Hawking radiation from a Reissner-Nordstrom black hole with a global monopole via covariantanomalies and effective action

Sunandan Gangopadhyay*

S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India(Received 25 March 2008; published 13 August 2008)

We adopt the covariant anomaly cancellation method as well as the effective action approach to obtain

the Hawking radiation from the Reissner-Nordstrom blackhole with a global monopole falling in the class

of the most general spherically symmetric charged blackhole (ffiffiffiffiffiffiffi�g

p� 1), using only covariant boundary

conditions at the event horizon.

DOI: 10.1103/PhysRevD.78.044026 PACS numbers: 04.70.Dy, 03.65.Sq, 04.62.+v

I. INTRODUCTION

Hawking radiation is an important and prominent quan-tum effect arising from the quantization of matter fields ina background space-time with an event horizon. The ra-diation is found to have a spectrum with Planck distribu-tion giving the black holes one of its thermodynamicproperties. Apart from the original derivation byHawking [1,2], there is a tunneling picture [3,4] based onpair creations of particles and antiparticles near the horizonwhich calculates WKB amplitudes for classically forbid-den paths. A common feature in these derivations is theuniversality of the radiation: i.e. Hawking radiation isdetermined universally by the horizon properties (if weneglect the gray body factor induced by the effect ofscattering outside the horizon).

Recently, Robinson and Wilczek [5] proposed an inter-esting approach to derive Hawking radiation from aSchwarzschild-type black hole through a gravitationalanomaly. The method was soon extended to the case ofcharged black holes [6]. Further applications of this ap-proach may be found in [7–13]. The basic idea in [5,6] isthat the effective theory near the horizon becomes twodimensional and chiral. This chiral theory is anomalous.Using the form for two-dimensional consistent gauge/gravitational anomaly, Hawking fluxes are obtained.However, the boundary condition necessary to fix theparameters are obtained from a vanishing of covariantcurrent and energy-momentum tensor at the horizon.Soon after, the analysis of [5,6] was reformulated in[14,15], in terms of covariant expressions only. The gen-eralization of this approach to a higher spin field has beendone in [16].

An alternative derivation of a Hawking flux based oneffective action using only covariant anomaly has beendiscussed in [17–19]. This approach is particularly usefulsince only the exploitation of a known structure of effectiveaction near the horizon is sufficient to determine theHawking flux. An important ingredient in this method isonce again to realize that the effective theory near the event

horizon becomes two dimensional and chiral. Anotherimportant aspect in this approach is the imposition ofcovariant boundary conditions only at the horizon.In this paper, we first adopt the covariant anomaly

cancellation approach [14] to discuss Hawking radiationfrom a Reissner-Nordstrom blackhole with a global mono-pole [20] which is an example of the most general spheri-cally symmetric charged black hole space-time(

ffiffiffiffiffiffiffi�gp

� 1). Finally we adopt the effective action approach[17] to reproduce the same result. However, as in [18], weshall once again ignore effects to the Hawking flux due toscatterings by the gravitational potential, for example, thegreybody factor [21].

II. HAWKING RADIATION FROM REISSNER-NORDSTROM BLACKHOLE WITH A GLOBAL

MONOPOLE

The metric of a general nonextremal Reissner-Nordstrom blackhole with a global monopoleOð3Þ is givenby [20]

ds2string ¼ pðrÞdt2 � 1

hðrÞdr2 � r2d�2; (1)

where,

A ¼ q

rdt; pðrÞ ¼ hðrÞ ¼ 1� �2 � 2m

rþ q2

r2; (2)

with m being the mass parameter of the black hole and �related to the symmetry breaking scale when the globalmonopole is formed during the early universe soon afterthe big-bang [22]. The event horizon for the above blackhole is situated at

rH ¼ ð1� �2Þ�1

�mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � ð1� �2Þq2

q �: (3)

It has been argued in [11] that since the metric (1) is nolonger asymptotically flat, the well-known formula1

*sunandan@bose.res.in

1Note that a spherically symmetric asymptotically boundedspace-time metric without any loss of generality, can be cast inthe form ds2 ¼ gttdt

2 þ grrdr2 þ r2d�2.

