Thermodynamics and Hawking radiation of five-dimensional rotating charged Gödel black holes

0.22 MB·English

by  Shuang-Qing Wu

Thermodynamics and Hawking radiation of five-dimensional rotating charged G¨odel
black holes
Shuang-Qing Wu ∗ and Jun-Jin Peng †
College of Physical Science and Technology, HuaZhong Normal University,
Wuhan, Hubei 430079, People’s Republic of China
WestudythethermodynamicsofG¨odel-typerotatingchargedblackholesinfive-dimensionalmin-
imal supergravity. These black holes exhibit some peculiar features such as the presence of closed
time-likecurvesandtheabsenceofgloballyspatial-likeCauchysurface. Weexplicitlycomputetheir
energies, angular momenta, and electric charges that are consistent with thefirst law of thermody-
1 namics. Besides, Weextendthecovariant anomaly cancellation method,as well as theapproach of
1 the effective action, to derive their Hawking fluxes. Both the methods of the anomaly cancellation
0 and effective action give the same Hawking fluxes as those from Planck distribution for blackbody
2 radiation in thebackground of thecharged rotating G¨odel black holes. Ourresultsfurthersupport
that Hawking radiation is a quantumphenomenon arising at the eventhorizon.
n
a
PACSnumbers: 04.70.Dy,04.62.+v
J

8
2
I. INTRODUCTION

]
h
G¨odeluniverseisamodelthatdescribestheuniversewithaglobalrotation. Infourdimensions,theG¨odeluniverse
t
- is an exact solution of Einstein field equation with a negative cosmological constant and homogeneous pressure less
p
matter,foundbyG¨odelin1949[1]. Unliketheusualsolutionsingeneralrelativity,thissolutionpossessessomepeculiar
e
h features such as the allowance of closed time-like curves and the absence of globally spatial-like Cauchy surface. It
[ is of great importance for the conceptual development of general relativity. In recent years, much of interest has

been focused on various generalizations of the four dimensional G¨odel universe, particularly in the context of the
1
five dimensional minimal supergravity theory [2–8]. Just as in G¨odel’s original four dimensional solution, all the
v
4 higher dimensionalgeneralizedsolutionspresentclosedtime-like curvesfor alltimes. Furthermore,they canbe easily
7 uplifted to M theory. A remarkable observation also showed that the maximally supersymmetric analogues of the
4 G¨odel universe in [2] are T-dual to pp-waves [3].
5 Among all the G¨odel-type generalizations in five dimensional minimal supergravity, one solution describing a sta-
.
1 tionary Kerr black hole embedded in the rotating G¨odel universe was recently found by Gimon and Hashimoto [4].
0 This solution is not required to preserve any supersymmetry, compared with the supersymmetric one in [2]. Its var-
1 ious properties have been intensively investigated in [9–14, 21]. Particularly in [9], Barnich and Comp`ere proposed
1 an effective method to calculate conserved charges in the G¨odel-type background. They obtained the Kerr G¨odel
:
v black hole’s conserved charges that fulfill the first law of thermodynamics. The charged generalization of the Kerr
i G¨odel black hole has been found by one of the authors [8]. Such a solution is an analytic solution in five dimensional
X
Einstein field equation coupled with Maxwell and Chern-Simons terms in G¨odel background. We shall refer to it as
r Einstein-Maxwell-Chern-Simons-G¨odel (EMCS-G¨odel) black hole. After getting this black hole solution, it is very
a
necessary to study its thermodynamical properties. In this paper, we explicitly compute the mass, angular momenta
and electric charge of the EMCS-Go¨del black hole along the lines of [9]. These conserved charges satisfy the differ-
ential first law and the generalized integral Smarr formula of black hole thermodynamics. However, unlike work [9],
to close the integralSmarrformula, the G¨odelparameter is seen as a thermodynamicalvariable [8]. For the extremal
EMCS-Go¨del black holes, their microscopic entropies can be derived through Kerr/CFT correspondence [14].
