Schwinger Effect and Hawking Radiation in Charged Black Holes* Sang - PowerPoint PPT Presentation

Schwinger Effect and Hawking Radiation in Charged Black Holes* Sang Pyo Kim Kunsan National University HTGRG-2, Quy Nhon, Vietnam August 10-15, 2015 *Similar talks at ICGC&4 th GX, 12th ICGAC, 14 th IK 1/3 rd new material

Outline • Introduction • Effective Actions in In-Out Formalism • Road to QED in Charged Black Holes • Schwinger Effect in Near-Extremal BHs • Extremal Micro-BH, Extremal BH Entropy and Evolution • Conclusion

Spontaneous Pair Production and Vacuum Polarization

Hawking Radiation & Schwinger Effect • Hawking emission • Schwinger emission formula in charged BH formula in E-field [PR [CMP (‘74)] (‘51)] −  Γ m   = ω = j lm exp N   N ω − Φ S H q   T j S T  1 e H   1 qE =   − T 2 2 1 M Q π S   = 2 ( ) m T π H 2 2 + − 2 2 M M Q • Heisenberg-Euler, Weisskopf, Schwinger • No Hawking radiation QED actions when Q = M

One-Loop Effective Actions: Black Hole vs QED [SPK, Hwang (‘11)] Notation QED in E Schwarzsch ild BH κ 1 ( / ) qE m = k T β B π π 2 2 ω 2 1 k ∑ ∫ ∑ d m ∑ ∫ d ∫ ⊥ # of States β π β π σ 2 , , J l m p 2 ( 2 ) 2 k m ⊥ − β + ∑ ∫ ∑ ∫ ( ) − βω ± ±   2 2 Vac. Persistenc e ( ) ln( 1 ) ( ) ln( 1 m ) e e J J s s {cos( )} [cos( )] 2 s k m s ⊥ 1 − βω − β + ∞ ds ds ∑ ∫ ∑ ∫ ( ) ∫ 2 2 ± π π  Vac. Polarizati on ( ) 2 ( ) 2 2 2 m e e s s 2 J J 0 s s sin( ) sin( ) 2 2

Schwinger Effect in Charged Black Holes Zaumen, Nature (‘74) Carter, PRL (‘74) Gibbons, CMP (‘75) Damour, Ruffini (‘76) ⋮ Khriplovich (‘99) Gabriel (‘01) SPK, Page (‘04), (‘05), (‘08) Ruffini, Vereshchagin, Xue (‘10) Chen et al (‘12); Kerr-Newman BH, in preparation (‘15) Ruffini, Wu, Xue (‘13) SPK (‘13) Cai, SPK (‘14) SPK, Lee, Yoon (‘15); SPK (‘15) Cai, SPK, in preparation (‘15)

Spontaneous Emission of Bosons from Supercritical Point Charges • Mean number of charged bosons produced from supercritical point charges [SPK (‘13)] − π + 4 C 1 e = − π λ − 2 ( ) C N e − π λ − + C 2 ( ) C 1 e 2  + α  1 / 2 l Z = α − λ =   1 , C Z ( ) α   − ω Z 2 1 / m • Vacuum persistence (twice of the imaginary action) ( ) ( ) − π λ − − π λ + = + − + 2 ( ) 2 ( ) C C ln 1 ln 1 W e e               leading Schwinger formula charged vacuum

Boson Emission from Extremal RN BH • Including only the leading terms (effective charge and angular momentum) for the KG equation in an RN BH 2 m M = + ' Q Q ω q 2    +  2 2 1 1 m M m M   + = + +   ' 2 l l qQ   ω ω   2 2   • Mean number is the same as that for the Coulomb field ( 𝐷 ′ = 𝐷 invariant), and that for extremal RN black hole.

Effective Actions in In-Out Formalism

In-Out Formalism for QED Actions • In the in-out formalism, the vacuum persistence amplitude gives the effective action [Schwinger (‘51); DeWitt (‘75), (‘03)] and is equivalent to the Feynman integral 1 / 2 ∫ − D i ( g ) d xL iW = = e e 0, out | 0, in = eff • The complex effective action and the vacuum persistence for particle production 2 − = = ± ± ∑ 2 Im W 0, out | 0, in , 2 Im ln( 1 ) e W VT N k k

Effective Actions at T=0 & T • Zero-temperature effective actions in proper-time integral via the gamma-function regularization [SPK, Lee, Yoon (‘08), (‘10); SPK (‘11)]; the gamma-function & zeta-function regularization [SPK, Lee (‘14)]; quantum kinematic approach [Bastianelli, SPK, Schubert, in preparation (‘15)] ( ) ∑ ∑ ∑ = ± α = ± Γ + * ln ln ( k ) W i i a ib k l l k k l • finite-temperature effective action [SPK, Lee, Yoon (‘09), (‘10)] [ ] + ρ Tr ( ) U ∫ + = β β = 3 in exp 0, , in 0, , in i d xdtL U ρ eff Tr ( ) in

