution of a massless scalar field perturbation around this 4-dimensional EGB-AdS black hole.A massless scalar field Φ( r, t,
Ω) = φ ( r ) e − iωt Y lm (Ω), obeys Klein-Gordon equation ∇ µ Φ( t, r, Ω) = 1 √− g ∂ µ (cid:0) √− gg µν ∂ ν Φ( t, r, Ω) (cid:1) = 0 . (19)Then radial equation for the function φ ( r ) is obtained as φ (cid:48)(cid:48) ( r ) + f (cid:48) ( r ) f ( r ) φ (cid:48) ( r ) + (cid:18) ω f ( r ) − l ( l + 1) r f ( r ) − f (cid:48) ( r ) rf ( r ) (cid:19) φ ( r ) = 0 , (20)where ω are complex numbers ω = ω r + iω im , corresponding to the QNM frequencies of theoscillations describing the perturbation.Near the horizon r + , we can impose the boundary condition of the scalar field, φ ( r ) → ( r − r + ) iω πT .Then we define φ ( r ) as ϕ ( r ) exp [ − i (cid:82) ωf ( r ) dr ], where the exp [ − i (cid:82) ωf ( r ) dr ] asymptoticallyapproaches to ingoing wave near horizon, we can rewrite Eq. (20) into ϕ (cid:48)(cid:48) ( r ) + (cid:18) f (cid:48) ( r ) f ( r ) − iωf ( r ) (cid:19) ϕ (cid:48) ( r ) − (cid:18) f (cid:48) ( r ) rf ( r ) + l ( l + 1) r f ( r ) (cid:19) ϕ ( r ) = 0 . (21)For Eq. (21), we have ϕ ( r ) = 1 in the limit of r → r + . At the AdS boundary ( r → ∞ ), weneed ϕ ( r ) = 0. Under these boundary conditions, we will numerically solve Eq. (21) to find QNMfrequencies by adopting the shooting method.In the left panel of Fig. 3, we plot T − r + diagram of charged AdS black holes with fixed pressure P = 0 . < P c = 0 .
219 in four dimensional EGB gravity. For
P < P c there is an inflection pointand the behavior is reminiscent of the Van der Waals system. The critical point can be got from ∂T∂r + (cid:12)(cid:12)(cid:12) P = P c ,r + = r c = ∂ T∂r (cid:12)(cid:12)(cid:12) P = P c ,r + = r c = 0 . (22)The behavior of the Gibbs free energy is plotted in the right panel of Fig. 3. The cross point “5”between the solid line marked as “1-5” and the solid line denoted as “4-5” shows that the Gibbsfree energy and P coincide for small and large black holes. In the left panel of Fig. 3, the point“5” is separated into “L5” and “R5” for the same Gibbs free energy and the chosen T ∗ ≈ . T * r + T T * T G FIG. 3: T − r + (left panel) and G − T (right panel) diagrams of 4D EGB AdS black holes with α = 0 . , Q =0 . P (cid:39) . In TABLE.I, we further list the QNM frequencies of massless scalar perturbation (for l = 0 and1) for small and large charged black holes near SBH/LBH phase transition point. With regard tosmall black hole phase, the radius of black hole becomes smaller and smaller when the temperaturedecreases from phase transition temperature T ∗ . In this process the absolute values of imaginarypart of QNM frequencies decrease, while the real part frequencies change very little. On the otherhand, when temperature for large black hole phase increases from the phase transition temperature T ∗ , the black hole gets bigger. The QNM frequencies increase in both the real part and the absolutevalue of imaginary part. It means that the massless scalar perturbation outside the black hole getsmore oscillations but it decays faster. These results are consistent with the overall discussionsreported in [51, 52]. Fig. 4 and Fig. 5 respectively illustrates the QNM frequencies with l = 0 and l = 1 for small and large black hole phases. Increase in the black hole’s size is indicated by thearrows.In addition, at the critical position P = P c , with P c (cid:39) .
06, a second-order phase transitionoccurs. The QNM frequencies of the small and large black hole phases (for l = 0 and l = 1) areplotted in Fig. 6. We can see that QNM frequencies of two black hole phases possess the same TABLE I: The QNM frequencies of massless scalar perturbation with the change of black hole temperaturewith α = 0 .
01 and Q = 0 .
1. The upper part, above the horizontal line, is for the small black hole phase,while the lower part is for the large black hole phase.
T r + ω ( l = 0) ω ( l = 1)0.1855 0.24719 1.67171-0.342286I 3.30238-0.488428I0.186 0.24833 1.67141-0.343580I 3.30097-0.491701I0.187 0.25072 1.67077-0.346294I 3.29793-0.498354I0.188 0.25325 1.67009-0.349273I 3.29450-0.505050I0.189 0.25594 1.66937-0.352404I 3.29115-0.512268I0.190 0.91129 2.72028-0.982813I 4.02092-0.993442I0.191 0.93516 2.74446-0.992160I 4.05526-1.003787I0.192 0.95733 2.76751-1.001339I 4.08825-1.013420I0.193 0.97819 2.78981-1.010033I 4.11994-1.022668I0.194 0.99800 2.81150-1.018480I 4.15075-1.031497I behavior as the black hole horizon increases at the critical point. ● ● ● ● ● - - - - - - - ( ω ) I m ( ω ) Small BH ● ● ● ● ● - - - - - ( ω ) I m ( ω ) Large BH
FIG. 4: The behavior of QNMs for large and small black holes in the complex- ω with Q = 0 . l = 0.The arrow indicates the increase of black hole horizon. In the neutral case, we can obtain similar T − r + and G − T diagrams to the charged case.For instance, the coexistence temperature T ∗ equals to 0 . P = 0 .
