Hawking radiation from the (2 + 1)-dimensional BTZ black holes Qing-Quan Jiang a, Shuang-Qing Wu b and Xu Cai a

arXiv:hep-th/0701048v2 9 Jan 2007

a Institute b College

of Particle Physics, Central China Normal University, Wuhan, Hubei 430079, People’s Republic of China

of Physical Science and Technology, Central China Normal University, Wuhan, Hubei 430079, People’s Republic of China

Abstract Motivated by Robinson-Wilczek’s recent viewpoint that Hawking radiation can be treated as a compensating flux to cancel gravitational anomaly at the horizon of a Schwarzschild-type black hole, we investigate Hawking radiation from the rotating (2 + 1)-dimensional BTZ black hole and the charged (2 + 1)-dimensional BTZ black hole via gauge or gravitational anomaly at the horizon. To restore gauge invariance or general coordinate symmetry to hold in the effective theory, one must introduce a gauge current or energy momentum tensor flux to cancel gauge or gravitational anomaly at the horizon. The results show that the values of these compensating fluxes are exactly equal to that of (1+1)-dimensional blackbody radiation at the Hawking temperature. Key words: gauge anomaly, gravitational anomaly, Hawking radiation, BTZ black hole PACS: 04.70.Dy; 97.60.Lf; 05.30.Ch

Email addresses: [email protected] (Qing-Quan Jiang), [email protected] (Shuang-Qing Wu), [email protected] (Xu Cai).

Preprint submitted to Elsevier

6 March 2008

1

Introduction

Classically a black hole is black, but according to quantum mechanics it can emit any kinds of particles via thermal radiation. This is known as Hawking evaporation soon after Stephen Hawking first discovered it [1]. Heuristically there exist several derivations of Hawking radiation [2,3]. Among them, one method to derive Hawking radiation is attributed to the trace anomaly in the conformal symmetry [4], where the authors determined the full momentum energy tensor by using symmetry arguments and the conservation law of the energy momentum tensor together with the trace anomaly. Although their derived result is in quantitative agreement with Hawking’s result, it is difficult to generalize their method to the cases of higher dimensional black holes by using partial wave analysis since Hawking flux is connected with the anomaly through an integral over the whole spacetime. In addition, their observation was based upon the assumptions that the fields were massless and there was no back-scattering effect. So the method appears to be rather special. Recently, Robinson and Wilczek [5] have proposed a new derivation of Hawking radiation via gravitational anomaly at the horizon of a Schwarzschild-type black hole. Here, gravitational anomaly stems from the fact that one want to integrate out the horizon-skimming modes, whose contributions would give a divergent energy momentum tensor because these modes make the time coordinate ill-defined at the horizon. Thus, the effective theory formed outside the black hole horizon to exclude the offending modes at the horizon is chiral here, but contains an anomaly with respect to general coordinate symmetry. To restore general coordinate covariance at the quantum level, one must introduce a energy momentum tensor flux to cancel gravitational anomaly at the horizon. The result shows that the compensating energy momentum tensor flux has an equivalent form to that of Hawking radiation. In this derivation, a key is to introduce a dimensional reduction technique, which generates the physics near the horizon of the original higher dimensional black holes can be effectively described by an infinite collection of two-dimensional fields. This is mainly because gravitational anomaly can only occur in theories with chiral matter coupled to gravity in spacetimes of dimension 4n + 2 where n is an integer, here taking the simplest case for a scalar field in (1 + 1) dimensions. Obviously the derivation of Hawking radiation via the anomalous point of view is only dependent on the anomaly at the horizon, so the method may be more universal. Following that, by considering gauge and gravitational anomalies at the horizon, Iso et al. [6] have investigated Hawking radiation of charged particles from the Reissner-Nordstr¨om black holes. The result shows that the electric current and energy momentum tensor fluxes, required to cancel gauge and gravitational anomalies at the horizon, are exactly equal to that of Hawking radiation. Subsequently, there have been considerable efforts to generalize this work to other cases, and all the obtained results are very successful to 2

