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&4th GX, 12th ICGAC, 14th IK 1/3rd 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

formula in charged BH [CMP (‘74)]

• No Hawking radiation

when Q = M

• Schwinger emission

formula in E-field [PR (‘51)]

• Heisenberg-Euler,

Weisskopf, Schwinger QED actions

      =        − = m qE T T m N

S S S

π 2 1 exp

( )

2 2 2 2 2

2 1 1 Q M M Q M T e N

H T q lm j H

H j

− + − = Γ =

Φ −

π

ω ω



One-Loop Effective Actions:

Black Hole vs QED

[SPK, Hwang (‘11)]

) 2 sin( )] 2 [cos( ) ( ) 2 sin( )} 2 {cos( ) ( 2 1

• n

Polarizati Vac. ) 1 ln( ) ( ) 1 ln( ) ( e Persistenc Vac. ) 2 ( k 2 1 States

• f

# 2 ) / ( E in QED 2 BH ild Schwarzsch 1 Notation

2 ) 2 2 ( 2 ) 2 2 ( 2 2 , , B

2 2

s s e s ds s s e s ds e e d m d m qE T k

s m m k J s J m m k J J p m l J π β π βω β βω σ

π β π ω β π π κ β

+ − ∞ − + − − ⊥

⊥ ⊥

∑ ∫ ∫ ∑ ∫ ∑ ∫ ∑ ∫ ∑ ∫ ∑ ∫ ∑ ∫

± ± ± =   

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)]

• Vacuum persistence (twice of the imaginary action)

( )

2 2 ) ( 2 4 ) ( 2

/ 1 , 2 / 1 1 1 1 ω α λ α α

λ π π λ π

m Z Z l Z C e e e N

C C C C

− =       + − = + + =

− − − − −

( ) ( )

             

vacuum charged ) ( 2 formula Schwinger leading ) ( 2

1 ln 1 ln

C C

e e W

+ − − −

+ − + =

λ π λ π

Boson Emission from Extremal RN BH

• Including only the leading terms (effective charge and

angular momentum) for the KG equation in an RN BH

• Mean number is the same as that for the Coulomb field

(𝐷′ = 𝐷 invariant), and that for extremal RN black hole.

        + +       + = + + = ω ω ω M m qQ M m l l q M m Q Q

2 2 2 2

2 2 1 2 1 ' '

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 =

• The complex effective action and the vacuum persistence for

particle production

in 0, |

• ut

0,

eff

2 / 1

) (

= =

∫ −

xL d g

D

i iW

e e

∑ ± ± = =

− k Im 2 2

) 1 ln( Im 2 , in 0, |

• ut

0,

k W

N VT W e

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)]

• finite-temperature effective action [SPK, Lee, Yoon (‘09),

(‘10)]

( )

∑ ∑ ∑

+ Γ ± = ± =

k k * k

) k ( ln ln

l l l

ib a i i W α

[ ]

) ( Tr ) ( Tr in , 0, in , 0, exp

in in eff 3

ρ ρ β β

+ +

= =

∫

U U xdtL d i

Road to QED in Charged BHs

Why Schwinger Effect in (A)dS2?

Near-Horizon Geometry of RN BHs

RN Black Holes Near- extremal BH AdS2×S2 Nonextremal BH Rindler2 ×S2 Rotating BH in dS S-scalar wave QED in dS2

Schwinger Effect in (A)dS

[Cai, SPK (‘14)]

Vacuum Fluctuations

QED Schwinger Mechanism/ Unruh Effect de Sitter Gibbons- Hawking Radiation

Effective Temperature for Unruh Effect in (A)dS

[Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)] Effective Temperature

Unruh Effect TU = a/2π (A)dS R = 2H2

• r -2K2

2 2 U eff

8π R T T + =

Schwinger formula in (A)dS

• (A)dS metric and the gauge potential for E
• Schwinger formula for scalars in dS2 [Garriga (‘94); SPK,

Page (‘08)] and in AdS2 [Pioline, Troost (‘05); SPK, Page (‘08)]

