Physics Letters B 661 (2008) 300–306

www.elsevier.com/locate/physletb

Covariant anomalies and Hawking radiation from charged rotatingblack strings in anti-de Sitter spacetimes

Jun-Jin Peng, Shuang-Qing Wu ∗

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

Received 31 December 2007; received in revised form 10 February 2008; accepted 14 February 2008

Available online 20 February 2008

Editor: M. Cvetic

Abstract

Motivated by the success of the recently proposed method of anomaly cancellation to derive Hawking fluxes from black hole horizons of space-times in various dimensions, we have further extended the covariant anomaly cancellation method shortly simplified by Banerjee and Kulkarnito explore the Hawking radiation of the (3 + 1)-dimensional charged rotating black strings and their higher dimensional extensions in anti-deSitter spacetimes, whose horizons are not spherical but can be toroidal, cylindrical or planar, according to their global identifications. It should beemphasized that our analysis presented here is very general in the sense that the determinant of the reduced (1 + 1)-dimensional effective metricfrom these black strings need not be equal to one (

√−g �= 1). Our results indicate that the gauge and energy–momentum fluxes needed to cancelthe (1 + 1)-dimensional covariant gauge and gravitational anomalies are compatible with the Hawking fluxes. Besides, thermodynamics of theseblack strings are studied in the case of a variable cosmological constant.© 2008 Elsevier B.V. All rights reserved.

PACS: 04.70.Dy; 04.62.+v; 11.30.-j

Keywords: Covariant anomaly; Hawking radiation; Black string; Thermodynamics

1. Introduction

Hawking’s discovery [1] that a black hole is not completelyblack but can emit radiation from its horizon is an interestingand significant quantum effect arising in the background space-time with an event horizon. This effect is named as the famousHawking radiation. It plays an important role in catching onsome clues to the complete theory of Quantum Gravity. Thisfact provides a strong motivation for understanding the essenceof Hawking radiation since Hawking discovered it more thanthirty years ago. During the past years, one prominent successis that a lot of new methods to derive Hawking radiation havebeen developed. For example, Robinson and Wilczek (RW) [2]recently presented a novel method to derive the Hawking tem-perature of a Schwarzschild black hole via the cancellation of

* Corresponding author.E-mail address: sqwu@phy.ccnu.edu.cn (S.-Q. Wu).

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.02.023

(1 + 1)-dimensional consistent gravitational anomaly. Subse-quently, this work was extended to the cases of a charged blackhole [3] and rotating charged black holes [4] by consideringboth gauge and gravitational anomalies. Following this method,a lot of work [5–15] appeared soon. In Refs. [10], the originalwork of [2,3] was generalized to the general case where the met-ric determinant

√−g �= 1, and a peculiar case in which√−g

vanishes at the horizon has also been investigated [11] in de-tails. As far as the case of non-spherical topology is concerned,only a little work [8,10] appeared so far. Therefore it is of spe-cial interest to further investigate Hawking radiation of blackobjects with non-spherical topology in other cases.

However, the original anomaly cancellation method pro-posed in Refs. [2,3] encompasses not only the consistent anom-aly but also the covariant one. Quite recently, Banerjee andKulkarni [12] suggested to simplify this model by consider-ing only the covariant gauge and gravitational anomalies. Theirsimplification further cleans the description of the anomaly can-cellation method totally in terms of the covariant expressions,

J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306 301

thus making the analysis more economical and conceptuallycleaner. An extension of their work to the case where the de-terminant

√−g �= 1 was done in [13]. Based upon these devel-opments, some direct applications soon appeared in [14,15].

On the other hand, with the discovery of anti-de Sitter/con-formal-field-theory (AdS/CFT) correspondence [16], it is ofgreat interest to consider rotating charged generalizations ofblack holes in AdS spaces. The correspondence between the su-pergravity in asymptotically AdS spacetimes and CFT makes itpossible to get some insights into the thermodynamic behaviorof some strong coupling CFTs by studying thermodynamics ofthe asymptotically AdS spacetimes. Specifically speaking, ac-cording to the AdS/CFT correspondence, rotating black holes inAdS spaces are dual to certain CFTs in a rotating space, whilethe charged ones are dual to CFTs with a chemical potential.

What is more, higher dimensional generalizations (with orwithout a cosmological constant) of rotating black holes andtheir properties have attracted considerable attention in recentyears, in particular, in the context of string theory, and with theadvent of brane-world theories [17], raising the possibility ofdirect observation of Hawking radiation and of as probes oflarge spatial extra dimensions in future high energy colliders.The recent brane-world scenarios that predict the emergenceof a TeV-scale gravity in the higher dimensional theory haveopened the door towards testing Hawking effect and exploringextra dimensions through the decay of mini black holes to highenergy particles.