support the Robinson-Wilczek’s prescription[7,8,9,10,11]. Recently, (2 + 1)-dimensional BTZ black holes have received a lot of attention by physicists. This is mainly because in (3 + 1) or even higher dimensional gravity, the black hole’s properties at the quantum level remain until now as a mystery, it is believed that the black holes in (2 + 1)-dimensional will provide a good relatively simple laboratory and a better understanding for analyzing general aspects of black hole physics. On the other hand, it is due to the AdS/CFT duality [12], which relates thermal properties of black holes in AdS space to that of dual CFT. Combined with these reasons, it has been attracted much attention in recent years to study the thermal properties of (2+ 1)-dimensional black holes especially in the background geometry with nonvanishing cosmological constant. In this paper, our main focus is on studying Hawking radiation of (2+1) Banados-Teitelboim-Zanelli (BTZ) black holes via gauge and gravitational anomalies, especially that of the rotating BTZ black hole in the dragging coordinate system and that of the charged unrotating BTZ black hole. In fact, since the metric of the rotating black holes has an azimuthal symmetry, the angular momentum is conserved. After a dimensional reduction technique at the horizon, it generates a U(1) gauge symmetry for each partial wave in the effective two-dimensional theory. Now each partial wave of quantum fields in the black hole can be effectively described by a twodimensional charged field with a U(1) charge m, where m is the azimuthal quantum number. One then adopt the same procedure as the charged black holes to obtain Hawking radiation of the rotating black holes via gauge and gravitational anomalies. However, the matter field in the ergosphere near the horizon must be dragged by the gravitational field with an azimuthal angular momentum because there exists a frame dragging effect of the coordinate in the rotating spacetime, so the observer rested in the dragging coordinate system will not find a U(1) gauge flux since the U(1) gauge symmetry is no longer incorporated in each partial wave for the two-dimensional theory due to the dragging effect. As to the charged black hole, the effective twodimensional theory poses gauge symmetry with respect to the Maxwell field of the original black hole. In the two cases, if the effective field theory is formulated outside the black hole horizon to exclude the offending modes at the horizon, it contains an anomaly with these symmetries. In this paper, we are interested in studying Hawking radiation from the Banados-Teitelboim-Zanelli (BTZ) black holes via gauge and gravitational anomalies at the horizon. The result supports the Robinson-Wilczek’s viewpoint, and shows that Hawking radiation can be derived from the anomalous cancellation condition and the regularity requirement at the horizon. For a rotating black hole in the dragging coordinate system, the effective action becomes anomalous in general coordinate symmetry when omitting the classically irrelevant ingoing modes at the horizon. To restore diffeomorphism covariance at the quantum level, the compensating flux of energy momentum 3

tensor is exactly equal to that of Hawking radiation whose Hawking distribution is given by the Planckian distribution without a chemical potential for an azimuthal angular momentum. In the case of a charged BTZ black hole, the effective two-dimensional quantum field near the horizon endows with the gauge and general coordinate symmetries, where the gauge symmetry is originated from the electric field of the charged black hole. Thus in order to restore gauge invariance and general coordinate covariance at the quantum level, a charge current and energy momentum tensor fluxes, which are required to cancel gauge and gravitational anomalies at the horizon, have an equivalent form to that of Hawking radiation. The paper is outlined as follows. In Sec. 2, we, with slightly different from Ref.[5], review the Robinson-Wilczek’s method in the case of a Schwarzschildtype black hole. Adopting the dragging coordinate system, Sec. 3 is devoted to proving that the energy momentum tensor flux, derived by cancelling the anomaly with respect to general coordinate covariance at the horizon of the rotating BTZ black holes, is equal to that of Hawking radiation. In Sec. 4, we show that gauge and gravitational anomalies at the horizon of the charged BTZ black hole can be cancelled by the electric and energy momentum tensor flux of Hawking radiation. Finally, Sec. 5 ends up with our brief discussions.