In the above, we have mentioned that the EMCS-Go¨del black hole exhibits thermodynamical characters. Thus
there must exist Hawking radiation at its event horizon. This quantum phenomenon is actually very universal and
can be found in any geometry background with event horizons. It is regarded as a clue for seeking the theory
of quantum gravity. Although Hawking radiation has not yet been observed on laboratory, it has been verified by
severaldifferentapproachessinceHawkingdiscoveredthiseffectmorethanthirtyyearsago. Recently,Wilczekandhis
collaboratorsproposedanewderivationofHawkingradiationfromfourdimensionalblackholesviagravitationaland
gauge anomalies [16–18]. In their works, Hawking radiation is treated as a compensating flux to cancel gravitational
and gauge anomalies at the horizon, which arise since the effective field theory becomes two dimensional and chiral
∗ Electronicaddress: [email protected]
† Electronicaddress: [email protected]
2
after performing a procedure of dimensional reduction near the horizon of a black hole. This anomaly cancellation
method supports that Hawking radiation is a common property of the horizon. It is very universal and has been
successfully applied to black objects in various dimensions [19–30]. Noticing that the anomalous energy momentum
tensors and currents encompass two types of forms in the two dimensional chiral effective theory, apart from the
consistent form in [16–18], the other type is the covariant one. In [27], it was argued that Hawking fluxes of energy
momentum tensors and gauge currents can be obtained by cancelling the covariant gravitationalanomaly and gauge
anomaly atthe horizon. Suchanargumentmakesthe originalanomalycancellationmethod [16–18]more economical
and conceptually cleaner. Based on development in [27], several extensions can be found in [28–30]. Especially in
[29], Hawking radiation of black strings in four and higher dimensions has been studied via covariant anomalies.
A notablefeature ofthe anomalycancellationmethodis thatthe boundaryconditionsatthe eventhorizonplayan
important role in determining the Hawking fluxes. Indeed, in [31], by only imposing the boundary condition that the
covariantenergy momentum tensor and the covariantgauge current vanish at the horizon,the chiral effective action,
which describes the two dimensional chiral theory near the horizon, has been used to compute the Hawking fluxes
of charged spherically symmetric black holes. This effective action method is very universal and holds true for other
black holes [21, 25, 33]. In addition to the chiral effective action, the normal effective action that induces anomaly
freeenergymomentumtensorsandgaugecurrentshasalsoreproducedtheHawkingfluxesoftheReissner-Nordstro¨m
black hole [18]. A lot of works on applying the effective action to study Hawking effect can be found in [34–37].
In this paper, we investigate the thermodynamics of the EMCS-Go¨del black hole and then generalize the covariant
anomaly cancellation method, as well as the effective action approach, to study its Hawking radiation. Both the
methods present the same Hawking fluxes. Our results support that Hawking radiation is a universal quantum
phenomenon arising at the event horizon. The remainder of this paper goes as follows. In section II, we calculate
the mass, the angular momenta and the electric charge of the EMCS-Go¨del black hole, which satisfy the first law of
thermodynamics. In section III, we compute the Hawking fluxes by treating them as compensating fluxes to cancel
the covariantgravitationaland gauge anomalies near the horizon. In section IV, we reproduce the Hawking fluxes of
theEMCS-Go¨delblackholeviatheapproachoftheeffectiveaction,includingthenormaleffectiveactioninsubsection
IVA and the chiral effective action in subsection IVB. The last section is our conclusions.
II. THERMODYNAMICS OF THE EMCS-GO¨DEL BLACK HOLE
In this section, we study the thermodynamics of the EMCS-Go¨del black hole [8]. Although the main results were