Owing to the above two aspects, therefore it is of great im-portance to study Hawking radiation of higher dimensional AdSspacetimes, especially those of non-spherical topology. In thisarticle, we will employ the covariant anomaly method to inves-tigate Hawking radiation of the (3 + 1)-dimensional chargedrotating black strings in AdS spacetimes and their higher di-mensional extensions [18]. Our aim is to derive Hawking fluxesfrom these black strings via covariant gauge and gravitationalanomalies. Our motivation is also due to the fact that the hori-zon topology of these black strings is not spherical but canbe toroidal, cylindrical or planar, depending on their globalidentifications. It should be pointed that our covariant anom-aly analysis is very general because the determinant of the re-duced (1 + 1)-dimensional effective metric need not be equal toone (

√−g �= 1). Our results show that the gauge and energy–momentum fluxes needed to cancel the (1 + 1)-dimensionalcovariant gauge and gravitational anomalies are in agreementwith the Hawking thermal fluxes. In addition, thermodynamicsof these black strings are studied in the case of a variable cos-mological constant.

Our Letter is organized as follows. In Section 2, we shallinvestigate thermodynamics of the (3 + 1)-dimensional blackstring and generalize the first law to the case of a variable cos-mological constant. Then we derive the fluxes of the chargeand energy–momentum tensor via covariant anomalies. A par-allel analysis in higher dimensions is presented in Section 3,where we have investigated Hawking radiation from the higher-dimensional black strings. Section 4 is our summary. Appen-dix A is supplemented to address the non-uniqueness of two-dimensional effective and physically equivalent metrics in the

process of dimensional reduction, taken the four-dimensionalcase as an explicit example.

2. Hawking radiation of the (3 + 1)-dimensional blackstring via covariant anomalies

We start with the action of Einstein–Maxwell theory ind-dimensional spacetimes with a negative cosmological con-stant, which reads

(1)

Sd = − 1

16π

∫ddx

√−g

[R + (d − 1)(d − 2)

l2− FμνFμν

],

where l is the radius of AdS spaces. Under this action, a(3 + 1)-dimensional rotating charged black string solution inasymptotically AdS spaces was presented in [18]. In terms ofBoyer–Lindquist-like coordinates, its line element and the cor-responding gauge potential take the following forms

ds2 = −f (r)(Ξ dt − a dφ)2 + dr2

f (r)

(2)+ r2

l4

(a dt − Ξl2 dφ

)2 + r2 dz2,

(3)A = Q

r(Ξ dt − a dφ),

where

(4)f (r) = r2

l2− 2M

r+ Q2

r2, Ξ =

√1 + a2

l2,

in which M , Q, and a are parameters related to the mass, thecharge, and the angular momentum, respectively. When the co-ordinate z assumes the values −∞ < z < +∞, the metric (2)describes a stationary black string with a cylindrical horizon.On the other hand, if the coordinate z is compactified to theregion 0 � z < 2π , one can get a closed black string with atoroidal horizon.

We first investigate thermodynamics of the black string.There exists an event horizon at r+, which is the largest pos-itive root of the equation f (r) = 0. The angular velocity andelectrostatic potential of the horizon are

(5)Ω = a

Ξl2, Φ = Q

Ξr+.

The Hawking temperature T = κ/(2π) can be obtained via thesurface gravity on the horizon, which is given by

(6)κ = f ′(r+)

2Ξ= 3r4+ − Q2l2

2Ξl2r3+.

For convenience, we assume the length of the black string alongthe z-direction is one. Then the Bekenstein–Hawking entropyfor the horizon is S = πΞr2

H /2. It can be verified that the mass,the angular momentum and the electric charge

(7)M= 1

4

(3Ξ2 − 1

)M, J = 3

4ΞMa, Q = 1

2ΞQ,

302 J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306

satisfy the integral and differential expressions for the firstlaw [19]

(8)M= 2T S + 2ΩJ + ΦQ+ Θl,

(9)δM = T δS + ΩδJ + ΦδQ+ Θδl,

where Θ = −(2r3+ +3Ma2)/(4l3) is the generalized force con-jugate to the variable state parameter l.

