Thermal behavior of a radially deformed black hole spacetime
TThermal behavior of a radially deformed black hole spacetime
Subhajit Barman ∗ Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India
Sajal Mukherjee † Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Pune-411007, India
In the present article, we study the Hawking effect and the bounds on greybody factor in a spacetimewith radial deformation. This deformation is expected to carry the imprint of a non-Einsteiniantheory of gravity, but shares some of the important characteristics of general relativity (GR). Inparticular, this radial deformation will restore the asymptotic behavior, and also allows for theseparation of the scalar field equation in terms of the angular and radial coordinates — making itsuitable to study the Hawking effect and greybody factors. However, the radial deformation wouldintroduce a change in the locations of the horizon, and therefore, the temperature of the Hawkingeffect naturally alters. In fact, we observe that the deformation parameter has an enhancing effecton both temperature and bounds on the greybody factor, which introduces a useful distinction withthe Kerr spacetime. We discuss these effects elaborately, and broadly study the thermal behaviorof a radially deformed spacetime.
I. INTRODUCTION
The Kerr metric is one of the remarkable findings ofGR [1]. From the weak field to strong field regime, Kerrsolution has passed all tests with flying colors [2–5]. Allof these success stories make a strong case of GR, andeven constrained some of the alternative theories of grav-ity [6, 7]. Moreover, with the gravitational wave (GW)astronomy coming to the fore, these studies emerge withbrighter possibilities [8–11]. Besides these successes ofGR, there are also strong limitations which motivates toseek for alternatives. It is also known that GR fails toexplain both the small and large scale structure of the nature , and therefore, a modification of the theory is re-quired [12, 13].If we attempt to modify GR, it is also bound to hap-pen that the resultant spacetime may loose some of itsuseful properties, such as axis-symmetry or separabilitycondition. Nonetheless, if we assure that these condi-tions are built-in restored, and still aim to modify space-time structure, we may end up with constraining itsmetric functions [14]. The deviation from GR may becoded within these constraints. One of such possibilitiescomes into play if we modify ∆, which in Kerr case is∆ = r − rr s + a , and r s = 2 M with M being the massof the black hole (BH). In the present article, we will beconcerned with this specific example, where we modify ∆by adding a r -dependent term to it [15]. Note that thisdeformation is only radial, and does not effect angulardistribution of the spacetime.The motivation to study a spacetime which mimics aradially deformed Kerr spacetime is two folded. First,it provides a simple yet useful extension of GR, whichis well-grounded with GW data [16]. Therefore, it can ∗ [email protected] † [email protected] be a potential candidate of alternative theories of grav-ity, and studying along this line can be beneficial. Themetric corresponding to these deformed BH spacetimeshas the same asymptotic features as the original onesfrom Einstein gravity. Secondly, the radial deformationmakes it possible for the field equation to be separable interms of the radial and angular coordinates, which pavesthe way for the formulation of semi-classical analysis inthese spacetimes. Furthermore, the horizon structure dif-fers, as the position of the horizon is now changed due tothe introduction of the deformation. Then the effects ofthese deformations will also be felt through the predic-tions of semi-classical gravity, which concerns the hori-zon structure. In this regard, the Hawking effect [17] is amajor arena to venture in, which states that an asymp-totic observer in a BH spacetime will realize a Planckianthermal distribution of particles with temperature pro-portional to the surface gravity of the BH’s event hori-zon. In the deformed BH spacetime, the distortion in thehorizon structure is expected to change its surface grav-ity, which naturally affects the spectrum of the perceivedHawking radiation.Another important thing to note, is that the spectrumof the Hawking effect as should practically be seen by anasymptotic observer is not an absolute blackbody distri-bution, rather it is a greybody distribution. This grey-body distribution is characterized by the transmissioncoefficient through the effective potential of the consid-ered field. Greybody factor also contains the informa-tion regarding different BH parameters. Here also onecan expect prominent effects of the deformation param-eter. However, a straight forward exact estimation ofthese greybody factors is an insuperable job analytically,though one can seek the help of numerical methods [18–25]. Analytically these estimations can be performed inasymptotic frequency regimes [19, 26–33], i.e., for veryhigh or low frequencies of the field wave modes. Onthe other hand, there are methods that deal with takingextremal limit to evaluate these quantities, see [34–36], a r X i v : . [ g r- q c ] F e b or analytically estimating the bounds on these greybodyfactors, see [37–43]. These bounds have the advantageof being predicted in all frequency regimes, including theintermediate frequency regimes, and also for all valuesof the angular momentum quantum number. In partic-ular, we are going to consider a massless minimally cou-pled scalar field in the radially deformed BH spacetime,and estimate these bounds to study the spectrum of theHawking effect with the greybody factors. Especially ourmotivation is understanding the changes caused by theinclusion of the radial deformation parameter in a sta-tionary and rotating BH spacetime.In Section (II), we begin by providing a detailed inves-tigation of the horizon structure in the radially deformedBH spacetime. In Section (III), we consider a masslessminimally coupled scalar field in this radially deformedBH spacetime, and obtain the scalar field equation ofmotion. Decomposition of the scalar field in terms of thespheriodal harmonics provides one with a Schr¨odingerwave like equation, namely the Teukolsky equation forstationary Kerr BHs, in terms of the radial tortiose co-ordinate. In particular, from this equation the structureof the effective potential can be perceived. Subsequently,in Section (IV), a study of the Hawking effect and thecorresponding temperature and spectrum of the Hawk-ing quanta are provided. Furthermore, in Section (V),we study the bounds on the greybody factors in these ra-dially deformed BH spacetimes considering the effectivepotential from Section (III). We conclude our analysiswith a discussion in Section (VI).