) 1 )( / ( , ) 1 )( / ( ,

2 2 2 2 1 2 2 2 2

− − = + − = − − = + − =

Kx Kx Ht Ht

e K E A dx dt e ds e H E A dx e dt ds

                 − −       − =         − − +       = =

−

4 2 4 2

2 2 2 2 2 2

K m K qE K qE K S H qE H m H qE H S e N

AdS dS S

π π

Effective Temperature for Schwinger formula

• Effective temperature for accelerating observer in (A)dS

[Narnhofer, Peter, Thirring (‘96); Deser, Levin (‘97)]

• Effective temperature for Schwinger formula in (A)dS

[Cai, SPK (‘14)]

) 2 ( 2 , 8 ,

2 2 2 2 U eff /

eff

K H R R T T e N

T m

− = + = =

−

π

U 2 2 U AdS U 2 GH 2 U dS GH U 2 /

8 ; 2 , , 8 ,

eff

T R T T T T T T H T m qE T R m m e N

T m

+ + = + + = = = − = =

−

π π

Scalar QED Action in dS2

• Pair production and vacuum polarization from the in-out

formalism [Cai, SPK (‘14)]

( ) ( )

2 2 2 2 / n subtractio Schwinger 2 / ) ( 2 2 (1) dS dS ) 1 ( dS 2 2 ) ( dS

2 , 4 1 2 6 2 ) 2 / sin( ) 2 / cos( 12 2 ) 2 / sin( 1 2 4 1 ln Im 2 , 1 H qE S H m H qE S s s s s e s s s e s ds P S H L N W e e e N

s S s S S S S S S

π π π

λ µ π π µ

µ λ µ µ µ λ µ

= −       +       =                  − − −                         + − = + = − + =

− ∞ − − − − − −

∫

    

Scalar QED Action in AdS2

• Pair production and vacuum polarization

( ) ( )

2 2 2 2 2 / 2 2 (1) AdS AdS ) 1 ( AdS ) ( ) ( ) ( AdS

2 , 4 1 2 12 2 ) 2 / sin( 1 ) 2 / cosh( 2 4 1 ln Im 2 , 1 K qE S K m K qE S s s s s S e s ds P S K L N W e e e N

s S S S S S S S

π π π π

κ ν ν π ν

κ ν κ ν κ ν κ

= −       −       =       − − − = + = + − =

∫

∞ − + − + − − −

Spinor QED Action in dS2

• Pair production and vacuum polarization [SPK (‘15)]

( )

( )

( )

2 2 2 2 / 2 / ) ( 2 2 sp dS sp ds ) 1 ( dS 2 2 ) ( sp ds

2 , 2 6 2 ) 2 cot( 2 2 1 ln Im 2 , 1 H qE S H m H qE S s s s e e s ds P S H L N W e e e N

s S s S S S S S S

π π π

λ µ π π µ

µ λ µ µ µ λ µ

=       +       =       + − − − = − − = − − =

∫

∞ − − − − − − −

Spinor QED Action in AdS2

• Pair production and vacuum polarization [SPK (‘15)]

( )

( )

( )

2 2 2 2 2 / ) ( 2 / ) ( 2 2 sp AdS sp AdS sp AdS ) ( ) ( ) ( sp AdS

2 , 2 6 2 ) 2 cot( 2 2 1 ln Im 2 , 1 K qE S K m K qE S s s s e e s ds P S K L N W e e e N

s S S s S S S S S S S S

π π π

κ ν π π ν

ν κ ν κ ν κ ν κ ν κ

=       −       =       + − − − = − − = − − =

∫

∞ + − − − + − + − − −

Bosonic or Fermionic Current in (A)dS2

• Current in 2nd quantized field theory (in curved spacetime)

= (2 charge: 2q) × (density of states along E: D/H) × (mean number: N)

• 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)].

( )

( )

( )

( )

AdS 2 AdS dS 2 dS

2 4 1 2 ) 2 ( 2 4 1 2 ) 2 ( N KS q J N HS q J         + =         + = π σ π σ

ν µ

Schwinger Effect in D-dimensional dS

• The Schwinger effect in a constant E in a D-dimensional dS

should be independent of 𝑢 and 𝑦‖ due to the symmetry of spacetime and the field, and the integration of 𝑙‖ gives the density of states D.