presented in [8], here we give the explicit calculations by adopting the gauge field whose electric-static potential
vanishes at infinity. Our starting point is the EMCS-Go¨del black hole, which is a non-extremal charged rotating
G¨odel-type black hole solution in five-dimensional ungauged minimal supergravity. The relevant Einstein-Maxwell
Lagrangianwith Chern-Simons term reads
√ g 1
L= − (R F Fµν) ǫλρσµνA F F , (1)
µν λ ρσ µν
16π − − 24π√3
where ǫλρσµν is the five-dimensional tensor density with ǫ01234 = 1, and F = ∂ A ∂ A denotes the abelian
µν µ ν ν µ
− −
field-strength tensor. The Einstein and gauge field equations of motion derived from Lagrangian(1) are
1 1
R g R=2 F F α g F Fρσ ,
µν − 2 µν (cid:18) µα ν − 4 µν ρσ (cid:19)
1
Fµν + ǫµνλρσA F =0. (2)
ν λ ρσ
∇ (cid:18) √3√ g (cid:19)
−
Parameterized by four constants (µ,a,q,j), which correspond to the mass, the angular momentum, the electric
charge and the scale of the G¨odel background, respectively, the EMCS-Go¨del black hole satisfying Eq. (2) takes the
form [8]
h(r) 2 1
ds2 = f(r) dt+ (dφ+cosθdψ) + r2(dθ2+sin2θdψ2)
− h f(r) i 4
dr2 r2V(r)
+ + (dφ+cosθdψ)2, (3)
V(r) 4f(r)
A = B(r)dt+C(r)(dφ+cosθdψ), (4)
3
where
2µ q2
f(r) = 1 + ,
− r2 r4
(2µ q)a q2a
h(r) = jr2+3jq+ − ,
2r2 − 2r4
2µ 8j(µ+q) a+2j(µ+2q) 2(µ q)a2
V(r) = 1 + + −
− r2 (cid:2) r2 (cid:3) r4
q2 1 16ja 8j2(µ+3q)
+ − − ,
(cid:2) r4 (cid:3)
√3q √3 qa
B(r) = , C(r)= jr2+2jq . (5)
2r2 2 (cid:16) − 2r2(cid:17)
In the aboveequations,the Euleranglesθ, ψ andφ runoverthe ranges0to π, 0 to 2π and0 to 4π,respectively. The
line element (3) is the charged generalization of the Kerr G¨odel black hole. It is asymptotically rotating. Just as its
unchargedcounterpart,itexhibitsthepeculiarfeaturessuchasthepresenceofclosedtime-likecurvesandtheabsence
of globally spatial-like Cauchy surface. When the electric charge parameter q =0, it returns to the Kerr G¨odel black
hole in [4], whose Hawking radiation has been investigated via the covariantanomalies and effective action [21]. The
angular velocities and the electro-static potential of the EMCS-G¨odel black hole are given by
Ω(r) = Ω =h(r)/U(r), Ω =0, (6)
φ ψ
Φ = ℓµA =B(r)+Ω C(r), (7)
µ φ
where
r2V(r) 4h2(r)
U(r) = −
4f(r)
(µ q)a2 q2a2 r2
= j2r2(r2+2µ+6q)+3jqa+ − + , (8)
− 2r2 − 4r4 4
and the corotating vector ℓ = ∂ + Ω(r)∂ . With help of this vector, the surface gravity κ is defined by κ2 =
t φ
1ℓ ℓµ;ν , where the outside event horizon r is determined by equation V(r )=0 and reads
−2 µ;ν |r=r+ + +
r2 = µ 4j(µ+q)a 8j2(µ+q)(µ+2q)+√δ,
+ − −
δ = [µ q 8j2(µ+q)2]
− −
[µ+q 2a2 8j(µ+2q)a 8j2(µ+2q)2]. (9)
× − − −
Hence Hawking temperature via the surface gravity formula is read off as
κ r V′(r )
+ +
T = = . (10)
H
2π 8π U(r )
+
p
Here, and in what follows, the prime ′ denotes the derivative with respect to the radial coordinate r. The entropies
via the Bekenstein-Hawking area law are
S =π2r2 U(r ). (11)
+ +
p
It is worth noting that the electro-static potential (7) is not zero at infinity, but Φ = √3/2, since the G¨odel
∞
−
universe possessesa globalrotation at infinity. In order to make the electro-static potential vanish at infinity, we can
rescale the gauge field (4) as
A=Bˆ(r)dt+C(r)(dφ+cosθdψ), (12)
where Bˆ(r)=B(r)+√3/2. We shall adopt Eq. (12) for all the calculations related to gauge fields.
Now, we compute the mass, angular momenta, and electric charge of the EMCS-Go¨del black hole. Because of
the presence of closed time-like curves and the special asymptotical structure of the G¨odel-type black hole, naive
applicationofthetraditionalapproaches,suchasthemethodsofKomarintegral,theusualAbbott-Deserconstruction
andthe covariantphasespace [40],fails to giveconservedchargesin agreementwith the firstlawofthermodynamics.
4
In[9],anewmethod,basedoncohomologicaltechniques[15],hasbeensuccessfullyusedtoderivetheconservedcharges
oftheKerrG¨odelblackhole. ThismethodisalsoapplicabletotheEMCS-Go¨delblackhole. Ourcomputationfollows
work [9]. Here we only give the formulas closely relevant to our calculations. For more details see [9, 15].
