Gravitational Quasinormal Modes of Regular Phantom Black …

Research ArticleGravitational Quasinormal Modes of RegularPhantom Black Hole

Jin Li12 Kai Lin34 Hao Wen1 and Wei-Liang Qian56

1College of Physics Chongqing University Chongqing 401331 China2State Key Laboratory of Theoretical Physics Institute of Theoretical Physics Chinese Academy of Sciences Beijing 100190 China3Instituto de Fısica e Quımica Universidade Federal de Itajuba 37500-903 Itajuba MG Brazil4Department of Astronomy China West Normal University Nanchong Sichuan 637002 China5Escola de Engenharia de Lorena Universidade de Sao Paulo Sao Paulo SP Brazil6Faculdade de Engenharia de Guaratingueta Universidade Estadual Paulista Guaratingueta SP Brazil

Correspondence should be addressed to Jin Li cqstarvhotmailcom

Received 24 August 2016 Accepted 9 October 2016 Published 16 February 2017

Academic Editor Christian Corda

Copyright copy 2017 Jin Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We investigate the gravitational quasinormal modes (QNMs) for a type of regular black hole (BH) known as phantom BH which isa static self-gravitating solution of a minimally coupled phantom scalar field with a potential The studies are carried out for threedifferent spacetimes asymptotically flat de Sitter (dS) and anti-de Sitter (AdS) In order to consider the standard odd parity andeven parity of gravitational perturbations the corresponding master equations are derived The QNMs are discussed by evaluatingthe temporal evolution of the perturbation field which in turn provides direct information on the stability of BH spacetime It isfound that in asymptotically flat dS and AdS spacetimes the gravitational perturbations have similar characteristics for both oddand even parities The decay rate of perturbation is strongly dependent on the scale parameter 119887 which measures the couplingstrength between phantom scalar field and the gravity Furthermore through the analysis of Hawking radiation it is shown that thethermodynamics of such regular phantom BH is also influenced by 119887 The obtained results might shed some light on the quantuminterpretation of QNM perturbation

1 Introduction

As a major topic in cosmology the accelerated expansion ofour universe has caused widespread concern in the scientificcommunity Since the effect of gravity causes the expansionspeed to slow down the accelerated expansion of the universeimplies the existence of an unknown form of energy in theuniverse The latter provides a repulsive force to push theexpansion of the universe Such unknown energy is calleddark energy (DE) Subsequently a large number of DEmodels have been proposed among which the one withcosmological constant is the most famous Even though themodel of DE with the cosmological constant is reasonable inphysical theory and consistent with most observations twodifficulties still remain unsolved namely how to deriveldquovacuum energyrdquo from quantum field theory and why the

magnitudes of present DE and dark matter are of the sameorder

Many modern astrophysics observations indicated thepossibility of pressure to density ratio119908 lt minus1 For example amodel-free data analysis from 172 type Ia supernovae (SNIa)resulted in a range of minus12 lt 119908 lt minus1 for our presentepoch [1] According to the WMAP data during 7 years119908 = minus110+014minus014(1120590) [2] By using the data from Chandratelescope an analysis of the hot gas in 26 X-ray luminousdynamically relaxed galaxy clusters gives 119908 = minus120+024minus028[3] The data on SNIa from the SNLS3 sample estimates119908 = minus1069+0091minus0092 [4] In fact several DE models with asupernegative equation of state provide better fits to the abovedata [5ndash8] And all these approaches are in favor of phantomDE scenario [9ndash13] in which a constant equation of stateparameter is used [14 15] This implies that the phantom

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 5234214 19 pageshttpsdoiorg10115520175234214

2 Advances in High Energy Physics

model might be meaningful for in-depth understanding ofDE

In the phantom model the signature of the metric is +2and the action of the model reads119878 = intradicminus1198921198894119909 [119877 + 120598119892120583]120597120583120601120597]120601 minus 2119881 (120601)] (1)

where 119877 is the scalar curvature 119881(120601) is the potential of thescalar field and 120598 = minus1 corresponds to a phantom scalar fieldwhile 120598 = +1 is for a normal canonical scalar field

Bronnikov and Fabris first investigated the properties ofBHwith phantom scalar field in vacuum and derived a phan-tom regular BH solution 10 years ago [16] Inside the eventhorizon of such phantom BH there is no singularity similarto the case of regular BHs with nonlinear electrodynamicssources [17] Outside the event horizon the properties of aphantom BH are similar to those of a Schwarzschild BH Dueto the absence of the singularity such phantom regular BHsolution has attracted much attention from researchers

On the other hand the research of BH perturbation hasalways been an important issue in BH physics The firstwork on QNM in AdS spacetime was about scalar wave inSchwarzschild-AdS spacetime [18] which is then followedby a study on scalar wave in topological AdS spacetime[19] There are a large number of works on regular BHrsquosQNMs [17 20ndash25] Among various types of perturbationgravitational perturbation is generally considered to be themost important form due to its practical significance Theintrinsic properties and the stability of a BH can be unfoldedthrough its corresponding gravitational perturbation In thefifties of last century Regge and Wheeler began to study thegravitational perturbations of static spherically symmetricBHs It was pointed out later that [26] the higher dimen-sional gravitational perturbations can be classified into threetypes namely scalar-gravitational vector-gravitational andtensor-gravitational perturbations The first two types areassociated with odd (vector-gravitational) and even (scalar-gravitational) parity in accordance with the spatial inversionsymmetry of the perturbations and are of great physicalinterest [27] These findings significantly simplify the studyof gravitational perturbation of BH Subsequently peopledeveloped many new methods and further studies on thegravitational perturbations result in a large number of masterequations for various forms of BHs in 4-dimensions [27ndash30]in higher dimensions [26] and for stationary BHs [31 32]In fact gravitational perturbations of a BH may generaterelatively strong gravitational waves (GWs) Recently theGWs from a binary BH system have been detected by LIGO[33] so BH is proven to be the most probable source of GWsby modern technology Meanwhile since many alternativetheories of gravity can produce the same GW signal withinthe present accuracy in far field the reported GW detectionstill leaves a window for alternative gravity theories [34]which included the theory of phantom BHs Therefore theproperties of QNMs of gravitational perturbation near thehorizon of phantomBHmay provide us essential informationon the underlying physics of gravity theory This is the mainpurpose of the present study

The thermodynamics of BHs is also an important subjectin BH physics Some works indicated that Hawking radiationcan be considered as an effective quantum thermal radia-tion around the horizon [35 36] where the correspondingHawking temperature can be derived from the tunneling rate[35ndash40] Furthermore a natural correspondence betweenHawking radiation and QNM has been established recently[35ndash37 39 41]Therefore in this work wewill also investigatethe Hawking radiation of regular phantom BH

The paper is organized as follows In Section 2 we brieflyreview the regular phantom BH solutions and discuss theirproperties in three different spacetimes namely asymptot-ically flat de Sitter (dS) and anti-de Sitter (AdS) In thiswork we focus on the odd parity and even parity gravita-tional perturbations As the main component of this paperSection 3 includes two subsections In Section 31 we derivethe master equation for odd parity gravitational perturbationand analyze the corresponding temporal evolution of theperturbed metric in Section 32 corresponding studies arecarried out for the even parity gravitational perturbationIn Section 4 we calculate the Hawking radiation of theregular phantom BH We summarize our results and drawconcluding remarks in Section 5

2 The General Metric for RegularPhantom Black Holes

In this section we discuss the phantom (120598 = minus1) regularBH solution by considering the following static metric withspherical symmetry

1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903) + 119901 (119903)2 (1198891205792 + sin21205791198891205932) (2)

According to the action (1) the field equation for a self-gravitating minimally coupled scalar field with an arbitrarypotential 119881(120601) can be expressed as

119866120583] = 119877120583] minus 119892120583]2 (119877 minus 120598120601120572120601120572 minus 2119881 (120601)) minus 120598120601120583120601] = 0 (3)

By combining the scalar field equation

120598120601120572120572 minus 119889119881 (120601)119889120601 = 0 (4)

a regular phantom BH solution can be obtained as

119891 (119903) = 1198881198872 + 11199012 (119903) + 31198981198873 [ 1198871199031199012 (119903) + arctan(119903119887)]sdot 1199012 (119903) (5)

where119901 (119903) = radic1198872 + 1199032 (6)

Advances in High Energy Physics 3

119881 (120601 (119903)) = minus 1198881198872 1199012 + 211990321199012minus 31198981198873 31198871199031199012 + 1199012 + 211990321199012 arctan(119903119887) 120601 (119903) = radic2120598 arctan(119903119887) + 1206010

(7)

119898 is the Schwarzschild mass defined in the usual wayand 119888 and 119887 are integration constant and scale parameterrespectively Then it is necessary to determine the possiblekinds of spacetime for such phantom BH which can beclassified as a regular infinity (119903 rarr infin) to be flat de Sitter

(dS) or anti-de Sitter (AdS) The corresponding parameters119888 119887 119898 should be restricted in each spacetimeFor the asymptotically flat spacetime in accordance with

(5) one has 119888 = minus31205871198982119887 and119898= 211988733 [1205871198872 minus 2119887119903ℎ + 1205871199032ℎminus 21198872 arctan (119903ℎ119887) minus 21199032

ℎarctan (119903h119887)] (8)

In this case the spacetime is asymptotically flat namely 119903 rarrinfin 119891(119903) rarr 1 And 119903ℎ is the event horizon of the phantomBH We note when 119887 rarr 0 that (5) becomes Schwarzschildflat spacetime

For the de Sitter spacetime one has

119888 = minus 1198872 [(1198872 + 1199032119888 ) arctan (119903119888119887) minus (1198872 + 1199032ℎ) arctan (119903ℎ119887) + 119887 (119903119888 minus 119903ℎ)](1198872 + 1199032119888 ) (1198872 + 1199032ℎ) arctan (119903119888119887) minus (1198872 + 1199032119888 ) (1198872 + 1199032

