Hawking Radiation and Covariant Anomalies

0.13 MB·

by  Rabin Banerjee

Hawking Radiation and Covariant Anomalies
Rabin Banerjee∗ and Shailesh Kulkarni†
Satyendra Nath Bose National Centre for Basic Sciences,
Block-JD, Sector-III, Salt Lake, Kolkata 700 098, India
Generalising themethod of Wilczek and collaborators weprovideaderivation ofHawking radia-
tionfromchargedblackholesusingonlycovariantgaugeandgravitationalanomalies. Thereliability
and universality of theanomaly cancellation approach to Hawking radiation is also discussed.
Introduction:- the covariant form of the gauge and gravitational
Hawking radiation is an important quantum effect in anomaly, respectively [22, 23, 24, 25, 26, 27, 28].
8
black hole physics. Specifically, it arises in the back- The consistent current and anomaly satisfy the Wess
0
ground spacetime with event horizons. The radiation Zumino consistency condition but do not transform
0
has a spectrum with Planck distribution giving the covariantly under a gauge transformation. Expres-
2
black holes one of its thermodynamic properties that sions for the covariant current and anomaly, on the
n makeitconsistentwiththerestofphysics. Hawking’s contrary,transform covariantly under gauge transfor-
a
originalresult[1] has since beenrederivedin different mation but do not satisfy the Wess Zumino condi-
J
ways thereby reinforcing the conclusion to a certain tion. Similar conclusions also hold for the gravita-
8
extent. However, the fact that no one derivation is tional case, except that currents are now replaced by
1
truly clinching has led to open problems leading to energy-momentumtensorsandgaugetransformations

alternative approaches with fresh insights. by general coordinate transformations. In[3, 4] the
]
h An anomaly in quantum field theory is a break- chargeandtheenergymomentumfluxoftheHawking
-t down of some classical symmetry due to the process radiation is obtained by a cancellation of the consis-
p of quantization (for reviews, see [25, 26, 28]). Specif- tent anomaly. However the boundary condition nec-
e ically, for instance, a gauge anomaly is an anomaly essary to fix the parameters are obtained from a van-
h in gauge symmetry, taking the form of nonconserva- ishing of the covariant current at the event horizon.
[ tion of the gauge current. Such anomalies charac- In this paper we generalise the method of [3, 4]
3 terise a theoreticalinconsistency, leading to problems by presenting a unified description totally in terms of
v with the probabilistic interpretation of quantum me- covariant expressions. This discussion is specifically
9 chanics. The cancellation of gauge anomalies gives done for Hawkingradiationfrom chargedblackholes.
4 strong constraints on model building. Likewise, a The charge flux is determined by a cancellation of
4 gravitational anomaly [22, 23] is an anomaly in gen- the covariant gauge anomaly while the energy mo-
2 eralcovariance,takingthe formofnonconservationof mentum flux is fixed by cancellation of the covariant
7. the energy-momentum tensor. There are other types gravitational anomaly. These are the only inputs.
0 of anomalies but here we shall be concerned with Also, we show that the analysis of [3, 4] is resilient
7 onlygaugeandgravitationalanomalies. The simplest and the results are unaffected by taking more general
0 case for these anomalies which is also relevant for the expressions for the consistent anomaly which occur
: present analysis, occurs for 1+1 dimensional chiral due to peculiarities of two dimensional spacetime.
v
fields.
i
X General discussion on covariant and consistent
LongbackChristiansenandFulling[2]reproduced
anomalies:-
r Hawking’s result by exploiting the trace anomaly in
a Here we briefly summarise some results on anoma-
the energy momentum tensor of quantum fields in
lies highlighting the peculiarities of two dimensional
a Schwarzchild black hole background. The use of
spacetime. First, the consistent gauge anomaly as
anomalies, though in a different form, has been pow-
taken in [3, 4] is considered,
erfully resurrected recently by Robinson and Wilczek
[3]. They observed that effective field theories be-
e2 e2
come two dimensional and chiral near the event hori- Jµ = ǫ¯ρσ∂ A = ǫρσ∂ A (1)
µ ρ σ ρ σ
zon of a Schwarzchild black hole. This leads to a ∇ ±4π ±4π√ g
−
two dimensional gravitational anomaly. The exis-
where +( ) corresponds to left(right)-handed fields,
tence of energy flux of Hawking’s radiation is nec-
−
respectively. Here g is the two dimensional (r t)
essary to cancel this anomaly. The method of [3] µν
−
partofthecompleteReissner-Nordstrommetricgiven
was soon extended to charged black holes [4] by us-
by [3, 4]
ing the gauge anomaly in addition to the gravita-
tional anomaly. Further advances and applications
1
of this approach may be found in a host of papers ds2 =f(r)dt2 dr2 r2dΩ2 . (2)
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], − f(r) − (d−2)
including a recent review [21].