Road to QED in Charged BHs

Why Schwinger Effect in (A)dS 2 ? Near-Horizon Geometry of RN BHs Near- AdS 2 × S 2 extremal BH RN Black Holes Nonextremal Rindler 2 × S 2 BH Rotating S-scalar QED in dS 2 BH in dS wave

Schwinger Effect in (A)dS [Cai, SPK (‘14)] Schwinger QED Mechanism/ Unruh Effect Vacuum Fluctuations Gibbons- de Sitter Hawking Radiation

Effective Temperature for Unruh Effect in (A)dS [Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)] Unruh T U = a/2 π Effect Effective Temperature R = 2H 2 (A)dS or -2K 2 R = + 2 T T 8 π eff U 2

Schwinger formula in (A)dS • (A)dS metric and the gauge potential for E = − + = − − 2 2 2 2 Ht Ht , ( / )( 1 ) ds dt e dx A E H e 1 = − + = − − 2 2 2 2 Kx Kx , ( / )( 1 ) ds e dt dx A E K e 0 • Schwinger formula for scalars in dS 2 [Garriga (‘94); SPK, Page (‘08)] and in AdS 2 [Pioline, Troost (‘05); SPK, Page (‘08)]    2 π   2 2 qE H qE    = + − −   2 S m  dS     4 H H H    − = S  N e    π 2   2 2 qE qE K   = − − −   2  S m AdS     4 K K K    

Effective Temperature for Schwinger formula • Effective temperature for accelerating observer in (A)dS [Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)] R − = = + = − / 2 2 2 m T , , 2 ( 2 ) N e T T R H K eff π eff U 2 8 • Effective temperature for Schwinger formula in (A)dS [Cai, SPK (‘14)] R qE H − = = − = = / 2 m T , , , N e m m T T eff π U GH 8 2 m R = + + = + + 2 2 2 ; T T T T T T T π dS U GH U AdS U U 2 8

Scalar QED Action in dS 2 • Pair production and vacuum polarization from the in-out formalism [Cai, SPK (‘14)] − − − ( ) + 2 S S S µ λ µ e e ( ) = = + ( 1 ) , 2 Im ln 1 N W N − dS dS dS − 2 S µ 1 e    Schwinger subtractio n         2   H S 1 2 ∞ ds s   ∫ − − π µ = − +  ( ) / 2 S S s   (1) µ λ L P e ( )   dS π 2    sin( / 2 ) 12 4 2 0 s s s            cos( / 2 ) 2 s s − π −  − −  / S s   µ  e      sin( / 2 ) 6  s s  2 2     1 qE m qE = π + − = π     2 , 2 S S µ λ  2    2 4 H H H

Scalar QED Action in AdS 2 • Pair production and vacuum polarization − − − + − ( ) ( ) S S S S κ ν κ ν e e ( ) = = + ( 1 ) , 2 Im ln 1 N W N − + + AdS AdS AdS ( ) S S 1 κ ν e   2 1 2 ∞ K S ds s ∫ − π = − π − − ν / 2 (1) S s cosh( / 2 ) κ   L P e S s ( ) ν AdS π 2   sin( / 2 ) 12 4 2 0 s s s 2 2     1 qE m qE = π − − = π     2 , 2 S S ν κ  2    2 4 K K K

Spinor QED Action in dS 2 • Pair production and vacuum polarization [SPK (‘15)] − − − − ( ) 2 ( ) S S S µ λ µ e e = = − − sp ( 1 ) sp , 2 Im ln 1 N W N − ds dS ds − 2 S µ 1 e ( ) 2   H S 2 ∞ ds s s ∫ − − π − π µ = − − − + ( ) / 2 / S S s S s   sp µ λ µ cot( ) L P e e ( ) dS π 2   2 6 2 2 0 s s 2 2     qE m qE π + = π =     2 , 2 S S µ λ  2    2 H H H

Spinor QED Action in AdS 2 • Pair production and vacuum polarization [SPK (‘15)] − − − + − ( ) ( ) ( ) S S S S κ ν κ ν e e = = − − sp sp sp , 2 Im ln 1 N W N − + AdS − AdS AdS ( ) S S 1 κ ν e ( )   2 2 K S ∞ ds s s ∫ − − π − + π = − − − + ν   sp ( ) / 2 ( ) / 2 S S s S S s κ ν κ ν cot( ) L P e e ( ) AdS π 2   2 6 2 2 0 s s 2 2     qE m qE = π − = π     2 , 2 S S ν κ  2    2 K K K

Bosonic or Fermionic Current in (A)dS 2 • Current in 2 nd quantized field theory (in curved spacetime) = (2 charge: 2q) × (density of states along E: D/H) × (mean number: N) ( )  σ +  2 1 HS   µ = ( 2 ) J q N   ( ) dS dS π 2  4 2  ( )  σ +  2 1 KS   ν = ( 2 ) J q N   ( ) AdS AdS π 2  4 2  • Consistent with the current from Frob et al (‘14); Stahl, Strobel (‘15); Stahl, Strobel, Xue (‘15) in D = 2. • Magnetogenesis and IR hyperconductivity [Frob et al (‘14)].