ℎ) arctan (119903ℎ119887) + 119887 (119903119888 minus 119903ℎ) (1198872 minus 119903119888119903ℎ) 119898 = 1198873 (1199032119888 minus 1199032ℎ)3 [(1198872 + 1199032119888 ) (1198872 + 1199032

ℎ) arctan (119903119888119887) minus (1198872 + 1199032119888 ) (1198872 + 1199032

ℎ) arctan (119903ℎ119887) + 119887 (119903119888 minus 119903ℎ) (1198872 minus 119903119888119903ℎ)] (9)

where 119903119888 119903ℎ are the cosmological horizon and event horizonrespectively We note when 119887 rarr 0 that (5) becomesSchwarzschild dS metric

For the anti-de Sitter spacetime we choose119891(119903) rarr 1199032 with119903 rarr infin without loss of generality By expanding (5) aroundinfinity one finds

119888 = 21198873 minus 31205871198982119887 119898= 21198873 (1198872 + 1199032ℎ + 1)3 [1205871198872 minus 21198872 arctan (119903ℎ119887) minus 21199032

ℎarctan (119903ℎ119887) minus 2119887119903ℎ + 1205871199032

ℎ]

(10)

where 119903ℎ is the event horizon of AdS spacetime We notewhen 119887 rarr 0 that (5) can be returned to Schwarzschild-AdS spacetime In this context the parameter 119887 measuresthe coupling strength between phantom scalar field and thegravity for all three spacetimes

Since the parameters 119888 119898 can be expressed in terms of 119887119903ℎ and 119903119888 (dS) the structures of the regular phantom BHspacetime are completely determined by 119887 119903ℎ and 119903119888 (dS) (cfFigure 1) One can readily verify that all the spacetimesare indeed nonsingular even at 119903 = 0 As for any asymp-totically flat spacetime such flat regular phantom BH hasa Schwarzschild-like structure However its tendency ofapproaching flat spacetime at infinity becomes slower withincreasing 119887 In the dS case the spacetime is bounded by twohorizons that is 119903ℎ lt 119903 lt 119903119888 There is a maximum for 119891(119903)and it decreases with increasing 119887 For the AdS phantom BH119891(119903) rarr 1199032 when 119903 rarr infin and for larger 119887 the approach to theasymptotic solution becomes slower

3 Gravitational Quasinormal Frequenciesfor Regular Phantom Black Holes

As proposed by Regge-Wheeler two important gravitationalperturbations are of odd and even parity The perturbationgauge ℎ120583] for each type has its own definition In this sectionwe choose the Regge-Wheeler-Zerilli gauge to discuss themaster equation for each perturbation type Here we take themagnetic quantum 119872 = 0 to make 120593 disappear completelybecause all values of 119872 lead to the same radial equation[27] Since the total number of equations in the even paritycase is bigger than that in odd parity case the derivationof the master equation for even parity perturbation is thusmore complicated Once the master equation is derivedthe effective potential and the corresponding quasinormalmodes can be obtained We will first discuss the odd paritygravitational quasinormal frequencies in asymptotically flatdS and AdS spacetimes and then study the case for evenparity In our work we consider the metric perturbations notonly of the Ricci curvature tensor and scalar curvature (thelhs of the Einstein field equation) but also of the energymomentum tensor (the rhs of the Einstein field equation)On the other hand we will not consider the perturbations ofthe phantom scalar field This is because such perturbationscan be canceled out through an appropriate choice of119881 in theaction (see Appendix for details)

In order to discuss gravitational perturbation one maywrite down 119892120583] = 119892120583] + ℎ120583] (11)

where the small perturbation ℎ120583] will be divided into odd andeven modes in the subsequent sections


r

2 4 6 8 10

b = 1

b = 3

b = 5

minus20

minus15

minus10

f(r)

minus05

05

10

(a) Asymptotically flat

r

2 4 6 8 10

b = 2

b = 3

b = 5

minus15

minus10

minus05

f(r)

05

(b) dS

r

1 2 3 4 5

b = 01

b = 3

b = 5

minus40

minus20

f(r)

0

20

40

(c) AdS

Figure 1 The structures of the regular phantom BHrsquos metric function 119891(119903) for different values of 119887 (a) asymptotically flat spacetime with119903ℎ = 1 (b) de Sitter spacetime with 119903ℎ = 1 119903119888 = 10 (c) anti-de Sitter spacetime with 119903ℎ = 131 Master Equation and Quasinormal Modes for Odd ParityPerturbation The odd parity perturbation ℎ120583] has the formas [27]

ℎ120583] = ( 0 0 0 ℎ0 (119903 119905)0 0 0 ℎ1 (119903 119905)0 0 0 0ℎ0 (119903 119905) ℎ1 (119903 119905) 0 0 )119876119901 (120579) (12)

where 119876119901(120579) = sin 120579119889119875119897(cos 120579)119889120579 (119875119897(cos 120579) is the Legendrefunction) which satisfies11987610158401015840119901 minus cot (120579) 1198761015840119901 (120579) = minus119896119876119901 (120579) (13)

where 119896 = 119897(119897 + 1) and 119897 is the angular quantum numberThen the separation of variables can be carried out by

writing ℎ0(119903 119905) = exp(minus119894120596119905)ℎ0(119903) ℎ1(119903 119905) = exp(minus119894120596119905)ℎ1(119903)

By substituting (5)ndash(7) (11)ndash(13) into the field equation (3)and only keeping the first order perturbation terms we obtainthe independent perturbation equations as follows

12057511986613 = ℎ10158400 minus 2ℎ01199011015840119901 minus 11205961199012 119894ℎ1 (minus12059621199012+ 119891 (minus2 + 119897 + 1198972 + 211990111989110158401199011015840 + 1199012 (2119881 (120601) + 11989110158401015840))+ 1198912119901 (12059811990112060110158402 + 211990110158401015840)) = 0

(14)

12057511986623 = 119894120596ℎ01198912 + ℎ11198911015840119891 + ℎ10158401 = 0 (15)

Equation (15) implies ℎ0 = minus(1198912119894120596)(ℎ11198911015840119891 + ℎ10158401) Substi-tuting ℎ0 into (14) we get the master equation for odd parity


00

01

02

03

04

V(r)

05

06

50 15 2010

r

b = 05

b = 1

b = 3

b = 5

l = 2

(a)

r

0 10 20 30 40 50 60 70000

005

010

015

020

025

030

V(r)

035

l = 2

l = 3

l = 4

l = 5

b = 5

(b)

Figure 2 The effective potential for odd parity gravitational perturbation in asymptotically flat spacetime with 119903ℎ = 1 119897 = 2 (a) and with119903ℎ = 1 119887 = 5 (b)perturbationℎ101584010158401 + (31198911015840119891 minus 21199011015840119901 )ℎ10158401 + ℎ111989121199012 1199012 (1205962 + 11989110158402)minus 119891 (minus2 + 119897 + 1198972 + 21199012119881 (120601) + 411990111989110158401199011015840)minus 1198912119901 (12059811990112060110158402 + 211990110158401015840) = 0 (16)

Finally we renormalize ℎ1 byℎ1 (119903) = 119861 (119903)Φ (119903) 119861 (119903) = 119901 (119903)119891 (119903) (17)

By substituting (17) into the master equation (16) and usinga tortoise coordinate 119903lowast the Schrodinger-type wave equationfor this case can be expressed as1198892Φ1198891199032lowast + (1205962 minus 119881119900 (119903))Φ = 0 (18)

where the effective potential for odd parity perturbations119881119900(119903) is119881119900 (119903) = 1198911199012 [minus2 + 119897 + 1198972 + 211989111990110158402+ 1199012 (2119881 (120601) + 12059811989112060110158402 + 11989110158401015840) + 119901 (11989110158401199011015840 + 11989111990110158401015840)] (19)

Equation (19) can be used to describe the effective poten-tial of ldquooddrdquo-type perturbation119881119900 in different spacetimes andbe utilized to discuss the relationship between the effectivepotential andmodel parameters such as the angular harmonicindex 119897 and the parameter 119887

(I) Figures 2 3 and 4 show the potential functions tem-poral evolution of the gravitational perturbation andquasinormal frequency obtained by WKB method inasymptotically flat spacetime

(i) The form of the effective potential as a functionof 119903 for different values of 119897 and 119887 is shown inFigure 2 From the left plot one sees that as119887 increases the shape of the effective potentialbecomes smoother The maximum of the effec-tive potential decreases with increasing 119887 andthe position of the peak shifts to the right Fromthe right plot we see that with the increase ofangular quantumnumber 119897 themaximumof theeffective potential also increases 119881(119903) is alwaysfound to be positive definite outside the eventhorizon which indicates that the correspondingQNMs are likely to be stable

(ii) We adopt the finite differencemethod to analyzethe stability of such BH By applying the coordi-nate transformation (119905 119903) rarr (120583 ]) with 120583 = 119905 minus119903lowast ] = 119905+119903lowast to (16) and integrating numericallyusing the finite difference method [24 42ndash44]we obtain the differential equation for ℎ1(120583 ])Figure 3 shows the stability of a regular phantomBH in asymptotically flat spacetime with 119903ℎ =1 The temporal evolution of each mode inFigure 3 corresponds to a corresponding casein Figure 2 Since 119881(119903) decreases significantlywith the growth of 119887 the decay rate (|Im(120596)|)becomes smaller and oscillation frequency (ieRe(120596)) also drops as 119897 increases the valuesof 119881(119903) are raised correspondingly so thatthe oscillation frequency (ie Re(120596)) slightlyincreases together with the decay rate (|Im(120596)|)

(iii) We employ the WKB approximation [45 46]to evaluate the quasinormal frequencies


t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5

Asymptotically flat (l = 2)

(a)

t

0 20 40 60 80 100

Asymptotically flat (b = 5)

l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)

Figure 3The temporal evolution of odd parity gravitational perturbation in asymptotically flat phantom BH with 119903ℎ = 1 119897 = 2 (a) and with119903ℎ = 1 119887 = 5 (b)The complex frequency 120596 is determined by[47]1205962 = [1198810 + (minus2119881101584010158400 )12 119875]minus 119894 (119899 + 12) (minus2119881101584010158400 )12 (1 + Ω) (3rd order)