2

Hawking radiation from a Schwarzschild-type black hole

The metric of a d-dimensional Schwarzschild-type black hole takes the form ds2 = f (r)dt2 −

dr 2 − r 2 dΩ2(d−2) , f (r)

(1)

where, f (r) is dependent on the matter distribution. The horizon (r = rH ) is determined by f (rH ) = 0. The surface gravity at the horizon is given by κ = (1/2)f,r |r=rH . Near the horizon, after performing the partial wave decomposition in the terms of the sperical harmonics and then transforming to the tortoise coordinate defined by dr∗ /dr ≡ f −1 (r), the effective radial potentials or the other interactions vanish exponentially. Now, the quantum field in the d-dimensional black holes can be effectively described by an infinite collection of quantum fields in the (1 + 1)-dimensional spacetime ds2 = f (r)dt2 −

dr 2 , f (r)

(2)

together with a background dilaton field Ψ = r (D−2) , whose effect on the flow of the energy momentum tensor is dropped since the involved background is static. 4

It is well-known that an anomaly often occurs if the symmetry or the corresponding conservation law, which is valid in the classical theory, is violated in the quantized version. In the static Schwarzschild-type black hole, there is a global Killing vector, which generates general coordinate symmetry of the spacetime. If we formulate the effective field theory outside the horizon to integrate out the classically irrelevant ingoing modes at the horizon, the effective theory is chiral here since there is not a divergent energy due to a pile up of the offending modes, but it contains an anomaly with respect to general coordinate symmetry (gravitational anomaly), which often takes the form of the nonconservation of the energy momentum tensor. Gravitational anomaly can only occur in theories with chiral matter coupled to gravity in spacetimes of dimension 4n + 2 where n is an integer, here taking the simplest case for a scalar field in (1 + 1) dimensions. In 1 + 1 dimensions, the consistent anomaly for the energy momentum tensor reads [13] 1 1 √ ǫβµ ∂µ ∂α Γναβ , ∇µ Tνµ ≡ √ ∂µ Nνµ = −g 96π −g

(3)

where ǫ01 = 1. To restore general coordinate covariance at the quantum level, one must introduce a compensating flux of energy momentum tensor whose contributions exactly cancel gravitational anomaly at the horizon. We expect that the compensating flux of energy momentum tensor is precisely equal to that of Hawking radiation, thus the effective theory is self-consistent. Now, we will specifically determine the flux of energy momentum tensor. This flux is required to cancel gravitational anomaly at the horizon due to omitting the quantum effect of the classically irrelevant ingoing modes, so we expect that it is exactly equal to that of Hawking radiation. Outside the horizon of the black hole, the effective field theory is formulated to exclude the horizonskimming modes. Splitting the region outside the horizon into two patches: Near the horizon rH ≤ r ≤ rH + ε, when integrating out the ingoing modes, the effective theory is chiral here, but contains gravitational anomaly. The nonconservation of energy momentum tensor in this region specifically satisfies r the anomalous equation as ∂r T(H)t = ∂r Ntr (r), where Ntr (r) =

1 (f 2 + f,rr f ). 192π ,r

(4)

In the other region rH + ε ≤ r, since the number of the ingoing and outgoing modes is always identical as it is in the fundamental theory, there is no anomaly with respect to general coordinate symmetry, and the energy mor mentum tensor should be conserved here, which means ∂r T(o)t = 0. In classical theory, general coordinate covariance of the classical action tells us its variR 2 √ ation must satisfy −δλ S = d x −g(2) λν ∇µ Tνµ = 0, where λν is a variation parameter. But in our cases, the variation of the effective action, under general coordinate transformation, can be read off 5