PHYSICAL REVIEW D 78, 044026 (2008)

1550-7998=2008=78(4)=044026(7) 044026-1 � 2008 The American Physical Society

�¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffi�grr

gtt

sðgtt;rÞjr¼rH ; gtt ¼ pðrÞ; grr ¼�hðrÞ

(4)

for computing the surface gravity for a general sphericallysymmetric asymptotically flat metric becomes problematicto be applied in the case described by the metric (1) as itdoes not correspond to the normalized timelike Killingvector. The correct surface gravity of the metric (1) is

� ¼ 1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p p0ðrHÞ (5)

since it corresponds to the normalized timelike Killingvector

l�ðtÞ ¼ ð1� �2Þ�1=2ð@tÞ�: (6)

It is for this reason that the anomaly cancellation method aswell as the effective action approach cannot be immedi-ately used to obtain the consistent formula of the Hawkingtemperature for the metric (1). Nevertheless, we can do thesame analysis in another different way. By rescaling t !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

pt, we can rewrite the metric (1) as

ds2 ¼ fðrÞdt2 � hðrÞ�1dr2 þ r2d�2;

fðrÞ ¼ ð1� �2ÞhðrÞ; hðrÞ ¼ 1� �2 � 2m

r� q2

r2

(7)

and immediately derive the expected result for the

Hawking temperature T ¼ f0ðrHÞ=ð4�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p Þ. Hence,we shall apply the anomaly cancellation method and theeffective action approach to the above form of the metric(7). The important point to note is the determinant of theabove metric

ffiffiffiffiffiffiffi�gp

� 1.

III. ANOMALY CANCELLATION APPROACH

With the aid of dimensional reduction procedure, onecan effectively describe a theory with a metric given by the‘‘r� t’’ sector of the full space-time metric (7) near thehorizon.

Now we divide the space-time into two regions anddiscuss the gauge/gravitational anomalies separately.

IV. GAUGE ANOMALY

Since the space-time has been divided into two regions,we divide the current J� into two parts. The current outsidethe horizon denoted by J

�ðoÞ is anomaly free and hence

satisfies the conservation law

r�J�ðoÞ ¼ 0: (8)

Near the horizon there are only outgoing (right-handed)fields and the current becomes covariantly anomalous andsatisfies [6]

r�J�ðHÞ ¼ � e2

4�����F�� ¼ e2

2�ffiffiffiffiffiffiffi�g

p @rAt; (9)

where, ���� ¼ ���=ffiffiffiffiffiffiffi�g

pand ���� ¼ ffiffiffiffiffiffiffi�g

p��� are two-

dimensional antisymmetric tensors for the upper and lowercases with �tr ¼ �rt ¼ 1.Now outside the horizon, the conservation equation (8)

yields the differential equation

@rð ffiffiffiffiffiffiffi�gp

JrðoÞÞ ¼ 0; (10)

whereas in the region near the horizon, the anomaly equa-tion (9) leads to the following differential equation:

@rð ffiffiffiffiffiffiffi�gp

JrðHÞÞ ¼e2

2�@rAt: (11)

Solving (10) and (11) in the region outside and near thehorizon, we get

JrðoÞðrÞ ¼coffiffiffiffiffiffiffi�g

p ; (12)

JrðHÞðrÞ ¼1ffiffiffiffiffiffiffi�g

p�cH þ e2

2�

Z r

rH

@rAtðrÞ�

¼ 1ffiffiffiffiffiffiffi�gp

�cH þ e2

2�½AtðrÞ � AtðrHÞ�

�; (13)

where, co and cH are integration constants. Now as in [6],writing JrðrÞ as

JrðrÞ ¼ JrðoÞðrÞ�ðr� rH � �Þ þ JrðHÞðrÞHðrÞ; (14)

where, HðrÞ ¼ 1� ðr� rH � �Þ, we find

r�J� ¼ @rJ

rðrÞ þ @r

�ln

ffiffiffiffiffiffiffi�gp ÞJrðrÞ

¼ 1ffiffiffiffiffiffiffi�gp @rð ffiffiffiffiffiffiffi�g

pJrðrÞ

�

¼ 1ffiffiffiffiffiffiffi�gp

�� ffiffiffiffiffiffiffi�gp ðJrðoÞðrÞ � JrðHÞðrÞÞ þ

e2

2�AtðrÞ

�

� ðr� rþ � �Þ þ @r

�e2

2�AtðrÞHðrÞ

��: (15)