119894 1205962 minus 1198810radicminus2119881101584010158400 minus 119875 minus Ω minus 1198754 minus 1198755 minus 1198756 = 119899 + 12 (6th order) (20)

where 119881(119899)0 = (119889119899119881119889119903119899lowast)|119903lowast=119903lowast(119903ℎ) 119875 Ω 1198754 1198755and 1198756 are presented in [47 48]By making use of (19) we evaluate the QNMfrequencies by employing the 6th order WKBmethod (see Figure 4) Figure 4 shows thatthe fundamental quasinormal modes (119899 = 0)have the smallest imaginary parts as the modesdecay the slowest As 119899 increases the imaginarypart of the corresponding quasinormal modebecomes bigger for given 119897 and 119887 For a givenprincipal quantum number 119899 both the real andthe imaginary parts of the frequency decreasewith increasing 119887 on the other hand the realpart of the frequency increases significantlywith the angular quantum number 119887 whilethe imaginary part also increases slightly withincreasing 119887

(II) Figures 5 6 and 7 show the potential function bet-ween the event horizon 119903ℎ and cosmological horizon

minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

Asymptotically flat 6th WKB

n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)

Figure 4 Calculated quasinormal frequencies of odd parity grav-itational perturbation for 119903ℎ = 1 in asymptotically flat spacetimeEach group of dots from top-right to bottom-left corresponds toQNM frequencies obtained by assuming different values of 119887 =03 04 05 07 08 09 10 12 respectively

119903119888 temporal evolution of the gravitational field andquasinormal frequency obtained by WKB method indS spacetime

(i) The form of the effective potential in dS space-time is similar to that in asymptotically flatspacetime Figure 5 indicates that for given119897 and 119903 the effective potential decreases with


r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)

Figure 5 The effective potential for odd parity gravitational perturbation in dS spacetime with 119903ℎ = 1 119903119888 = 10 119897 = 2 (a) and with 119903ℎ =1 119903119888 = 10 119887 = 5 (b)dS (l = 2)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)

Figure 6 The temporal evolution of odd parity gravitational perturbation for dS phantom BH with 119903ℎ = 1 119897 = 2 (a) and with 119903ℎ = 1 119887 = 5(b)

increasing 119887 and for given 119887 and 119903 it increaseswith increasing angular quantum number 119897 Inthe range 119903ℎ lt 119903 lt 119903119888 119881(119903) is also positivedefinite which implies that the correspondingQNM is likely to be stable

(ii) Figure 6 studies the stability of a regular phan-tom with 119903ℎ = 1 119903c = 10 Similar to the casein asymptotically flat spacetime it is found thatwith increasing 119887 the oscillation frequency (ieRe(120596)) decreases while the decay rate (|Im(120596)|)becomes smaller Therefore for smaller value of

119887 the BH returns to its stable state more quicklyas small perturbation dies out faster

(iii) In accordancewith the results of finite differencemethod the frequencies presented in Figure 7by WKB method also show that for given119899 and 119897 the decay rate of perturbation (ie|Im(120596)|) decreases with increasing 119887 Moreoverit illustrates that for a given 119887 the fundamentalmode can be found at 119899 = 0 119897 = 2 Due tothe smallness of the real parts of the frequenciesthese modes of oscillation persist for longertime

8 Advances in High Energy Physicsminus

Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)

Figure 7 Calculated quasinormal frequencies of odd parity grav-itational perturbation with 119903ℎ = 1 119903119888 = 10 in dS spacetimeEach group of dots from top-right to bottom-left corresponds toQNM frequencies obtained by assuming different values of 119887 =01 02 03 04 05 06 07 08 respectively

(III) Figures 8 and 9 show the potential function beyondthe event horizon 119903 gt 119903ℎ and the temporal evolutionof the gravitational perturbation in AdS spacetime

(i) The form of the effective potential in AdSspacetime is quite different from that in asymp-totically flat and dS spacetime Figure 8 indicatesthat the value of119881(119903) is divergent as 119903 rarr infinTheeffects of 119887 and 119897 on the effective potential aresimilar to previous cases for a given 119903 smaller 119887or larger 119897 leads to bigger 119881

(ii) Figure 9 studies the stability of AdS regularphantom BH spacetime with 119903ℎ = 1 Theresults are consistent with the above calculatedpotential function It is again inferred that thefundamental mode of gravitational perturba-tion in odd parity occurs for 119897 = 2 and larger 119887since such kind of QNM will take a longer timeto be stable

32 Master Equation and Quasinormal Modes for Even ParityPerturbation Another canonical form for the gravitationalperturbations is of even parity After applying the separationof variables it can be expressed as [27 28]

ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)

where 1198670(119903)1198671(119903)1198672(119903) 119870(119903) are unknown functions forthe even parity perturbation It is noted that these functionsare not independent

Now we derive the first order perturbation equationsby substituting (5)ndash(7) (11) and (21) into (3) and find therelationships among 1198670(119903)1198671(119903)1198672(119903) 119870(119903)12057511986634 = 0 lArrrArr1198670 = 1198672 (22)

12057511986612 = 0 lArrrArr1205941 (119903)119894120596 1198671 minus 1199011015840119901 1198670 + 1198701015840 + (minus 11989110158402119891 + 1199011015840119901 )119870 = 0 (23)

12057511986611 = 0 lArrrArrminus 1205940 (119903)1198670 minus 119901101584011986710158400119901 minus (minus2 + 119897 + 1198972)11987021198911199012

+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)

12057511986623 = 0 lArrrArrminus 1198941205961198671119891 minus 11989110158401198670119891 minus 11986710158400 + 1198701015840 = 0 (25)

12057511986622 = 0 lArrrArr21198941205961198671119891 + 11986710158400 minus (1 + 119891101584011990121198911199011015840)1198701015840+ (minus2 + 119897 + 1198972) 119891 minus 212059621199012211989121199011199011015840 119870minus minus2 + 119897 + 1198972 + 21199012119881 (120601)21198911199011199011015840 1198670 = 0

(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)

Figure 8The effective potential for odd parity gravitational perturbation in AdS spacetime with 119903ℎ = 1 119897 = 2 (a) and with 119903ℎ = 1 119887 = 1 (b)AdS-odd (l = 2)

log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)

Figure 9 The temporal evolution of odd parity gravitational perturbation in AdS phantom BH spacetime with 119903ℎ = 1 119897 = 2 (a) and with119903ℎ = 1 119887 = 1 (b)where

1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)


Solving (23)1198670 can be expressed as

1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)

We define a function Ψ(119903) satisfying1198671 = minus119894120596 119903119891 (Ψ + 119870) (29)

By observing (22) (28) and (29) it turns out that we need toexpress119870 by Ψ in order to express all perturbation functions1198670(119903) 1198671(119903) 1198672(119903) and 119870(119903) in terms of Ψ This can beachieved by substituting (28) and (29) into (25) and (26)and evaluating the subtraction (25)-(26) and one eventuallyobtains the following expression(minus2 + 119897 + 1198972 + 21199012119881 (120601) + 311990111989110158401199011015840)1198701015840211989111990110158402minus (11990312059621198912 + 1205942 (119903))Ψ

minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where

1205942 (119903) = 119903 (minus2 + 119897 + 1198972 + 21199012119881 (120601) + 211990111989110158401199011015840)2119891211990110158402 1205941 (119903) 1205943 (119903) = 14119891211990110158402 (1198911015840 (minus411989111990110158402 + 1198972 + 119897 + 21199012119881 (120601) minus 2)

+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)

Substituting (28)ndash(31) into (24)119870 can be solved as

119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)

where 1198601(119903) 1198602(119903) are1198601 (119903) = minus (2119903119891119901212059411199011015840 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2)) 4119901120594311990110158402 (11990110158401015840119901 + 11990110158402 minus 11990121205940) 1198913 minus 211990121199011015840 (1205940 (1198972 + 119897+ 21199012119881 (120601) + 311990111989110158401199011015840 minus 2) minus 21205943 (1199031205941 + 1198911015840) 11990110158402) 1198912 + (2119881 (120601) 1205940 (21199031205941 + 1198911015840) 1199015 + (1205940 (41205962 + 311989110158402 + 611990312059411198911015840) 1199011015840+ 2119881 (120601) (1199011015840 (211990312059410158401 + 11989110158401015840) minus 119891101584011990110158401015840 + 21205941 (1199011015840 minus 11990311990110158401015840))) 1199014 + (1205940 (minus4119903120596211990110158402 + 2 (1198972 + 119897 minus 2) 1199031205941 + (1198972 + 119897 minus 2) 1198911015840)+ 1199011015840 (minus4119901101584010158401205962 + 2119881 (120601) (21199031205941 + 1198911015840) 1199011015840 + 61199031198911015840119901101584012059410158401 + 31198911015840119901101584011989110158401015840 minus 31198911015840211990110158401015840 + 612059411198911015840 (1199011015840 minus 11990311990110158401015840))) 1199013 + (minus4120596211990110158403+ 31198911015840211990110158403 + 411990312059621199011015840101584011990110158402 minus 2 (1198972 + 119897 minus 2)119881 (120601) 1199011015840 + 21198972119903120594101584011199011015840 + 2119897119903120594101584011199011015840 minus 4119903120594101584011199011015840 + 1198972119891101584010158401199011015840 + 119897119891101584010158401199011015840 minus 2119891101584010158401199011015840 minus 1198972119891101584011990110158401015840minus 119897119891101584011990110158401015840 + 2119891101584011990110158401015840 + 21205941 (3119903119891101584011990110158403 + (1198972 + 119897 minus 2) 1199011015840 minus (1198972 + 119897 minus 2) 11990311990110158401015840)) 1199012 + 211990110158402 (2119903120596211990110158402 + (1198972 + 119897 minus 2) 1199031205941minus (1198972 + 119897 minus 2) 1198911015840) 119901 minus (1198972 + 119897 minus 2)2 1199011015840)119891 minus 11990121199011015840 (21199031205941 (21199011015840 (119901 minus 1199031199011015840) 1205962 + (1198972 + 119897 + 21199012119881 (120601) minus 2)1198911015840 + 3119901119891101584021199011015840)