II. HORIZON STRUCTURE OF A RADIALLYDEFORMED SPACETIME
We start with the following Kerr metric written in amore generic form [15]: ds = − N ( r, θ ) − W ( r, θ ) sin θK ( r, θ ) dt − rW ( r, θ ) sin θdtdφ + r K ( r, θ ) sin θdφ + σ ( r, θ ) (cid:18) B ( r, θ ) N ( r, θ ) dr + r dθ (cid:19) , (1)with, N ( r, θ ) = ∆ r , K ( r, θ ) = Σ r σ ( r, θ ) ,W ( r, θ ) = ar s ( r + a cos θ ) − ,B ( r, θ ) = 1 , Σ = ( r + a ) − ∆ a sin θ ,σ ( r, θ ) = r − ( r + a cos θ ) , ∆ = r − rr s + a . (2)In the above expressions, r s = 2 M , where M denotesthe mass of the BH, and a is the angular momentumper unit mass. In order to inject the radial deforma-tion, we use the substitution r s → r s + η/ ( r ). Thissubstitution would not change any of the built-in prop-erties of the spacetime including the separability con- dition of the Klein-Gordon equation. The only differ-ence that distinguishes the deformed spacetime from theKerr metric of Eq. (1), is in ∆, which now becomes,¯∆ = r − rr s + a − η/r , and also Σ changes to ¯Σ with∆ is replaced by ¯∆ in the expressions of Eq. (2).Due to the presence of deformation parameter η , thelocations of the horizons, given as N ( r, θ ) = 0, woulddiffer from the usual Kerr case. Moreover, as η is clearlycoupled with r , new solutions may also emerge. To bespecific, the locations of the horizons are obtained from, r − r r s + a r − η = 0 , (3)and we will apply Descartes’ sign rule to estimate thenumber of solution(s). Note that with η <
0, there isno positive solution for the above case, and the nakedsingularity always exists. With η >
0, Eq. (3) can eitherhave one or three positive solution(s). In the later case,the event horizon, r H and these inner horizons, r and r (assume r < r ), can be expressed in terms of the otherBH parameters as r H = r s − (3 a − r s )3 A / + A / × ,r = r s i √ a − r s )3 × A / − (1 − i √ A / × ,r = r s − i √ a − r s )3 × A / − (1 + i √ A / × , (4)where, A = β + 3 √ β , with the expression of β and β given by β = − a r s + 2 r s + 27 η and β = (cid:0) a − a r s − a r s η + 4 r s η + 27 η (cid:1) / .We may now employ the above expressions to havea deeper understanding about the horizon structure inpresence of η . Based on the properties of A , whether itis positive, negative or complex, we encounter differentoutcomes. In case of (a) A > r H is always positive anddescribe the event horizon, while r and r are complexconjugate to each others. For (b) A <
0, say A = − α where α >
0, Eq. (4) becomes: r H = r s − / exp( − iπ/ a − r s )3 α / + α / exp( iπ/ × / ,r = r s / (3 a − r s )3 × α / − α / × / ,r = r s − / exp( iπ/ a − r s )3 α / + α / exp( − iπ/ × / , (5)which now readily gives r H and r are complex conjugateto each other, and r now becomes the outer horizon. In-terestingly, if we use the η = 0 limit, and the extremalitycondition a = r s /
2, we gather α = − r s /
4. The aboveequations now gives r H = r = r s / r = 0, which isthe usual Kerr case. This serves as an useful validationof our solution. Now we consider the last case, i.e., (c) A = β + i √ (cid:112) − β . In this case, we further simplify A as, A = 2( − a + r s ) / exp( iα ), where tan α = (cid:8) √ − a + a r s + 18 a r s η − ηr s − η ) (cid:9) / ( − a r s + 2 r s + 27 η ) . (6)As it can be shown that in this case, the imaginary partof all of the above expressions would identically vanish,and r H continue to be the event horizon.To summarize, the presence of η manifests an addi-tional horizon other than the event and Cauchy horizon.Except for the A < r H and r become zero), r H continue to be the event hori-zon, where r and r are the inner horizons. For a clearexposition of this horizon structure in presence of η , weillustrate the horizon structure in Fig. (1). . . . A n g u l a r m o m e n t u m a Locations of the horizon η = 0 η = 0 . η = 0 . FIG. 1. In the above figure, we demonstrate the contours ofthe horizon structure in a radially deformed spacetime. De-pending on the deformation parameter η , the plots changeaccordingly and can have maximum three positive real solu-tions. Beside the inner and outer horizon, the presence of η introduces another real root which always exists. This wouldindicate that for no ranges of BH parameters the singularitycan be naked. Here we set r s = 1. The other intrinsic property that is associated with arotating spacetime is known as the ergoregion, where noobserver can be kept stand still. This is given by thecondition g tt = 0, which according to Eq. (1) becomes N = W sin θ . Ergoregion is closely related with framedragging and zero angular momentum observer (ZAMO) ,both are well known relativistic effects [44]. Given thatZAMO is relevant for the present purpose, we shall brieflydiscuss the same as follows. For the present spacetime,the angular velocity of ZAMO at a given ( r, θ ) isΩ = g tφ g tt = a ( r r s + η ) r ¯Σ . (7)If one considers the maximally extended case of θ = 0 or θ = π for the expression of this angular velocity, thenΩ = Ω m ( r ) = a ( r + a ) (cid:0) r + a − ¯∆ (cid:1) , (8) which becomes Ω H = a/ ( r H + a ), on the event horizon.Now we move on to understand the nature of Ω m asa function of r , which we will see to be appearing in thescalar field equation of motion in the next section. In or-der to prove that Ω m is a monotonic function in r outsidethe event horizon, which is often required in the contextof greybody factor, we may recall the case with η = 0first. It can be shown that in the Kerr case, Ω m hasa peak at r = a/ √
3, which always lies inside the outerevent horizon and henceforth, the function is monotoni-cally decreasing outside the horizon. Unfortunately, withthe η being present, the analysis becomes more involvedand a straightforward solution to r turns out to be un-likely. However, we carry out an approximate analysis toincorporate the effects of η up to the linear order terms.We note that the peak of Ω m as r varies, located at r peak ,has now become r peak = a/ √ − aη √ a r s + 5 η ) + O ( η ) , (9)which shifts close to the singularity compared to the Kerrcase. Similarly, we can find that the expression of thelocation of outer horizon changes as: r H | η (cid:28) = r s (cid:112) r s − a (cid:16) η/ (cid:112) r s − a (cid:17)(cid:16) r s + (cid:112) r s − a (cid:17) + O ( η ) . (10)The above equation clearly states that the outer hori-zon shifts away from the singularity. Therefore, we con-clude that the addition of η will shift the outer horizonaway from singularity, and move the angular velocity’speak close to the singularity. This essentially assuresthat outside the outer horizon, Ω m is montonic in r . Foran illustration of this incident, we plot Ω m for variousvalues of η in Fig. (2). As it can be realized, with η (cid:54) = 0,say η = 10 − , the maxima in Ω m is covered within theouter event horizon. Larger the value of η , the peak getsshifted close to the singularity, and the outer horizonshifts away from singularity. This indicates that for anynonzero value of η , the monotonicity of Ω m is retainedoutside the event horizon. III. THE SCALAR FIELD EQUATION OFMOTION
In this deformed spacetime we consider a massless min-imally coupled free scalar field Φ( x ) described by the ac-tion S Φ = (cid:90) d x (cid:20) − √− gg µν ∂ µ Φ( x ) ∂ ν Φ( x ) (cid:21) . (11)The variation of this action with respect to the scalar fieldΦ provides one with the scalar filed equation of motion (cid:3) Φ( x ) 1 √− g ∂ µ (cid:16) √− gg µν ∂ ν Φ( x ) (cid:17) = 0 . (12) . . . A n g u l a r v e l o c i t y Ω m ( r ) Radial distance η = 0 η = 0 . η = 0 .