The term in the total derivative is cancelled by quantumeffects of classically irrelevant ingoing modes. Hence thevanishing of the Ward identity under gauge transformationimplies that the coefficient of the delta function is zero,leading to the condition

JrðoÞðrÞ � JrðHÞðrÞÞ þe2

2�ffiffiffiffiffiffiffi�g

p AtðrÞ ¼ 0: (16)

Substituting (12) and (13) in the above equation, we get

co ¼ cH � e2

2�AtðrHÞ: (17)

The coefficient cH vanishes by requiring that the covariant

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current JrðHÞðrÞ vanishes at the horizon. Hence the charge

flux corresponding to JrðrÞ is given by

co ¼ ffiffiffiffiffiffiffi�gp

JrðoÞðrÞ ¼ � e2

2�AtðrHÞ ¼ � e2q

2�rH: (18)

This is precisely the charge flux obtained in [11] using theRobinson-Wilczek method of cancellation of consistentgauge anomaly.

V. GRAVITATIONAL ANOMALY

In this case also, since the theory is free from anomaly inthe region outside the horizon, we have the energy-momentum tensor satisfying the conservation law

r�T�ðoÞ� ¼ F��J

�ðoÞ: (19)

However, the omission of the ingoing modes in the regionr 2 ½rþ;1� near the horizon leads to an anomaly in theenergy-momentum tensor there. As we have mentionedearlier, in this paper we shall focus only on the covariantform of the d ¼ 2 gravitational anomaly given by [5,6]

r�T�ðHÞ� ¼ F��J

�ðHÞ þ

1

96�����@

�R ¼ F��J�ðHÞ þA�:

(20)

It is easy to check that for the metric (1), the two-dimensional Ricci scalar R is given by

R ¼ hf00

fþ f0h0

2f� f02h

2f2; (21)

and the anomaly is purely timelike with

A r ¼ 0; At ¼ 1ffiffiffiffiffiffiffi�gp @rN

rt ; (22)

where,

Nrt ¼ 1

96�

�hf00 þ f0h0

2� f02h

f

�: (23)

We now solve the above Eqs. (19) and (20) for the � ¼ tcomponent. In the region outside the horizon, the conser-vation equation (19) yields the differential equation

@rð ffiffiffiffiffiffiffi�gp

TrðoÞtÞ ¼

ffiffiffiffiffiffiffi�gp

FrtJrðoÞðrÞ ¼ co@rAt; (24)

where we have used Frt ¼ @rAt and (12). Integrating theabove equation leads to

TrðoÞtðrÞ ¼

1ffiffiffiffiffiffiffi�gp ðao þ coAtðrÞÞ; (25)

where, ao is an integration constant. In the region near thehorizon, the anomaly equation (20) leads to the followingdifferential equation:

@rð ffiffiffiffiffiffiffi�gp

TrðHÞtÞ ¼

ffiffiffiffiffiffiffi�gp

FrtJrðHÞðrÞ þ @rN

rt ðrÞ

¼�cH þ e2

2�½AtðrÞ � AtðrHÞ�

�@rAtðrÞ

þ @rNrt ðrÞ

¼ @r

�e2

2�

�1

2A2t ðrÞ � AtðrHÞAtðrÞ

�þ Nr

t ðrÞ�;

(26)

where we have used (13) in the second line and set cH ¼ 0in the last line of the above equation. Integration of theabove equation leads to

TrðHÞtðrÞ ¼

1ffiffiffiffiffiffiffi�gp

�bH þ

Z r

rH

@r

�e2

2�

�1

2A2t ðrÞ

� AtðrHÞAtðrÞ�þ Nr

t ðrÞ��

¼ 1ffiffiffiffiffiffiffi�gp

�bH þ e2

4�½A2

t ðrÞ þ A2t ðrHÞ�

� e2

2�AtðrHÞAtðrÞ þ Nr

t ðrÞ � Nrt ðrHÞ

�; (27)

where, bH is an integration constant.Writing the energy-momentum tensor as a sum of two

contributions [6]

