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Hawking Radiations and Anomalies

Satoshi Iso (KEK)

based on collaborations

while I was staying at MIT(05/03-06/01)

with Hiroshi Umetsu (OIQP) and Frank Wilczek (MIT)

hep-th/0602146

hep-th/0603???

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Hawking radiation is the most prominent quantum effectto arise for quantum fields in a background space-time with an event horizon. Hawking (1975) : calculate Bogoliubov coefficients for particle creations between in- and out- states in a collapsing star.

[1] Introduction

(1) Vacuum in curved backgrounds is not unique.

a(n)|vac> =0 How can we identify annihilation ops.?

(2) Only outgoing modes come out of the horizon. Ingoing modes are decoupled from the exterior world. decoherence(thermal distribution)

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ＢＨ

Basic facts about black holes

M(mass) Q(charge) a (angular mom.)

Schwarzshild 　　　 (Q=a=0)Reissner-Nordstrom 　 (with Q)Kerr 　　　　　　　　　　　　 (with a)Kerr-Newman 　　　 (with Q and a)

Schwarzshild

r

t=Schwartshild time

r=2M

light cone at each radius

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Kruskal coordinates U,V : regular coordinates around horizon

where

r=const

tr=0

r=0

II: BH

IV: WH

I: exterior region

U V

III

U=0, V=0 at horizon

U=0 future horizonV=0 past horizon

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Horizon is not a singular point but a null hypersurface. No information comes out of the horizon.

Physical picture of Hawking radiation

BH

××

virtual pair creationof particles

E-E

××-E E

Hawking radiation

real pair creation

Hawking temperature

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Various derivations of Hawking radiation

(1)Hawking (1975) calculate Bogoliubov coefficinents Unruh (1976) B.coeff. in eternal BH, Unruh effect

(2) Euclidean method (Gibbons Hawking 1977)

Periodicity of the metric along the imaginary time direction=KMS condition

(4) Christensen Fulling (1977)

(3) Tunneling (Parikh Wilczek 2000)

calculate WKB amplitude for classically forbidden trajectories

Obtain each component of EM tensor in Schwarzshild BHusing conformal anomalies.

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Christensen Fulling method in d=2

Symmetries (stationary, rotational inv.) Conservation law of EM tensor

where

Then we need to determine 2 constants K, Q.

Trace of EM tensor is known from trace anomaly.

restrict the form of EM tensor as

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Determination of K and Q Impose 2 conditions

(1) regularity at future horizon

EM tensor should be regular at future horizon. Q=0

(2) No ingoing flux at r → ∞

Typical form of EM for radiation from blackbody with temp. T is

Hence K can be determined by asymptotic form of H2(r).

Flux of Hawking radiation

92 constants K, Q and 2 functions trace(r), Θ(r)

D=4 case is more complicated and we can not determine all the components.

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Determination of Hawking flux in d=4 needs non-universal function Θ(r).

Hawking radiation is a universal phenomena and the Hawking flux should be determined only by a few macroscopic parameter of BH.

Furthermore, it is much more complicated to extend the treatmentto Reissner-Nordstrom or Kerr BH.

Instead of conformal anomaly, we will use gauge or gravitational anomalies to determine the Hawking flux.

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Plan of the talk

[2] Basic idea

[3] Reissner-Nordstrom black hole

[4] Kerr or Kerr-Newman black hole

[5] Effective action approach to Hawking radiation

[5] Summary and Discussions

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[2] Basic idea

r=0

r=0

BH

(1)Near horizon, each partial wave of d-dim quantum field behaves as d=2 massless free field.

Quantum fields in black holes.

Outgoing modes = right moving Ingoing modes = left moving

Effectively 2-dim conformal fields

(different from Robinson-Wilczek 2005)

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(2) Ingoing modes are decoupled once they are inside the horizon.

These modes are classically irrelevant for the physics in exterior region.

So we first neglect ingoing modes near the horizon.

The effective theory becomes chiral in the two-dimensional sense.

gauge and gravitational anomalies = breakdown of gauge and general coordinate invariance

(3) But the underlying theory is NOT anomalous.

Anomalies must be cancelled by quantum effects of the classically irrelevant ingoing modes. 　　 ( ～ Wess-Zumino term)

flux of Hawking radiation

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Analogy with anomaly inflow mechanism

Quantum Halldroplet

Chern-Simons term for gauge potential is induced in the bulk.

Gauge symmetry will be broken at the boundary.