We now turn to investigate Hawking radiation from the blackstring via the covariant anomaly cancellation method. SinceHawking radiation is the quantum effect due to the horizon, itis very important for us to analyze the physics near the hori-zon. To this end, we first show that a free scalar field theory inthe background (2) can be reduced to the (1 + 1)-dimensionaleffective theory near the horizon. For simplicity, we considerthe action of a massless scalar field with the minimal electro-magnetic coupling term

S[ϕ] = 1

2

∫d4x ϕDμ

(√−ggμνDνϕ)

= 1

2

∫dt dr dφ dz r2ϕ

{−Ξ2r2 − a2f (r)

r2f (r)

×[Dt + r2/l2 − f (r)

Ξ2r2 − a2f (r)aΞDφ

]2

(10)

+ 1

Ξ2r2 − a2f (r)D2

φ + 1

r2∂r

[r2f (r)∂r

] + 1

r2∂2z

}ϕ,

where Dμ = ∂μ + ieAμ. After performing a partial wave de-composition ϕ = ∑

m ϕm(t, r) exp(imφ + ikzz), and only keep-ing the dominant terms near the horizon, the action (10) be-comes

S[ϕ] � 1

2

∑m

∫dt dr r2ϕm

{−Ξ2r2 − a2f (r)

r2f (r)

×[∂t + ieQΞr

Ξ2r2 − a2f (r)

(11)+ imaΞr2/l2 − f (r)

Ξ2r2 − a2f (r)

]2

+ f (r)∂2r

}ϕm.

Therefore, the physics near the horizon can be described by aninfinite set of (1 + 1)-dimensional effective massless fields inthe background of spacetime with the metric and the gauge po-tential (see Appendix A for the non-uniqueness of the reducedtwo-dimensional effective metrics)

(12)ds2 = −h(r) dt2 + dr2

f (r), h(r) = r2f (r)

Ξ2r2 − a2f (r),

(13)

At = eA(0)t + mA

(1)t

= eQΞr

Ξ2r2 − a2f (r)+ maΞ

r2/l2 − f (r)

Ξ2r2 − a2f (r),

together with the dilaton Ψ = r√

Ξ2r2 − a2f (r), which doesnot contribute to the anomalies. In this effective theory, A

(1)t

can be interpreted as an induced U(1) gauge field with thecharge given by the azimuthal quantum number m for each par-tial wave.

Note that the two-dimensional effective metric (12) is ther − t sector of the four-dimensional black string spacetime (2)in the dragging coordinate system with a dragging velocitydφ/dt = −gtφ/gφφ = A

(1)t , and its metric determinant

(14)√−g = √

h(r)/f (r) = r√Ξ2r2 − a2f (r)

is, in general, not equal to one. It is regular at the horizon(√−g = 1/Ξ when r = r+), and approaches to one at spa-

tial infinity (√−g → 1 when r → ∞). The latter asymptotic

property is very useful to compute the charge flux and energyflux which are obtained by taking the r → ∞ limits of thegauge current and energy–momentum tensor, respectively. Inthe remanding part of this section, we will extend the methodin [12] to analyze Hawking radiation of the black string throughthe covariant gauge and gravitational anomalies in the effectivebackground spacetime (12).

As a warm-up, we briefly review the basic idea of the anom-aly cancellation method in which the quantum fields near thehorizon can be effectively described in terms of an infinite col-lection of (1 + 1)-dimensional massless fields and the underly-ing effective theory should be invariant. Specifically speaking,if we omit the classically irrelevant ingoing modes near thehorizon, the (1 + 1)-dimensional effective field theory becomeschiral, leading to gauge and gravitational anomalies. Therefore,in order to preserve the invariance of the underlying theoryunder gauge and general coordinate transformations, the anom-alies must be cancelled by a compensating current, which iscompatible with the Hawking fluxes of charge and energy–momentum.

Consider first the charge flux of the black string. It is wellknown that the covariant gauge anomaly can be read off as [20]

(15)∇μ

1

eJ (0)μ = ∇μ

1

mJ(1)μ = −1

4π√−g

εαβFαβ,

where εαβ is an antisymmetry tensor density with εtr = εrt = 1and Fαβ = ∂αAβ − ∂βAα .