TrtðrÞ ¼ Tr

ðoÞtðrÞðr� rH � �Þ þ TrðHÞtðrÞHðrÞ; (28)

we find

r�T�t ¼ @rT

rtðrÞ þ @rðln ffiffiffiffiffiffiffi�g

p ÞTrtðrÞ

¼ 1ffiffiffiffiffiffiffi�gp @rð ffiffiffiffiffiffiffi�g

pTr

tðrÞÞ

¼ 1ffiffiffiffiffiffiffi�gp

�� e2

2�AtðrHÞ@rAtðrÞ þ

� ffiffiffiffiffiffiffi�gp ðTr

ðoÞtðrÞ

� TrðHÞtðrÞÞ þ

e2

4�A2t ðrÞ þ Nr

t ðrÞ�ðr� rþ � �Þ

þ @r

��e2

4�A2t ðrÞ þ Nr

t ðrÞ�HðrÞ��

; (29)

where we have substituted the value of c0 from (18) in thelast line.Now the first term in the above equation is a classical

effect coming from the Lorentz force. The term in the totalderivative is once again cancelled by quantum effects ofclassically irrelevant ingoing modes. The quantum effect tocancel this term is the Wess-Zumino term induced by theingoing modes near the horizon. Hence the vanishing of theWard identity under diffeomorphism transformation im-plies that the coefficient of the delta function in the aboveequation vanishes

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TrðoÞt � Tr

ðHÞt þ1ffiffiffiffiffiffiffi�g

p�e2

4�A2t ðrÞ þ Nr

t ðrÞ�¼ 0: (30)

Substituting (25) and (27) in the above equation, we get

ao ¼ bH þ e2

4�A2t ðrHÞ � Nr

t ðrHÞ: (31)

The integration constant bH can be fixed by imposing thatthe covariant energy-momentum tensor vanishes at thehorizon. From (27), this gives bH ¼ 0. Hence the totalflux of the energy-momentum tensor is given by

ao ¼ e2

4�A2t ðrHÞ � Nr

t ðrHÞ

¼ e2q2

4�r2Hþ 1

192�f0ðrHÞh0ðrHÞ

¼ e2q2

4�r2Hþ 1

192�

f02ðrHÞð1� �2Þ : (32)

This is precisely the Hawking flux obtained in [11] usingthe Robinson-Wilczek method of cancellation of consistentanomaly.

VI. EFFECTIVE ACTION APPROACH

As we mentioned earlier, with the aid of dimensionalreduction technique, the effective field theory near thehorizon becomes a two-dimensional chiral theory with ametric given by the ‘‘r� t’’ sector of the full space-timemetric (7) near the horizon.

We now adopt the methodology in [17]. For a two-dimensional theory the expressions for the anomalous(chiral) and normal effective actions are known [23]. Weshall use only the anomalous form of the effective actionfor deriving the charge and the energy flux. The current andthe energy-momentum tensor in the region near the horizonis computed by taking an appropriate functional derivativeof the chiral effective action. Next, the parameters appear-ing in the solution are fixed by imposing the vanishing ofthe covariant current and energy-momentum tensor at thehorizon. Once these are fixed, the charge and the energyflux are obtained by taking the asymptotic (r ! 1) limit ofthe chiral current and energy-momentum tensors. We alsouse the expression for the normal effective action to estab-lish a connection between the chiral and the normal currentand energy-momentum tensors.

With the above methodology in mind, we write down theanomalous (chiral) effective action (describing the theorynear the horizon) [23]

�ðHÞ ¼ �13zð!Þ þ zðAÞ; (33)

where A� and !� are the gauge field and the spin connec-

tion and

zðvÞ ¼ 1

4�

Zd2xd2y���@�v�ðxÞ��1

g ðx; yÞ� @�½ð��� þ ffiffiffiffiffiffiffi�g

pg��Þv�ðyÞ�; (34)

where �g ¼ r�r� is the Laplacian in this background.

