1. 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/
hep-th/0603???

2. Hawking radiation is the most prominent quantum effect
[1] Introduction
Hawking radiation is the most prominent quantum effect
to 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) 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)

3. Basic facts about black holes
Schwarzshild 　　　 (Q=a=0)
Reissner-Nordstrom 　(with Q)
Kerr　　　　　　　　　　　　 (with a)
Kerr-Newman 　　　(with Q and a)
M(mass)
Q(charge)
a (angular mom.) ＢＨ
Schwarzshild
t=Schwartshild time
light cone at each radius r
r=2M

4. Kruskal coordinates U,V : regular coordinates around horizon
where V t
U=0, V=0 at horizon U
r=0
U=0 future horizon
V=0 past horizon
II: BH
III
I: exterior region
IV: WH
r=const
r=0

5. Horizon is not a singular point but a null hypersurface.
No information comes out of the horizon.
Physical picture of Hawking radiation
virtual pair creation
of particles BH × × E -E × ×
Hawking radiation -E E
real pair creation
Hawking temperature

6. Various derivations of Hawking radiation
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
(3) Tunneling (Parikh Wilczek 2000)
calculate WKB amplitude for classically forbidden trajectories
(4) Christensen Fulling (1977)
Obtain each component of EM tensor in Schwarzshild BH
using conformal anomalies.

7. Christensen Fulling method in d=2
Symmetries (stationary, rotational inv.)
Conservation law of EM tensor
restrict the form of EM tensor as
where
Trace of EM tensor is known from trace anomaly.
Then we need to determine 2 constants K, Q.

8. 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

9. D=4 case is more complicated and we can not determine all the components.
2 constants K, Q and 2 functions trace(r), Θ(r)

10. Determination of Hawking flux in d=4 needs
non-universal function Θ(r).
Furthermore, it is much more complicated to extend the treatment
to Reissner-Nordstrom or Kerr BH.
Hawking radiation is a universal phenomena and
the Hawking flux should be determined only by a
few macroscopic parameter of BH.
Instead of conformal anomaly, we will use gauge or
gravitational anomalies to determine the Hawking flux.

11. 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

12. [2] Basic idea r=0 (different from Robinson-Wilczek 2005) BH
Quantum fields
in black holes.
r=0
Near horizon, each partial wave of d-dim quantum field
behaves as d=2 massless free field.
Outgoing modes = right moving
Ingoing modes = left moving
Effectively 2-dim conformal fields

13. (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

14. Analogy with anomaly inflow mechanism
Chern-Simons term for gauge
potential is induced in the bulk.
Quantum Hall
droplet
Gauge symmetry will be broken
at the boundary.
chiral edge
current
Chiral edge currents along the boundary
rescue the gauge invariance.

15. [3] Hawking radiation from charged black holes
via gauge and gravitational anomalies
IUW
hep-th/
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

16. Note that
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.

17. Hawking radiation from RN BH.
Planck distribution with a chemical potential
for fermoins
e: charge of radiated particles
Q: charge of BH
Fluxes of current and EM tensor are given by
( Extremal BH radiates charged particles~ Schwinger mechnism )

18. Gauge current and gauge anomaly
If we neglect ingoing modes in region H
the theory becomes chiral there. H O
horizon
Gauge current has anomaly in region H. ε
consistent current
We can define a covariant current by
which satisfies

19. In region O,
In near horizon region H,
consistent current
= current at infinity
= value of consistent
current at horizon
are integration constants.
Current is written as a sum of two regions.
where

20. Variation of the effective action under gauge tr.
Using anomaly eq.
impose =0
cancelled by WZ term

21. ・Determination of
We assume that the covariant current
to vanish at horizon.
Unruh vac.
Reproduces the correct Hawking flux

22. EM tensor and Gravitational anomaly
Effective d=2 theory contains background of
graviton, gauge potential and dilaton.
Under diffeo. they transform
Ward id. for the partition function
=anomaly

23. Gravitational anomaly
consistent current
covariant current
In the presence of gauge and gravitational anomaly, Ward id. becomes
non-universal

24. Solve component of Ward.id.
(1) In region O
(2) In region H
(near horizon)
Using

25. Variation of effective action under diffeo.
(1)
(2)
(3)
(1) classical effect of background electric field
(2) cancelled by induced WZ term of ingoing modes
(3) Coefficient must vanish.

26. 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

27. [4] Rotating black holes
(IUW, to appear)
Basic idea
Kerr=axial symmetric
isometry KK
diffeo in
axial direction
U(1) gauge symmetry
in d=2
a part of metric
background electric field
partial wave
with m
charge m

28. Kerr black hole
scalar field in Kerr geometry

29. 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.

30. Results
Flux of angular momentum
Flux of energy
where
(angular velocity at horizon)
These results are consistent with those for Hawking radiation.

31. [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 action
of quantum fields in black hole background.
EM tensor or current can be explicitly obtained.

32. Effective action of charged fields in electric and gravitational bkg.
gravity
gauge
The induced EM tensor and current are given by

33. We need to impose boundary condition for
(Leutwyler 85)
where
We need to impose boundary condition for

34. 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

35. ・U(1) gauge current　in RN BH
B satisfies
It can be solved as
where
constant
hence or

36. Boundary condition
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

37. Hence U(1) current is completely determined
Flux of U(1) charge

38. Similarly EM tensor can be also determined.
Boundary conditions
EM tensor can be fully determined and the flux becomes

39. [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 for
quantum field in a background space-time with event-horizon.
quantum effect of classically
irrelevant ingoing modes at horizon.
(though anomaly)
ingoing
outgoing

40. (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 ?
anomaly for each frequency ?
RG analysis near horizon ?
(We have neglected the effect of grey body factor.)

41. (4) Entropy of BH and Membrane paradigm
Quantum effect of ingoing modes
effective modes at horizon
cf. Carlip
Horizon constraints
entropy of BH as diffeo
on horizon keeping the constraint