005 1
FIG. 2. The above figure demonstrates the variation of theangular velocity (given by the curves) with the radial distance,and the vertical lines denote the locations of the event hori-zons. For different values of η , it can be shown that the peaksof Ω m always reside inside the event horizon. Therefore, out-side the horizon, angular velocity of ZAMO is a monotonicallydecreasing function of r . We take, a = r s / . By substituting the metric components from Eq. (1) andEq. (2) with the expressions of ¯∆ and ¯Σ corresponding tothe deformed spacetime, the equation of motion becomes − ¯Σ sin θ ¯∆ ∂ t Φ + ¯∆ − a sin θ ¯∆ sin θ ∂ φ Φ − a sin θ ¯∆ (cid:0) r + a − ¯∆ (cid:1) ∂ t ∂ φ Φ + ∂ r (cid:0) ¯∆ sin θ∂ r Φ (cid:1) + ∂ θ (sin θ∂ θ Φ) = 0 . (13)Like the Kerr BH here also the metric components areindependent of time t and azimuthal angle φ , which sug-gests a field decomposition of the form Φ( t, r, θ, φ ) =exp( − iωt + imφ ) R ( r ) S ( θ ) / √ r + a , where S ( θ ) denotesspheroidal harmonics. Then using this field decompo-sition and the tortoise coordinate r (cid:63) , defined from theexpression dr (cid:63) = r + a ¯ (cid:52) dr , (14)the previous scalar field equation of motion from Eq. (13)can be expressed as ∂ r ∗ R + (cid:104) ω − V ( r ) (cid:105) R = 0 . (15)In this deformed geometry, one may express the effectivepotential V ( r ) as V ( r ) = ¯∆( r + a ) (cid:110) A ωlm + ( r ¯∆) (cid:48) r + a − r ( r + a ) (cid:111) − a m ( r + a ) + 2 maω ( r + a ) ( r + a − ¯∆) , (16)where, A ωlm denotes the eigenvalue corresponding to thespheroidal harmonics equation. In slow rotation limitthis eigenvalue can be expressed as A ωlm = l ( l + 1) − m aω + O (cid:104) ( aω ) (cid:105) . (17) This Schr¨odinger wave like equation of Eq. (15) in de-formed BH spacetime resembles the Teukolsky equation[45] from general Kerr BH spacetime. It should be notedthat the from this equation the Regge-Wheeler [40, 46]like equation corresponding to a deformed Schwarzschildspacetime can be obtained quite easily making a = 0.In this non-rotating limit, i.e., a = 0, the expression ofpotential V ( r ) from Eq. (16) becomes V ( r ) = (1 − r s /r − η/r ) (cid:26) l ( l + 1) r + r s r + 3 ηr (cid:27) . (18)Here we have used the fact that when a = 0 the eigen-value A ωlm of spheroidal harmonics equation signifiesspherical harmonics with expression A lm = l ( l + 1). Inthe geometrical optics limit, i.e., l (cid:29)
1, we gather thatthe above equation encounters an maxima at the photonorbit, r ph , which is a solution of the equation:2 r − r s r − η = 0 . (19)With η = 0, we arrive ar r ph = 3 r s /
2, which is the lo-cation of photon orbit in Schwarzschild BH [47, 48]. InFig. (3), we depict the potential V ( r ) from Eq. (16) fordifferent values of the parameter η . As it can be evidentthat there exist a maxima, r peak , in each of these figures,which shifts based on the values of η . .
510 1 1 . . . . P o t e n t i a l V ( r ) Radial distance η = 0 η = 0 . η = 0 . FIG. 3. The above figure illustrates the scalar field potential V ( r ) from Eq. (16) for different values of η . We set the otherparameters at a = 0 . l = 2, ω = 3, m = 0, r s = 1. In passing, we should also touch upon an importantconsequence of the above potential. It can be sensed fromFig. (3) that in either side of the peak (at r = r peak ), thefunction V ( r ) is monotonic. In particular, from the eventhorizon r H to r peak , it is monotonically increasing, andfrom r peak to ∞ , it is monotonically decreasing. It issimilar to saying that there will be a single peak outsidethe event horizon. In the presence of both a and η , theconcern may arise weather this feature remains intact ornot. Consequently, even with η set to zero, it is unlikelyto guess from Eq. (16) that the potential will have a sin-gle peak outside the event horizon. Therefore, an ana-lytical proof is beyond expectation. What we found thatwithin the weak rotation approximation ( aω (cid:28)
1) [41],this property is always valid for a wide range of variousBH parameters, and different modes.
IV. HAWKING EFFECT IN RADIALLYDEFORMED SPACETIME
In the original work [17], the thermal nature of theHawking effect is realized through the usage of Bogoli-ubov transformation between the ingoing and outgoingfield modes described in terms of the null coordinates.The Hawking effect can be realized through other variousmeans, like using tunnelling formalism [49–55], path inte-gral approach [56], conformal symmetry [57], via anoma-lies [58–60], canonical formulation [61–64], and as an ef-fect of near horizon local instability [65–67]. However,the conclusion remains the same, i.e., an asymptotic ob-server will perceive the BH horizon with some tempera-ture proportional to its surface gravity. In particular, inthe case of a deformed BH spacetime, the number den-sity of the Hawking quanta perceived by an asymptoticobserver with frequency ω and angular momentum quan-tum number m will be given by N ω = Γ( ω ) e π ( ω − m Ω H ) /κ H − , (20)where, κ H , Ω H , and Γ( ω ) denote the surface gravity,angular velocity at the event horizon, and the grey-body factor respectively. For modes with m Ω H > ω this expression gives rise to the so called super-radiancephenomenon [68]. From the Planckian distribution ofEq. (20) the characteristic temperature corresponding tothe Hawking effect is T H = κ H / π . In a radially de-formed BH spacetime (with r s → r s + η/r deformation)the surface gravity at the outer horizon r = r H , whichgives the temperature of the horizon, can be found outto be κ H = ( r H − r )( r H − r )2 r H ( r H + a ) , (21)the evaluation of which is given in Appendix A. The in-ner horizons, r and r can exist if they are real or ceaseto exist if they become imaginary for certain values of η .From Eq. (4), Eq. (5) it is clear that the position of thehorizon now has a signature of the deformation whichwill also be apparent in the spectrum of the Hawkingeffect. Another quantity is the angular velocity at theouter horizon from Eq. (7), which will also carry the sig-nature of the deformation in the spectrum of Eq. (20).In our following discussion we study the nature of sur-face gravity and angular velocity at the outer horizon, inparticular, we observe how it changes with varying η .In Fig. (4) we have plotted the surface gravity cor-responding to the outer horizon r = r H in a deformedspacetime with respect to varying η . The figure is ob-tained considering the other BH parameters to be r s = 1and a = 0 .
45. In the same figure the zero η situation is . . . . .
45 0 0 .
02 0 .
04 0 .
06 0 .
08 0 . s u r f a ce g r a v i t y η κ DK κ Kerr FIG. 4. In the above figure the event horizon’s surface gravity κ H in a deformed BH spacetime, denoted by κ DK , is plottedwith respect to η . We have set the other parameters at a =0 . r s = 1. The plot also depicts the case when η = 0, i.e.,the Kerr BH scenario, denoted by κ Kerr . . . . . . . .
55 0 0 . . . . . s u r f a ce g r a v i t y aη = 0 η = 0 . η = 0 . η = 0 . FIG. 5. In the above figure the surface gravity in a deformedspacetime is plotted with respect to varying a with different η . We have set the other parameter at r s = 1. also depicted by a dash-dotted line. From this figure itcan be observed that as the value of the deformation η increases the value of the horizon’s surface gravity alsoincreases, signifying an increase in the characteristic tem-perature of the Hawking effect.In Fig. (5), we have plotted the event horizon’s sur-face gravity, which gives the Hawking temperature, withrespect to varying angular momentum parameter a fordifferent values of the deformation parameter η . Fromthis figure, we note that as the value of η increases, sur-face gravity departs further from the Kerr case and alsoincreases corresponding to different fixed values of a .Similarly, in Fig. (6), we demonstrate how the angu-lar velocity of the event horizon changes with deforma-tion parameter. Note that Ω H is implicitly affected by η through the value of r H . . . .
75 0 0 .