Let ϕi = (g ,A ) denote the fields of the five-dimensional ungauged minimal supergravity. ϕ¯i = (g¯ ,A¯ ) is
µν µ µν µ
any fixed reference solution of the motion equtions in Eq. (2). Consider the linearized theory for the variables
δϕi =ϕi ϕ¯i =(δg ,δA )=(h ,a ). The equivalence classes of conserved3-forms of this linearizedtheory are in
µν µ µν µ
correspon−dencewithequivalenceclassesoffielddependentgaugeparametersξµ(x)andΛ(x)satisfyingthereducibility
equations [9]
g¯ =0,
ξ µν
L
A¯ +∂ Λ=0. (13)
ξ µ µ
L
Each pair of solutions (ξ,Λ) of Eq. (13) is associated with a conserved 3-form k [δϕ,ϕ¯] that can be obtained by
ξ,Λ
computingtheweaklyvanishingNoethercurrentsrelatedtothegaugetransformations. Obviously,whenξ isaKilling
vector ξ¯of the background ϕ¯ and Λ is a constant c, Eq. (13) holds. For the solutions (ξ¯,0), the conserved 3-form
kξ,Λ can be decomposed as kξ¯,0 =kξg¯r +kξe¯m+kξC¯S, where kξg¯r, kξe¯m and kξC¯S are the contributions from gravitation,
electromagnetism and the Chern-Simons term, respectively. kgr is defined by
ξ¯
kgr[h,g¯]= δKK ξ¯ Θgr, (14)
ξ¯ − ξ¯ − ·
where the Komar 3-form
√ g
KK = − ( µξ¯ν νξ¯µ)ǫ dxλ dxρ dxσ, (15)
ξ¯ 192π ∇ −∇ µνλρσ ∧ ∧
√ g¯
Θgr = − (¯ hµα ¯µh)ǫ dxν dxλ dxρ dxσ, (16)
α µνλρσ
384π ∇ −∇ ∧ ∧ ∧
and ξ¯ =ξ¯µ ∂ . The electromagnetic contribution kem is similar with kgr, which reads
· ∂(dxµ) ξ¯ ξ¯
kem[a,h;A¯,g¯]= δQem ξ¯ Θem, (17)
ξ¯ − ξ¯,0 − ·
where
√ g
Qem = − [(ξ¯αA +c)Fµν]ǫ dxλ dxρ dxσ, (18)
ξ¯,c 48π α µνλρσ ∧ ∧
√ g¯
Θem = − (F¯αµa )ǫ dxν dxλ dxρ dxσ. (19)
α µνλρσ
96π ∧ ∧ ∧
The contribution from the Chern-Simons term is
1
kCS[a,A¯]= (ξ¯αA¯ )a F¯ dxρ dxµ dxν. (20)
ξ¯ 2√3π α ρ µν ∧ ∧
For the solution (0,1), which corresponds to the contribution from the electric charge, the conserved 3-form
k [a,h;A¯,g¯]= δ(Qem+J), (21)
0,1 − 0,1
where
1
J= A F dxρ dxµ dxν. (22)
ρ µν
−4√3π ∧ ∧
Takeintoaccountapathγ inthespaceofsolutionsthatinterpolatesbetweenagivensolutionϕandthebackground
solution ϕ¯. Let d ϕ be a one-formin the field space. As long as the pair (ξ¯,c) satisfy Eq. (13) for all solutions along
V
this path, we can get a closed 3-form
Kξ¯,c =Z kξ¯,c[dVϕ;ϕ], (23)
γ
i.e. dKξ¯,c =0 in a four-dimensional hypersurface Σ. Using Eq. (23), one can define conservedcharges
Qξ¯,c =I Kξ¯,c, (24)
S
5
where the three-dimensional closed surface S is the boundary of the hypersurface Σ.
Next we turn our attention to calculate the conserved charges of the EMCS-Go¨del black hole via (24). The mass
is computed as
M = K
I ∂/∂t,0
S
3
= π(m+q) 4π(m+q)ja 8π(m+2q)(m+q)j2. (25)
4 − −
The angular momentum along the φ direction
J = K
φ −I ∂/∂φ,0
S
1 q
= π a m 2(m q)aj 8(m2+mq 2q2)j2
2 n h − 2 − − − − i
3jq2+8(3m+5q)j2q2 , (26)
− o
while the one with respect to the coordinate ψ is zero. The electric charge is given by
Q = K
0,1
I
S
√3
= π[q 4(m+q)aj 8(m+q)qj2]. (27)
2 − −
The electric charge can also be computed through
1 1 1
Q= √ gFαβǫ A F dxρ dxµ dxν, (28)
αβρµν ρ µν
4π Z (cid:18)12 − − √3 (cid:19) ∧ ∧
S3
where the integration is performed on the 3-sphere at infinity. All the conservedcharges are consistent with the first
law of thermodynamics
dM = T dS+Ω dJ +Φ dQ+Wdj, (29)
H + φ +
2 2 1
M = T S+Ω J + Φ Q Wj, (30)
H + φ +
3 3 − 3
where Ω = Ω(r ) and Φ = Bˆ(r )+Ω(r )C(r ) are the angular velocity and the electro-static potential at the
+ + + + + +
event horizon, and
W =2π(m+q)[a+2j(m+2q)]
is the generalizedforceconjugateto the G¨odelparameterj sincewe haveconsideredj as athermodynamicalvariable
to close the expression of the integral Bekenstein-Smarr formula.