+ 1198911015840 (41199011015840 (119901 minus 1199031199011015840) 1205962 + (1198972 + 119897 + 21199012119881 (120601) minus 2) 1198911015840 + 3119901119891101584021199011015840))minus11198602 (119903) = minus (2119901 (2120594211990110158402 (11990110158401015840119901 + 11990110158402 minus 11990121205940) 1198913 + 21199011205942 (1199031205941 + 1198911015840) 119901101584031198912 + (2119903119881 (120601) 120594012059411199014+ (31199031205940120594111989110158401199011015840 + 2119881 (120601) (119903119901101584012059410158401 + 1205941 (1199011015840 minus 11990311990110158401015840))) 1199013+ (1199031205940 ((1198972 + 119897 minus 2) 1205941 minus 2120596211990110158402) + 1199011015840 (2119903119881 (120601) 12059411199011015840 + 31198911015840 (119903119901101584012059410158401 + 1205941 (1199011015840 minus 11990311990110158401015840)))) 1199012+ (1199031199011015840 (21199011015840119901101584010158401205962 + (1198972 + 119897 minus 2) 12059410158401) + 1205941 (3119903119891101584011990110158403 + (1198972 + 119897 minus 2) 1199011015840 minus (1198972 + 119897 minus 2) 11990311990110158401015840)) 119901+ 11990311990110158402 (2120596211990110158402 + (1198972 + 119897 minus 2) 1205941)) 119891 + 1199031199011199011015840 (21205962119891101584011990110158402 minus 1205941 (3119901119901101584011989110158402 + (1198972 + 119897 + 21199012119881 (120601) minus 2)1198911015840 minus 2119903120596211990110158402))))


sdot 4119901120594311990110158402 (11990110158401015840119901 + 11990110158402 minus 11990121205940) 1198913 minus 211990121199011015840 (1205940 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2) minus 21205943 (1199031205941 + 1198911015840) 11990110158402) 1198912 + (2119881 (120601)sdot 1205940 (21199031205941 + 1198911015840) 1199015 + (1205940 (41205962 + 311989110158402 + 611990312059411198911015840) 1199011015840 + 2119881 (120601) (1199011015840 (211990312059410158401 + 11989110158401015840) minus 119891101584011990110158401015840 + 21205941 (1199011015840 minus 11990311990110158401015840))) 1199014+ (1205940 (minus4119903120596211990110158402 + 2 (1198972 + 119897 minus 2) 1199031205941 + (1198972 + 119897 minus 2) 1198911015840)+ 1199011015840 (minus4119901101584010158401205962 + 2119881 (120601) (21199031205941 + 1198911015840) 1199011015840 + 61199031198911015840119901101584012059410158401 + 31198911015840119901101584011989110158401015840 minus 31198911015840211990110158401015840 + 612059411198911015840 (1199011015840 minus 11990311990110158401015840))) 1199013 + (minus4120596211990110158403+ 31198911015840211990110158403 + 411990312059621199011015840101584011990110158402 minus 2 (1198972 + 119897 minus 2)119881 (120601) 1199011015840 + 21198972119903120594101584011199011015840 + 2119897119903120594101584011199011015840 minus 4119903120594101584011199011015840 + 1198972119891101584010158401199011015840 + 119897119891101584010158401199011015840 minus 2119891101584010158401199011015840 minus 1198972119891101584011990110158401015840minus 119897119891101584011990110158401015840 + 2119891101584011990110158401015840 + 21205941 (3119903119891101584011990110158403 + (1198972 + 119897 minus 2) 1199011015840 minus (1198972 + 119897 minus 2) 11990311990110158401015840)) 1199012 + 211990110158402 (2119903120596211990110158402 + (1198972 + 119897 minus 2) 1199031205941minus (1198972 + 119897 minus 2) 1198911015840) 119901 minus (1198972 + 119897 minus 2)2 1199011015840)119891 minus 11990121199011015840 (21199031205941 (21199011015840 (119901 minus 1199031199011015840) 1205962 + (1198972 + 119897 + 21199012119881 (120601) minus 2)1198911015840 + 3119901119891101584021199011015840)+ 1198911015840 (41199011015840 (119901 minus 1199031199011015840) 1205962 + (1198972 + 119897 + 21199012119881 (120601) minus 2) 1198911015840 + 3119901119891101584021199011015840))minus1

(33)

The resulting master equation can be derived by evaluat-ing 12057511986623 + 12057511986622= 1198941205961198671119891 + (minus2 + 119897 + 1198972) 119891 minus 212059621199012211989121199011199011015840 119870 minus 1199011198911015840119870101584021198911199011015840

minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)

By substituting (28) (29) and (32) into the above equationthe corresponding master equation is given by

Ψ10158401015840 (119903) + Ψ1015840 (119903) 11986021198601 + 119860101584011198601 + minus21199031205941 minus 11989110158402119891 + 1199011015840 (119901 (21205962 + 2119891119881 (120601) + 11990312059411198911015840 + 11989110158402) + 2 (minus1199031205962 + 1198911198911015840) 1199011015840)119891119875V + Ψ (119903)sdot minus119903 (1 + 1198602) 1205941 (119875V minus 11990111989110158401199011015840)1198601119891119875V+ 121198601119891119875V (minus4119903120596211990110158402 + 1198602 (minus119875V1198911015840 + 1199011015840 (119901 (41205962 + 11989110158402) minus 411990312059621199011015840)) + 2119891 (119875V11986010158402 + 211986021199011015840 (119901119881 (120601) + 11989110158401199011015840))) = 0

(35)

where 119875V(119903) = minus2 + 119897 + 1198972 + 21199012119881(120601) + 311990111989110158401199011015840 Finally onedefines Ψ(119903) = 1198611(119903)1198612(119903)Φ(119903) and 119891(119903) = 119876(119903)119865(119903) (where119876(119903) is the coefficient of 1205962) the master equation can besimplified into the following equation

1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)

where 119889119903lowast = 119889119903119865(119903)119881119890 = 119865 (119903) (2119876 (119903) (1198651015840 (119903) 1198761015840 (119903) + 119865 (119903) 11987610158401015840 (119903)) minus 119865 (119903) 1198761015840 (119903)2) + 4 (119903)4119876 (119903)2 (37)


where

119876 (119903) = 12radic1198601 (311990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2)2 41198911199011015840 (119903 (11986010158401 minus 2) 1199011015840 minus 11990111986010158401 + 1198602 (119901 minus 1199031199011015840)) (311990111989110158401199011015840 + 1198972 + 119897+ 21199012119881 (120601) minus 2) + 1198601 (119903212059421 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2)2+ 2 (1199031199011015840 (4 (1198972 + 119897 minus 2) 11989111990110158401015840 minus 11989110158401199011015840 (211989111990110158402 + 3 (1198972 + 119897 minus 2))) + 1199013 (119881 (120601) (611989110158401199011015840 minus 411989111990110158401015840) + 4119891119901101584012060110158401198811015840 (120601))+ 119901 (minus2119891 (31199031198911015840101584011990110158403 + (1198972 + 119897 minus 2) 11990110158401015840 + 211990311990110158403119881 (120601)) + 11989110158401199011015840 (21198911199011015840 (311990311990110158401015840 + 1199011015840) + 3 (1198972 + 119897 minus 2)) minus 81199031198911015840211990110158403)+ 211990121199011015840 (119891 (1199011015840 (311989110158401015840 minus 211990312060110158401198811015840 (120601)) + 2119881 (120601) (211990311990110158401015840 + 1199011015840)) + 11989110158401199011015840 (41198911015840 minus 3119903119881 (120601)))))12 (119903) = 11611986021 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2)2 (minus16 (1199012 (11990110158402 minus 211990111990110158401015840)119881 (120601)2 + (2119901119891101584011990110158403 minus (1198972 + 119897 minus 119901211989110158401015840 minus 2) 11990110158402minus 411990121198911015840119901101584010158401199011015840 minus (1198972 + 119897 minus 2) 11990111990110158401015840)119881 (120601) + 1199011015840 ((211990110158403 minus 3119901119901101584011990110158401015840) 11989110158402 minus (11990110158401198811015840 (120601) 12060110158401199012 + 2 (1198972 + 119897 minus 2) 11990110158401015840) 1198911015840minus (1198972 + 119897 minus 2) (1199011198811015840 (120601) 1206011015840 + 119901101584011989110158401015840))) 1198912 minus 4 (12119881 (120601)2 119891101584010158401199014 + 21198911015840 (21199011015840119881 (120601)2 + (16119901101584011989110158401015840 minus 119891101584011990110158401015840)119881 (120601)+ 119891101584011990110158401198811015840 (120601)Ψ1015840) 1199013 + 2 (12119891101584021198911015840101584011990110158402 + 119881 (120601) (51198911015840211990110158402 + 6 (1198972 + 119897 minus 2) 11989110158401015840)) 1199012 + 1198911015840 (41198911015840211990110158403+ 2 (1198972 + 119897 minus 2)119881 (120601) 1199011015840 + 16 (1198972 + 119897 minus 2) 119891101584010158401199011015840 minus (1198972 + 119897 minus 2) 119891101584011990110158401015840) 119901 + (1198972 + 119897 minus 2) 1198911015840211990110158402 + 3 (1198972 + 119897 minus 2)2 11989110158401015840)119891+ (8119901119901101584011989110158402 + 3 (1198972 + 119897 + 21199012119881 (120601) minus 2) 1198911015840)2)11986021 + 41198602119891 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2) (2119901119901101584011989110158402 + (1198972 + 119897 minus 411989111990110158402+ 21199012119881 (120601) minus 2)1198911015840 minus 4119891119901119881 (120601) 1199011015840)1198601 minus 4119891 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2) (119860101584011198911015840 (1198972 + 119897 + 21199012119881 (120601) + 211990111989110158401199011015840minus 2) + 2119891 (minus119860101584010158401 1198972 minus 119860101584010158401 119897 minus 211986010158401119891101584011990110158402 + 11986010158402 (119903) (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2) minus 21199012119881 (120601)119860101584010158401 + 2119860101584010158401minus 1199011199011015840 (2119881 (120601)11986010158401 + 31198911015840119860101584010158401)))1198601 + 4119860221198912 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840 minus 2)2 minus 41198912119860101584021 (1198972 + 119897 + 21199012119881 (120601) + 311990111989110158401199011015840minus 2)2