05 0 . Ω H η η = 0 . η = 0 . FIG. 6. Angular velocity of the horizon is shown for a = 0 . r s = 1. We retrieve the kerr case for η = 0 limit, whichis shown as the curve constant along x -axis. V. THE BOUNDS ON THE GREYBODYFACTOR
The spectrum of the Hawking effect Eq. (20) is givenby a greybody distribution as perceived by an asymp-totic future observer. The greybody factor arises fromthe transmission amplitude of the field modes throughthe effective potential outside the horizon, for the modesnearly escaping the formation of the horizon and travel-ling from the near-horizon region to an asymptotic ob-server. In this section we estimate the bounds on thegreybody factor, see [37, 39, 41], which can be analyti-cally expressed in all frequency range, for certain generalfield momenta and general spacetime dimensionalities.These bounds on the greybody factor can be expressedas Γ( ω ) ≥ sech Θ , (22)whereΘ = (cid:90) ∞−∞ (cid:112) ( h (cid:48) ) + ( ω − V ( r ) − h ) h dr (cid:63) . (23)Here, V ( r ) denotes the effective potential correspondingto a massless minimally coupled scalar field and ω cor-responds to the frequency of field mode. Furthermore, h ≡ h ( r (cid:63) ) is some positive function satisfying the condi-tion h ( −∞ ) = h ( ∞ ) = ω . The tortoise coordinate r (cid:63) isobtained from the expression of Eq. (14). From Eq. (16)one can obtain the effective potential correspond to amassless minimally coupled free scalar field in a deformedBH spacetime and express ω − V ( r ) in another conve-nient form as ω − V ( r ) = ( ω − m Ω m ) + U ( r ) = ( ω − m Ω m ) + m a ¯ (cid:52) ( r + a ) (cid:16) ¯ (cid:52) + ηr n +1 + 2 rr s (cid:17) − ¯ (cid:52) ( r + a ) (cid:110) A ωlm + ( r ¯ (cid:52) ) (cid:48) ( r + a ) − (cid:52) r ( r + a ) (cid:111) . (24) Now one can consider the simplest choice of the pos-itive function h ( r (cid:63) ) = ω , as also done in [37] forSchwarzschild BHs. However, it will only be fruitful forthe case of m = 0, as the first quantity from Eq. (24) con-tributes to a diverging term in the integration of Eq. (23)for the calculation of the bound for m (cid:54) = 0. We thenconsider the evaluation of these bounds on the greybodyfactor in a case by case manner as done in [41]. In par-ticular, in separate situations for the angular momentumquantum number m = 0 and m (cid:54) = 0. A. The case of m=0 :
This particular case of m = 0 is the simplest and pro-vides an overall picture of the bounds on the greybodyfactor. In this case we are going to consider the positivefunction to be h = ω , i.e., h (cid:48) = 0. Now as one makesthe choice of m = 0 in the expression of the scalar fieldeffective potential from Eq. (24), one gets ω − V = ω − ¯ (cid:52) ( r + a ) (cid:20) A ωl + ( r ¯ (cid:52) ) (cid:48) ( r + a ) − (cid:52) r ( r + a ) (cid:21) . (25)Then with this expression of the potential and the abovechoice of the positive function h one can represent thebound on the greybody factor to beΓ( ω ) ≥ sech (cid:18) I ωl ω (cid:19) , (26)where, I ωl = I ωlm ( m = 0) and the quantity I ωlm in gen-eral is defined by the integral I ωlm = (cid:90) ∞ r H dr (cid:20) A ωlm r + a + ( r ¯ (cid:52) ) (cid:48) ( r + a ) − (cid:52) r ( r + a ) (cid:21) . (27)One can evaluate this integral by step by step carrying . . .
02 0 .
04 0 .
06 0 .
08 0 . B o und o n g r e y b o d y f a c t o r η ω = 1 ω = 1 . ω = 1 . ω = 1 . FIG. 7. In the above figure the lower bound on the greybodyfactor in a deformed spacetime for m = 0 is plotted withrespect to varying η for different fixed ω . We have set theother parameters at l = 1, r s = 1, and a = 0 .
05. We note thatthe quantity aω (cid:28) . . .
751 0 1 2 3 B o und o n g r e y b o d y f a c t o r ω η = 0 η = 0 . η = 0 . η = 0 . FIG. 8. In the above figure the lower bound on the greybodyfactor in a deformed BH spacetime for m = 0 is plotted withrespect to varying ω for different fixed η . We have set theother parameters at l = 1, r s = 1, and a = 0 . out all of its components which we describe in AppendixB. Utilizing these expressions, we obtain the integral ofEq. (27) for the case of m = 0 as I ωl = A ωl I + a I + r s I − a r s I + 3 η I , (28)where the expressions for I , I , I , and I are givenin Appendix B. With this expression of the integral I ωl from Eq. (28) and putting it in Eq. (26) one can find outthe bound on the greybody factor in a deformed space-time for m = 0. We have further plotted this boundin Fig. (7) for varying η with different fixed ω , and inFig. (8) for varying ω with different fixed η . From Fig. (7)it can be observed that as the value of the deformationparameter increases the bound on the greybody factoralso increases. However, it never goes beyond the upperlimit 1. The bound also increases with increasing fre-quency ω of the wave mode, which can be observed fromboth Fig. (7) and Fig. (8). B. The case of m (cid:54) = 0 As already observed from Fig. (2) and the discussionsrelated to Eq. (9) and Eq. (10), we consider the angularvelocity of ZAMO to be monotonic, in particular mono-tonic decreasing, in the entire range between the outerhorizon and the asymptotic infinity. We are going toconsider this property and find out the bounds on thegreybody factor. Because of this feature then one canalso consider the form of the positive function h in termsof Ω m ( r ), as done in [41], which can be made monotonicin the region between the event horizon and the asymp-totic infinity. With this consideration from Eq. (22) thebound on the greybody factor becomes Γ( ω ) ≥ sech (cid:110) (cid:82) ∞−∞ dr (cid:63) | h (cid:48) | h + (cid:82) ∞−∞ dr (cid:63) | ω − V − h | h (cid:111) , (29)where, we have sought the help of the triangle inequality to express the numerator of Eq. (23). This expression further simplifies to Γ( ω ) ≥ sech (cid:110) (cid:12)(cid:12)(cid:12) ln (cid:104) h ( ∞ ) h ( −∞ ) (cid:105)(cid:12)(cid:12)(cid:12) + (cid:82) ∞−∞ dr (cid:63) | ω − V − h | h (cid:111) . (30)We consider Ω H ≡ Ω( r H ) to be the angular velocityat the event horizon. In regard to the monotonicity ofΩ m ( r ), it should be noted that Ω m ( r ) is always smallerthan Ω H in the region outside the outer horizon, whichcan be used for the monotonicity of h . Furthermore, thismonotonicity can be achieved in two different regions,namely in the non super-radiant regimes ω > m Ω H or m < m (cid:63) with m (cid:63) = ω/ Ω H , and for the case of the super-radiant modes of m > m (cid:63) . In the following discussion weshall consider the case with the non super-radiant andsuper-radiant modes separately.
1. Non super-radiant modes m < m (cid:63)
First we consider the non super-radiant modes, i.e., m < m (cid:63) , where we can have another two possibilitieseither m < m ∈ (0 , m (cid:63) ). The function h in bothcases are considered to be h = ω − m Ω m ( r ), which ispositive. Let us check for the first quantity of Eq. (30)in these two scenarios. In particular we have h ( ∞ ) h ( −∞ ) = 11 − m Ω H ω < , when, m < > , when, m ∈ (0 , m (cid:63) ) . (31)On the other hand, using the expression of ( ω − V ( r ))from Eq. (24) the second quantity of Eq. (30) becomes12 (cid:90) ∞−∞ dr (cid:63) | ω − V ( r ) − h | h = 12 (cid:90) ∞−∞ dr |U ( r ) | ω − m Ω m ( r ) > (cid:90) ∞ r H dr |U ( r ) | ω − m Ω H , when m < > (cid:90) ∞ r H dr |U ( r ) | ω , when m ∈ (0 , m (cid:63) ) , (32)where, the expression U ( r ) = U ( r )( r + a ) / ¯∆, and U ( r )is obtained from Eq. (24). Then for the scenario m < m < m (cid:63) , the bound onthe greybody factor from Eq. (30) can be obtained usingthe outcomes of Eq. (31) and Eq. (32) asΓ( ω ) ≥ sech (cid:26)
12 ln (cid:18) − m Ω H ω (cid:19) + I ωlm ω − m Ω H ) − m a ω − m Ω H ) (cid:0) I + r s I + η I (cid:1)(cid:27) . (33)On the other hand, for the scenario m ∈ (0 , m (cid:63) ) ofthe non super-radiant modes the bound on the greybody . . . . . .