III. HAWKING FLUXES AND COVARIANT ANOMALIES
In this section, we shall investigate Hawking radiation of the EMCS-Go¨del black hole [8] via the covariantgravita-
tional and gauge anomaly cancellation method [27] developed on basis of [16–18]. The same results will be obtained
if we adopt the consistent anomaly cancellation method in [16–18]. Before our proceeding, it is necessary for us to
briefly reviewthis approach. By performingthe technique ofdimensionalreduction,the masslessscalarfieldnearthe
horizoncanbe effectively describedby a collectionofscalarfields in the backgroundof(1+1)-dimensionalspacetime.
Thereby we can treat the higher dimensional theory as a (1+1)-dimensional effective theory near the horizon. If we
omit the classically irrelevant ingoing modes inside the horizon, the two dimensional effective theory becomes chiral.
Such a chiral theory exhibits covariant gravitational and gauge anomalies. Imposing the boundary condition that
the covariant energy momentum tensor and current vanish at the horizon, we can get fluxes that just cancel these
anomalies and are identified with Hawking fluxes for the energy momentum tensor and charges.
6
Wefirstimplementaprocessofdimensionalreductionbyconsideringthefreepartoftheactionforascalarmassless
complex field in the background of metric (3) and gauge field (12). We have
1
S[ϕ] = d5xϕ∗ √ ggµν ϕ
µ ν
2Z D − D
(cid:0) (cid:1)
= 1 dtdrdθdφdψsinθϕ∗ 4rU(r) +Ω(r) 2+∂ r3V(r)∂
t φ r r
16Z n− V(r) (cid:0)D D (cid:1) (cid:2) (cid:3)
r3 1 (∂ cosθ∂ )2
+ 2 +4r ∂ (sinθ∂ )+ ψ − φ ϕ, (31)
U(r)Dφ hsinθ θ θ sin2θ io
where =∂ +ieA . After performing a partialwavedecompositionϕ= ϕ (t,r)exp(imφ+inψ)Θ (θ),
Dµ µ µ lmn lmn lmn
where the spin-weighted spheroidal functions Θlmn(θ) satisfy P
1 (n mcosθ)2
∂ (sinθ∂ ) − +l(l+1) m2 Θ (θ)=0, (32)
hsinθ θ θ − sin2θ − i lmn
and only keeping the dominant terms near the horizon, the action (31) becomes
1 1
S[ϕ] dtdr r2 U(r)ϕ∗ ∂ +ie Bˆ(r)+Ω(r)C(r)
≃ 8XlmnZ p lmnn− F(r)(cid:2) t (cid:0) (cid:1)
2
+imΩ(r) +∂ [F(r)∂ ] ϕ . (33)
r r lmn
(cid:3) o
In Eq. (33), we have defined 2F(r) = rV(r)U(r)−1/2. Thereby the physics near the horizon can be described by an
infinite set of effective massless fields on a (1+1)-dimensional spacetime with the metric and the gauge potential
dr2
ds2 = F(r)dt2+ , (34)
− F(r)
= e (0)+m (1) =e Bˆ(r)+Ω(r)C(r) +mΩ(r), =0, (35)
At At At Ar
(cid:2) (cid:3)
where ( ) = 0, and F( ) = F′( ) = F′′( ) = 0. In such a two dimensional effective theory, the t-component
t
A ∞ ∞ ∞ ∞
(0)
of the gauge field contains two types of U(1) fields. the gauge field comes from the originalelectric field (12),
A At
while (1) canbeinterpretedasaninducedU(1)gaugefieldfromtheaxialisometryintheφdirection. Theazimuthal
At
quantum number m for each partial wave serves as charges of the gauge field (1).
At
Next, we pay our attention to derive the currents of the gauge field (35) via covariant gauge anomaly. In our
case, there are two U(1) gauge symmetries yielding two gauge currents J(0)r and J(1)r, corresponding to the gauge
potentials (0) and (1), respectively. Except for different types of charges,both the gauge potentials are essentially
consistentAwitth eachAotther. Thus we only give an explicit derivation of the current J(0)r. J(1)r can be obtained by a
similar procedure.
