Effect of GUP on Hawking radiation of BTZ black hole
EEffect of GUP on Hawking radiation of BTZ black holeT . Ibungochouba Singh , Y . Kenedy Meitei , and I . Ablu Meitei , Department of Mathematics, Manipur University, Canchipur, 795003, India Department of Physics, Modern College, Imphal, Manipur, 795050, India
Abstract
The Hawking radiation of BTZ black hole is investigated based on generalizeduncertainty principle effect by using Hamilton-Jacobi method and Dirac equation.The tunneling probability and the Hawking temperature of the spin-1/2 particlesof the BTZ black hole are investigated using modified Dirac equation based on theGUP. The modified Hawking temperature for fermion crossing the back hole horizonincludes the mass parameter of the black hole, angular momentum, energy and alsooutgoing mass of the emitted particle. Besides, considering the effect of GUP intoaccount, the modified Hawking radiation of massless particle from a BTZ black holeis investigated using Damour and Ruffini method, tortoise coordinate transformationand modified Klein-Gordon equation. The relation between the modified Hawkingtemperature obtained by using Damour-Ruffini method and the energy of the emittedparticle is derived. The original Hawking temperature is also recovered in the absenceof quantum gravity effect.
PACS numbers : 04.70.Dy, 04.20.Gz, 03.65.-w
Key-words : BTZ black hole; Generalized Klein-Gordon equation; Generalized Diracequation; Damour and Ruffini method.1.
Introduction:
Hawking [1, 2] discovered the black body radiation of scalar particle based on semi-classical calculation by applying the Wick Rotation method. Refs. [3, 4] showed thatthe entropy of black hole is proportional to the horizon area of the black hole. Ref [5]proposed the Hawking radiation as a semiclassical tunneling process and their aim isto find well-behaved co-ordinate system at the event horizon to calculate the emissionrate. They have shown that Hawking radiation of black hole is not pure thermal if theself gravitation is taken into account. The change of Bekenstein-Hawking entropy wasrelated to the tunneling rate for the Schwarzschild black hole.The Hawking radiation as a tunneling of extremal and rotating black hole was investi-gated using Hamilton-Jacobi method and WKB approximation [6] which is an extensionof complex path analysis [7]. Refs. [8-10] investigated the tunneling of spin-1/2 andspin-3/2 particle near the black hole horizon applying the WKB approximation. Ref.[11] investigated the tunneling of Dirac particles from a stationary black hole using PauliSigma matrices, WKB approximation and Feynman prescription. By choosing appropri-ate gamma matrices and wave function, the action of radiant particle can be obtainedfrom the Dirac equation which ‘in turn’ is related to the Boltzmann factor of emissionaccording to the semiclassical WKB approximation. Since then, the Hawking radiationsof black holes in different space-times were investigated [12-15] using Hamilton-JacobimethodThe presence of minimal length of the order of Plank scale was predicted in differenttheories of quantum gravity such as string theory, loop quantum gravity, doubly special1 a r X i v : . [ phy s i c s . g e n - ph ] S e p heory of relativity and Gedanken experiments [16-24]. This observable minimal lengthleads us to obtain the generalized uncertainty principle(GUP) through modified commu-tation relation. Refs. [25, 26] derived the expression of GUP as ∆ x ∆ p ≥ ¯ h [1 + β (∆ p ) ],where β = β (cid:96) p ¯ h , β is