02 0 .
04 0 .
06 0 .
08 0 . B o und o n g r e y b o d y f a c t o r η ω = 1 ω = 1 . ω = 1 . ω = 1 . FIG. 9. In the above figure the lower bound on the greybodyfactor in a radially deformed BH spacetime is plotted withrespect to varying η for different fixed ω . We have set theother parameters at l = 2, m = − r s = 1, and a = 0 . m (cid:54) = 0. factor from Eq. (30) isΓ( ω ) ≥ sech (cid:26) −
12 ln (cid:18) − m Ω H ω (cid:19) + I ωlm ω − m a ω (cid:0) I + r s I + η I (cid:1)(cid:27) , (34)where, the expression of I ωlm is given in Eq. (27). The ex-plicit evaluation of this quantity I ωlm and the other quan-tities I , I , and I can be obtained from the evaluatedintegrals of Appendix B(in particular from Eq. (B.1)).Now it can be noticed that one can get the bound on thegreybody factor for m = 0 case(the expression obtainedby putting the result of Eq. (28) in Eq. (26)) from thesebounds of Eq. (33) and Eq. (34) by simply making m = 0in these expressions. Here the contribution of non zero m comes through specific two quantities in the bound, andthey also contain the effect of the deformation parameter η for m (cid:54) = 0. In Fig. (9) we have plotted this bound withrespect to varying η for fixed parameters l = 2, m = − a = 0 .
05 and different values of the frequency ω . Notehere we have kept the quantity aω (cid:28)
2. super-radiant modes m ≥ m (cid:63) The super-radiant modes are those for which m Ω H > ω or m ≥ m (cid:63) . In this scenario the expression of the boundon the greybody factor from Eq. (30) is further simplifiedusing the triangle inequality and is expressed as Γ( ω ) ≥ sech (cid:110)(cid:82) ∞−∞ (cid:104) | h (cid:48) | h + U ( r )2 h + | ( h − ω − m Ω m ) ) | h (cid:105) dr (cid:63) (cid:111) , (35)where, U ( r ) is obtained from Eq. (24). One can carry out this integral by considering two regions m ∈ [ m (cid:63) , m (cid:63) )and m ∈ [2 m (cid:63) , ∞ ), where the value of h can be taken tobe positive and with a reasonable boundary condition,see [41]. Let us discuss these two situations in a case bycase manner. Case I ( m ∈ [ m (cid:63) , m (cid:63) )): In this case we consider thethe function to be h ( r ) = max { ω − m Ω m , m Ω H − ω } ,which satisfies the requirement for h ( r ) to be positiveand also its asymptotic behaviors, see [41]. Then thefirst quantity of the right hand side of Eq. (35) can beevaluated to be (cid:90) ∞−∞ | h (cid:48) | h dr (cid:63) = | ln h ( r ) | ∞ r H = ln (cid:18) ωm Ω H − ω (cid:19) . (36)The second quantity in the same equation becomes (cid:90) ∞−∞ U ( r )2 h dr (cid:63) ≤ (cid:90) ∞−∞ U ( r )2( m Ω H − ω ) dr = I lm m Ω H − ω ) , (37)and the third quantity (cid:90) ∞−∞ | h − ( ω − m Ω m ) ) | h dr (cid:63) = J m , ( say )= (cid:90) r −∞ | ( m Ω H − ω ) − ( ω − m Ω m ) ) | m Ω H − ω ) dr (cid:63) , (38)where, r is obtained from equation ω − m Ω m ( r ) = m Ω H − ω . Case II ( m ∈ [2 m (cid:63) , ∞ )): In this case the function canbe chosen to be h ( r ) = max { m Ω m − ω, ω } , which satis-fies the requirement for h ( r ) to be positive and also itsasymptotic behaviors, see [41]. Then the first quantityof the right hand side of Eq. (35) can be evaluated to be (cid:90) ∞−∞ | h (cid:48) | h dr (cid:63) = | ln h ( r ) | ∞ r H = ln (cid:18) m Ω H − ωω (cid:19) . (39)The second quantity in the same equation becomes (cid:90) ∞−∞ U ( r )2 h dr (cid:63) ≤ (cid:90) ∞−∞ U ( r )2 ω dr = I lm ω , (40)and the third quantity (cid:90) ∞−∞ | h − ( ω − m Ω m ) ) | h dr (cid:63) = J m , ( say )= (cid:90) ∞ r (cid:48) | ω − ( ω − m Ω m ) ) | m Ω H − ω ) dr (cid:63) , (41)where, r (cid:48) is obtained from equation m Ω m ( r (cid:48) ) − ω = ω .In these two cases also major contributions fromEq. (37) and Eq. (40), in the bound on the greybodyfactor, are analogous to the results obtained from nonsuper-radiant and m = 0 cases. However, here the ef-fects of deformation also come from the first and thirdquantities of the integral in Eq. (35). One may choose asuitable range of parameter values and depict this casealso in a figure by plotting the lower bound on Γ( ω ) withrespect to η . However, that plot does not impart anynew information. C. The bound on the greybody factor when a = 0 The case of a = 0 is particularly significant as itdenotes the radially deformed static Schwarzschild BHspacetime. In this case one does not need to considerthe intricacies coming from the super-radiant frequency,and the contribution from different angular momentumquantum number m becomes irrelevant in the calculationof the bound on the greybody factor. Then one can con-sider the expressions provided for m = 0 in Eq. (26) withEq. (27) for the evaluation of these bounds. . . . . .
02 0 .
04 0 .
06 0 .