• dS radiation in E=0 limit and Schwinger effect in H=0 limit

( )

( ) ( )

        + =               − −       +       =         − ± + =

⊥ − − − − − ⊥ −

∫

2 2 2 2 2 2 2 2 2 ) ( 2 2 2 2 || dS 2

/ / 2 , 2 1 2 1 ) 2 ( 2 4 1 2 k H qE H qE H qE S D H m H qE S e e e k d S H dtdx N d

S S S S D D

 π π π π σ

λ µ µ

µ µ λ µ

Schwinger Effect in Near- Extremal Charged Black Hole

Near-Horizon Geometry of RN BH

Inner Outer Horizons

AdS2×S2

• G. t’Hooft & A. Strominger, “conformal symmetry near the

horizon of BH,” MG14, July 2015. magnification

Near-Horizon Geometry

• Charged RN black hole: metric and potential
• Near-horizon geometry AdS2×S2 of near-extremal BH

[Bardeen, Horowitz (‘99)]

r Q A r Q r M r f d r f dr dt r f ds = + − = Ω + + − =

2 2 2 2 2 2 2

, 2 1 ) ( , ) ( ) (

( )

Q B Q M Q t Q Q r d Q dr B Q d Q B ds 2 , , ,

2 2 2 2 2 2 2 2 2 2 2 2 2

ε τ ε ρ τ ρ = − = = − Ω + − + − − =

Vacua for QED in AdS

FH PH

• Boundary condition for

quantum field for Schwinger effect in AdS2×S2

• Hamilton-Jacobi

equation (or phase integral)

) , (

) (

ϕ ϑ

ρ ω lm iS t i

Y e e−

Hawking Radiation vs Schwinger Effect

• Hawking radiation (pole at outer horizon) suppressed:
• Schwinger formula for charged scalars and fermions in

spherical harmonics (poles at outer horizon and infinity) [Chen et al (’12); (‘15)]

( )

B Q M B M Q M T e N

H T qA H

H

ε ε π

ω 2

2 / 2 / 1 4 2 / 2 / 1 , + + + = =

− −

B Q S l Q m q S qQ S e e e e e N

c b a S S S S S S S S S S NBH

b c a c b a b a b a

2 2 2 2 2

2 , ) 2 / 1 ( ) ( 2 , 2 , 1 1 1 ω π π π = + − − = =         + ×         ± − =

+ − + − − − − − + −



Interpretation of Schwinger Effect

• Thermal interpretation of Schwinger formula for

charged scalars (upper signs) and fermions (lower signs) in spherical harmonics [SPK, Lee, Yoon (‘15)]

Q m q m qE T Q T T T Q T T T e e e e e N

H U U U RN U U RN T m T qA T qA T m T m T m NBH

RN H H RN RN RN

π π π π

ω ω

2 2 / ) 2 1 ( , ) 2 1 ( , 1 1 1

2 2 2 2 ) (

= = − − = − + =           + ×           ± − =

+ − − − − − − −



Schwinger Effect and Hawking Radiation

• Thermal interpretation of Schwinger formula for charged

scalars and fermions [SPK, Lee, Yoon (‘15); SPK (‘15)]

                

charges

• f

Radiation Hawking ('00) AP Spindel & Gabriel Space Rindler in Effect Schwinger ('14) JHEP SPK & Cai AdS in Effect Schwinger

2

1 ) 1 ( 1           + ×           ± − × =

− − − − − − − − −

RN H H RN RN RN RN RN

T m T qA T qA T m T m T m T m T m NBH

e e e e e e e e N

ω ω

Scalar QED Action in BH

• Vacuum polarization and persistence

( )

sc ) 1 ( sc 2 / 2 / (1) sc

1 ln ) 1 2 ( Im 2 12 2 ) 2 / sin( 1 2 sinh ) 1 2 ( 12 2 ) 2 / sin( 1 2 sinh ) 1 2 ( N l W s s s s S e s ds P l s s s s S e s ds P l W

l b s S l b s S l

c a

+ + =         − −       + −         − −       + − =

∑ ∫ ∑ ∫ ∑

∞ − ∞ −

π π

π π

Spinor QED Action in BH

• Vacuum polarization and persistence

( )