(38)

Here we study the QNMs for even parity perturbation inasymptotically flat dS and AdS spacetimes Figures 10 11and 12 show the effective potential functions and corre-sponding temporal evolution of even parity perturbationin asymptotically flat dS and AdS spacetimes respectivelyWe consider the potential in the range 119903 gt 119903ℎ in asymp-totically flat and AdS spacetimes and 119903ℎ lt 119903 lt 119903119888 indS spacetime Here the potential functions are all positivedefinite so that the corresponding regular phantom BHs arelikely to be stable Furthermore in asymptotically flat anddS spacetimes the relationships between Re(120596) Im(120596) and119887 119897 are similar to those of odd parity the differences are indetails However the potential function in AdS spacetimeis mostly a convex function and therefore it approachesinfinity faster than that of odd parity Nevertheless theresulting QNMs in AdS spacetime are also found to bestable

4 Hawking Radiation of the RegularPhantom Black Hole

As mentioned before in Section 2 the interior structures ofregular BHs are quite different from those of traditional onesIn order to achieve a better understanding of the propertiesof such regular phantom BHs it is meaningful to investigatethe correspondingHawking radiation which is considered tobe a promisingmethod to study the thermodynamics of BHs

From a quantum mechanics viewpoint QNM carriesinformation on the quantization of the system around BHrsquoshorizon [35ndash37 39 41] The Hawking radiation spectrumprovides an interpretation of such quantization at an effectivetemperature [25 35 36] In this paper we adopt a simple andeffective method (ie Hamilton-Jacobi equation) to analyzethe Hawking radiation of the regular phantom BH Basedon the tunneling theory of Hawking radiation [49 50] this


V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)

Figure 10 (a) The effective potential of even parity perturbation in asymptotically flat spacetime (b) The corresponding temporal evolutionof even parity gravitational perturbation field in asymptotically flat spacetime Calculations are carried out by using 119903ℎ = 1method was first put forward by Srinivasan in 1999 [51] andplays an important role in recent studies [52 53]

Up to this point we have been using the natural units inthe calculations of QNM However since the Planck constantℏ is very important in the study of tunneling radiation we

will explicitly write it down here in the field functions Inother words ℎ0 and ℎ1 of (12) are now rewritten as ℎ0(119903 119905) =exp(minus119894120596119905ℏ)ℎ0(119903) ℎ1(119903 119905) = exp(minus119894120596119905ℏ)ℎ1(119903) and (21)becomes

ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)

Figure 11 (a) The effective potential of even parity perturbation in dS spacetime (b) The corresponding temporal evolution of even paritygravitational perturbation In the calculations we choose 119903ℎ = 1 119903119888 = 10The corresponding Schrodinger-type equation can be ex-pressed as 1198892Φ1198891199032lowast + (1205962ℏ2 minus 119881 (119903))Φ = 0 (40)

Since the Hawking radiation reflects BHrsquos radial proper-ties in the vicinity of the horizon it is worthwhile to explicitlydiscuss radial field equations According to (40) near thehorizons 119903 rarr 119903ℎ one has 119881119900 rarr 0 and 119881119890 rarr 0 so that thefield equation can be simplified to1198892Φ (119903)1198891199032lowast + 1205962ℏ2 Φ (119903) = 0 (41)

By introducingΦ(119903) = 119862119890minus(119894ℏ)119877(119903) and using the semiclassicalapproximation to neglect the terms ofO(ℏ) as ℏ is small oneobtains minus119891211987710158402 + 1205962 = 0 (42)

Therefore 1198771015840(119903) = plusmn120596119891 where plusmn represents ingoing oroutgoing mode The above equation is no other than theHamilton-Jacobi equation at the horizon We note that theabove argument also applies to the case of asymptoticallydS spacetime where the event horizon 119903ℎ and cosmologicalhorizon 119903119888 should be both considered In fact according to theHamilton-Jacobi equation 119892120583](120597119878120597119909120583)(120597119878120597119909])+1198982 = 0 theradialHamilton-Jacobi equation can be transformed into (42)at the horizon so we can use this result to study the Hawkingtunneling radiation under semiclassical approximation [54ndash56]

Owing to the coordinate singularity at the horizon theintegration from inner surface of horizon to outer surfaceshall be carried out by using ResidueTheorem

119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)

Figure 12 (a) The effective potential of even parity perturbation in AdS spacetime (b)The corresponding temporal evolution of even paritygravitational perturbation in AdS spacetime In the calculations we choose 119903ℎ = 1Then the tunneling rate of Hawking radiation is [57 58]Γ = exp (minus2 Im (119877+))

exp (minus2 Im (119877minus)) = exp(minus 41205871205961198911015840 (119903ℎ)) (44)

According to the relationship between tunneling rate Γ andthe Hawking temperature the temperature of the BH in thevicinity of the event horizon 119903ℎ is119879119867 = 1198911015840 (119903ℎ)4120587 (45)

From the inside of the cosmological horizon 119903119888 in dS space-time we consider the incident wave solution of Hawkingradiation Therefore there is an extra minus sign in theresulting expression of the Hawking temperature in theneighborhood of the cosmological horizon [59] as follows119879119888 = minus1198911015840 (119903119888)4120587 (46)

Figure 13 shows the impact of the parameter 119887 on thethermodynamics of a regular phantom BH for differentspacetimes In asymptotically flat spacetime the Hawkingtemperature 119879119867 decreases monotonously with increasing 119887and a given 119903ℎ and for a given 119887 119879119867 becomes smaller withincreasing 119903ℎ In asymptotically AdS spacetime howeverthe Hawking temperature 119879119867 increases monotonously withincreasing 119887 and a given 119903ℎ and for a given 119887 119879119867 becomeslarger with increasing 119903ℎ In asymptotically dS spacetimethe relationship between 119879119867 and the event horizon 119903ℎ issimilar to that in asymptotically flat case while the Hawkingtemperature 119879119862 in the vicinity of the cosmological horizonℎ119888 also decreases monotonously with increasing 119887 and itdecays faster than other cases The different dependenceof Hawking temperature on 119887 probably results from thedistinctive spacetime structure of AdS phantom BH fromothers


Asymptotically flat Asymptotically AdS

Asymptotically dS around rh Asymptotically dS around rc

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b

Figure 13 The Hawking temperature as a function of 119887 in asymptotically flat and dS spacetimes

5 Conclusions and Remarks

We studied static spherically symmetric solutions of regularphantom BH in asymptotically flat dS and AdS spacetimesand then investigated the QNMs of gravitational perturba-tions as well as Hawking radiations for these BH spacetimesIn our calculations besides the metric perturbation 120575119892120583] ofthe Ricci curvature tensor and scalar curvature its effecton the energy momentum tensor of the matter field is alsoconsidered In the derivation of the master equation foreven parity we made use of the method proposed in [26]However the BH metric in this paper is a self-gravitatingsolution of a minimally coupled scalar field with an arbitrarypotential rather than the Lovelock equations so that it doesnot satisfy the relation of Eq (54) in [26] As a result theobtained master equation is more complicated It is foundthat the calculated effective potential of AdS spacetime in thelimit 119903 rarr infin is very different from those of asymptoticallyflat and dS spacetimes For the asymptotical AdS spacetimethe effective potentials diverge at infinity which implies

that the wave function Φ should vanish in this limit [60ndash64] At the outside of the event horizon on the otherhand the corresponding wave function must be an incomingwave The distinct nature of AdS BH spacetime leads tothe fact that some traditional numerical methods such asWKB approximation continuous fraction method cease tobe valid In our calculations we therefore employed the finitedifferencemethod to numerically calculate the temporal evo-lution of small gravitational perturbations The relationshipsbetween Hawking temperature and parameters such as 119887 119903ℎin different spacetime are also studied

Owing to the importance of the parameter 119887 whichcarries the physical content of the coupling strength betweenphantom scalar field and the gravity we studied the depen-dence of various physical quantities on this specific param-eter among others From the above calculated results wedraw the following conclusions For odd parity in asymptoticflat and dS spacetime the perturbation frequency (Re(120596))becomes smaller and the decay rate (|Im(120596)|) decreases when119887 increases The perturbation frequency (Re(120596)) and the


decay rate (|Im(120596)|) both increase with increasing 119897 For evenparity the shape of the effective potential in the vicinity ofthe event horizon is very different from that for odd parityIn particular 119881119890(119903) decreases monotonically in the range 119903 gt119903ℎ which is caused by the metric transformation 119865(119903) =119891(119903)119876(119903) However our numerical calculations show thatthe resultant gravitational perturbations are stable in all threedifferent spacetime backgrounds The Hawking temperature119879119867 (and 119879119888) of the phantom BH decreases with increasing 119887in asymptotically flat and dS spacetimes On the contrary inAdS spacetime the temperature 119879119867 increases with increasing119887 These results reflect distinct natures of the horizons ofdifferent spacetimes

Recently another type of regular phantom BH which isa solution of Einstein-Maxwell equations was proposed byLemos and Zanchin [65] The properties of interior as wellas exterior structures were investigated It is meaningful toinvestigate the stability of such BH solution and compare itwith the phantom BH in the present paper