08 0 . B o und o n g r e y b o d y f a c t o r η ω = 1 ω = 1 . ω = 1 . ω = 1 . FIG. 10. In the above figure the lower bound on the greybodyfactor in a radially deformed Schwarzschild BH spacetime isplotted with respect to varying η for different fixed ω . Wehave set the other parameters at l = 1, and r s = 1. In particular, the expression of Eq. (27) in a = 0 casecan be evaluated to provide I ωlm = I lm = (cid:90) ∞ r H dr (cid:20) l ( l + 1) r + r s r + 3 ηr (cid:21) = l ( l + 1) r H + r s r H + 3 η r H , (42)where, we have used the fact that in the a = 0 case A ωlm denotes the eigenvalue corresponding to the Spher-ical harmonics equation, i.e., A ωlm | a =0 = l ( l + 1), whichcan also be seen from Eq. (17). Then the bound on thegreybody factor can be estimated using Eq. (26) asΓ( ω ) ≥ sech (cid:26) ωr H (cid:18) l ( l + 1) + r s r H + 3 η r H (cid:19)(cid:27) . (43)This ensures that at very high frequency the boundhas the formΓ( ω ) ≥ − (cid:16) l ( l + 1) + r s r H + η r H (cid:17) (2 ωr H ) + O (cid:20) ω (cid:21) , (44)in agreement with the result from Born approximation[69]. In Fig. (10) we have plotted the bound of Eq. (43)and observed that in this radially deformed SchwarzschildBH spacetime also the value of the bound increases withincreasing value of the deformation parameter and thefrequency of the wave mode. VI. CONCLUSION
In this article, we have discussed the thermal behaviorof a BH spacetime which is deformed from the Kerr so-lution. We have mentioned that this deformation is onlyradial and therefore, the separability of scalar field waveequation and asymptotic properties remain the same asthe Kerr spacetime. However, the position of the hori-zons and near horizon geometry differs from the Kerrcase. This, in fact, motivated us to pursue the analy-sis concerning the Hawking radiation and bounds on thegreybody factors in this spacetime.The addition of the deformation parameter η makesthe horizon equation a cubic one, and there can be eitherone or three real positive solution(s), as seen in Section(II). In this deformed BH spacetime, we observed thatother than the event horizon, i.e., the outer horizon pro-vided by the largest positive real root, there are othertwo inner horizons. This would make sure that the sin-gularity is never naked, as it is covered by at least onehorizon all the time. This is in stark contrast with KerrBH case, where the extremality condition exists, and noreal positive root of the horizon equation can be foundfor a > r s / η could change the thermal behaviorof a BH solution. In Fig. (4) and Fig. (5), the event hori-zon’s surface gravity of this deformed BH spacetime isdepicted for varying η , which imply an increasing Hawk-ing temperature with increasing deformation parameter.In order to describe the bounds on the greybody fac-tors in presence of η , which we discussed in Section (V),we closely followed the analytical treatment as given in[37, 39, 41]. Note that these bounds are the lower bounds,while the upper bound is always 1. In the case wherebound becomes 1, all the modes can escape to infin-ity unaltered, while the case of the lower bound near-ing zero signify that much smaller amount of the wavemodes transmit to the infinity. In Section (A), we con-sidered the m = 0 case, and the bounds shown in Fig. (7)and Fig. (8) imply an increase in the greybody factor forincreasing deformation parameter η or wave mode fre-quency ω . We observed in m (cid:54) = 0 case, a significant con-tribution in the bound coming from an integral Eq. (27)similar to the m = 0 case, which keeps the dependenceof the bounds on η also same here, see Fig. (9). There-fore, it is apparent that the radial deformation enhancesthe transmission probability for the modes to travel toinfinity, which is also evident from Fig. (3). From thisfigure, we observed the height of the effective potential V ( r ) decreases with increasing the deformation parame-ter η , signifying a greater possibility for the field modes totransmit through the potential barrier. Moreover, fromEq. (20), it can be observed that, at least for the m = 0case when the horizon’s angular velocity does not con-tribute, the number density of Hawking quanta corre-sponding to a certain wave mode of frequency ω in adeformed BH spacetime will be higher compared to the0Kerr case. Finally, in the last part of Section (V), wefound that for the Schwarzschild case also the bound onthe greybody factor increases with increasing η .In passing, we would like to remind that a more generaldeformation can be found if we use r s → r s + η/r n +1 with n ≥
1, which also provides a spacetime where the fieldequation of motion is separable with respect to the radialand angular coordinates. Note that based on the valueof n , different horizon structure emerges, and we end upwith different number of horizons. However, as far asthe thermal behavior is concerned, we are doubtful howmuch impact does n can cause on the overall numbers.This is because as n increases, the spacetime becomesmore Kerr-like, and η loses its essence. Therefore, thedominant contribution from η would only come in the n = 1 case, while for all n >
1, the contribution fromthe deformation becomes dimmer. However, it remainsan interesting arena to venture further.
ACKNOWLEDGEMENT
The authors acknowledge Bibhas Ranjan Majhi andGolam Mortuza Hossain for useful discussions concern-ing the current topic. S.M. wishes to thank Departmentof Science and Technology (DST), Government of India,and S.B. thanks Indian Institute of Technology Guwahati(IIT Guwahati) for financial support.
Appendix A: Surface gravity at the outer horizon
In this part of the appendix we estimate the surfacegravity at the outer horizon of the deformed BH. Onecan construct a vector null at the outer horizon [70] as ξ µ = t µ + Ω H φ µ , (A.1)where, Ω H is the angular velocity of the outer horizon.The surface gravity κ H at the horizon can be obtainedfrom the expression κ ξ µ = 12 ( − ξ ν ξ ν ) ; µ . (A.2)In deformed spacetime this quantity ξ ν ξ ν can be evalu-ated to be represented in terms of a simplified form givenby ξ ν ξ ν = ¯Σ sin θρ (Ω H − Ω) − ρ ¯∆¯Σ , (A.3)where, the different quantities are given by¯Σ = ( r + a ) − a ¯∆ sin θ , and, ¯∆ = ( r + a − rr s ) − ηr , (A.4) and the expression of Ω, which is the angular velocityof ZAMO, is taken from Eq. (7). Now in Eq. (A.3) thequantities (Ω H − Ω) and ¯∆ both vanishes at the outerhorizon. In particular, one can express the quantity ¯∆ =( r − r H )( r − r )( r − r ). Then the contribution of it inthe surface gravity will be ( − ξ ν ξ ν ) ; µ = (cid:16) ρ ¯∆ ,r ¯Σ (cid:17) (cid:12)(cid:12)(cid:12) r = r H ∂ µ r = (cid:16) ρ ( r − r )( r − r )¯Σ (cid:17) (cid:12)(cid:12)(cid:12) r = r H ∂ µ r . (A.5)Furthermore, the vector ξ µ can also be cast into the form ξ µ = (cid:2) ρ / ( r + a ) (cid:3) | r = r H ∂ µ r at the outer horizon. Thenthe surface gravity at the outer horizon can be readilyfound from the Eq. (A.2) with the help of Eq. (A.5) as κ = ( r H − r )( r H − r )2 r H ( r H + a ) . (A.6) Appendix B: Evaluation of the integrals necessaryfor estimating the bounds of Section (V)
In this part of the Appendix we evaluate the necessaryintegrals, which are used in Eq. (28) for the estimationof the greybody factor, and provide their explicit expres-sions. I = (cid:90) ∞ r H drr + a = π − − (cid:0) r H a (cid:1) a I = (cid:90) ∞ r H dr ( r + a ) = π − − (cid:0) r H a (cid:1) a − ar H a ( r H + a ) I p = (cid:90) ∞ r H r dr ( r + a ) p = (cid:0) r H + a (cid:1) − p p − I p = (cid:90) ∞ r H r dr ( r + a ) r p = r − p − H (cid:20) − pa r H + ( p − r H ( r H + a ) + 2 p ( p + 2) p + 4 F (cid:18) , p + 42 , p + 62 , − a r H (cid:19)(cid:21) I p = (cid:90) ∞ r H dr ( r + a ) r p +1 = r − p − H (cid:20) − r H ( a ( p +2)+ pr H )( r H + a ) + 2( p + 2)( p + 4) p + 6 F (cid:18) , p + 62 , p + 82 , − a r H (cid:19)(cid:21) , (B.1)where, q F s ( m ; n ; z ) denotes the generalized Hypergeometric function . It should also be noted that for ourcalculations the required expressions of the integrals I p ,and I p always have p = 0.1 [1] R. P. Kerr, “Gravitational field of a spinning mass as anexample of algebraically special metrics,” Phys. Rev.Lett. (1963) 237–238.[2] R. A. Hulse and J. H. Taylor, “Discovery of a pulsar ina binary system,” Astrophys. J. (1975) L51–L53.[3]
COBE
Collaboration, G. F. Smoot et al. , “Structure inthe COBE differential microwave radiometer first yearmaps,”
Astrophys. J. (1992) L1–L5.[4]
Supernova Cosmology Project
Collaboration,S. Perlmutter et al. , “Measurements of Omega andLambda from 42 high redshift supernovae,”
Astrophys.J. (1999) 565–586, arXiv:astro-ph/9812133[astro-ph] .[5] C. W. F. Everitt et al. , “Gravity Probe B: Final Resultsof a Space Experiment to Test General Relativity,”
Phys. Rev. Lett. (2011) 221101, arXiv:1105.3456[gr-qc] .[6] S. Mukherjee and S. Chakraborty, “Horndeski theoriesconfront the Gravity Probe B experiment,”
Phys. Rev.