The energy-momentum tensor is computed from a varia-tion of this effective action. To get their covariant forms inwhich we are interested, one needs to add appropriate localpolynomials [23]. Here we quote the final result for thechiral covariant energy-momentum tensor and the chiralcovariant current [23]:

T�� ¼ e2

4�D�BD�B

þ 1

4�

�1

48D�GD�G� 1

24D�D�Gþ 1

24�� R

�;

(35)

J� ¼ � e2

2�D�B; (36)

where D� is the chiral covariant derivative

D� ¼ r� � ����r� ¼ � ����D�: (37)

Also BðxÞ and GðxÞ are given by

BðxÞ ¼Z

d2yffiffiffiffiffiffiffi�g

p��1

g ðx; yÞ ����@�A�ðyÞ; (38)

GðxÞ ¼Z

d2y��1g ðx; yÞ ffiffiffiffiffiffiffi�g

pRðyÞ; (39)

and satisfy

r�r�B ¼ �@rAtðrÞ; r�r�G ¼ R: (40)

The solutions for B and G read

B ¼ BoðrÞ � atþ b; @rBo ¼ 1ffiffiffiffiffiffifh

p ðAtðrÞ þ cÞ; (41)

G ¼ GoðrÞ � 4ptþ q; @rGo ¼ � 1ffiffiffiffiffiffifh

p0@ ffiffiffi

h

f

sf0 þ z

1A;

(42)

where a, b, c, p, q, z are constants of integration.By taking the covariant divergence of (35) and (36), we

get the anomalous Ward identities (9) and (20).In the region away from the horizon, the effective theory

is given by the standard effective action � of a conformalfield with a central charge c ¼ 1 in this black hole back-ground [23] and reads

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� ¼ 1

96�

Zd2xd2y

ffiffiffiffiffiffiffi�gp

RðxÞ 1

�g

ðx; yÞ ffiffiffiffiffiffiffi�gp

RðyÞ

þ e2

2�

Zd2xd2y���@�A�ðxÞ 1

�g

ðx; yÞ���@�A�ðyÞ:

(43)

The covariant energy-momentum tensor T��ðoÞ and the

covarant gauge current J�ðoÞ in the region outside the hori-

zon are given by

T��ðoÞ ¼ 2ffiffiffiffiffiffiffi�gp �

g��

¼ 1

48�ð2g��R� 2r�r�Gþr�Gr�G

� 1

2g��r�Gr�GÞ

þ e2

�

�r�Br�B� 1

2g��r�Br�B

�; (44)

J�ðoÞ ¼�

A� ¼ e2

�����@�B; (45)

and satisfy the normal Ward identities (8) and (19).

VII. CHARGE AND ENERGY FLUX

In this section we calculate the charge and the energyflux by using the expressions for the anomalous covariantgauge current (36) and anomalous covariant energy-momentum tensor (35). We will show that the results arethe same as those obtained by the covariant anomalycancellation method.

Using the solution for BðxÞ (37) and (38) the � ¼ rcomponent of the anomalous (chiral) covariant gauge cur-rent (36) becomes

JrðrÞ ¼ e2

2�

ffiffiffih

f

s½AtðrÞ þ aþ c�: (46)

Now implementation of the boundary condition, namely,the vanishing of the anomalous (chiral) covariant gaugecurrent at the horizon, JrðrHÞ ¼ 0, leads to

aþ c ¼ �AtðrHÞ: (47)

Hence the expression JrðrÞ reads

JrðrÞ ¼ e2

2�

ffiffiffih

f

s½AtðrÞ � AtðrHÞ�: (48)

Now the charge flux is given by the asymptotic (r ! 1)limit of the anomaly free covariant gauge current (45).Now from (9), we observe that the anomaly vanishes inthis limit. Hence the charge flux is abstracted by taking theasymptotic limit of the above equation multiplied by anoverall factor of

ffiffiffiffiffiffiffi�gp

. This yields

c0 ¼ ð ffiffiffiffiffiffiffi�gp

JrÞðr ! 1Þ ¼� ffiffiffi

f

h

sJr�ðr ! 1Þ

¼ � e2

2�AtðrHÞ ¼ � e2q

2�rH; (49)

which agrees with (18).We now consider the normal (anomaly free) covariant

gauge current (45) to establish its relation with the chiral(anomalous) covariant gauge current (48). The � ¼ rcomponent of J�ðoÞ is given by

JrðoÞðrÞ ¼e2

�

ffiffiffih

f

sa: (50)

The asymptotic form of the above Eq. (50) must agree withthe asymptotic form of (46).2 This yields [using (47)]

a ¼ c ¼ �12AtðrHÞ: (51)