Appendix

We note that 119881(120601(119903)) in (7) is just the 0-order solutionHowever when we investigate the perturbation in the BHbackground the potential could in principle be expandedto higher order terms Since the higher order terms of 119881(120601)have not been determined yet we can freely choose the formof such higher order terms to cancel out the contributionsof the perturbation of phantom scalar field In the Appendixwe carry out explicit calculations to show that this is indeedpossible

We consider the following metric and phantom scalarfield 119892120583] = 119892120583] + ℎ120583]120601 = 120601 + 1206011 (A1)

where119892120583] 120601 are backgroundmetric and phantom scalar fieldrespectively and 1206011 represents the first order perturbation ofphantom scalar field Then119892120583] = 119892120583] minus ℎ120583] + ℎ120583120572ℎ]120572 (A2)

The corresponding metric and potential of phantom scalarfield can be expressed asradicminus119892 = radicminus119892 + radicminus1198921 + radicminus1198922 + sdot sdot sdot 119877 = 119877 + 1198771 + 1198772 + sdot sdot sdot 119881 (120601) = 119881 + 1198811 + 1198812 + sdot sdot sdot (A3)

where the superscript ndash and subscripts 1 2 represent thebackground quantities and the first and the second order per-turbations respectively Therefore the action of the gravitywith the phantom scalar field (1) can be expanded as119878 = 119878 + 1198781 + 1198782 + sdot sdot sdot (A4)

where119878 = int1198894119909radicminus119892 (119877 + 120598119892120583]120601120583120601] minus 2119881) 1198781 = int1198894119909 radicminus119892 [1198771 + 120598 (21206011205831206011205831 minus ℎ120583]120601120583120601])minus 21198811] + radicminus1198921 (119877 + 120598119892120583]120601120583120601] minus 2119881) 1198782 = int1198894119909 radicminus119892 [1198772+ 120598 (12060111205831206011205831 minus 2ℎ120583]1206011205831206011] + ℎ120583120572ℎ]120572120601120583120601]) minus 21198812]+ radicminus1198921 [1198771 + 120598 (21206011205831206011205831 minus ℎ120583]120601120583120601]) minus 21198811]+ radicminus1198922 (119877 + 120598119892120583]120601120583120601] minus 2119881)

(A5)

In the background equation there is no effect of 1198811 and 1198812while in the perturbed equation we can always choose theform of the perturbed potential 1198811 1198812 as1198811 = 1205981206011205831206011205831 1198812 = 120598212060111205831206011205831 minus 120598ℎ120583]1206011205831206011] (A6)

to cancel out the contributions from the phantom scalar fieldperturbation 1206011 Therefore in the text we do not explicitlyconsider the perturbation of the phantom field

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by Natural Science Foundationof China Grants no 11205254 no 11178018 no 11573022no 11605015 and no 11375279 the Fundamental ResearchFunds for the Central Universities 106112016CDJXY300002and 106112015CDJRC131216 Chongqing Postdoctoral ScienceFoundation (no Xm2015027) and theOpen Project Programof State Key Laboratory of Theoretical Physics Institute ofTheoretical Physics Chinese Academy of Sciences China(no Y5KF181CJ1)The authors also acknowledge the financialsupport by Brazilian foundations Fundacao de Amparo aPesquisa do Estado de Sao Paulo (FAPESP) Fundacao deAmparo a Pesquisa do Estado de Minas Gerais (FAPEMIG)Conselho Nacional de Desenvolvimento Cientıfico e Tec-nologico (CNPq) and Coordenacao de Aperfeicoamento dePessoal de Nıvel Superior (CAPES) and the Chinese StateScholarship

References

[1] U Alam V Sahni T D Saini and A A Starobinsky ldquoIs theresupernova evidence for dark energy metamorphosisrdquoMonthlyNotices of the Royal Astronomical Society vol 354 no 1 pp 275ndash291 2004


[2] E Komatsu ldquoSeven-year wilkinson microwave anisotropyprobe (WMAP) observations cosmological interpretationrdquoTheAstrophysical Journal Supplement Series vol 192 article 18 2011

[3] S W Allen R W Schmidt H Ebeling A C Fabian and LVan Speybroeck ldquoConstraints on dark energy from Chandraobservations of the largest relaxed galaxy clustersrdquo MonthlyNotices of the Royal Astronomical Society vol 353 no 2 pp 457ndash467 2004

[4] M Sullivan J Guy A Conley et al ldquoSNLS3 constraints on darkenergy combining the supernova legacy survey three-year datawith other probesrdquo The Astrophysical Journal vol 737 no 2article 102 2011

[5] S Hannestad and E Mortsell ldquoProbing the dark side con-straints on the dark energy equation of state from CMB largescale structure and type Ia supernovaerdquo Physical Review D vol66 no 6 Article ID 063508 2002

[6] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 no 1-2 pp 23ndash292002

[7] A Melchiorri L Mersini C J Odman and M Trodden ldquoThestate of the dark energy equation of staterdquo Physical Review Dvol 68 no 4 Article ID 043509 2003

[8] J Weller and A M Lewis ldquoLarge-scale cosmic microwavebackground anisotropies and dark energyrdquo Monthly Notices ofthe Royal Astronomical Society vol 346 no 3 pp 987ndash9932003

[9] R Gannouji D Polarski A Ranquet and A A StarobinskyldquoScalar-tensor models of normal and phantom dark energyrdquoJournal of Cosmology and Astroparticle Physics vol 2006 no9 p 16 2006

[10] A Vikman ldquoCan dark energy evolve to the phantomrdquo PhysicalReview D vol 71 no 2 Article ID 023515 14 pages 2005

[11] R R Caldwell M Kamionkowski and N N WeinbergldquoPhantom energy dark energy with 119908 lt minus1 causes a cosmicdoomsdayrdquo Physical Review Letters vol 91 no 7 Article ID071301 2003

[12] Z-Y Sun and Y-G Shen ldquoPhantom cosmology with non-minimally coupled real scalar fieldrdquo General Relativity andGravitation vol 37 no 1 pp 243ndash251 2005

[13] L P Chimento and R Lazkoz ldquoConstructing phantom cos-mologies from standard scalar field universesrdquo Physical ReviewLetters vol 91 no 21 p 211301 2003

[14] I Maor R Brustein J McMahon and P J Steinhardt ldquoMeasur-ing the equation of state of the universe pitfalls and prospectsrdquoPhysical Review D vol 65 no 12 Article ID 123003 2002

[15] J-M Virey P Taxil A Tilquin A Ealet D Fouchez and CTao ldquoProbing dark energy with supernovae bias from the timeevolution of the equation of staterdquo Physical Review D vol 70no 4 Article ID 043514 2004

[16] KA Bronnikov and J C Fabris ldquoRegular phantomblack holesrdquoPhysical Review Letters vol 96 no 25 Article ID 251101 2006

[17] A Flachi and J P S Lemos ldquoQuasinormal modes of regularblack holesrdquo Physical Review D vol 87 Article ID 024034 2013

[18] J S F Chan and R B Mann ldquoScalar wave falloff in asymptoti-cally anti-de Sitter backgroundsrdquo Physical Review D vol 55 no12 pp 7546ndash7562 1997

[19] J S F Chan and R B Mann ldquoScalar wave falloff in topologicalblack hole backgroundsrdquo Physical Review D vol 59 no 6Article ID 064025 23 pages 1999

[20] KA Bronnikov RAKonoplya andA Zhidenko ldquoInstabilitiesof wormholes and regular black holes supported by a phantomscalar fieldrdquo Physical Review DmdashParticles Fields Gravitationand Cosmology vol 86 no 2 Article ID 024028 2012

[21] K Lin J Li and S Z Yang ldquoQuasinormal modes of Haywardregular black holerdquo International Journal of Theoretical Physicsvol 52 no 10 pp 3771ndash3778 2013

[22] S Fernando andT Clark ldquoBlack holes inmassive gravity quasi-normal modes of scalar perturbationsrdquo General Relativity andGravitation vol 46 no 12 article 2014

[23] C F B Macedo and L C B Crispino ldquoAbsorption of planarmassless scalar waves by Bardeen regular black holesrdquo PhysicalReview D vol 90 no 6 Article ID 064001 2014

[24] J Li H Ma and K Lin ldquoDirac quasinormal modes inspherically symmetric regular black holesrdquo Physical Review DmdashParticles Fields Gravitation and Cosmology vol 88 no 6Article ID 064001 2013

[25] J Li K Lin andN Yang ldquoNonlinear electromagnetic quasinor-mal modes and Hawking radiation of a regular black hole withmagnetic chargerdquoThe European Physical Journal C vol 75 no3 article 131 2015

[26] T Takahashi and J Soda ldquoMaster equations for gravitationalperturbations of static Lovelock black holes in higher dimen-sionsrdquo Progress ofTheoretical Physics vol 124 no 5 pp 911ndash9242010

[27] T Regge and J A Wheeler ldquoStability of a Schwarzschildsingularityrdquo Physical Review vol 108 no 4 pp 1063ndash1069 1957

[28] F J Zerilli ldquoEffective potential for even-parity Regge-Wheelergravitational perturbation equationsrdquo Physical Review Lettersvol 24 no 13 pp 737ndash738 1970

[29] V Moncrief ldquoGravitational perturbations of spherically sym-metric systems II Perfect fluid interiorsrdquoAnnals of Physics vol88 pp 343ndash370 1974

[30] T Takahashi and J Soda ldquoHawking radiation from fluctuatingblack holesrdquo Classical and Quantum Gravity vol 27 no 17Article ID 175008 39 pages 2010

[31] S Chandrasekar The Mathematical Theory of Black HolesOxford University Press Oxford UK 1983

[32] S A Teukolsky ldquoPerturbations of a rotating black hole IFundamental equations for gravitational electromagnetic andneutrino-field perturbationsrdquo The Astrophysical Journal vol185 pp 635ndash647 1973

[33] B P Abbott R Abbott T D Abbott et al ldquoObservation ofgravitational waves from a binary black hole mergerrdquo PhysicalReview Letters vol 116 no 6 Article ID 061102 16 pages 2016