of order unity and (cid:96) p is the Plank length respectively. The modifiedcommutation relation can be written as[ x α , p γ ] = i ¯ hδ αγ [1 + βp ] , (1)where x α and p γ are the modified position and momentum operators respectively, whichare also defined by x α = x α p γ = p γ (1 + βp γ ) , (2) x α and p γ satisfy the usual commutation relation [ x α , p γ ] = i ¯ hδ αγ . It has also beenextended the GUP based on doubly special relativity known as DSR-GUP [27, 28]. TheUnrugh effect based on modified form of GUP has been investigated in [29]. Refs. [30,31] discussed the black hole thermodynamics and the tunneling rate was obtained in [32,33] applying modified from of GUP. Ref. [34] studied creation of scalar particles by anelectric field in the presence of quantum gravity effect. Ref. [35] discussed the tunnelingof massless scalar particle from Schwarzschild black hole by considering quantum gravityinto account influenced by DSR-GUP and Parikh and Wilczek method. Fermion tunnel-ing from Schwarzschild black hole was investigated by applying generalised form of Diracequation and the resulting remnant of the black hole was discussed in [36]. Quantumgravity effects on the black holes have been discussed in [37-48].Ref. [49] investigated the Hawking radiation using tortoise co-ordinate transforma-tion in which gravitational field is assumed to be independent of time. The Klein-Gordonscalar field is reduced to a standard form of wave equation near the horizon. By trans-forming outgoing wave from outside into inside of the horizon, the thermal radiationspectrum of stationary and non-stationary black hole can be obtained. Ref. [50], basedon the generalised treatment of barrier penetration proposed by Damour and Ruffini,investigated the tunneling of fermions and bosons crossing black hole horizon and theexact Hawking temperature of Schwarzschild black hole is recovered. Following theirwork, the Hawking temperature in different space-times have been studied in [51-57].The aim of this paper is to investigate the tunneling of fermions crossing the horizonof a BTZ black hole by taking quantum gravity effects into account. The correction toHawking temperature is recovered using generalized Klein-Gordon equation and gener-alized Dirac equation influenced by GUP.The paper is organized as follows. In section 2, the correction of Hawking temperatureof BTZ back hole is derived using Dirac equation influenced by GUP. In section 3, utilizinggeneralized tortoise co-ordinate transformation and Klein-Gordon equation, the Hawkingtemperture of BTZ black hole has also been discussed with and without the influence ofGUP. Some conclusions are given in the last section.2. BTZ black hole
The line element of BTZ black hole in (2 + 1) dimensional space time is given by [58] ds = ∆ dt − dr − r dφ , (3)where ∆ = − M + r (cid:96) . The line element (3) has a singularity at ∆ = 0 and the radius ofthe black hole is given by r h = √ M (cid:96), (4)where M stands for Arnowwitt-Deser-Misner(ADM) mass and expression of ADM isgiven by M = r h (cid:96) . (5)The Hawking temperature of BTZ black hole is given by T = √ M π(cid:96) . (6)3. Tunneling of Dirac particles