so that g = detg = 1 and dΩ2 is the line
The approach of [3, 4] is based on the fact that − − µν (d−2)
a two dimensional chiral (gauge and/or gravity) the- element on the (d 2) sphere. The gauge potential is
−
ory is anomalous. Such theories admit two types of defined as A= Qdt.
−r
anomalous currents and energy momentum tensors; Now a word regarding our conventions. As is ev-
the consistent and the covariant [25, 26, 28]. The ident from (1) the antisymmetric tensor ǫ¯ρσ differs
covariant divergence of these currents and energy- from its numerical counterpart ǫρσ (ǫ01 = ǫ = 1)
01
−
momentum tensors yields either the consistent or by the factor √ g. Since here √ g = 1, the two get
− −
2
identified. Henceforth we shall always use ǫρσ, omit- Nowwewillconcentrateourattentiononthegrav-
ting the √ g factor. ity sector. If we omit the ingoing modes the en-
The cur−rent Jµ in (1) is called the consistent cur- ergy momentum tensor near the horizonwill notcon-
rent and satisfies the Wess-Zumino consistency con- serve, while there is no difficulty in the region out-
dition. Effectively this means that the following inte- side the horizon. The analysis [3, 4] for obtaining the
grability condition holds [24, 25]; flow of energy momentum tensor was done by using
the minimal form of the consistent d = 2 anomaly
δJµ(x) δJν(y) [22, 23, 26, 27], for right handed fields,
= . (3)
δA (y) δA (x)
ν µ
1
Tµ = ǫβδ∂ ∂ Γα , (10)
The covariant divergence of the consistent current ∇µ ν 96π δ α νβ
yields the consistent anomaly. The structure appear-
ing in (1) is the minimal form, since only odd parity Hereweconsiderthegeneralformford=2consistent
termsoccur. Howeveritispossiblethatnormalparity gravitational anomaly. It is worthwhile to point out
terms appear in (1). Indeed, as we now argue, such thattheconsistentgravitationalanomalyandthecon-
a term is a natural consequence of two dimensional sistentgaugeanomalyareanalogoussatisfyingsimilar
properties. consistencyconditions. Thisiseasilyobservedhereby
To fix our notions, consider the interaction La- comparing (10) with (1) where the affine connection
grangian for a chiral field ψ in the presence of an plays the role of the gauge potential. We therefore
external gauge potential Aµ in 1+1 dimensions, omit the details and write the generalised anomaly
by an inspection of (6) on how to include the normal
1 γ parity term. The result is,
=ψ¯ ± 5 γ Aµψ. (4)
I µ
L (cid:18) 2 (cid:19)
1
T¯µ = ∂ ∂ (ǫβδ+gβδ)Γα = . (11)
Using the property of two dimensional γ- matrices, ∇µ ν 96π δ α νβ Aν
(cid:2) (cid:3)
γ γµ = ǫµνγ , (5) The covariant energy momentum tensor, on the
5 ν
− other hand, has the divergence anomaly,
it is found that A couples as a chiral combination
µ 1
t(gitµyν (±5)ǫµhνo)Aldνs.dNueotetothtahtetshpeecuisfiucalstflrautctsupraeceofidtehne- ∇µT˜νµ = 96πǫνµ∂µR=A˜ν. (12)
two dimensional metric. Hence the expressionfor the
This is called the covariant anomaly as distinct from
anomaly in (1) generalises to,
the consistent anomaly (10).