Using the above solutions in (46) and (50) yields (48) and

ffiffiffiffiffiffiffi�gp

JrðoÞðrÞ ¼ffiffiffif

h

sJrðoÞðrÞ ¼ � e2

2�AtðrHÞ: (52)

The above expressions (48) and (52) yield the equationbetween the chiral (anomalous) and the normal energy-momentum tensors (16).Now we focus our attention on the gravity sector. Using

the solutions for BðxÞ (38) and GðxÞ (39), the r� t com-ponent of the anomalous (chiral) covariant energy-momentum tensor (35) becomes

TrtðrÞ ¼ e2

4�

ffiffiffih

f

s½AtðrÞ � AtðrHÞ�2

þ 1

12�

ffiffiffih

f

s �p� 1

4

� ffiffiffih

f

sf0 þ z

��2

þ 1

24�

ffiffiffih

f

s � ffiffiffih

f

sf0�p� 1

4

� ffiffiffih

f

sf0 þ z

��

þ 1

4hf00 � f0

8

�h

ff0 � h0

��: (53)

Now implementing the boundary condition, namely, thevanishing of the covariant energy-momentum tensor at thehorizon, Tr

tðrHÞ ¼ 0, leads to

p ¼ 1

4

�z�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0ðrHÞh0ðrHÞ

q �; f0ðrHÞ � f0ðr ¼ rHÞ:

(54)

Using either of the above solutions in (53) yields

2This is true since the anomaly in the asymptotic limit (r !1) vanishes as can be readily seen from (9).

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TrtðrÞ ¼ e2

4�

ffiffiffih

f

s½AtðrÞ � AtðrHÞ�2 þ 1

192�

�ffiffiffih

f

s �f0ðrHÞh0ðrHÞ � 2h

ff02 þ 2hf00 þ f0h0

�:

(55)

The energy flux is now given by the asymptotic (r ! 1)limit of the anomaly free energy-momentum tensor (44).Now from (20), we observe that the anomaly vanishes inthis limit. Hence the energy flux is abstracted by taking theasymptotic limit of the above equation multiplied by anoverall factor of

ffiffiffiffiffiffiffi�gp

. This yields

a0 ¼ ð ffiffiffiffiffiffiffi�gp

TrtÞðr ! 1Þ ¼

� ffiffiffif

h

sTr

t

�ðr ! 1Þ

¼ e2

4�AtðrHÞ þ 1

192�f0ðrHÞh0ðrHÞ

¼ e2q2

4�r2Hþ 1

192�f0ðrHÞh0ðrHÞ; (56)

which correctly reproduces the energy flux (32).We now consider the normal (anomaly free) energy-

momentum tensor (44) to establish its relation with thechiral (anomalous) energy-momentum tensor (55). Ther� t component of T�

�ðoÞ is given by

TrtðoÞðrÞ ¼

e2

�

ffiffiffih

f

sa½AtðrÞ þ c� � 1

12�

ffiffiffih

f

szp

¼ � e2

2�

ffiffiffih

f

sAtðrHÞ

�AtðrÞ � 1

2AtðrHÞ

�

� 1

12�

ffiffiffih

f

szp; (57)

where we have used (51). Once again since the anomaly inthe asymptotic limit (r ! 1) vanishes as can be readilyseen from (20), the asymptotic form of the above Eq. (57)must agree with the asymptotic form of (53). This yields

p ¼ � z

4: (58)

Solving (54) and (58) gives two solutions for p and z:

p ¼ 1

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0ðrHÞh0ðrHÞ

q; z ¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0ðrHÞh0ðrHÞ

q;

p ¼ � 1

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0ðrHÞh0ðrHÞ

q; z ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0ðrHÞh0ðrHÞ

q:

(59)

Using either of the above solutions in (53) and (57) yields(55) and

ffiffiffiffiffiffiffi�gp

TrtðoÞðrÞ ¼

ffiffiffif

h

sTr

tðoÞðrÞ

¼ � e2

2�AtðrHÞ

�AtðrÞ � 1

2AtðrHÞ

�

þ 1

192�f0ðrHÞh0ðrHÞ: (60)

The above expressions (55) and (60) yield the equationbetween the chiral (anomalous) and the normal energy-momentum tensors (30).It should be noted that in the above computation, the

greybody factor (backreaction) has been neglectedthroughout. The effect of the backreaction can, however,be included in the following way in both the approachesdiscussed in this paper. One can show by simple scalingarguments [24] that the horizon can be defined by fðrþÞ ¼hðrþÞ ¼ 0 [in (7)] where rþ is the modified horizon radiusgiven by [25]

rþ ¼ rH

�1þ �

m2

�; (61)

and the constant � is related to the trace anomaly coeffi-cient taking into account the degrees of freedom of thefields [24]. Once this effect is taken into account, it is rþwhich replaces rH in the expressions for the charge flux(49) and the energy flux (56).