In our case, there are two U(1) gauge symmetries yieldingtwo gauge currents J (0)r and J (1)r , corresponding to the gaugepotentials A

(0)t and A

(1)t , respectively. We shall adopt Eq. (15) to

derive the current with respect to the gauge potential A(0)t . Near

the horizon, the current becomes anomalous while it is still con-served outside the horizon. Thus, if we split the region outsidethe horizon into two parts: [r+, r+ + ε) and [r+ + ε,+∞),and introduce two step functions Θ(r) = Θ(r − r+ − ε) andH(r) = 1 − Θ(r), then we can write the total current in termsof the conserved one J

(0)μ

(O) and the anomalous one J(0)μ

(H) as

(16)J (0)μ = J(0)μ

(O) Θ(r) + J(0)μ

(H) H(r),

where J(0)μ

(O) and J(0)μ

(H) satisfy ∇μJ(0)μ

(O) = 0 and Eq. (15), re-spectively. Both equations can be solved as

√−gJ(0)r(O) = c

(0)O ,

(17)√−gJ

(0)r(H) = c

(0)H + e

2π

[At −At (r+)

],

J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306 303

where the charge flux c(0)O and c

(0)H are two integration constants.

Substituting Eq. (16) into Eq. (15), the Ward identity becomes

∂r

[√−gJ (0)r] = ∂r

(e

2πAtH

)+

{√−g[J

(0)r(O) − J

(0)r(H)

](18)+ e

2πAt

}δ(r − r+ − ε).

In order to make the current anomaly free under the gauge trans-formation, the first term must be cancelled by the classicallyirrelevant ingoing modes and the second term should vanish atthe horizon, which reads

(19)c(0)O = c

(0)H − e

2πAt (r+).

Imposing the boundary condition that the covariant current van-ishes at the horizon, which means c

(0)H = 0, then the charge flux

with respect to the gauge potential A(0)t is given by

(20)c(0)O = − e

2πAt (r+).

Similarly, the current corresponding to the gauge potential A(1)t

is

(21)c(1)O = − m

2πAt (r+).

From Eq. (15), one can see that J (0)r and J (1)r are not in-dependent for there exists the relation 1

eJ (0)r = 1

mJ (1)r = J r

between them, where J μ satisfies the covariant gauge anomalyequation

(22)∇μJ μ = −1

4π√−g

εαβFαβ.

By analogy, the charge flux at spatial infinity can be obtained as

J r (r → ∞) = cO = − 1

2πAt (r+)

(23)= − 1

2π

(eQ

Ξr++ ma

Ξl2

).

With the expression of the charge flux in hand, we now payour attention to the energy–momentum flux. Near the horizon, ifwe omit the quantum effect of the ingoing modes, the effectivefield theory will exhibit a gravitational anomaly. For the right-handed fields, the (1 + 1)-dimensional covariant gravitationalanomaly is expressed as [20]

(24)∇μT μν = −1

96π

√−gεμν∂μR = 1√−g

∂μNμν = Aν.

In the case of a background spacetime with the effective met-ric (12), the anomaly is timelike (Ar = 0), and

At = 1√−g∂rN

rt ,

(25)Nrt = 1

96π

(f h′′ − f h′2

h+ 1

2h′f ′

).

Now we take the effect from the charged field into account. Theenergy–momentum outside the horizon satisfies the Lorentz

force law

(26)∇μTμ

(O)ν =FμνJ μ

(O),

while the energy–momentum near the horizon obeys the anom-alous Ward identity after adding the gravitational anomaly,

(27)∇μTμ

(H)ν =FμνJ μ

(H) + Aν.

Writing the total energy–momentum tensor as

(28)T μν = T

μ

(O)νΘ(r) + Tμ

(H)νH(r),

we solve both equations for the ν = t component and get

(29)√−gT r

(O)t = aO + cOAt ,

(30)√−gT r

(H)t = aH +(

cOAt + 1

4πA2

t + Nrt

)∣∣∣∣r

r+,

where aO and aH are two constants. Similar to the case of thecharge current, we insert Eq. (28) into (27) and only considerthe ν = t component as before, then we arrive at

√−g∇μTμt = cO∂rAt + ∂r

[(1

4πA2

t + Nrt

)H

]

+[√−g

(T r

(O)t − T r(H)t

)(31)+ 1

4πA2

t + Nrt

]δ(r − r+ − ε).

The first term is the classical effect of the background electricfield for constant current flow. The second term should be can-celled by the quantum effect of the classically irrelevant ingoingmodes. In order to keep energy–momentum tensor anomalyfree, the third term must vanish at the horizon, which yields

aO = aH + 1

4πA2

t (r+) − Nrt (r+)

(32)= aH + 1

4πA2

t (r+) + 1

192πh′(r+)f ′(r+).

This equation is not enough to fix aO completely. It is necessaryto impose the boundary condition that the covariant energy–momentum tensor vanishes at horizon, i.e., aH = 0. Therefore,the Hawking flux of the total energy–momentum tensor at spa-tial infinity is

(33)T rt (r → ∞) = aO = 1

4πA2

t (r+) + κ2

48π,

from which the Hawking temperature can be determined as

(34)T = κ

2π= 1

4π

√h′(r+)f ′(r+) = f ′(r+)

4πΞ.