− δλ W = =

Z

Z

d2 xλt ∂r Ttr 2

d xλ

t



h

∂r Ntr (r)H(r)

i

+

h

r T(o)t

−

r T(H)t

+

Ntr (r)

i 



δ r − rH − ǫ (5),

where the energy momentum tensor combines contributions from the two rer r gions outside the horizon, that is, Ttr = T(o)t Θ+ (r) + T(H)t H(r), in which Θ+ (r) = Θ(r − rH − ǫ) is the scalar step function and H(r) = 1 − Θ+ (r) is a scalar “top hat” function. Obviously, here we have not taken the quantum effect of the ingoing modes into account. If it is incorporated, the first item should be cancelled since its contribution to the energy momentum tensor is −Ntr (r)H(r). At the same time, in order to restore general coordinate covariance at the quantum level, the coefficient of the delta-function should vanish in the effective action, that is ao = aH − Ntr (rH ) ,

(6)

r r where ao = T(o)t is an integration constant and aH = T(H)t + Ntr (rH ) − Ntr (r) is the value of the energy flow at the horizon. To ensure the regularity of the physical quantities at the horizon, the covariant energy momentum tensor should vanish at the horizon, which means r r T˜(H)t = T(H)t +





1 f f,rr − 2f,r2 = 0 . 192π

(7)

The validity of this condition has been clarified in Refs. [7,8]. Thus we have aH = 2Ntr (rH ) =

κ2 , 24π

(8)

where κ = f,r (rH )/2 is the surface gravity at the black hole horizon. The total flux of energy momentum tensor is ao = Ntr (rH ) =

π 2 T , 12 H

(9)

where TH = κ/(2π) = f,r (rH )/(4π) is Hawking temperature of the black hole. Next, we shall prove the total compensating flux of energy momentum tensor is exactly equal to that of Hawking radiation. For fermions, Hawking radiant spectrum of a Schwarzschild-type black hole is given by the Planckian distribution N(ω) = 1/[exp( TωH ) + 1], so the energy momentum tensor flux of Hawking radiation from the black hole is FH =

Z

0

∞

π ω N(ω)dω = TH2 . π 12

(10)

Obviously, the flux of energy momentum tensor, required to cancel gravitational anomaly at the horizon and restore general coordinate covariance at 6

the quantum level, is precisely equal to that of Hawking radiation. This shows that Hawking radiation can be also derived from the conditions of anomaly cancellation and regularity requirement at the horizon of the black hole. To verify its validity, Iso etc. have subsequently studied Hawking radiation from the stationary rotating Kerr and Kerr-Newman black holes via gauge and gravitational anomalies[7]. The stationary rotating black holes are normally characterized by the angular momentum and energy momentum tensor conservation. If we use the anomaly language, these physical quantities become anomalous because gauge and general coordinate symmetries are destroyed by omitting the classically irrelevant ingoing modes at the horizon. To restore these symmetries, one must introduce a gauge current and energy momentum tensor flux to cancel gauge or gravitational anomalies. The result shows these compensating fluxes are exactly equal to that of Hawking radiation. However, provided that rotating black holes live in the dragging coordinate system, a free falling observer would not, upon passing the horizon, find a gauge flux since the dragging coordinate system behaves like the locally non-rotating coordinate system, which results the effective theory does not possess the U(1) gauge symmetry originated from the axisymmetry of the black hole. In the subsequent section, we will extend their analysis to the case of the rotating BTZ black hole in the dragging coordinate system. For holographic description on gravitational anomalies of the (2 + 1)-dimensional BTZ black holes see Ref.[14], for example.

3

Hawking radiation from a rotating BTZ black hole

The rotating BTZ black holes is an exact solution to Einstein field equation in a (2 + 1)-dimensional gravity theory S=

Z

√ dx3 −g((3) R − 2Λ),

(11)

with a negative cosmological constant Λ = −1/l2 , its explicit expression takes the form [15] 

2

dr 2 J ds = f (r)dt − − r 2 dφ − 2 dt f (r) 2r 2

2

,

(12)

where the lapse function is f (r) = −M +

r2 J2 + , l2 4r 2

(13)

M and J are the ADM mass and angular momentum of the rotating BTZ black hole, respectively. The outer horizon (r+ ) and the inner horizon (r− ) are given 7

by the equation f (r) = 0. The surface gravity and the angular momentum of the outer horizon are easily evaluated as r2 − r2 1 ∂f |r=r+ = + 2 − , 2 ∂r r+ l r− J , Ω+ = 2 |r=r+ = 2r r+ l κ=