e2
J¯µ =∂ J¯µ = ∂ [(ǫαβ gαβ)A ]. (6) Covariant gauge anomaly and charge flux:-
µ µ α β
∇ ±4π ± The current is conserved outside the horizon so that
J˜µ = ∂ J˜µ = ∂ J˜r = 0. Near the horizon
Thisisanon-minimalformfortheconsistentanomaly ∇µ (o) µ (o) r (o)
dictated by the symmetry of the Lagrangian,and has there are only outgoing (right-handed) fields and the
appeared earlier in the literature [23]. It is clear that current becomes (covariantly) anomalous (9),
if Jµ is a consistent current then J¯µ, which is given
by, e2 e2
∂ J˜r = F = ∂ A . (13)
r (H) 2π rt 2π r t
e2
J¯µ =Jµ+ Aµ (7)
4π The solution in the different regions is given by,
is also a consistentcurrentsince the extra piece satis- J˜r =c , (14)
(o) o
fies the integrability condition (3).
e2
It is possible to modify the new consistent current J˜r =c + [A (r) A (r )], (15)
(7), by adding a local counterterm, such that it be- (H) H 2π t − t +
comes covariant,
where c and c are integration constants.
o H
e2 The current is now written as a sum of two con-
J˜µ =J¯µ∓ 4πAα(ǫαµ±gαµ). (8) tributions from the two regions,J˜µ =J˜(µo)Θ(r−r+−
ǫ)+J˜µ H, where H = 1 Θ(r r ǫ). Then by
The current J˜µ yields the gauge covariant anomaly, (H) − − + −
using the conservation equations, the Ward identity
becomes,
e2
J˜µ = ǫαβF . (9)
µ αβ
∇ ±4π e2
∂ J˜µ =∂ J˜r =∂ A H +
µ r r t
Notethatthecovariantcurrent(8)doesnotsatisfythe (cid:18)2π (cid:19)
Wess-Zumino consistency condition since the coun- e2
δ(r r ǫ)(J˜r J˜r + A ). (16)
tertermviolatestheintegrabilitycondition(3). More- − +− (o)− (H) 2π t
over the gauge covariant anomaly (9) has a unique
formdictatedbythegaugetransformationproperties. To make the currentanomalyfree the firstterm must
Thisiscontrarytothe consistentanomalywhichmay be canceled by quantum effects of the classically in-
have a minimal (1) or non-minimal (6) structure. significant ingoing modes. This is the Wess-Zumino
3
term induced by these modes near the horizon. Ef- Writingtheenergy-momentumtensorasasumoftwo
fectively it implies a redefinition of the current as combinations T˜µ = T˜µ Θ(r r ǫ)+T˜µ H we
J˜′r = (J˜r − 2eπ2AtH) which is anomaly free provided find ν ν(o) − + − ν(H)
the coefficient of the delta function vanishes, leading
to the condition, T˜µ =∂ T˜r =c ∂ A (r)+∂ (e2A2+N˜r)H +
∇µ t r t o r t r(cid:20) 4π t t (cid:21)
e2
c =c A (r ). (17) e2
o H − 2π t + (T˜r T˜r + A2+N˜r)δ(r r ǫ)(.25)
t(o)− t(H) 4π t t − +−
The coefficient c is fixed by requiring the vanishing
H
The first term is a classical effect coming from the
of the covariant current at the horizon. This yields
Lorentz force. The second term has to be canceled
c =0 from (15). Hence the value of the charge flux
H
by the quantum effect of the incoming modes. As be-
is given by,
fore, it implies the existence of a Wess-Zumino term
e2 e2Q modifying the energy-momentum tensor as T˜t′µ =
co =−2πAt(r+)= 2πr . (18) T˜µ (e2A2+N˜r)H whichisanomalyfreeprovided
+ t −h 4π t t i
the coefficient of the last term vanishes. This yields
This is precisely the current flow of the Hawking
the condition,
blackbody radiation with a chemical potential [4].
e2
Covariant gravitational anomaly and energy- a =a + A2(r ) N˜r(r ). (26)
o H 4π t + − t +
momentum flux:-
Inthe presenceofa chargedfieldthe classicalenergy- wheretheintegrationconstanta isfixedbyrequiring
H
momentumtensorisnolongerconservedbutgivesrise
that the covariant energy momentum tensor vanishes
to the Lorentz force law, ∇µT˜νµ = FµνJ˜µ. The cor- at the horizon. From (24) this gives aH = 0. Hence
responding anomalous Ward identity for covariantly the totalfluxofthe energymomentumtensorisgiven
regularised quantities is then given by, by
∇µT˜νµ =FµνJ˜µ+A˜ν, (19) a = e2A2(r ) N˜r(r ). (27)
o 4π t + − t +
where ˜ is the covariant gravitation anomaly (12).