3. Hawking radiation of higher dimensional black stringsthrough covariant anomalies

In this section, we shall consider the case of higher dimen-sional black strings in AdS spacetimes. In d-dimensions, there

304 J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306

are, in general, n ≡ [(d − 1)/2] rotation parameters. Parame-terized by the mass M , the charge Q, and the rotation parame-ters ai , the metric and gauge potential for the higher dimen-sional black string solution take the following forms [18]

ds2 = −[Ξ2f (r) + (

1 − Ξ2) r2

l2

]dt2

− 2

[r2

l2− f (r)

]Ξai dt dφi

+{[

r2

l2− f (r)

]aiaj + r2δij

}dφi dφj

+ dr2

f (r)+ r2δab dza dzb,

(35)A = Q

rd−3

√d − 2

2d − 6

(Ξ dt − ai dφi

),

where

f (r) = r2

l2− 2M

rd−3+ Q2

r2d−6,

(36)Ξ =(

1 +n∑

i=1

a2i

l2

)1/2

.

In the above equations, indices i, j = 1, . . . , n, and a, b =1, . . . , d −2−n. δab dza dzb is the Euclidean metric on (d −2−n)-dimensional sub-manifold with the volume Vd−2−n, whichcan be set to unit for simplicity. The coordinates φi assume thevalues 0 � φi < 2π . The metric (35) can be regarded as blackstrings or black branes with toroidal, cylindrical or planar hori-zons, relying on its global identifications.

As before, we first study thermodynamics of black stringsin higher dimensions. The horizon of the black strings is de-termined by the equation f (r) = 0, namely, the largest positiveroot of f (r+) = 0. The angular velocities and electrostatic po-tential on the horizon are

(37)Ωi = ai

Ξl2, Φ =

√d − 2

2d − 6

Q

Ξrd−3+,

while the Bekenstein–Hawking entropy and the surface gravityof the horizon are given by

S = Vd−2

4Ξrd−2+ ,

(38)κ = f ′(r+)

2Ξ= (d − 1)r2d−4+ − (d − 3)Q2l2

2Ξl2r2d−5+,

where Vd−2 = (2π)nVd−2−n is the volume of the hypersurfacewith constant t and r .

It is of no difficulty to check that the mass, the angularmomenta and the electric charge of black strings in higher di-mensions

M= Vd−2

8π

[(d − 1)Ξ2 − 1

]M,

J i = (d − 1)Vd−2

8πΞMai,

(39)Q = √2(d − 2)(d − 3)

Vd−2ΞQ,

8π

obey the differential and integral Bekenstein–Smarr mass for-mulae [19]

(40)δM = T δS + ΩiδJ i + ΦδQ+ Θδl,

(41)d − 3

d − 2M= T S + ΩiJ i + d − 3

d − 2ΦQ+ 1

d − 2Θl.

In order to close the first law of thermodynamics, we have in-troduced the generalized force

(42)Θ = − (d − 2)rd−1+ + (d − 1)Ml2(Ξ2 − 1)

8πl3Vd−2,

which is conjugate to a variable cosmological constant l. Re-markably, the above relations provide a strong support for thedefinitions of the mass, the angular momenta and the electriccharge for the black strings.

Now we turn to perform a dimension reduction of the met-ric (35). As usual, we will consider the action of a masslessscalar field near the horizon

S[ϕ] = 1

2

∫dt dr dnφ dd−2−nz rd−2ϕ

×{−Ξ2r2 + l2(1 − Ξ2)f (r)

r2f (r)

×[Dt + r2/l2 − f (r)

Ξ2r2 + l2(1 − Ξ2)f (r)ΞaiDφi

]2

+ 1

rd−2∂r

[rd−2f (r)∂r

]+ 1

r2

[δij − r2/l2 − f (r)

Ξ2r2 + l2(1 − Ξ2)f (r)aiaj

]

×DφiDφj + 1

r2δab∂za ∂zb

}ϕ

� 1

2

∑m

∫dt dr rd−3

√Ξ2r2 + l2

(1 − Ξ2

)f (r)ϕm

×{−

√Ξ2r2 + l2(1 − Ξ2)f (r)

rf (r)

×[∂t + ieQΞr5−d

Ξ2r2 + l2(1 − Ξ2)f (r)

√d − 2

2d − 6

+ imiaiΞ [r2/l2 − f (r)]Ξ2r2 + l2(1 − Ξ2)f (r)

]2

(43)+ rf (r)√Ξ2r2 + l2(1 − Ξ2)f (r)

∂2r

}ϕm.