Due to the anomaly cancellation method, the gauge current behaves differently in the range outside the horizon
and that near the horizon. In the former, namely, the range r [r +ε,+ ), the current J(0)µ is anomaly free and
∈ + ∞ (O)
takes the conserved form
J(0)µ =0, (36)
∇µ (O)
while in the range near the horizon (r [r ,r +ε]), because of the breakdownof the classical gauge symmetry, the
+ +
∈
current J(0)µ satisfies the anomaly Ward identity [17, 18, 27]
(H)
1 1
J(0)µ = − ǫαβ , (37)
∇µe (H) 4π√ g Fαβ
−
where ǫαβ is an antisymmetry tensor density with ǫtr = ǫ = 1 and = ∂ ∂ . Solving Eqs. (36) and
tr αβ α β β α
− F A − A
(37), we have
√ gJ(0)r = c(0),
− (O) O
e
√ gJ(0)r = c(0)+ (r) (r ) , (38)
− (H) H 2π At −At +
(cid:2) (cid:3)
7
where the charge flux c(0) and c(0) are two integration constants, which denote the current at infinity and the one at
O H
the horizon,respectively. Introducingtwostepfunctions Θ(r)=Θ(r r ε)andH(r)=1 Θ(r)to writethe total
+
− − −
current as
J(0)µ =J(0)µΘ(r)+J(0)µH(r), (39)
(O) (H)
we find that the Ward identity becomes
e e
∂ √ gJ(0)r =∂ H + √ g J(0)r J(0)r + δ(r r ε). (40)
r − r 2πAt − (O) − (H) 2πAt − +−
(cid:2) (cid:3) (cid:0) (cid:1) (cid:8) (cid:2) (cid:3) (cid:9)
In orderto make the currentpreservethe gauge symmetry, the first term in the aboveequation must be cancelled by
the classically irrelevant ingoing modes while the second term should vanish at the horizon, which yields
e
c(0) =c(0) (r ), (r )=e Bˆ(r )+Ω(r )C(r ) +mΩ(r ). (41)
O H − 2πAt + At + + + + +
(cid:2) (cid:3)
(0)
Further imposing the boundary condition that the covariant current vanishes at the horizon, namely, c = 0, then
H
the charge flux corresponding to the gauge potential (0) is given by
At
e
c(0) = (r ). (42)
O −2πAt +
Following the analysis of computing c(0) step by step, the current with respect to the gauge potential (1) reads
O At
m
c(1) = (r ). (43)
O −2πAt +
From Eq. (37), one can see that J(0)r and J(1)r are not independent for each oter but there exists the relation
1J(0)r = 1J(1)r = r between them, where µ satisfies the covariantgauge anomaly equation
e m J J
1
µ = − ǫαβ , (44)
∇µJ(H) 4π√ g Fαβ
−
near the horizon. By analogy, the current out of the horizon can be solved as
1
c = (r ). (45)
O t +
−2πA
With the expression of the charge flux in hand, we now consider the energy momentum flux in the way similar to
the gauge anomaly. Near the horizon, if we eliminate the quantum effect of the ingoing modes, the invariance under
general coordinate transformation will break down. Thus the two dimensional effective field theory will exhibit a
gravitational anomaly. For the right-handed fields, the covariant gravitationalanomaly has the form [27]
1 1
Tµ = ǫ ∂µR= ∂ Nµ . (46)
∇µ ν 96π√ g νµ √ g µ ν
− −
In the case of a backgroundspacetime with the effective metric (34), the anomaly is timelike ( Tµ =0), and
∇µ t
1
Nr = 2FF′′ F′2 . (47)
t 192π −
(cid:0) (cid:1)
Becauseofthe presenceofthe externalgaugefield , the energymomentumtensoroutside the horizondoesnottake
A
the conserved form but satisfies the Lorentz force law
Tµ = µ , (48)
∇µ (O)ν FµνJ(O)
while the energy momentum near the horizon obeys the anomalous Ward identity after adding the gravitational
anomaly,
1
Tµ = µ + ǫ ∂µR. (49)
∇µ (H)ν FµνJ(H) 96π√ g νµ
−
8
Solving both the equations (48) and (49) for the ν =t component, we get
√ gTr =a +c (r), (50a)
− (O)t O OAt
1 r
√ gTr =a + c (r)+ 2(r)+Nr , (50b)
− (H)t H h OAt 4πAt ti(cid:12)r+
(cid:12)
(cid:12)
where a and a are two constants, corresponding to the fluxes at infinity and horizon, respectively. Similar to
O H
the case of the gauge current, we express the total energy momentum tensor as a sum of two combinations Tµ =