[34] RKonoplya andA Zhidenko ldquoDetection of gravitationalwavesfrom black holes is there a window for alternative theoriesrdquoPhysics Letters B vol 756 pp 350ndash353 2016

[35] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 11 pages 2012

[36] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 2011 no 8 article 101 2011

[37] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics vol 2013 article 82013

[38] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014


[39] C Corda ldquoBlack hole quantum spectrumrdquo The EuropeanPhysical Journal C vol 73 article 2665 2013

[40] C Corda ldquoQuantum transitions of minimum energy for Hawk-ing quanta in highly excited black holes problems for loopquantum gravityrdquo Electronic Journal of Theoretical Physics vol11 no 30 pp 27ndash34 2014

[41] C Corda ldquoNon-strictly black body spectrum from the tun-nelling mechanismrdquo Annals of Physics vol 337 pp 49ndash54 2013

[42] C Gundlach R H Price and J Pullin ldquoLate-time behaviorof stellar collapse and explosions I Linearized perturbationsrdquoPhysical Review D vol 49 no 2 pp 883ndash889 1994

[43] K Lin J Li and N Yang ldquoDynamical behavior and non-minimal derivative coupling scalar field of Reissner-Nordstromblack hole with a global monopolerdquo General Relativity andGravitation vol 43 no 6 pp 1889ndash1899 2011

[44] J Li and Y H Zhong ldquoQuasinormal modes for electromagneticfield perturbation of the asymptotic safe black holerdquo Interna-tional Journal ofTheoretical Physics vol 52 no 5 pp 1583ndash15872013

[45] B F Schutz and C M Will ldquoBlack hole normal modesmdashasemianalytic approachrdquo The Astrophysical Journal vol 291 pL33 1985

[46] S Iyer and C M Will ldquoBlack-hole normal modes a WKBapproach I Foundations and application of a higher-orderWKB analysis of potential-barrier scatteringrdquo Physical ReviewD vol 35 no 12 pp 3621ndash3631 1987

[47] Y Zhang E-K Li and J-L Geng ldquoQuasinormal modes ofmassless scalar field perturbation of a Reissner-Nordstrom deSitter black hole with a globalmonopolerdquoGeneral Relativity andGravitation vol 46 no 5 article 1728 pp 1ndash11 2014

[48] RAKonoplya ldquoGravitational quasinormal radiation of higher-dimensional black holesrdquo Physical Review D Third Series vol68 no 12 124017 7 pages 2003

[49] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[50] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[51] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 20 pages 1999

[52] K Lin and S-Z Yang ldquoQuantum tunnelling in charged blackholes beyond the semi-classical approximationrdquo EurophysicsLetters vol 86 no 2 Article ID 20006 2009

[53] K Lin and S Z Yang ldquoFermion tunnels of cylindrical symmet-ric black hole and the corrected entropyrdquo International Journalof Theoretical Physics vol 48 no 10 pp 2920ndash2927 2009

[54] K Lin and S-Z Yang ldquoFermion tunneling from higher-dimensional black holesrdquo Physical Review D vol 79 no 6Article ID 064035 2009

[55] K Lin and S Z Yang ldquoFermions tunneling of higher-dimensional Kerr-anti-de Sitter black hole with one rotationalparameterrdquo Physics Letters B vol 674 no 2 pp 127ndash130 2009

[56] K Lin and S Yang ldquoFermion tunnels of higher-dimensionalanti-de Sitter SCHwarzschild black hole and its correctedentropyrdquo Physics Letters B vol 680 no 5 pp 506ndash509 2009

[57] R Di Criscienzo L Vanzo and S Zerbini ldquoApplications of thetunneling method to particle decay and radiation from nakedsingularitiesrdquo Journal of High Energy Physics vol 2010 article92 2010

[58] L Vanzo G Acquaviva and R Di Criscienzo ldquoTunnellingmethods and Hawkingrsquos radiation achievements andprospectsrdquo Classical and Quantum Gravity vol 28 no 18Article ID 183001 2011

[59] S Z Yang and K Lin ldquoTunneling radiation and entropycorrection of arbitrary spin particles in Kerr-de Sitter blackholerdquoThe Chinese Physical Society vol 59 no 8 2010

[60] J Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[61] E Witten ldquoAnti de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[62] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[63] O Aharony S S Gubser J Maldacena H Ooguri and YOz ldquoLarge N field theories string theory and gravityrdquo PhysicsReports vol 323 no 3-4 pp 183ndash386 2000

[64] G T Horowitz and V E Hubeny ldquoQuasinormal modes of AdSblack holes and the approach to thermal equilibriumrdquo PhysicalReview D vol 62 no 2 Article ID 024027 11 pages 2000

[65] J P S Lemos andV T Zanchin ldquoRegular black holes Guilfoylersquoselectrically charged solutionswith a perfect fluid phantomcorerdquoPhysical Review D vol 93 no 12 Article ID 124012 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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OpticsInternational Journal of



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model might be meaningful for in-depth understanding ofDE

In the phantom model the signature of the metric is +2and the action of the model reads119878 = intradicminus1198921198894119909 [119877 + 120598119892120583]120597120583120601120597]120601 minus 2119881 (120601)] (1)

where 119877 is the scalar curvature 119881(120601) is the potential of thescalar field and 120598 = minus1 corresponds to a phantom scalar fieldwhile 120598 = +1 is for a normal canonical scalar field

Bronnikov and Fabris first investigated the properties ofBHwith phantom scalar field in vacuum and derived a phan-tom regular BH solution 10 years ago [16] Inside the eventhorizon of such phantom BH there is no singularity similarto the case of regular BHs with nonlinear electrodynamicssources [17] Outside the event horizon the properties of aphantom BH are similar to those of a Schwarzschild BH Dueto the absence of the singularity such phantom regular BHsolution has attracted much attention from researchers

On the other hand the research of BH perturbation hasalways been an important issue in BH physics The firstwork on QNM in AdS spacetime was about scalar wave inSchwarzschild-AdS spacetime [18] which is then followedby a study on scalar wave in topological AdS spacetime[19] There are a large number of works on regular BHrsquosQNMs [17 20ndash25] Among various types of perturbationgravitational perturbation is generally considered to be themost important form due to its practical significance Theintrinsic properties and the stability of a BH can be unfoldedthrough its corresponding gravitational perturbation In thefifties of last century Regge and Wheeler began to study thegravitational perturbations of static spherically symmetricBHs It was pointed out later that [26] the higher dimen-sional gravitational perturbations can be classified into threetypes namely scalar-gravitational vector-gravitational andtensor-gravitational perturbations The first two types areassociated with odd (vector-gravitational) and even (scalar-gravitational) parity in accordance with the spatial inversionsymmetry of the perturbations and are of great physicalinterest [27] These findings significantly simplify the studyof gravitational perturbation of BH Subsequently peopledeveloped many new methods and further studies on thegravitational perturbations result in a large number of masterequations for various forms of BHs in 4-dimensions [27ndash30]in higher dimensions [26] and for stationary BHs [31 32]In fact gravitational perturbations of a BH may generaterelatively strong gravitational waves (GWs) Recently theGWs from a binary BH system have been detected by LIGO[33] so BH is proven to be the most probable source of GWsby modern technology Meanwhile since many alternativetheories of gravity can produce the same GW signal withinthe present accuracy in far field the reported GW detectionstill leaves a window for alternative gravity theories [34]which included the theory of phantom BHs Therefore theproperties of QNMs of gravitational perturbation near thehorizon of phantomBHmay provide us essential informationon the underlying physics of gravity theory This is the mainpurpose of the present study

The thermodynamics of BHs is also an important subjectin BH physics Some works indicated that Hawking radiationcan be considered as an effective quantum thermal radia-tion around the horizon [35 36] where the correspondingHawking temperature can be derived from the tunneling rate[35ndash40] Furthermore a natural correspondence betweenHawking radiation and QNM has been established recently[35ndash37 39 41]Therefore in this work wewill also investigatethe Hawking radiation of regular phantom BH

The paper is organized as follows In Section 2 we brieflyreview the regular phantom BH solutions and discuss theirproperties in three different spacetimes namely asymptot-ically flat de Sitter (dS) and anti-de Sitter (AdS) In thiswork we focus on the odd parity and even parity gravita-tional perturbations As the main component of this paperSection 3 includes two subsections In Section 31 we derivethe master equation for odd parity gravitational perturbationand analyze the corresponding temporal evolution of theperturbed metric in Section 32 corresponding studies arecarried out for the even parity gravitational perturbationIn Section 4 we calculate the Hawking radiation of theregular phantom BH We summarize our results and drawconcluding remarks in Section 5

2 The General Metric for RegularPhantom Black Holes

In this section we discuss the phantom (120598 = minus1) regularBH solution by considering the following static metric withspherical symmetry

1198891199042 = minus119891 (119903) 1198891199052 + 1198891199032119891 (119903) + 119901 (119903)2 (1198891205792 + sin21205791198891205932) (2)

According to the action (1) the field equation for a self-gravitating minimally coupled scalar field with an arbitrarypotential 119881(120601) can be expressed as

119866120583] = 119877120583] minus 119892120583]2 (119877 minus 120598120601120572120601120572 minus 2119881 (120601)) minus 120598120601120583120601] = 0 (3)

By combining the scalar field equation

120598120601120572120572 minus 119889119881 (120601)119889120601 = 0 (4)

a regular phantom BH solution can be obtained as

119891 (119903) = 1198881198872 + 11199012 (119903) + 31198981198873 [ 1198871199031199012 (119903) + arctan(119903119887)]sdot 1199012 (119903) (5)

where119901 (119903) = radic1198872 + 1199032 (6)



(7)




ℎarctan (119903h119887)] (8)





ℎ) arctan (119903119888119887) minus (1198872 + 1199032119888 ) (1198872 + 1199032





ℎarctan (119903ℎ119887) minus 2119887119903ℎ + 1205871199032

ℎ]

(10)








r

2 4 6 8 10

b = 1

b = 3

b = 5

minus20

minus15

minus10

f(r)

minus05

05

10


r

2 4 6 8 10

b = 2

b = 3

b = 5

minus15

minus10

minus05

f(r)