(14)

respectively. This coordinate system in Eq.(12) is described by the observer at the infinity. If transforming the dragging coordinate system, defined by ψ =φ−

J t, 2r 2

ξ = t,

(15)

and then performing a dimensional reduction technique as in Sec.2 on the action of the scalar field in the (2 + 1)-dimensional rotating BTZ black hole in the dragging coordinate system, one would find that physics near the horizon can be effectively described by an infinite collection of two-dimensional scalar fields on the metric ds2 = f (r)dξ 2 −

dr 2 , f (r)

(16)

and a dilaton background Ψ = r, whose contributions to the anomalous flux are dropped. Now, we will study the compensating flux and anomaly at the horizon. In our cases, the rotating (2+1)-dimensional BTZ black hole is rested in the dragging coordinate system, so only general coordinate symmetry existed in the two-dimensional reduction. When neglecting the offending modes at the horizon, the effective action becomes an anomaly with respect to general coordinate symmetry. To demand general coordinate covariance at the quantum level, one must introduce an energy momentum tensor flux to cancel gravitational anomaly. The effective field theory is formulated outside the horizon to integrate out the horizon-skimming modes: In the region r+ +ǫ ≤ r, there is no anomaly, and the energy momentum tensor satisfies the conservar tion equation ∂r T(o)t = 0, but in the near-horizon region r+ ≤ r ≤ r+ + ǫ, the effective theory contains gravitational anomaly since the offending modes are integrated out. In terms of the energy momentum tensor, it can be expressed r as ∂r T(H)t = ∂r Ntr (r), where Ntr (r) = (f,r2 +f,rr f )/(192π). Using the formalism in the preceding section and demanding general coordinate covariance at the quantum level to hold in the effective action, one have co = cH − Ntr (r+ ) = cH −

1 2 , f 192π ,r r=r+

(17)

r where co is an integration constant, and cH = T(H)t + Ntr (r+ ) − Ntr (r) is the value of the energy flow at the horizon. Imposing a condition that the covariant

8

energy momentum tensor vanishes at the horizon [see Eq. (7)], the total flux of energy momentum tensor is co = Ntr (r+ ) =

π 2 T , 12 +

(18)

where T+ =

2 r 2 − r− κ = + , 2π 2πr+ l2

(19)

is Hawking temperature of the black hole. So one must introduce an energy momentum tensor flux whose value is expressed by Eq.(18) to cancel gravitational anomaly and restore general coordinate covariance for the effective action. If we take the quantum effect of the classically irrelevant ingoing modes into account, the introducing energy momentum tensor flux should be equal to that of Hawking radiation. In the dragging coordinate system, the Hawking distribution for fermions in the rotating BTZ black hole is given by the distribution N(ω) = 1/[exp( Tω+ ) + 1], here the energy ω is carried by the observer in the dragging coordinates which is related by the formula ω = ω ′ − mΩ+ to the energy ω ′ measured at the infinity (m is an azimuthal angular quantum number), so the energy momentum tensor flux of Hawking radiation reads off FH =

Z

0

∞

π ω N(ω)dω = T+2 . π 12

(20)

Obviously, the right hand sides of Eqs. (18) and (20) coincide with each other, which shows that Hawking radiation can be also determined from by the anomalous point of view. Here, we take the rotating BTZ black hole in the dragging coordinate system as an example. In fact, transforming the dragging coordinates (ξ, ψ) into the original coordinates (t, φ), one can discover the same result that Hawking radiation can cancel gauge and gravitational anomalies at the horizon and restore gauge invariance and general coordinate covariance at the quantum level. Here, gauge anomaly arises from the destruction of the U(1) gauge symmetry along φ direction after excluding the ingoing modes at the horizon. To restore gauge symmetry of the effective theory, a U(1) gauge current flux must be introduced to cancel U(1) gauge anomaly. At the quantum level, this gauge current flux should be equal to the angular momentum flux of Hawking radiation. Noted that the Hawking distribution for fermions case is now given by the Planckian distribution with chemical potential for an ′ + azimuthal angular momentum m as N±m (ω ′ ) = 1/[exp( ω ∓mΩ ) + 1], thus T+ the angular momentum and energy momentum tensor fluxes are, respectively, given by 9