To study the tunneling of Dirac particles from BTZ black hole, the generalized Diracequation influenced by GUP is given by [36][ iγ ∂ + iγ i (1 − βm ) ∂ i + iγ i β ¯ h ( ∂ j ∂ j ) ∂ i + m ¯ h (1 − βm + β ¯ h ∂ j ∂ j ) − iγ µ Γ µ (1 + β ¯ h∂ j ∂ j − βm )] ψ = 0 , (7)where ψ is a Dirac spinner wave function. For the BTZ black hole in 2 + 1 dimensionalspace, γ a matrices in ( t, r, φ ) coordinate system are chosen as γ t = 1 √ ∆ (cid:32) − (cid:33) γ r = √ ∆ (cid:32) ii (cid:33) γ φ = 1 r (cid:32) − (cid:33) . (8)Using he following ansatz for the wave function ψ ( x ) = exp ( i ¯ h S ( t, r, φ )) (cid:32) A ( t, r, φ ) B ( t, r, φ ) (cid:33) , (9)where A ( t, r, φ ), B ( t, r, φ ) and S are functions of t , r φ and S is the action of emittedfermion. Ref. [60] showed that the decoupling of Dirac equation could be done onlyfor stationary space time or in the spherically symmetric Vaidya-Bonner black hole [61].To find the solution of Dirac equation, using Eqs. (3), (8) and (9) in Eq. (7), the twodecoupled equations are obtained as[ 1 √ ∆ ( ∂S∂t ) + m (1 − βm ) + βm ∆( ∂S∂r ) + βm r ( ∂S∂φ ) ] A
3[ (1 − βm ) r ( ∂S∂φ ) + β ∆ r ( ∂S∂r ) ( ∂S∂φ ) + βr ( ∂S∂φ ) ] B + i [(1 − βm ) √ ∆( ∂S∂r ) + β ∆ √ ∆( ∂S∂r ) + β √ ∆ r ( ∂S∂r )( ∂S∂φ ) ] B = 0 . (10)+[ (1 − βm ) r ( ∂S∂φ ) + β ∆ r ( ∂S∂r ) ( ∂S∂φ ) + βr ( ∂S∂φ ) ] A − i [(1 − βm ) √ ∆( ∂S∂r ) + β ∆ √ ∆( ∂S∂r ) + β √ ∆ r ( ∂S∂r )( ∂S∂φ ) ] A +[ 1 √ ∆ ( ∂S∂t ) − m (1 − βm ) − βm ∆( ∂S∂r ) − βm r ( ∂S∂φ ) ] B = 0 (11)The nontrivial solution of the above two equations for A ( t, r, φ ) and B ( t, r, φ ) will beobtained only when the determinant of the coefficient matrix is zero. Neglecting higherorders of β , the simplified form of equation is obtained from Eqs. (10) and (11) as1∆ (cid:18) ∂S∂t (cid:19) − ∆ (cid:18) ∂S∂r (cid:19) − r (cid:18) ∂S∂φ (cid:19) − m − β { r (cid:18) ∂S∂φ (cid:19) (cid:18) ∂S∂r (cid:19) + 2∆ (cid:18) ∂S∂r (cid:19) + 2 r (cid:18) ∂S∂φ (cid:19) − m } = 0 . (12)To investigate the fermions tunneling across the black hole horizon, we need to separatethe variables t, r, φ involved in the above equation. Taking S ( t, r, φ ) = − ωt + qφ + H ( r ),where ω and q are energy and angular momentum of the particle respectively, and H ( r ) = H ( r ) + βH ( r ) [59]. The integral of the radial action is obtained as H ± ( r ) = ± (cid:90) (cid:113) ω − ∆( q r + m )∆ [1 + β (2 m ∆ ω − ω )∆ { ω − ∆( q r + m ) } ] dr. (13)By taking contour as upper part of semi-circle and using Feynman prescription, theintegral of Eq. (13) is computed as H ± ( r ) = ± πiω(cid:96) √ M [1 + βY ] , (14)where H + ( r h ) and H − ( r h ) are outgoing and incoming wave solutions of radial part re-spectively and Y = m √ M(cid:96) [ Mω ( q M(cid:96) + m ) − − ω M [ Mω ( q M(cid:96) + m ) − − πω(cid:96) √ M (1 + βY )] . (15)The corrected Hawking temperature, T D of Dirac particle emitted from BTZ black holeafter neglecting higher order terms of β , is obtained as T D = T (1 − βY ) , (16)4here T = √ M π(cid:96) is the Hawking temperature of black hole in the absence of quantumgravity effect. The importance of quantum gravity effect is to lower Hawking temperatureof BTZ black hole and the modified Hawking temperature depends on the mass of theblack hole and also on the angular momentum, energy and the mass of the emittedDirac particle. From Eq. (16), we observe the quantum gravity effect prevents therise of Hawking temperature in BTZ black hole. When β = 0, the standard Hawkingtemperature of BTZ black hole is recovered.3. Tortoise coordinate transformation and tunneling of scalar particles:
Using tortoise coordinate transformation dr ∗ = ∆ − dr, (17)the conformally flat two dimensional BTZ black hole is given by ds = dt − dr ∗ . (18)From Eq. (2) the square of momentum is given by p γ p γ = − ¯ h [1 − β ¯ h ( ∂ γ ∂ γ )] ∂ γ [1 − β ¯ h ( ∂ a ∂ a )] ∂ γ (cid:39) − ¯ h [ ∂ b ∂ b − β ¯ h ( ∂ b ∂ b ) ∂ b ∂ b ] , (19)where the terms of higher order of β are neglected. The Klein-Gordon equation in1-spatial dimension is defined by p ψ ( t, x ) = ( E c − m c ) ψ ( t, x ) with E = i ¯ h∂ and p = p b p b . Then, the two dimensional Klein-Gordon equation of scalar particle can bewritten as (cid:34) ∂ ∂t − ∂ ∂r ∗ − β ¯ h ∂ ∂r ∗ (cid:35) ψ = 0 . (20)We observe that the space time metric given by Eq. (18) is stationary and for separationof variables in Eq. (20), the wave function ψ ( t, r ∗ ) can be written as ψ ( t, r ∗ ) = e − iwt R ( r ∗ ) (21)where w denotes the energy of radiating particle. Applying Eq. (21) into Eq. (20), weobtain 2 β ¯ h ∂ ∂r ∗ R ( r ∗ ) + d dr ∗ R ( r ∗ ) + w R ( r ∗ ) = 0 . (22)Then, the following two cases will be discussed:(1) For the case β = 0 :Eq. (22) becomes the standard wave equation as ∂ ∂r ∗ R ( r ∗ ) + w R ( r ∗ ) = 0 . (23)5ere R ( r ∗ ) can be written as R ( r ∗ ) = e ηr ∗ , (24)where η is given by η = ± iw. (25)The negative solution indicates the incoming wave. Then the outgoing wave solution canbe expressed as ψ ( t, r ∗ ) = e − iwt e iwr ∗ (26)If we define the advanced time coordinate v = t + r ∗ , the outgoing wave solution can bewritten as ψ ( v, r ∗ ) = e − iwv e iwr ∗ . (27)By integrating Eq. (17), the tortoise coordinates transformation is obtained as r ∗ = log (cid:32) r − √ M (cid:96)r + √ M (cid:96) (cid:33) (cid:96) √ M . (28)The incoming wave has no singularity at the horizon in Eddington coordinate systembut outgoing wave has a logarithmic singularity at the horizon. According to Damourand Ruffini [49] and Sannan [50], the outgoing wave function can continue analyticallyfrom outside of the horizon into the inside through the negative half of the complex plane.Therefore, inside the black hole horizon the outgoing wave function can be written as ψ out ( r < r h ) = e − iwu (cid:32) r − √ M (cid:96)r + √ M (cid:96) (cid:33) iw(cid:96) √ M . (29)The outgoing wave outside the horizon is ψ out ( r > r h ) = e − iwv e πw(cid:96) √ M (cid:32) r − √ M (cid:96)r + √ M (cid:96) (cid:33) iw(cid:96) √ M . (30)The tunneling probability across the horizon, r = r h isΓ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ out ( r < r h ) ψ out ( r > r h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − πw(cid:96) √ M . (31)Thus, the Hawking temperature of a BTZ black hole in the absence of GUP is recoveredas T = 1 β B = √ M π(cid:96) . (32)(2) For the case of β (cid:54) = 0 : 6sing Eq. (24) into Eq. (22), we get η = − ± (cid:113) − β ¯ h w β ¯ h . (33)Let us assume 8 β ¯ h w << β , we get η = − ± (1 − β ¯ h w )4 β ¯ h . (34)There are two values of η . For positive sign, one value is η = ± iω which is the same as β = 0 given in case 1. In such case the original Hawking temperature is recovered andthe influence of GUP will also vanish.Here, we are interested in the negative sign because it includes the influence of GUP.Then, the other value of η is η = ± iw (cid:115) β ¯ h w − . (35)The above equation has an upper bound for the energy of the particles i.e. ¯ hω < √ β . Inthis case there are two values of η . The positive/negative sign corresponds to outgoing/ingoing wave. Here, we will discuss only the outgoing wave function as ψ ( t, r ∗ ) = e − iwt e iwr ∗ (cid:113) β ¯ h w − . (36)The generalized tortoise coordinate transformation which is different from [52, 57] isˆ r ∗ = r ∗ (cid:115) β ¯ h w − . (37)Then the outgoing wave solution can be written ψ ( t, ˆ r ∗ ) = e − iwt e iw ˆ r ∗ . (38)By defining advanced time