Chiral edge currents along the boundary rescue the gauge invariance.

chiral edgecurrent

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[3] Hawking radiation from charged black holes via gauge and gravitational anomalies

Metric and gauge potential of charged black hole (Reissner-Nordstrom)

Charged fields in RN BH. Partial wave decomposition

Each partial wave behaves as a d=2 free massless field.

Infinite set of d=2 quantum fields

IUWhep-th/0602146

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Note that

(1)The effective d=2 current or EM tensor are given by integrating d-dimensional ones over (d-2)-sphere.

(2) The effective 2-dim theory contains a dilaton background in addition to the d=2 metric.

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Hawking radiation from RN BH.

Planck distribution with a chemical potential

for fermoins

Fluxes of current and EM tensor are given by

e: charge of radiated particlesQ: charge of BH

( Extremal BH radiates charged particles~ Schwinger mechnism )

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Gauge current and gauge anomaly

horizon

ε

If we neglect ingoing modes in region H the theory becomes chiral there.

Gauge current has anomaly in region H.

consistent current

We can define a covariant current by

which satisfies

OH

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In region O,

In near horizon region H,

are integration constants.

Current is written as a sum of two regions.

where

= current at infinity

= value of consistent current at horizon

consistent current

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Variation of the effective action under gauge tr.

Using anomaly eq.

=0

impose

cancelled by WZ term

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・ Determination of

We assume that the covariant current to vanish at horizon.

Unruh vac.

Reproduces the correct Hawking flux

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EM tensor and Gravitational anomaly

Under diffeo. they transform

Effective d=2 theory contains background of graviton, gauge potential and dilaton.

Ward id. for the partition function

=anomaly

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Gravitational anomaly

consistent current

covariant current

In the presence of gauge and gravitational anomaly, Ward id. becomes

non-universal

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Solve component of Ward.id.

(1) In region O

(2) In region H

Using

(near horizon)

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Variation of effective action under diffeo.

(1) classical effect of background electric field

(1) (2) (3)

(2) cancelled by induced WZ term of ingoing modes

(3) Coefficient must vanish.

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Determination of

We assume that the covariant current to vanish at horizon.

since

we can determine

and therefore flux at infinity is given by

Reproduces the flux of Hawking radiation

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[4] Rotating black holes

Basic ideaKerr=axial symmetric

isometry

U(1) gauge symmetryin d=2

diffeo in axial direction

KK

partial wavewith m

charge m

a part of metric background electric field

(IUW, to appear)

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Kerr black hole

scalar field in Kerr geometry

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Near horizon, each partial wave is decoupled and can be treated as free massless d=2 field.

dilaton

metric

gauge potential

U(1) charge of is m.

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Results

Flux of angular momentum

Flux of energy

where(angular velocity at horizon)

These results are consistent with those for Hawking radiation.

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[5] Effective action approach to Hawking radiation (IU, to appear)

Quantum fields in BH background can be described by d=2 conformal fields near horizon.

For free d=2 free fields, we can calclate the effective actionof quantum fields in black hole background.

EM tensor or current can be explicitly obtained.

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Effective action of charged fields in electric and gravitational bkg.

gravity

gauge

The induced EM tensor and current are given by

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where

We need to impose boundary condition for

(Leutwyler 85)

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Boundary condition (for Unruh vacuum)

(1) Physical quantities must be regular at the future horizon.(2) There are no ingoing fluxes at infinity.

RN BH

tortoise coordinate

conformal metric

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・ U(1) gauge current　 in RN BH

B satisfies

It can be solved as

where

constant

hence

or

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Boundary condition

(1)Current is regular at future horizon in Kruskal coordinate

Metric is regular at outer horizon

regularity of JU at horizon imposes Since

(2) Ingoing current vanish at infinity

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Hence U(1) current is completely determined

Flux of U(1) charge

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Similarly EM tensor can be also determined.

Boundary conditions

EM tensor can be fully determined and the flux becomes

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[6] Summary and Discussions

(1) Hawking flux can be universally determined by demanding cancellation of gauge or gravitational anomalies at horizon.

Hawking radiation is a quantum effect to arise forquantum field in a background space-time with event-horizon.

quantum effect of classicallyirrelevant ingoing modes at horizon.(though anomaly)

outgoing

ingoing

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(2) The treatment can be applied to any type of black holes.

i.e. Schwarzshild Reissner-Nordstrom Kerr Kerr-Newman

Nonabelian gauge field?

(3) Planck distribution ?

(We have neglected the effect of grey body factor.)

anomaly for each frequency ?

RG analysis near horizon ?

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(4) Entropy of BH and Membrane paradigm

Horizon constraints entropy of BH as diffeo on horizon keeping the constraint

Quantum effect of ingoing modes

effective modes at horizon

cf. Carlip