ν
Since tAhe current J˜µ itself is anomalous one might Since f(r+) = 0 we find from (20) that N˜tr(r+) =
envisage the possibility of an additional term in (19) (f′)2|r+. Using the surface gravity of the black hole
ppreonpsoirntitohnealWtaordthiedegnatuitgyefaonrocmoanlsyi.steInndtleyedretghuilsarhiasepd- −κ=1922ππ= (f′)|r+,thefinalresultisexpressedinterms
β 2
objects[4]. Suchatermisruledoutherebecausethere of the inverse temperature β as
isnosuchcovariantpiecewiththecorrectdimensions,
having one free index. e2Q2 π
For the metric (2) the covariantanomalyis purely ao = 4πr2 + 12β2. (28)
time-like ( ˜ =0) while, +
r
A
This is just the energy flux from blackbody radiation
[ff′′ (f′)2] with a chemical potential [4].
˜ =∂ N˜r; N˜r = − 2 . (20)
At r t t 96π
Generalised consistent anomaly and flux:-
Next,theWardidentityissolvedfortheν =tcompo- Here we show that the conclusions of [3, 4] remain
nent. In the exterior region there is no anomaly and unaffectedbytakingthegeneralformoftheconsistent
the Ward identity reads, anomaly (6, 11). Instead of repeating their analysis
we just point out the reasons for this robustness.
∂ T˜r =F J˜r . (21) For static configuration and for the specific choice
r t(o) rt (o)
of the potential (A = 0), it is clear that the normal
r
Using (14)this is solved as parity term in (6) vanishes. Likewise the normal par-
ityterminthecounterterm(8)alsovanishessinceonly
T˜r =a +c A (r), (22) the µ = r component in Jµ is relevant. Hence , ef-
t(o) o o t fectivelythesamestructuresoftheconsistent(gauge)
anomaly and the counterterm relating the consistent
where a is an integration constant. Near the hori-
o
and covariantcurrents,asused in[4], arevalid. Since
zontheanomalousWardidentity,obtainedfrom(19),
these were the two basic inputs the results concern-
reads
ingthechargefluxassociatedwithHawkingradiation
∂ T˜r =F J˜r +∂ N˜r, (23) remain intact.
r t(H) rt H r t
Identical conclusions also hold for the gravita-
Using J˜r from (15) yields the solution tional case. Although not immediately obvious, a
(H) little algebra shows that the normal parity term in
(11) vanishes. Hence the energy momentum flux
T˜tr(H) =aH +Z r dr∂r(cid:20)coAt+ 4eπ2A2t +N˜tr(cid:21). (24) (Agtiven by Ttr) remains as before.
r+
4
Discussions:- with Kerr-(Newman) metric.
This work was based on [4]but with a different pro-
cedure and emphasis. The flow of charge and energy
momentumfromchargedblackholehorizonswereob-
tained by a cancellation of the covariant anomalies.
Since the boundary condition involved the vanishing ∗ Electronic address: [email protected]
ofthecovariantcurrentatthehorizon,allcalculations † Electronic address: [email protected]
involved only covariant expressions. Neither the con- [1] S. Hawking, Commun. Math. Phys. 43, 199 (1975).
sistent anomaly nor the counterterm relating the dif- [2] S.Christensen and S.Fulling, Phys.Rev.D15, 2088
(1977).
ferentcurrents,whichwereessentialinputsin[4],were
[3] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95,
required. Consequently our analysis was economical
011303 (2005) [gr-qc/0502074].
and, we feel, also conceptually clean. We would here
[4] S. Iso, H. Umetsu and F.Wilczek, Phys. Rev. Lett.
like to mention that the interplay of covariant versus 96, 151302 (2006) [hep-th/0602146].
consistentanomalies,asoccurringin[3,4,5],hasbeen [5] S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74,
specifically discussed in the appendix of [10]. 044017 (2006) [hep-th/0606018].
It should be pointed out that the flux is identi- [6] K. Murata and J. Soda, Phys. Rev. D 74, 044018
fied with Jr or Tr which are the expressions for (2006) [hep-th/0606069].