05

(b) dS

r

1 2 3 4 5

b = 01

b = 3

b = 5

minus40

minus20

f(r)

0

20

40

(c) AdS


ℎ120583] = ( 0 0 0 ℎ0 (119903 119905)0 0 0 ℎ1 (119903 119905)0 0 0 0ℎ0 (119903 119905) ℎ1 (119903 119905) 0 0 )119876119901 (120579) (12)






(14)

12057511986623 = 119894120596ℎ01198912 + ℎ11198911015840119891 + ℎ10158401 = 0 (15)



00

01

02

03

04

V(r)

05

06

50 15 2010

r

b = 05

b = 1

b = 3

b = 5

l = 2

(a)

r

0 10 20 30 40 50 60 70000

005

010

015

020

025

030

V(r)

035

l = 2

l = 3

l = 4

l = 5

b = 5

(b)











t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5


(a)

t

0 20 40 60 80 100


l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)





minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6


n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)





r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(7)




ℎarctan (119903h119887)] (8)





ℎ) arctan (119903119888119887) minus (1198872 + 1199032119888 ) (1198872 + 1199032





ℎarctan (119903ℎ119887) minus 2119887119903ℎ + 1205871199032

ℎ]

(10)








r

2 4 6 8 10

b = 1

b = 3

b = 5

minus20

minus15

minus10

f(r)

minus05

05

10


r

2 4 6 8 10

b = 2

b = 3

b = 5

minus15

minus10

minus05

f(r)

05

(b) dS

r

1 2 3 4 5

b = 01

b = 3

b = 5

minus40

minus20

f(r)

0

20

40

(c) AdS


ℎ120583] = ( 0 0 0 ℎ0 (119903 119905)0 0 0 ℎ1 (119903 119905)0 0 0 0ℎ0 (119903 119905) ℎ1 (119903 119905) 0 0 )119876119901 (120579) (12)






(14)

12057511986623 = 119894120596ℎ01198912 + ℎ11198911015840119891 + ℎ10158401 = 0 (15)



00

01

02

03

04

V(r)

05

06

50 15 2010

r

b = 05

b = 1

b = 3

b = 5

l = 2

(a)

r

0 10 20 30 40 50 60 70000

005

010

015

020

025

030

V(r)

035

l = 2

l = 3

l = 4

l = 5

b = 5

(b)











t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5


(a)

t

0 20 40 60 80 100


l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)





minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6


n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)





r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




r

2 4 6 8 10

b = 1

b = 3

b = 5

minus20

minus15

minus10

f(r)

minus05

05

10


r

2 4 6 8 10

b = 2

b = 3

b = 5

minus15

minus10

minus05

f(r)

05

(b) dS

r

1 2 3 4 5

b = 01

b = 3

b = 5

minus40

minus20

f(r)

0

20

40

(c) AdS


ℎ120583] = ( 0 0 0 ℎ0 (119903 119905)0 0 0 ℎ1 (119903 119905)0 0 0 0ℎ0 (119903 119905) ℎ1 (119903 119905) 0 0 )119876119901 (120579) (12)






(14)

12057511986623 = 119894120596ℎ01198912 + ℎ11198911015840119891 + ℎ10158401 = 0 (15)



00

01

02

03

04

V(r)

05

06

50 15 2010

r

b = 05

b = 1

b = 3

b = 5

l = 2

(a)

r

0 10 20 30 40 50 60 70000

005

010

015

020

025

030

V(r)

035

l = 2

l = 3

l = 4

l = 5

b = 5

(b)











t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5


(a)

t

0 20 40 60 80 100


l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)





minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6


n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)





r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

01

02

03

04

V(r)

05

06

50 15 2010

r

b = 05

b = 1

b = 3

b = 5

l = 2

(a)

r

0 10 20 30 40 50 60 70000

005

010

015

020

025

030

V(r)

035

l = 2

l = 3

l = 4

l = 5

b = 5

(b)











t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5


(a)

t

0 20 40 60 80 100


l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)





minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6


n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)





r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




t

0 20 40 60 80 100minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 05

b = 1

b = 3

b = 5


(a)

t

0 20 40 60 80 100


l = 2

l = 3

l = 4

l = 5

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

Φ(t)

minus1

minus05

0

(b)





minusIm

(120596)

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6

l = 2 l = 3 l = 4 l = 5 l = 6


n = 0

n = 1

n = 2

02

04

06

08

1 215 2505

Re(120596)





r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




r

2 4 6 8 10

minus01

01

02

03

04

V(r)

05

06

b = 001

b = 05

b = 1

b = 3

b = 5

dS (l = 2)

(a)

r

2 4 6 8 10

minus01

01

02

V(r)

03

l = 2

l = 3

l = 4

l = 5

dS (b = 5)

(b)


t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

Φ(t)

minus3

minus2

minus1

0

b = 001

b = 05

b = 1

b = 3

b = 5

(a)

dS (b = 5)

t

0 20 40 60 80 100minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

Φ(t)

minus05

0

l = 2

l = 3

l = 4

l = 5

(b)







Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Im(120596

)

dS 6th WKBl = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8

n = 0

n = 1

n = 2

01

02

03

04

05

06

07

08

09

1

1 2 3 40

Re(120596)






ℎ120583] = exp (minus119894120596119905)((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579))

119875119897 (cos 120579) (21)





+ ( 11989110158402119891 + 31199011015840119901 )1198701015840 + 11987010158401015840 = 0 (24)



(26)


V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




V(r)

r

2 4 6 8 10

AdS-odd (l = 2)

1

2

3

4

5

b = 02

b = 05

b = 1

b = 3

(a)

V(r)

AdS-odd (b = 1)

0

10

20

30

40

50

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

(b)


log Φ

(t)

10minus15

10minus10

10minus5

5 15 2010

t

b = 02

b = 05

b = 1

b = 3

(a)

AdS-odd (b = 1)

log Φ

(t)

10minus10

100

5 15 2010

t

l = 2

l = 3

l = 4

l = 5

(b)


1205941 (119903) = minus2 + 119897 + 1198972 + 211989111990110158402 + 1199012 (2119881 (120601) + 12059811989112060110158402) + 2119901 (11989110158401199011015840 + 211989111990110158401015840)21199012 1205940 (119903) = minus2 + 119897 + 1198972 + 411989111990110158402 + 21199012 (119881 (120601) + 12059811989112060110158402) + 4119901 (11989110158401199011015840 + 211989111990110158401015840)21198911199012 (27)



1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198670 = 119901 (1205941 (119903) 119894120596)1198671 + 1198701015840 minus (11989110158402119891 minus 1199011015840119901)1198701199011015840 (28)



minus (1199031199011015840 (119903) minus 119901 (119903)119891 (119903)2 1199011015840 (119903) 1205962 + 1205943 (119903))119870 = 0(30)

where


+ 21199031205941 (211990111989110158401199011015840 + 1198972 + 119897 + 21199012119881 (120601) minus 2) + 2119901119891101584021199011015840minus 41198911199011199011015840119881 (120601)) (31)


119870 (119903) = 1198601 (119903) Ψ1015840 (119903) + 1198602 (119903) Ψ (119903) (32)





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(33)


minus minus2 + 119897 + 1198972 + 21199012119881 (120601) + 21199011198911015840119901101584021198911199011199011015840 1198670 = 0(34)



(35)


1198892Φ (119903)1198891199032lowast + 1205962 minus 119881119890 (119903)Φ (119903) = 0 (36)



where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where


(38)






V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




V(r)


0

1

2

3

4

5

4 6 82 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

r

0 2 4 6 8 10


00

01

02

03

04

05

06

b = 001

b = 05

b = 1

b = 2

(a)

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

b = 001

b = 05

b = 1

b = 2

Φ(t)

t

0 50 100


minus10

minus8

minus6

minus4

minus2

l = 2

l = 3

l = 4

l = 5

(b)




ℎ120583] = exp(minus119894120596119905ℏ )(((((((

1198670 (119903) 119891 (119903) 1198671 (119903) 0 01198671 (119903) 1198672 (119903)119891 (119903) 0 00 0 119901 (119903)2119870 (119903) 00 0 0 119901 (119903)2119870 (119903) sin2120579

)))))))

119875119897 (cos 120579) (39)


dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




dS (b = 1)

dS (l = 2)

V(r)

00

02

04

06

08

10

12

14

2 4 6 80 1210

r

b = 001

b = 05

b = 1

b = 2

V(r)

0

1

2

3

4

2 4 6 80 1210

r

l = 2

l = 3

l = 4

l = 5

(a)

dS (b = 1)

dS (l = 2)

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

l = 2

l = 3

l = 4

l = 5

Φ(t)

t

0 20 40 60 80 100minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

b = 001

b = 05

b = 1

b = 2

(b)






119877 (119903ℎ) = plusmn 1198941205871205961198911015840 (119903ℎ) (43)


AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




AdS (b = 1)

AdS (l = 2)

V(r)

0

50

100

150

200

2 4 6 80 10

r

l = 2

l = 3

l = 4

l = 5

V(r)

2 4 6 80 10

r

0

50

100

150

200

b = 001

b = 1

b = 2

b = 3

(a)

AdS-even (b = 1)

AdS-even (l = 2)

log Φ

(t)

10minus20

10minus10

5 1510

t

l = 4

l = 5

l = 2

l = 3

10minus20

10minus15

10minus10

10minus5

log Φ

(t)

4 6 82 12 1410

t

b = 001

b = 1

b = 2

b = 3

(b)









TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

0030

0035

50 15 2010

b

Tc

rc = 8

rc = 10

rc = 20

0005

0010

0015

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0

500

1000

1500

2000

50 15 2010

b

TH

rh = 1

rh = 5

rh = 10

0005

0010

0015

0020

0025

50 15 2010

b









Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Appendix







(A5)



Competing Interests


Acknowledgments


References









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Gravitational Quasinormal Modes of Regular Phantom Black …