Z





m2 r− 1 Nm (ω ′) − N−m (ω ′) dω ′ = , Fm = m 2π 2πr+ l 0  Z ∞ ′ 2 m2 r− ω π ′ ′ Nm (ω ) + N−m (ω ) dω ′ = FH = + T+2 . 2 2 2π 4πr+ l 12 0 ∞

(21)

These fluxes of Hawking radiation are capable of cancelling U(1) gauge and gravitational anomalies at the black hole horizon, respectively[10]. Here the compensating flux to cancel U(1) gauge anomaly is exactly equal to the angular momentum flux of Hawking radiation whose particle energy ω ′ is measured at the infinity.

4

Hawking radiation from a charged BTZ black hole

A static, charged BTZ black hole found for the action (see Refs.[15]) is written in the following form as ds2 = f (r)dr 2 −

dr 2 − r 2 dφ2 , f (r)

(22)

where the lapse function r2 r2 2 f (r) = −M + 2 − Q ln 2 , l l

(23)

M and Q are the ADM mass and the total charge of the black hole, respectively. The horizon of the black hole (r = r+ ) is determined by f (r+ ) = 0, and the surface gravity is given by 2r 3 − l4 Q2 1 , κ = ∂r f |r=r+ = + 2 2 2 2l r+

(24)

the nonvanishing component for the gauge potential of the Maxwell field is Φt (r) = −Q ln(r/l). It should be stressed that the expressions presented here for the lapse function and the electric potential obey not only the differential but also integral forms of the first law of (2 + 1) BTZ black hole thermodynamics with a variable cosmological constant, recently discussed in Ref. [16]. Anomaly often takes place in the spacetime with dimension 4n + 2, where n is an integer. For the sake of simplicity, we take a chiral scalar field in (1 + 1) dimensions. In the case of (2 + 1)-dimensional charged BTZ black hole, if performing the partial wave decomposition, and then transforming to the tortoise coordinate defined by dr∗ = dr/f (r), one finds that the physics near the horizon can be effectively described by an infinite collection of (1+1)dimensional fields, each propagating in the (1+1)-dimensional spacetime given 10

by ds2 = f (r)dt2 −

dr 2 , f (r)

(25)

a dilaton background Ψ = r and a gauge field Φt (r), where the gauge field is originated from the electric field of the charged BTZ black hole. Now the effective two-dimensional theory for each partial wave has gauge and general coordinate symmetries. If the effective theory is formulated outside the horizon to integrate out the ingoing modes at the horizon, it is chiral here, but contains the anomalies with respect to gauge and general coordinate symmetries, normally named as gauge and gravitational anomalies, respectively. Now, we study the charge flux and gauge anomaly at the horizon. As the effective two-dimensional chiral theory contains gauge anomaly at the horizon, a gauge current flux must be introduced to restore gauge invariance. At the quantum level, we expect that the compensating flux is equal to the charge flux of Hawking radiation. The effective field theory is still formulated outside the horizon to exclude the horizon-skimming modes. If we divide the region outside the horizon into two parts, in the region r+ ≤ r ≤ r+ +ǫ, when omitting the classically irrelevant ingoing modes, the effective theory becomes chiral here, but the gauge current exhibits an anomaly and satisfies the anomalous r equation, ∂r J(H) = e2 ∂r Φt (r)/(4π); in the other region r+ + ǫ ≤ r, there is no anomaly in gauge symmetry, so the gauge current is conserved there, and r then defined by ∂r J(o) = 0. Applying the analysis in Sec. 2, the variation of the effective action under gauge transformations satisfies −δλ W =