D97 no. 12, (2018) 124007, arXiv:1712.00562[gr-qc] .[7] E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani,U. Sperhake, L. C. Stein, N. Wex, K. Yagi, T. Baker, et al. , “Testing general relativity with present andfuture astrophysical observations,”
Classical andQuantum Gravity no. 24, (2015) 243001.[8] LIGO Scientific, Virgo
Collaboration, B. Abbott et al. , “Observation of Gravitational Waves from aBinary Black Hole Merger,”
Phys. Rev. Lett. no. 6,(2016) 061102, arXiv:1602.03837 [gr-qc] .[9]
LIGO Scientific, Virgo
Collaboration, B. Abbott et al. , “GW170817: Observation of Gravitational Wavesfrom a Binary Neutron Star Inspiral,”
Phys. Rev. Lett. no. 16, (2017) 161101, arXiv:1710.05832 [gr-qc] .[10]
LIGO Scientific, Virgo
Collaboration, B. Abbott et al. , “GWTC-1: A Gravitational-Wave TransientCatalog of Compact Binary Mergers Observed by LIGOand Virgo during the First and Second ObservingRuns,”
Phys. Rev. X no. 3, (2019) 031040, arXiv:1811.12907 [astro-ph.HE] .[11] LIGO Scientific, Virgo
Collaboration, R. Abbott et al. , “GW190814: Gravitational Waves from theCoalescence of a 23 Solar Mass Black Hole with a 2.6Solar Mass Compact Object,”
Astrophys. J. Lett. no. 2, (2020) L44, arXiv:2006.12611 [astro-ph.HE] .[12]
Supernova Search Team
Collaboration, A. G. Riess et al. , “Observational evidence from supernovae for anaccelerating universe and a cosmological constant,”
Astron. J. (1998) 1009–1038, arXiv:astro-ph/9805201 [astro-ph] .[13] S. W. Hawking, “Breakdown of Predictability inGravitational Collapse,”
Phys. Rev.
D14 (1976)2460–2473.[14] R. A. Konoplya, Z. Stuchl´ık, and A. Zhidenko,“Axisymmetric black holes allowing for separation ofvariables in the Klein-Gordon and Hamilton-Jacobiequations,”
Phys. Rev. D no. 8, (2018) 084044, arXiv:1801.07195 [gr-qc] .[15] R. Konoplya and A. Zhidenko, “Detection ofgravitational waves from black holes: Is there a windowfor alternative theories?,” Phys. Lett. B (2016) 350–353, arXiv:1602.04738 [gr-qc] .[16]
Virgo, LIGO Scientific
Collaboration, B. P. Abbott et al. , “GW151226: Observation of Gravitational Wavesfrom a 22-Solar-Mass Binary Black Hole Coalescence,”
Phys. Rev. Lett. no. 24, (2016) 241103, arXiv:1606.04855 [gr-qc] .[17] S. W. Hawking, “Particle creation by black holes,”
Comm. Math. Phys. no. 3, (1975) 199–220.[18] C. M. Harris and P. Kanti, “Hawking radiation from a(4+n)-dimensional black hole: Exact results for theSchwarzschild phase,” JHEP (2003) 014, arXiv:hep-ph/0309054 [hep-ph] .[19] J. V. Rocha, “Evaporation of large black holes in AdS:Greybody factor and decay rate,” JHEP (2009) 027, arXiv:0905.4373 [hep-th] .[20] M. Catal´an, E. Cisternas, P. A. Gonz´alez, andY. V´asquez, “Quasinormal modes and greybody factorsof a four-dimensional Lifshitz black hole with z=0,” Astrophys. Space Sci. no. 6, (2016) 189, arXiv:1404.3172 [gr-qc] .[21] R. B´ecar, P. A. Gonz´alez, and Y. V´asquez, “Fermionicgreybody factors of two and five-dimensional dilatonicblack holes,”
Eur. Phys. J.
C74 no. 8, (2014) 3028, arXiv:1404.6023 [gr-qc] .[22] R. Dong and D. Stojkovic, “Greybody factors for ablack hole in massive gravity,”
Phys. Rev.
D92 no. 8,(2015) 084045, arXiv:1505.03145 [gr-qc] .[23] T. Pappas, P. Kanti, and N. Pappas, “Hawkingradiation spectra for scalar fields by ahigher-dimensional Schwarzschild–de Sitter black hole,”
Phys. Rev.
D94 no. 2, (2016) 024035, arXiv:1604.08617 [hep-th] .[24] F. Gray, S. Schuster, A. Van–Brunt, and M. Visser,“The Hawking cascade from a black hole is extremelysparse,”
Class. Quant. Grav. no. 11, (2016) 115003, arXiv:1506.03975 [gr-qc] .[25] J. Abedi and H. Arfaei, “Fermionic greybody factors indilaton black holes,” Class. Quant. Grav. no. 19,(2014) 195005, arXiv:1308.1877 [hep-th] .[26] T. Harmark, J. Natario, and R. Schiappa, “GreybodyFactors for d-Dimensional Black Holes,” Adv. Theor.Math. Phys. no. 3, (2010) 727–794, arXiv:0708.0017 [hep-th] .[27] U. Keshet and A. Neitzke, “Asymptotic spectroscopy ofrotating black holes,” Phys. Rev.
D78 (2008) 044006, arXiv:0709.1532 [hep-th] .[28] W. Kim and J. J. Oh, “Greybody Factor and HawkingRadiation of Charged Dilatonic Black Holes,”
J. KoreanPhys. Soc. (2008) 986, arXiv:0709.1754 [hep-th] .[29] P. Gonzalez, E. Papantonopoulos, and J. Saavedra,“Chern-Simons black holes: scalar perturbations, massand area spectrum and greybody factors,” JHEP (2010) 050, arXiv:1003.1381 [hep-th] .[30] C. Campuzano, P. Gonzalez, E. Rojas, and J. Saavedra,“Greybody factors for topological massless black holes,” JHEP (2010) 103, arXiv:1003.2753 [gr-qc] .[31] P. Kanti, T. Pappas, and N. Pappas, “Greybody factorsfor scalar fields emitted by a higher-dimensionalSchwarzschild–de Sitter black hole,” Phys. Rev.
D90 no. 12, (2014) 124077, arXiv:1409.8664 [hep-th] . [32] A. Sporea Ciprian, “Greybody factors for (4 + n )-dimSchwarzschild-de Sitter black holes: Spin 1/2 case,” AIP Conf. Proc. no. 1, (2019) 020005.[33] G. Panotopoulos and A. Rinc´on, “Greybody factors fora minimally coupled scalar field in three-dimensionalEinstein-power-Maxwell black hole background,”
Phys.Rev.
D97 no. 8, (2018) 085014, arXiv:1804.04684[hep-th] .[34] M. Cvetic and F. Larsen, “General rotating black holesin string theory: Grey body factors and eventhorizons,”
Phys. Rev.