In the process of dimension reduction, we have implementeda partial wave decomposition ϕ = ∑

m ϕm(t, r) exp(imiφi +ikza za), in which m = {mi}. The above procedure demonstratesthat the physics near the horizon can be effectively depicted byan infinite collection of massless scalar fields in terms of thefollowing background metric and n + 1 gauge potentials

ds2 = −F(r) dt2 + dr2

F(r),

(44)F(r) = rf (r)√Ξ2r2 + l2(1 − Ξ2)f (r)

,

J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306 305

At = eA(0)t + miA

(i)t

= eQΞr5−d

Ξ2r2 + l2(1 − Ξ2)f (r)

√d − 2

2d − 6

(45)+ miaiΞ [r2/l2 − f (r)]Ξ2r2 + l2(1 − Ξ2)f (r)

,

together with the dilaton Ψ = rd−3√

Ξ2r2 + l2(1 − Ξ2)f (r),which makes no contribution to the anomalies. In the effectivegauge potential (45), A

(i)t are the induced U(1) gauge fields

originated from the n independent axial isometries. Each in-duced gauge field has an azimuthal quantum number serving asthe corresponding charge for each partial mode. So there are, intotal, the n + 1 gauge potentials yielding n + 1 gauge currentsJ (0)r and J (i)r which satisfy Eq. (15).

It is a position to carry out the anomaly analysis for thehigher dimensional black strings. The analysis is completely inparallel with that in the preceding section, but it is simpler thanever, since

√−g = 1 for the effective metric (44) in the caseconsidered now. Therefore, following the procedure in the lastsection, it is easy to obtain the gauge currents

(46)c(0)O = − e

2πAt (r+), c

(i)O = −mi

2πAt (r+).

With the aid of the definition J r = 1eJ (0)r = 1

mi J(i)r , the

charge flux can be computed as

(47)J r (r → ∞) = cO = − 1

2πAt (r+) = − 1

2π

(eΦ + miΩi

).

Analogously, the energy flux of energy–momentum tensor atspatial infinity can be derived as

(48)T rt (r → ∞) = aO = 1

4πA2

t (r+) + κ2

48π,

from which the Hawking temperature is deduced as

(49)T = κ

2π= F ′(r+)

4π= f ′(r+)

4πΞ.

4. Summary

In this Letter, we have studied thermodynamical and ther-mal properties of the (3+1)-dimensional rotating charged blackstrings and their higher dimensional extensions in AdS space-times. Firstly, we have generalized the first law of thermody-namics to the case of a variable cosmological constant. Thenwe applied the covariant anomaly cancellation method to de-rive the charge and energy–momentum fluxes from the horizonof these black strings, whose horizons are non-spherical but canbe toroidal, cylindrical or planar, according to their global iden-tifications. In both cases of the black strings, after performinga partial wave decomposition and a dimension reduction, thescalar field theory near the horizon of the original spacetimecan be effectively described by the (1 + 1)-dimensional chiraltheory, in which each rotation symmetry induces an effectiveU(1) gauge potential with the azimuthal quantum number act-ing as the charge of this gauge symmetry. Therefore, the (1+1)-dimensional field theory can be regarded as that of the charged

particles interacting with the effective U(1) background gaugefield.

Our results imply that the method of anomaly cancellationcan succeed in deriving the Hawking fluxes from arbitrary di-mensional AdS spacetimes with non-spherical horizons. Thederivation of Hawking radiation only relies on the covariantanomalies for gauge current and energy–momentum tensor, to-gether with the regular requirements of the covariant chargecurrent and energy–momentum flux at the horizon. Besides, ouranomaly analysis presented in this article is very general sincewe do not restrict ourselves to the simple case where the deter-minant of the reduced (1 + 1)-dimensional effective metric isequal to one.

Acknowledgements

S.-Q. Wu was partially supported by the Natural ScienceFoundation of China under Grant No. 10675051.

Appendix A. Non-uniqueness of the reducedtwo-dimensional effective metrics

In this appendix, we take the four-dimensional case as an ex-ample to show that the resultant two-dimensional effective andphysically equivalent metrics are non-unique in the process ofdimensional reduction, and they can differ by a regular, confor-mal factor.