ν
Tµ Θ(r)+Tµ H(r). Using Eqs. (50a) and (50b), we find
(O)ν (H)ν
1
√ g Tµ = c ∂ +∂ 2+Nr H
− ∇µ t O rAt rh(cid:0)4πAt t(cid:1) i
1
+ √ g Tr Tr + 2+Nr δ(r r ε). (51)
h − (O)t− (H)t 4πAt ti − +−
(cid:0) (cid:1)
In the above equation, the first term is the classical effect of the background U(1) gauge field for constant current
flow. The secondtermshould be cancelledby the quantum effect ofthe classicallyirrelevantingoingmodes. In order
toguaranteetheenergymomentumtensorisinvariantundergeneralcoordinatetransformations,thethirdtermmust
vanish at the horizon, which yields
1 1
a =a + 2(r )+ F′2(r ), (52)
O H 4πAt + 192π +
where we have used Nr(r )= F′2(r )/(192π). As what we have done to evaluate the gauge current at infinity, to
t + − +
fix a completely, we requireto impose the boundaryconditionthat the covariantenergymomentumtensorvanishes
O
at horizon, i.e., a = 0. We will see that such a boundary condition is compatible with the Unruh vacuum in the
H
next section. Therefore, the total flow of energy momentum tensor is
1 κ2 1 r V′(r )
a = 2(r )+ , κ= F′(r )= + + , (53)
O 4πAt + 48π 2 + 4 U(r )
+
p
Forthesakeofcomparingthetotalenergymomentumflux(53)withtheHawkingone,weconsiderHawkingradiation
with the Fermionic Plank distribution Ne,m(ω)=1/(e[ω−eΦˆ+−mΩ(r+)]/TH +1) in the backgroundof the EMCS-Go¨del
black hole, where T is the Hawking temperature (10) via surface gravityformula, Φˆ =Bˆ(r )+Ω(r )C(r ) is the
H + + + +
electricchemicalpotentialofthegaugefield(12)atthe horizonandΩ(r )isthe angularvelocityatthe horizon. The
+
Hawking flux with this distribution is
∞ dω 1 κ2
F = ω[N (ω)+N (ω)]= 2(r )+ , (54)
M Z 2π e,m −e,−m 4πAt + 48π
0
which takes the same form as Eq. (53). This implies that we have reproduced the Hawking temperature (10) via the
covariant anomaly cancellation method.
IV. HAWKING FLUXES AND EFFECTIVE ACTION
In this section, we will use the effective action method to exploit Hawking radiation of the EMCS-Go¨del black
hole in background of the two dimensional metric (34) and gauge field (35). In two dimensional effective theory,
there exist normal effective action and chiral effective action. The former describes the effective theory away from
the event horizon. The energy momentum tensor and gauge current induced from this action are anomaly free
and take consistent forms. The normal effective action has been used to derive the Hawking fluxes of the Reissner-
Nordstro¨mblackhole[18]. Ontheotherhand,thechiraleffectiveaction[31,32]depictsthechiraltheory,inwhichthe
energymomentum tensorandgaugecurrentarenotconservedbut covariantlyanomalous. By adoptingthe covariant
boundary condition at the event horizon, this effective action can be applied to compute the Hawking fluxes of black
holes [31]. In our work [25], the chiral effective action method has been extended to reproduce the Hawking fluxes of
theSchwarzschildblackholesintheisotropiccoordinateswherethedeterminantofthemetricvanishesatthehorizon.
9
A. Normal effective action and Hawking fluxes
Intwodimensionaleffectivetheory,thenormaleffectiveactionisobtainedbyfunctionalintegrationoftheconformal
anomaly[18,37]. Itconsistsofthegravitational(Polyakov)partandthegaugepart. Fromavariationofthiseffective
action, we get the energy momentum tensor and gauge current [18, 31]
1 1
T = g ρ
µν µ ν µν ρ
−π(cid:16)∇ B∇ B− 2∇ B∇ B(cid:17)
1 1
2 +g 2R ρ , (55)
µ ν µ ν µν ρ
−48πh∇ G∇ G− ∇ ∇ G (cid:0) − 2∇ G∇ G(cid:1)i
1
Jµ = ǫµν∂ , (56)
ν
π√ g B
−
where R= F′′(r) is the Ricci scalar of the metric (34), and the two auxiliary fields and satisfy
− B G
ǫµν
µ = , µ =R. (57)
µ µν µ
∇ ∇ B −2√ gF ∇ ∇ G
−
From Eqs. (55) and (56), we find that the gauge current takes the conserved form Jµ = 0 while the energy