sp ) 1 ( sp 2 / 2 / (1) sp

1 ln ) 1 2 ( Im 2 12 2 ) 2 / sin( 1 2 sinh ) 1 2 ( 6 2 ) 2 / sin( ) 2 / cos( 2 sinh ) 1 2 ( N j W s s s s S e s ds P l s s s s s S e s ds P l W

j b s S l b s S l

c a

− + − =         − −       + +         + −       + =

∑ ∫ ∑ ∫ ∑

∞ − ∞ −

π π

π π

Extremal Micro-BH, Extremal BH Entropy & Evolution

Micro-BH stable in QFT

• Breitenlohlner-Freedmann (BF) bound for boson (fermion)

stability in AdS2

• Compton wavelength of charge ≥ horizon of extremal BH

– Pairs cannot be created near the horizon (no Schwinger effect). – Extremal black holes emit neither Hawking radiation nor Schwinger pairs up to

𝑚𝑄𝑄 𝛽.

– Oppositely charged extremal BHs may form black atoms and remnants from Planckian regime. – Caveat: belongs to quantum gravity regime (loop gravity may check this).

      + ≤       4

2 2 2

K m K qE

Entropy of Extremal BH

• Hawking temperature of extremal BH, TH = κ/2π = 0.
• But, all black holes have the entropy SBH = A/4.
• Black hole thermodynamics dM = (κ/8π)dA +ΦHdQ gives a

null result: dM = dQ since ΦH = 1.

• A question at ICGC & 4th GX [KITPC, Beijing (‘15)] was

whether the effective temperature TCK gives the area-law and the answer is [SPK (‘15)]

• Rong-Gen Cai opposed against this entropy since the

Schwinger process is particle-dependent (m/e) while thermodynamics should rest on environmental quantities.

      ×       = 4 A e m SBH

Evolution of RN BH

[Hiscock, Weems (‘90)]

• Charge-loss rate?

(Schwinger formula)

• Total rate of mass loss

     − − =

2 H 2 H 2 3

/ exp r qQ m r q dt dQ π π   

?? H 4 H

dt dQ r Q aT dt dM + − = ασ

Evolution of RN BH

Q Extremal BH Q=M (No Hawking) ♦ ♦ Nea

Near-extre trema mal BH BH Nonextremal BH

Stability Line (no Schwinger) M

Naked Singularity

Conclusion

• The production of charged particles from an RN black hole

shows a strong interplay of the Schwinger effect and the Hawking radiation and has a thermal interpretation.

• Micro-black holes emit neither the Schwinger emission nor

Hawking radiation and are stable due to BF bound for extremal black holes up to 𝑚𝑄𝑚/ 𝛽 (quantum gravity models may check.)

• The vacuum polarization of QED in (A)dS and near-

extremal RN black hole exhibits the gravity-gauge relation (or AdS/CFT).

• The evolution and phase diagram of RN black holes should

properly include the Schwinger effect.

• The Schwinger effect in non-extremal black holes (Cai, SPK,

in preparation.)

CosPA 2015

• When: October 12-16, 2015
• Where: KAIST, Daejeon, Korea
• Ple

lena nary Spea eakers ers:

Martin Bucher, Asantha Cooray, Daniel Eisenstein, Raphael Flauger, Xiao-Gang He, Christopher Hirata, Donghui Jeong, Xiangdong Ji, Yeongduk Kim, Hye-Sung Lee, Simona Murgia, Hans Peter Nilles, Carsten Rott, Yannis K. Semertzidis, Gary Shiu, Yi Wang, Yvonne Wong, Jun’ichi Yokoyama, Yu-Feng Zhou

• Int

nterna natio iona nal Or l Organiz nizing ing Comitte tee

Pisin Chen, Kiwoon Choi, Richard Easther, Xiao-Gang He, Pauchy W-Y Hwang, Bo-Qiang Ma, Sandip Pakvasa, Raymond Volkas, David Wilshire, Jun’ichi Yokoyama *Kiwoon Choi (Chair), Sang Pyo Kim (Cochair)