The near-horizon limit of the action in Eq. (11) suggests thatit can be effectively replaced by the following action

S[ϕ] ≡ 1

2

∑m

∫dt dr Ψ ϕm

[−

√grr

−gtt

(∂t + iAt )2

(A.1)+√

−gtt

grr

∂2r

]ϕm

in the two-dimensional spacetime with an effective metricds2 = gtt dt2 + grr dr2. Compare Eq. (11) with (A.1), we get

Ψ

√grr

−gtt

= Ξ2r2 − a2f (r)

f (r),

(A.2)Ψ

√−gtt

grr

= r2f (r).

From these equations, we can solve the dilaton factor as Ψ =r√

Ξ2r2 − a2f (r), and obtain

(A.3)

√−gtt

grr

= rf (r)√Ξ2r2 − a2f (r)

.

With the help of the determinant√−g = √−gttgrr = G(r),

where G(r) is an arbitrary function regular at the horizon, themetric components can be obtained as follows

gtt = − rf (r)√Ξ2r2 − a2f (r)

G(r),

(A.4)grr =√

Ξ2r2 − a2f (r)G(r).

rf (r)

306 J.-J. Peng, S.-Q. Wu / Physics Letters B 661 (2008) 300–306

Thus the reduced two-dimensional effective metric can be writ-ten as a conformal form

ds2 = G(r)

[− rf (r)√

Ξ2r2 − a2f (r)dt2

(A.5)+√

Ξ2r2 − a2f (r)

rf (r)dr2

],

with√−g = G(r) acting as the conformal factor. Therefore,

the resultant two-dimensional effective metric (A.5) is non-unique in the process of dimensional reduction.

To ensure the physical equivalence of the reduced two-dimensional metric (A.5) with the original four-dimensionalone (2), the zeros and signs of the t t and rr components ofthe metric (2) in the dragging coordinate system should not bechanged in the process of dimensional reduction. This meansthat G(r+) �= 0. On the other hand, since the surface gravityκ ,

√−gJ r and√−gT r

t are all conformal invariant physicalquantities in two dimensions, one must demand G(r → ∞) = 1to obtain consistent physical results of the Hawking fluxesunder the conformal transformation. There are a lot of regu-lar functions

√−g = G(r) that posses these two properties.One of the alternatives is

√−g = r/√

Ξ2r2 − a2f (r) in Sec-tion 2. Another simple and natural choice is

√−g = 1, asdone in the higher dimensional case in Section 3. In thesetwo cases, the metrics differ by a regular conformal factor√−g = r/

√Ξ2r2 − a2f (r), both of them give the consistent

results in the anomaly analysis.

References

[1] S.W. Hawking, Nature (London) 248 (1974) 30;S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.

[2] S.P. Robinson, F. Wilczek, Phys. Rev. Lett. 95 (2005) 011303, gr-qc/0502074.

[3] S. Iso, H. Umetsu, F. Wilczek, Phys. Rev. Lett. 96 (2006) 151302, hep-th/0602146.

[4] S. Iso, H. Umetsu, F. Wilczek, Phys. Rev. D 74 (2006) 044017, hep-th/0606018;K. Murata, J. Soda, Phys. Rev. D 74 (2006) 044018, hep-th/0606069;S. Iso, T. Morita, H. Umetsu, JHEP 0704 (2007) 068, hep-th/0612286.

[5] E.C. Vagenas, S. Das, JHEP 0610 (2006) 025, hep-th/0606077;M.R. Setare, Eur. Phys. J. C 49 (2007) 865, hep-th/0608080;K. Xiao, W.B. Liu, H.B. Zhang, Phys. Lett. B 647 (2007) 482, hep-th/0702199;Z.B. Xu, B. Chen, Phys. Rev. D 75 (2007) 024041, hep-th/0612261;H. Shin, W. Kim, JHEP 0706 (2007) 012, arXiv: 0705.0265 [hep-th];W. Kim, H. Shin, JHEP 0707 (2007) 070, arXiv: 0706.3563 [hep-th];S. Das, S.P. Robinson, E.C. Vagenas, arXiv: 0705.2233 [hep-th].

[6] Q.Q. Jiang, S.Q. Wu, Phys. Lett. B 647 (2007) 200, hep-th/0701002;Q.Q. Jiang, S.Q. Wu, X. Cai, Phys. Lett. B 651 (2007) 58, hep-th/0701048;Q.Q. Jiang, S.Q. Wu, X. Cai, Phys. Lett. B 651 (2007) 65, arXiv:0705.3871 [hep-th];Q.Q. Jiang, S.Q. Wu, X. Cai, Phys. Rev. D 75 (2007) 064029, hep-th/0701235;S.Q. Wu, Phys. Rev. D 76 (2007) 029904, Erratum;Q.Q. Jiang, Class. Quantum Grav. 24 (2007) 4391, arXiv: 0705.2068 [hep-th].