VIII. DISCUSSIONS

In this paper, we studied the problem of Hawking radia-tion from a Reissner-Nordstrom black hole with a globalmonopole using the covariant anomaly cancellation tech-nique and effective action approach. The point to note inthe anomaly cancellation method is that Hawking radiationplays the role of cancelling gauge and gravitational anoma-lies at the horizon to restore the gauge/diffeomorphismsymmetry at the horizon. An advantage of this method isthat neither the consistent anomaly nor the countertermrelating the different (covariant and consistent) currents,which were essential ingredients in [6], were required.On the other hand, in the effective action technique, we

only need covariant boundary conditions, the importanceof which was first stressed in [17]. Another important inputin the entire procedure is the expression for the anomalous(chiral) effective action (which yields an anomalous Wardidentity having covariant gauge/gravitational anomaly).The unknown parameters in the covariant current andenergy-momentum tensor derived from this anomalouseffective action were fixed by a boundary condition,namely, the vanishing of the covariant current andenergy-momentum tensor at the event horizon of the blackhole. Finally, the charge and the energy flux were extractedby taking the r ! 1 limit of the chiral covariant currentand energy-momentum tensor. The relation between the

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chiral and the normal current and energy-momentum ten-sors is also established by requiring that both of themmatch in the asymptotic limit which is possible since thegauge/gravitational anomaly vanishes in this limit.

ACKNOWLEDGMENTS

The author would like to thank Professor R. Banerjeeand Mr. Shailesh Kulkarni for useful comments.

[1] S. Hawking, Commun. Math. Phys. 43, 199 (1975).[2] S. Hawking, Nature (London) 248, 30 (1974).[3] M. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042

(2000).[4] E. C. Vagenas, Phys. Lett. B 503, 399 (2001); Mod. Phys.

Lett. A 17, 609 (2002); Phys. Lett. B 533, 302 (2002).[5] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303

(2005).[6] S. Iso, H. Umetsu, and F. Wilczek, Phys. Rev. Lett. 96,

151302 (2006).[7] K. Murata and J. Soda, Phys. Rev. D 74, 044018 (2006).[8] M.R. Setare, Eur. Phys. J. C 49, 865 (2007).[9] S. Iso, T. Morita, and H. Umetsu, J. High Energy Phys. 04

(2007) 068.[10] Q. Q. Jiang and S.Q. Wu, Phys. Lett. B 647, 200 (2007).[11] S. Q. Wu and J. J. Peng, Classical Quantum Gravity 24,

5123 (2007).[12] S. Iso, T. Morita, and H. Umetsu, Phys. Rev. D 76, 064015

(2007).[13] S. Das, S. P Robinson, and E. C. Vagenas, Int. J. Mod.

Phys. D 17, 533 (2008).[14] R. Banerjee and S. Kulkarni, Phys. Rev. D 77, 024018

(2008).[15] S. Gangopadhyay and S. Kulkarni, Phys. Rev. D 77,

024038 (2008).[16] S. Iso, T. Morita, and H. Umetsu, Phys. Rev. D 77, 045007

(2008).[17] R. Banerjee and S. Kulkarni, Phys. Lett. B 659, 827

(2008).[18] S. Gangopadhyay, Phys. Rev. D 77, 064027 (2008).[19] S. Kulkarni, arXiv:0802.2456.[20] M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341

(1989).[21] W. Kim and J. J. Oh, arXiv:0709.1754.[22] J. P. Preskill, Phys. Rev. Lett. 43, 1365 (1979).[23] H. Leutwyler, Phys. Lett. B 153, 65 (1985).[24] J.W. York, Jr., Phys. Rev. D 31, 775 (1985).[25] C. O. Lousto and N. Sanchez, Phys. Lett. B 212, 411

(1988).

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