FIG. 5: The behavior of QNMs for large and small black holes in the complex- ω with Q = 0 . l = 1.The arrow indicates the increase of black hole horizon. ●●●●● ■■■■■ - - - - ( ω ) I m ( ω ) Small - Large BH (a) l = 0 ●●●●● ■■■■■ - - - - - ( ω ) I m ( ω ) Small - Large BH (b) l = 1 FIG. 6: The behavior of QNM frequencies for large (dashed) and small (solid) black holes in the complex- ω with Q = 0 .
1. The arrow indicates the increase of black hole horizon. hole phases are also qualitatively similar to the charged case.
IV. CLOSING REMARKS
In the 4-dimensional Einstein Gauss-Bonnet gravity, we have studied the P − V criticalityand phase transition of AdS black holes in the extended phase space. The VdW-like SBH/LBHphase transition could happen both in the charged and neutral cases. Then, we further calculatedthe QNMs of massless scalar perturbations under 4 situations (charged/uncharged and l = 0 or1). These results reveal that the slopes of the QNM frequency change drastically different in thesmall and large black hole phases as increasing of the horizon radius r + , when the Van der Waals0 TABLE II: The QNM frequencies of massless scalar perturbation with the change of black hole temperaturewith α = 0 .
01 and Q = 0. The upper part, above the horizontal line, is for the small black hole phase, whilethe lower part is for the large black hole phase. T r + ω ( l = 0) ω ( l = 1)0.214 0.18938 1.97357-0.340401I 2.86372-0.0557742I0.215 0.19088 1.97273-0.342787I 2.86116-0.0573858I0.216 0.19245 1.97185-0.345297I 2.85852-0.0591034I0.217 0.19408 1.97093-0.347917I 2.85572-0.0609143I0.218 0.19579 1.96997-0.350686I 2.85233-0.0626936I0.219 0.79874 2.42426-1.100808I 3.21754-1.100784I0.220 0.81586 2.44275-1.111345I 3.24017-1.110794I0.221 0.83201 2.46057-1.121425I 3.26214-1.120424I0.222 0.84739 2.47780-1.131115I 3.28349-1.129773I0.223 0.86211 2.49461-1.140502I 3.30429-1.138959I ● ● ● ● ● - - - - - - ( ω ) I m ( ω ) Small BH ● ● ● ● ● - - - - - ( ω ) I m ( ω ) Large BH
FIG. 7: The behavior of QNMs for large and small black holes in the complex- ω . The arrow indicates theincrease of black hole horizon. Q = 0 , l = 0 analogy SBH/LBH phase transition happens in the extended space. This clearly demonstrates thesignature of the phase transition between small and large black holes. In addition, at the criticalisobaric phase transitions, the QNM frequencies for both small and large black holes share the samebehavior, which showing that QNMs are not appropriate to probe the black hole phase transitionin the second order.This is one more example exhibits that the QNM can provide the dynamical physical phe-nomenon of the thermodynamic phase transition of black holes in 4D EGB gravity. Since theQNM is expected to be detected and has strong astrophysical interest. The ability of QNMs to1 ● ● ● ● ● - - - - ( ω ) I m ( ω ) Small BH ● ● ● ● ● - - - - - ( ω ) I m ( ω ) Large BH
FIG. 8: The behavior of QNMs for large and small black holes in the complex- ω . The arrow indicates theincrease of black hole horizon. Q = 0 , l = 1 reflect the thermodynamic phase transition is interesting, which is expected to disclose the obser-vational signature of the thermodynamic phase transition.This work is supported by the National Natural Science Foundation of China under GrantNos.11605152, 11675139 and 51802247, and Outstanding youth teacher programme from YangzhouUniversity. [1] C. Lanczos (1938) Annals Math. , no.8, 081301 (2020) doi:10.1103/PhysRevLett.124.081301[arXiv:1905.03601 [gr-qc]].[3] R. A. Konoplya and A. F. Zinhailo, [arXiv:2003.01188 [gr-qc]].[4] M. Guo and P. C. Li, Eur. Phys. J. C , no.6, 588 (2020) doi:10.1140/epjc/s10052-020-8164-7[arXiv:2003.02523 [gr-qc]].[5] P. G. S. Fernandes, Phys. Lett. B , 135468 (2020) doi:10.1016/j.physletb.2020.135468[arXiv:2003.05491 [gr-qc]].[6] A. Casalino, A. Colleaux, M. Rinaldi and S. Vicentini, [arXiv:2003.07068 [gr-qc]].[7] S. W. Wei and Y. X. Liu, [arXiv:2003.07769 [gr-qc]].[8] R. A. Konoplya and A. Zhidenko, Phys. Rev. D , no.8, 084038 (2020)doi:10.1103/PhysRevD.101.084038 [arXiv:2003.07788 [gr-qc]].[9] R. Kumar and S. G. Ghosh, JCAP , 053 (2020) doi:10.1088/1475-7516/2020/07/053[arXiv:2003.08927 [gr-qc]].[10] S. G. Ghosh and S. D. Maharaj, Phys. Dark Univ. , 100687 (2020) doi:10.1016/j.dark.2020.100687[arXiv:2003.09841 [gr-qc]]. [11] Y. P. Zhang, S. W. Wei and Y. X. Liu, [arXiv:2003.10960 [gr-qc]].[12] D. Lovelock (1971) J. Math. Phys.
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