Hawking Radiations and Anomalies

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/0602146 hep-th/0603???

2. [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. r=0 [2] Basic idea (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/0602146 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 Gauge current has anomaly in region H. horizon 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. (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 currentin 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