(o) t(o) [7] E. Vagenas and S. Das, JHEP 0610, 025 (2006)
the currents exterior to the horizon. Here these cur-
[hep-th/0606077].
rents are anomaly free implying that there is no dif- [8] M. R. Setare, Eur. Phys. J. C 49, 865 (2007)
ference between the covariant and consistent expres- [hep-th/0608080].
sions. Actually the germ of the anomaly lies in this [9] Z. Xu and B. Chen, Phys. Rev. D 75 024041 (2007)
difference[24,25]. Henceitbecomesessential,andnot [hep-th/0612261].
[10] S. Iso, T. Morita and H. Umetsu, JHEP 0704, 068
just desirable,to obtain the same flux in terms of the
(2007) [hep-th/0612286].
covariant expressions. In other words the Hawking [11] Q. Q. Jiang and S. Q. Wu, Phys. Lett. B 647, 200
flux must yield identical results whether one uses the
(2007) [hep-th/0701002].
consistentorthecovariantanomalies. Butthebound- [12] Q. Q.Jiang, S.Q. Wuand X.Cai, [hep-th/0701048].
ary condition must be covariant. This is consistent [13] Q. Q. Jiang, S. Q. Wu and X. Cai, Phys. Rev.D 75,
with the universality of the Hawking radiation and 064029 (2007) [hep-th/0701235].
gives further credibility to the anomaly cancellation [14] X.Kui,W.LiuandH.Zhang,Phys.Lett.B647,482
(2007) [hep-th/0702199].
approach.
[15] H. Shin and W. Kim, arXiv:0705.0265 [hep-th]
It was shown [3, 4], performing a partial wave
[16] J. J. Peng and S. Q.Wu, arXiv:0705.1225 [hep-th].
decomposition, that physics near the horizon is de-
[17] Q. Q.Jiang, arXiv:0705.2068 [hep-th].
scibed by an infinite collection of massless (1 + 1) [18] B. Chen and W. He, arXiv:0705.2984 [gr-qc].
dimensional fields, each partial wave propagating in [19] U. Miyamoto and K. Murata, arXiv:0705.3150 [hep-
spacetime with a metric given by the ‘r t′ sector th].
of the complete spacetime metric (2). T−his simpli- [20] S. Iso, T. Morita and H. Umetsu, arXiv:0705.3494
[hep-th].
fication, which effects a dimensional reduction from
[21] S. Das, S.P Robinson and E. C Vagenas, arXiv:
d-dimensions to d = 2 is also exploited here. It is
0705.2233
however noted that greybody factors have not been [22] L.Alvarez-GaumeandE.Witten,Nucl.PhysB234,
included. In that case dimensional reduction will not 269 (1984).
yield the real Hawking radiation for d > 2. For in- [23] W. A. Bardeen, B. Zumino, Nucl. Phys. B 244, 421
stance it is known [29] that for d = 4 reduction to (1984).
d=2 and keeping only the s-wave (i.e.l=0) reduces [24] H.Banerjee and R.Banerjee, Phys.Lett.B174, 313
(1986).
the Hawking flux with respect to its 2d value.
[25] H. Banerjee, R. Banerjee and P. Mitra, Z. Phys. C
Areasoninfavorofworkingwithcovariantanoma- 32, 445-454 (1986).
lies is the fact that their functional forms are unique,
[26] R. Bertlmann, ”Anomalies In Quantum Field The-
being governed solely by the gauge (diffeomorphism) ory,” (Oxford Sciences, Oxford, 2000).
transformation properties. This is not so for consis- [27] R. Bertlmann and E. Kohlprath, Ann. Phys. (N.Y.)
tent anomalies. They can and do have normal parity 288, 137 (2001).
terms, apart from the odd parity ones. In fact, the [28] K. Fujikawa and H. Suzuki, ”Path Integrals and
Quantum Anomalies, ” (Oxford Sciences, Oxford,
special property (5) of two dimensions yields a nat-
2004).
ural form for this anomaly which has normal parity
[29] R.Balbinot,A.Fabbri,P.Nicolini,V.Frolov,P.Sut-
terms. Our observation that the results of [3, 4] are ton,andA.Zelnikov,Phys.Rev.D63,084029(2001).
still valid lend further support to this scheme of de-
riving Hawking radiation. The present approach can
be easily extended to other (e.g rotating) black holes