[7] K. Murata, U. Miyamoto, Phys. Rev. D 76 (2007) 084038, arXiv:0707.0168 [hep-th].

[8] U. Miyamoto, K. Murata, Phys. Rev. D 77 (2008) 024020, arXiv:0705.3150 [hep-th];B. Chen, W. He, arXiv: 0705.2984 [gr-qc].

[9] Z.Z. Ma, arXiv: 0709.3684 [hep-th];C.G. Huang, J.R. Sun, X.N. Wu, H.Q. Zhang, arXiv: 0710.4766 [hep-th];X.N. Wu, C.G. Huang, J.R. Sun, arXiv: 0801.1347 [gr-qc];X.X. Zeng, D.Y. Chen, S.Z. Yang, Chin. Phys. B 17, in press;T.M. He, Y.J. Wang, P.H. Hu, J.H. Fan, Chin. Phys. B 17, in press.

[10] S.Q. Wu, J.J. Peng, Class. Quantum Grav. 24 (2007) 5123, arXiv:0706.0983 [hep-th];J.J. Peng, S.Q. Wu, arXiv: 0705.1225 [hep-th], Chin. Phys. B 17, in press.

[11] J.J. Peng, S.Q. Wu, arXiv: 0709.0044 [hep-th].[12] R. Banerjee, S. Kulkarni, Phys. Rev. D 77 (2008) 024018, arXiv:

0707.2449 [hep-th].[13] S.Q. Wu, Z.Y. Zhao, arXiv: 0709.4074 [hep-th].[14] J.J. Peng, S.Q. Wu, arXiv: 0709.0167 [hep-th];

K. Lin, X.X. Zeng, S.Z. Yang, Chin. Phys. Lett. 25 (2008) 390;X.X. Zeng, D.Y. Chen, S.Z. Yang, Gen. Relativ. Gravit. 40, in press;Y.W. Han, X.X. Zeng, S.Z. Yang, Int. J. Theor. Phys. 47, in press.

[15] R. Banerjee, S. Kulkarni, Phys. Lett. B 659 (2008) 827, arXiv: 0709.3916[hep-th];S. Gangopadhyay, S. Kulkarni, Phys. Rev. D 77 (2008) 024038, arXiv:0710.0974 [hep-th];S. Gangopadhyay, arXiv: 0712.3095 [hep-th].

[16] J.M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200;E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150;A. Strominger, JHEP 0110 (2001) 034, hep-th/0106113.

[17] K. Akama, Lect. Notes Phys. 176 (1982) 267, hep-th/0001113;V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 (1983) 136;V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 (1983) 139;M. Visser, Phys. Lett. B 159 (1985) 22, hep-th/9910093;G.W. Gibbons, D.L. Wiltshire, Nucl. Phys. B 287 (1987) 717, hep-th/0109093;P. Horava, E. Witten, Nucl. Phys. B 460 (1996) 506, hep-th/9510209;P. Horava, E. Witten, Nucl. Phys. B 475 (1996) 94, hep-th/9603142;N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263,hep-ph/9803315;I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett.B 436 (1998) 257, hep-ph/9804398;P. Argyres, S. Dimopoulos, J. March-Russell, Phys. Lett. B 441 (1998) 96,hep-th/9808138;L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hep-ph/9905221;L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hep-th/9906064;M. Gogberashvili, Mod. Phys. Lett. A 14 (1999) 2025, hep-ph/9904383;M. Gogberashvili, Europhys. Lett. 49 (2000) 396, hep-ph/9812365;N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Rev. D 59 (1999)086004, hep-ph/9807344;G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 484 (2000) 112, hep-th/0002190;G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485 (2000) 208, hep-th/0005016.

[18] A. Awad, Class. Quantum Grav. 20 (2003) 2827, hep-th/0209238.[19] S. Wang, S.Q. Wu, F. Xie, L. Dan, Chin. Phys. Lett. 23 (2006) 1096, hep-

th/0601147.[20] L. Alvarez-Gaume, E. Witten, Nucl. Phys. B 234 (1984) 269;

W.A. Bardeen, B. Zumino, Nucl. Phys. B 244 (1984) 421;R. Bertlmann, E. Kohlprath, Ann. Phys. (N.Y.) 288 (2001) 137, hep-th/0011067.