Z



dtdrλ ∂r

h e2

4π

i

Φt (r)H(r) +

h

r r J(o) −J(H) +



i e2 Φt (r) δ(r −r+ −ǫ) , 4π (26)

here we have not taken into account the quantum effect of the ingoing modes. If elevated the effective theory at the quantum level, the first term would be cancelled by the quantum effect of the classically irrelevant ingoing modes since its contribution to the total current is −e2 Φt (r)H(r)/(4π). To ensure gauge invariance of the total effective action, the coefficient of the delta function should also vanish, that is e2 Φt (r+ ) , D o = DH − 4π

(27)

r r where Do = J(o) is an integration constant and DH = J(H) − e2 [Φt (r) − Φt (r+ )]/(4π) is the value of the consistent gauge current at the horizon. To ensure the regularity requirement at the horizon, the covariant current formulated by the consistent one as Jer = J r + e2 Φt (r)H/4π should be vanished

11

here, which determines the value of the charge flux as Do = −

e2 e2 Q r+ Φt (r+ ) = ln , 2π 2π l

(28)

which shows that, in order to cancel gauge anomaly at the horizon, a gauge current flux expressed by Eq.(28) must be introduced in the anomalous point of view. In fact, this compensating flux is exactly equal to the electric current flux of Hawking radiation whose Hawking distribution is given by the Planckian distribution with a chemical potential for an electric charge of the field radiated from the black hole. Now, we concentrate on studying the total flux of energy momentum tensor and gravitational anomaly at the horizon. Apart from gauge symmetry, the effective two-dimensional theory for each partial wave has general coordinate symmetry. When it is formulated to exclude the offending modes at the horizon, the effective theory becomes anomalous with respect to general coordinate symmetry. To restore the symmetry under general coordinate transformations, the effective action must describe an energy momentum tensor flux to cancel gravitational anomaly at the horizon. Similar to the analysis in Sec.2, in the region r+ + ǫ ≤ r, since there is an effective background gauge potential, the energy momentum tensor satisfies the modified r r conservation equation ∂r T(o)t = J(o) ∂r Φt (r). In the other region (r+ ≤ r ≤ r+ + ǫ), when neglecting the classically irrelevant ingoing modes, the effective two-dimensional chiral theory contains gravitational anomaly at the horizon, and the energy momentum tensor should satisfy the anomalous equation as r r r ∂r T(H)t = J(H) ∂r Φt (r) + Φt (r)∂r J(H) + ∂r Ntr (r), where the second and third terms come, respectively, from gauge and gravitational anomalies at the horizon. Under the diffeomorphism transformation, the variation of the effective action becomes

− δλ W =



Z

dtdrλt Do ∂r Φt (r) + ∂r

+

h

r T(o)t



−

r T(H)t





e2 2 Φ (r) + Ntr (r) H 4π t



i e2 + Φ2t (r) + Ntr (r) δ(r − r+ − ǫ) , 4π



(29)

where Ntr (r) = f,r2 + f,rr f /(192π). In the above derivation, the effective action is obtained from integrating the outgoing modes. If the ingoing modes are incorporated into the action, the secondh term is cancelled by its i quantum 2 2 r effect whose contributions are given by − e Φt (r)/(4π) + Nt (r) H(r). The first term is the classical effect of the background electric for constant current flow, and the third one is nullified by demanding the effective action to be covariant at the horizon under the diffeomorphism transformation. We thus 12

have go = gH +

e2 2 Φt (r+ ) − Ntr (r+ ) . 4π

(30)

r where go = T(o)t − Do Φt (r) is an integration constant, and r gH = T(H)t −

Z

r

r+

h

dr∂r Do Φt (r) +

i e2 2 Φt (r) + Ntr (r) , 4π

is the value of energy momentum tensor flux at the horizon. Taken the boundary condition as expressed by Eq.(7) at the horizon, the compensating flux of energy momentum tensor reads go =

π e2 2 Φt (r+ ) + T+2 , 4π 12

(31)