µ
∇
momentum tensor obeys the Lorentz force law (48) and the trace anomaly
R
Tν = Jµ, Tµ = . (58)
∇µ ν Fµν µ −24π
In the background of metric (34) and gauge field (35), solving Eq. (57), we get
b 2K 1
∂ = a, ∂ = − , K = F′(r), (59)
t r
G G F(r) 2
β+ (r)
t
∂ = α, ∂ = A , (60)
t r
B B F(r)
whereparametersa,b,α,β areconstants. Theycanbedeterminedbyproperboundaryconditions. Asintheprevious
section, we still choose the boundary conditions that are compatible with the Unruh vacuum. Such a choice requires
us to express the energymomentum tensor andgaugecurrentin the Eddington-Finkelsteincoordinate system u,v ,
{ }
where u=t r , v =t+r , and dr =dr/F(r). We have
∗ ∗ ∗
−
1 1
T = (α β (r))2 (a b)2 4K2+4F(r)K′ , (61)
uu t
−4π − −A − 192π − −
(cid:2) (cid:3)
1
T = T = F(r)F′′(r), (62)
uv vu
−96π
1 1
T = (α+β+ (r))2 (a+b)2 4K2+4F(r)K′ , (63)
vv t
−4π A − 192π −
(cid:2) (cid:3)
1 1
J = (α β (r)), J = (α+β+ (r)). (64)
u t v t
2π − −A −2π A
Adopting the Unruh vacuum boundary conditions
J =0, T =0, r =r , (65a)
u uu +
J =0, T =0, r + , (65b)
v vv
→ ∞
the constants a, b, α, β can be solved as
1
a= b= κ, α= β = (r ). (66)
t +
− ± − 2A
Substituting the four constants into the (r,t)-component of the energy momentum tensor and the r-component of
the gauge current, which correspond to fluxes for Hawking radiation and the gauge field, respectively, we obtain
1 κ2 1
Tr = (r )[ (r ) 2 (r)]+ , Jr = 2(r ), (67)
t 4πAt + At + − At 48π −2πAt +
10
where Jr is a constant since the normal effective action describes the theory away from the horizon and the gauge
current is conserved. Taking the limit at infinity, we derive the charge flow and the Hawking fluxes
1
Jr(r ) = 2(r ), (68)
→∞ −2πAt +
1 κ2
Tr(r ) = 2(r )+ , (69)
t →∞ 4πAt + 48π
which agree with Eqs. (45) and (53) via the covariant anomalies in the previous section.
B. Chiral effective action and Hawking fluxes
Varying the chiral effective action, the covariantenergy momentum tensor Tµ and the covariantgauge current Jµ
ν
read [31, 32]
e e
1 1 1
T = D D D D D D +g R , (70)
µν µ ν µ ν µ ν µν
−4π B B− 96π(cid:16)2 G G− G (cid:17)
e 1
Jµ = Dµ , (71)
2π B
e
where the chiral covariant derivative D = √ gǫ Dν = +√ gǫ ν, J = √ gǫ Jν, and the two auxiliary
µ µν µ µν µ µν
− ∇ − ∇ −
fields and have been defined by Eq. (57). In the chiral effective theory, the covariant energy momentum tensor
B G e e
and gauge current satisfy the anomalous Ward identities,
1
Jµ = − ǫρσ , (72)
µ ρσ
∇ 4π√ g F
−
e 1
Tµ = Jµ+ ǫ ∂µR. (73)
∇µ ν Fµν 96π√ g νµ
−
e e
TheenergymomentumtensoralsoobeysthecovarianttraceanomalyTµ = R/(48π). Operatingthechiralcovariant
µ −
derivative on the auxiliary fields and , we get
G B e
D = F(r)D =a˜ ˜b+2K, (74)
t r
G − G −
D = F(r)D =α˜ β˜ (r), (75)
t r t
B − B − −A
wherea˜,˜b, α˜ andβ˜areconstants. Theirrelationswillbe determinedlater. Nowthe (r,t)-componentofthe covariant
energy momentum tensor and the covariant gauge current can be read off as
1 1
Tr = (α˜ β˜ (r))2+ (a˜ ˜b)2 4K2+4F(r)K′ , (76)
t 4π − −At 192π − −
(cid:2) (cid:3)
eJr = F(r)Jt = 1 (α˜ β˜ (r)). (77)
t
−2π − −A
e e
Here we do notpresentthe other componentsof the energymomentum tensorTµ, whichareuseless for computation
ν
of the Hawking flux. Finally, to derive the Hawking fluxes and currents for gauge fields, we need to impose the
e
covariant boundary conditions that the covariant energy momentum tensor and gauge current vanish at the horizon
[31], namely,
α˜ =β˜+ (r ), a˜=˜b 2κ. (78)
t +
A ±
Therefore, taking the asymptotic limit, we obtain the gauge currents and the fluxes for energy momentum tensor
1
Jr(r ) = 2(r ), (79)
→∞ −2πAt +
e 1 κ2
Tr(r ) = 2(r )+ . (80)
t →∞ 4πAt + 48π
e
They are in agreement with Eqs. (45) and (53).