D56 (1997) 4994–5007, arXiv:hep-th/9705192 [hep-th] .[35] M. Cvetic and F. Larsen, “Greybody Factors andCharges in Kerr/CFT,”
JHEP (2009) 088, arXiv:0908.1136 [hep-th] .[36] W. Li, L. Xu, and M. Liu, “Greybody factors inrotating charged Goedel black holes,” Class. Quant.Grav. (2009) 055008.[37] P. Boonserm and M. Visser, “Bounding the greybodyfactors for Schwarzschild black holes,” Phys. Rev.
D78 (2008) 101502, arXiv:0806.2209 [gr-qc] .[38] T. Ngampitipan and P. Boonserm, “Bounding theGreybody Factors for Non-rotating Black Holes,”
Int. J.Mod. Phys.
D22 (2013) 1350058, arXiv:1211.4070[math-ph] .[39] T. Ngampitipan and P. Boonserm, “Bounding thegreybody factors for the Reissner-Nordstr¨om blackholes,”
J. Phys. Conf. Ser. (2013) 012027, arXiv:1301.7527 [math-ph] .[40] P. Boonserm, T. Ngampitipan, and M. Visser,“Regge-Wheeler equation, linear stability, and greybodyfactors for dirty black holes,”
Phys. Rev.
D88 (2013)041502, arXiv:1305.1416 [gr-qc] .[41] P. Boonserm, T. Ngampitipan, and M. Visser,“Bounding the greybody factors for scalar fieldexcitations on the Kerr-Newman spacetime,”
JHEP (2014) 113, arXiv:1401.0568 [gr-qc] .[42] P. Boonserm, A. Chatrabhuti, T. Ngampitipan, andM. Visser, “Greybody factors for Myers-Perry blackholes,” J. Math. Phys. (2014) 112502, arXiv:1405.5678 [gr-qc] .[43] P. Boonserm, T. Ngampitipan, and P. Wongjun,“Greybody factor for black holes in dRGT massivegravity,” Eur. Phys. J.
C78 no. 6, (2018) 492, arXiv:1705.03278 [gr-qc] .[44] R. M. Wald,
General relativity . University of Chicagopress, 2010.[45] K. D. Kokkotas, R. A. Konoplya, and A. Zhidenko,“Quasinormal modes, scattering and Hawking radiationof Kerr-Newman black holes in a magnetic field,”
Phys.Rev. D (2011) 024031, arXiv:1011.1843 [gr-qc] .[46] P. P. Fiziev, “Exact solutions of Regge-Wheelerequation and quasi-normal modes of compact objects,” Class. Quant. Grav. (2006) 2447–2468, arXiv:gr-qc/0509123 .[47] E. Berti, “A Black-Hole Primer: Particles, Waves,Critical Phenomena and Superradiant Instabilities,” 10,2014. arXiv:1410.4481 [gr-qc] .[48] G. Mu˜noz, “Orbits of massless particles in theschwarzschild metric: Exact solutions,” AmericanJournal of Physics no. 6, (2014) 564–573.[49] M. K. Parikh and F. Wilczek, “Hawking radiation astunneling,” Phys. Rev. Lett. (2000) 5042–5045, arXiv:hep-th/9907001 [hep-th] . [50] M. Angheben, M. Nadalini, L. Vanzo, and S. Zerbini,“Hawking radiation as tunneling for extremal androtating black holes,” JHEP (2005) 014, arXiv:hep-th/0503081 [hep-th] .[51] R. Banerjee and B. R. Majhi, “Quantum TunnelingBeyond Semiclassical Approximation,” JHEP (2008)095, arXiv:0805.2220 [hep-th] .[52] K. Umetsu, “Hawking Radiation from Kerr-NewmanBlack Hole and Tunneling Mechanism,” Int. J. Mod.Phys.
A25 (2010) 4123–4140, arXiv:0907.1420[hep-th] .[53] A. Yale, “Exact Hawking Radiation of Scalars,Fermions, and Bosons Using the Tunneling MethodWithout Back-Reaction,”
Phys. Lett.
B697 (2011)398–403, arXiv:1012.3165 [gr-qc] .[54] L. Vanzo, G. Acquaviva, and R. Di Criscienzo,“Tunnelling Methods and Hawking’s radiation:achievements and prospects,”
Class. Quant. Grav. (2011) 183001, arXiv:1106.4153 [gr-qc] .[55] Z. Feng, Y. Chen, and X. Zu, “Hawking radiation ofvector particles via tunneling from 4-dimensional and5-dimensional black holes,” Astrophys. Space Sci. no. 2, (2015) 48, arXiv:1608.06377 [hep-th] .[56] J. B. Hartle and S. W. Hawking, “Path IntegralDerivation of Black Hole Radiance,”
Phys. Rev.
D13 (1976) 2188–2203.[57] I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker,“Hawking radiation by Kerr black holes and conformalsymmetry,”
Phys. Rev. Lett. (2010) 211305, arXiv:1006.4404 [hep-th] .[58] K. Murata and J. Soda, “Hawking radiation fromrotating black holes and gravitational anomalies,”
Phys.Rev.
D74 (2006) 044018, arXiv:hep-th/0606069[hep-th] .[59] S. Iso, H. Umetsu, and F. Wilczek, “Anomalies,Hawking radiations and regularity in rotating blackholes,”
Phys. Rev.
D74 (2006) 044017, arXiv:hep-th/0606018 [hep-th] .[60] Q.-Q. Jiang, S.-Q. Wu, and X. Cai, “Hawking radiationfrom the dilatonic black holes via anomalies,”
Phys.Rev.
D75 (2007) 064029, arXiv:hep-th/0701235[hep-th] .[61] S. Barman, G. M. Hossain, and C. Singha, “Exactderivation of the Hawking effect in canonicalformulation,”
Phys. Rev.
D97 no. 2, (2018) 025016, arXiv:1707.03614 [gr-qc] .[62] S. Barman and G. M. Hossain, “Consistent derivationof the Hawking effect for both nonextremal andextremal Kerr black holes,”
Phys. Rev.
D99 no. 6,(2019) 065010, arXiv:1809.09430 [gr-qc] .[63] G. M. Hossain and C. Singha, “New coordinates for asimpler canonical derivation of the Hawking effect,”
Eur. Phys. J. C no. 2, (2020) 82, arXiv:1902.04781[gr-qc] .[64] S. Barman, “The Hawking effect and the bounds ongreybody factor for higher dimensional Schwarzschildblack holes,” Eur. Phys. J. C no. 1, (2020) 50, arXiv:1907.09228 [gr-qc] .[65] S. Dalui, B. R. Majhi, and P. Mishra, “Horizon inducesinstability locally and creates quantum thermality,” Phys. Rev. D no. 4, (2020) 044006, arXiv:1910.07989 [gr-qc] .[66] S. Dalui and B. R. Majhi, “Near horizon localinstability and quantum thermality,”
Phys. Rev. D (2020) 124047, arXiv:2007.14312 [gr-qc] .[67] B. R. Majhi, “Is eigenstate thermalization hypothesisthe key for thermalization of horizon?,” arXiv:2101.04458 [gr-qc] .[68] T. Padmanabhan, Gravitation: Foundations andFrontiers . Cambridge University Press, 1 ed., 2010. [69] M. Visser, “Some general bounds for 1-D scattering,”
Phys. Rev.
A59 (1999) 427–438, arXiv:quant-ph/9901030 [quant-ph] .[70] E. Poisson,