2r 3 − l4 Q2 κ = + 2 2 , 2π 4πl r+

(32)

where T+ =

is the Hawking temperature of the black hole. To restore general coordinate symmetry of the effective action, the compensating flux of energy momentum tensor must takes the form as Eq.(31). At the quantum level, this flux corresponds to the energy momentum tensor flux of Hawking radiation whose Hawking distribution is given by the Planckian distribution with a chemical potential for an electric charge of the field radiated from the black hole. This is left to the following discussion. Now we turn to derive the Hawking flux of the black hole. Since the Hawking distribution for fermions of the charged BTZ black hole is given by the Planckian distribution n with an electrico chemical potential, specifically expressed as t (r+ ) N±e (ω) = 1/ exp[ ω±eΦ ] + 1 , the electric current and energy momentum T+ tensor fluxes of Hawking radiation are then given by

FQ = e FH =

Z

Z

0 ∞

0





e2 Q r+ 1 Ne (ω) − N−e (ω) dω = ln 2π 2π l   2 ω e 2 π Ne (ω) + N−e (ω) dω = Φt (r+ ) + T+2 . 2π 4π 12

∞

(33)

Comparing Eqs.(28) and (31), derived from the conditions of gauge and gravitational anomaly cancellations and the regularity requirement at the horizon, with Eq.(33), we easily find that the fluxes of Hawking radiation from the charged BTZ black hole are capable of cancelling gauge or gravitational anomaly at the horizon, and restoring gauge invariance or general coordinate covariance at the quantum level. 13

5

Concluding remarks

It is noted that an observer rest at the dragging coordinate system, which behaves like the locally non-rotating coordinate system, would not observe a U(1) gauge current flux (angular momentum flux) arising from the rotation of the black hole since he is co-rotating with the black hole. The absence of angular momentum in the dragging coordinate system is compensated by the frame-dragging effect, this does not contradict with the fact that the flux of angular momentum can be derived by using the Robinson- Wilczek’s method in the Boyer-Lindquist coordinate system. This is because the dragging coordinate system is related to the Boyer-Lindquist coordinate system by the transformation: ψ = φ − 2rJ2 t, and ξ = t. Thus the energy ω carried by the observer in the dragging coordinates is related by the formula ω = ω ′ − mΩ+ to the energy ω ′ measured at the infinity. If we restore the Boyer-Lindquist coordinate system from the dragging coordinate system, it is easy to calculate the flux of the angular momentum. In this paper, we have investigated Hawking radiation of the (2+1)-dimensional BTZ black holes from the anomalous point of view. Specifically in the cases of the rotating BTZ and charged BTZ black holes, as an anomaly often takes place in the spacetime with dimension 4n + 2 where n is an integer, we have reduced the (2 + 1)-dimensional theory to the two-dimensional effective theory by a dimensional reduction technique near the horizon. For the charged BTZ black hole, the scalar field in a (2 + 1)-dimensional theory can be effectively described by an infinite collection of two-dimensional quantum field with the gauge field background, whose gauge charge is the electric charge e. As to the rotating BTZ black hole, the effective theory can be interpreted as charged particles in a two-dimensional charged black hole, but the U(1) gauge charge m is now an azimuthal quantum number. However, if the rotating BTZ black hole is described in the dragging system, each partial wave for the effective theory also behaves as an independent two dimensional free massless scalar field, but here without a U(1) gauge field background. In the two-dimensional reduction, we formulate the effective field theory outside the horizon to integrate out the horizon-skimming modes, it is then chiral here and each partial wave exhibits gauge or gravitational anomaly, but only at the horizon. To restore gauge or general coordinate symmetry to hold in the effective theory, we have introduced a compensating gauge current or energy momentum tensor flux to cancel them. The result is that these compensating fluxes are exactly equal to that of Hawking radiation. Recently, this method has been applied in Ref.[17] to derive Hawking radiation. Acknowledgments: This work was partially supported by the Natural Science Foundation of China under Grant No. 10675051, 10635020, 70571027 and 14

a grant by the Ministry of Education of China under Grant No. 306022. S.Q.Wu was also supported in part by a starting fund from Central China Normal University.

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