coordinate v = t + ˆ r ∗ , the above equation can be written as ψ ( v, ˆ r ∗ ) = e − iwv e iw ˆ r ∗ . (39)Following Damour and Ruffini [49] and Sannan [50], the outgoing wave can be extendedinside the horizon as ψ out ( r < r h ) = e − iwv (cid:32) r − √ M (cid:96)r + √ M (cid:96) (cid:33) iw(cid:96) √ M (cid:113) β ¯ h w − . (40)The outgoing wave outside the horizon r = r h is ψ out ( r > r h ) = e − iwv (cid:32) √ M (cid:96) − r √ M (cid:96) + r (cid:33) iw(cid:96) √ M (cid:113) β ¯ h w − e πw(cid:96) √ M (cid:113) β ¯ h w − . (41)7 ω - T Figure 1: Temperature (T) vs energy of the emitted particle ( ω ). The blue line corre-sponds to scalar particles and red line corresponds to Dirac particles. Here, for simplicity,we have taken β = 1, (cid:96) = 1 , q = 1 , m = 1 and M = 20.The tunneling probability near the horizon r = r h at β (cid:54) = 0 isΓ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ out ( r < r h ) ψ out ( r > r h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = e − πw(cid:96) √ M (cid:113) β ¯ h w − . (42)From the thermal radiation spectrum of scalar particles, the Boltzmann factor is obtainedas β B = 2 π(cid:96) √ M (cid:115) β ¯ h w − . (43)The above expression is the inverse temperature. Assuming ε = 2 β ¯ h w , the correctedHawking temperature in Planck scale with the influence of GUP is given by T = √ M ε π(cid:96) (cid:110) ε O ( ε ) (cid:111) . (44)4. Conclusions and Discussion:
In this paper, we have investigated the quantum gravity effect of BTZ black hole byapplying the particle tunneling method. To take quantum gravity into account, we takethe generalised Dirac equation to discuss the tunneling of fermions crossing the blackhole horizon. The tunneling probability of the fermion is calculated and correspondinglythe modified Hawking temperature of BTZ black hole is derived. The modified Hawkingtemperature depends not only mass parameter of the black hole but also angular mo-mentum, energy and the mass of the emitted particles. We also discussed the modifiedHawking temperature of BTZ black hole using Damour and Ruffini method, tortoise co-ordinate transformation and generalized form of Klein-Gordon equation. In this case, itis shown that the corrected Hawking temperature is related to the energy of the emittedparticles.From Eq. (14), we observe that 8 if m √ Mω (cid:96) > − ω (cid:96) +2 √ Mm (cid:96)q + Mm (cid:96) , the corrected Hawking temperature of Dirac particleof BTZ black hole is higher than the standard Hawking temperature. • If m √ Mω (cid:96) = 1 − ω (cid:96) +2 √ Mm (cid:96)q + Mm (cid:96) , the GUP effect has been cancelled and the modifiedHawking temperature of Dirac particle reduces to the standard Hawking tempera-ture. • Lastly if m √ Mω (cid:96) < − ω (cid:96) +2 √ Mm (cid:96)q + Mm (cid:96) , the modified Hawking temperature of Diracparticle is lower than the standard Hawking temperature.In the above two cases, the presence of quantum gravity effect prevents the rise of Hawk-ing temperature of BTZ black hole in general. It is observed from Fig. (1) that theHawking temperature of BTZ black hole for the emission of the Dirac particles can benegative. The spin of the Dirac particles might be responsible for the negative Hawk-ing temperature. Spin systems with bound energy may have negative energy. NegativeHawking temperatures of certain classes of black holes are discussed in [62]. Acknowledgements : The YKM acknowledges CSIR (New Delhi) for giving finan-cial support.
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