pair production of scalar dyons in kerr-newman black holes · kerr-newman black holes chiang-mei...

Pair Production of Scalar Dyons inKerr-Newman Black Holes

Chiang-Mei Chen

Department of Physics, National Central University, [email protected]

16 November 2017@ 2nd East Asia Joint Workshop on Fields and Strings / KEK

Theory Workshop 2017, Tsukuba, Japan

Coworkers: Jia-Rui Sun, Fu-Yi Tang and Sang Pyo KimCMC, Kim, Sun, Tang, arXiv:1705.10629 [hep-th];

PRD 95 (2017) 044043 [arXiv:1607.02610 [hep-th]]

Pair Production of Scalar Dyons in Kerr-Newman Black Holes

Outline

MotivationScalar Dyons Pair ProductionThermal InterpretationDual CFT DescriptionSummary

RN black holes:CMC, Kim, Lin, Sun, Wu, PRD 85 (2012) 124041 [arXiv:1202.3224 [hep-th]]CMC, Sun, Tang, Tsai, CQG 32 (2015) 195003 [arXiv:1412.6876 [hep-th]]Kim, Lee, Yoon, arXiv:1503.00218 [hep-th]


Motivation

The spontaneous pair production from charged black holesmixtures two independent processes:

Schwinger mechanism by electromagnetic force

NS ∼ e− m

TS , TS = 2TU , TU =1

2π

qE

m

Hawking radiation by tunneling through horizon

NH ∼ e− ω

TH , TH =κ

2π

Near-extremal Kerr-Newman black holes: dual CFTdescription & exactly solvableNear-horizon region: GR solution & pair production occurs


Scalar Dyons Pair Production

dyonic Kerr-Newman Black Holes: M,Q,P, a

Near extremal: M2 ∼ Q2 + P 2 + a2

Near horizon geometry: exact solution, AdS structure



Klein-Groden equation of a complex scalar: m, q, p

“exact solution”: two integration constants c1, c2In/Out going modes at horizon and asymptotic: four fluxes

Outer boundary condition: no incoming flux at asymptotic

reflected

incident

B.C.

transmitted

inner boundary (horizon) outer boundary (asymptotic)

CMC, Kim, Lin, Sun, Wu, PRD 85 (2012) 124041 [arXiv:1202.3224 [hep-th]]



Flux conservation for bosonic particles

|Dincident| = |Dreflected|+ |Dtransmitted|

Bogoliubov relation

|α|2 − |β|2 = 1

Vacuum persistence amplitude |α|2 and Mean number ofproduced pairs |β|2

|α|2 ≡ Dincident

Dreflected, |β|2 ≡ Dtransmitted

Dreflected

Absorption cross section

σabs ≡Dtransmitted

Dincident=|β|2

|α|2


Dyonic Kerr-Newman Black Holes

Dyonic Kerr-Newman (KN) black holes:M, a = J/M, Q, P

ds2 = − Σ∆

(r2 + a2)2 −∆a2 sin2 θdt2 +

Σ

∆dr2 + Σdθ2

+(r2 + a2)2 −∆a2 sin2 θ

Σsin2 θ

[dϕ− a(2Mr−Q2−P 2)

(r2 + a2)2 −∆a2 sin2 θdt

]2A[1] =

Qr − Pa cos θ

Σ

(dt− a sin2 θdϕ

)+ P (cos θ∓1) dϕ

Σ = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2 +Q2 + P 2

dual potential: A[1] = A[1](Q→ P, P → −Q)

F[2] = dA[1], F[2] = dA[1] ⇒ F[2] =? F[2]


Near Horizon Geometry of Dyonic Kerr-Newman

Near horizon limits: ε→ 0, r20 ≡ Q2 + P 2 + a2

ϕ→ ϕ+a

r20+a2t, r → r0+εr, t→ r20+a2

εt, M → r0+ε2

B2

2r0

Near horizon of near extremal KN black holes

ds2 = (r20 + a2 cos2 θ)

[−(r2 −B2)dt2 +

dr2

r2 −B2+ dθ2

]+

(r20 + a2)2 sin2 θ

r20 + a2 cos2 θ

(dϕ+

2ar0r20 + a2

rdt

)2

A = −Q(r20 − a2 cos2 θ)− 2Pr0a cos θ

r20 + a2 cos2 θrdt

−(Qr0a sin2 θ + P (r20 + a2) cos θ

r20 + a2 cos2 θ∓P)dϕ



Klein-Gordon equation: A = A(Q→ P, P → −Q)

(∇α − iqAα − ipAα)(∇α − iqAα − ipAα)Φ−m2Φ = 0

ansatz: Φ(t, r, θ, ϕ) = e−iωt+i[n∓(qP−pQ)]ϕR(r)Θ(θ)

decoupled equations

∂r[(r2 −B2)∂rR

]+([

ω(r20 +a2)−(qQ+pP )(Q2+P 2)r+2nar0r

]2(r2

0 + a2)2(r2 −B2)−m2(r2

0 +a2)−λ

)R=0

1

sin θ∂θ(sin θ∂θΘ)−([

n(r20 +a2 cos2 θ)+(qQ+pP )ar0 sin2 θ−(qP−pQ)(r2

0 +a2) cos θ]2

(r20 + a2)2 sin2 θ

−m2a2 sin2 θ−λ

)Θ=0



Exact Solution:

R(r) = c1(r −B)i2 (κ−κ)(r +B)

i2 (κ+κ)

F

(1

2+ iκ+ iµ,

1

2+ iκ− iµ; 1 + iκ− iκ;

1

2− r

2B

)+ c2(r −B)−

i2 (κ−κ)(r +B)

i2 (κ+κ)

F

(1

2+ iκ+ iµ,

1

2+ iκ− iµ; 1− iκ+ iκ;

1

2− r

2B

)

three essential parameters

κ =ω

B, κ =

(qQ+ pP )(Q2 + P 2)− 2nar0

r20 + a2

,

µ =

√κ2 −m2(r2

0 + a2)− λ− 1

4



Expansion around r = B

RH(r) ≈ c(in)H (2B)i2 (κ+κ)(r−B)−

i2 (κ−κ)+c

(out)H (2B)

i2 (κ+κ)(r−B)

i2 (κ−κ)

parameters: c(in)H = c2 and c(out)H = c1

Expansion around r →∞

RB(r) ≈ c(in)B (r −B)−

12− i

2(κ+κ)−iµ(r +B)

i2

(κ+κ)

+c(out)B (r −B)−

12− i

2(κ+κ)+iµ(r +B)

i2

(κ+κ)

≈ c(in)B r−

12−iµ + c

(out)B r−

12

+iµ

Condition for existence of propagating modes

µ2 = κ2 −m2(r20 + a2)− λ− 1

4> 0



relations of parameters

c(in)B = c1(2B)

12

+iκ+iµ Γ(1 + iκ− iκ)Γ(−2iµ)

Γ(

12 − iκ− iµ

)Γ(

12 + iκ− iµ

)+c2(2B)

12

+iκ+iµ Γ(1− iκ+ iκ)Γ(−2iµ)

Γ(

12 + iκ− iµ

)Γ(

12 − iκ− iµ

)c

(out)B = c1(2B)

12

+iκ−iµ Γ(1 + iκ− iκ)Γ(2iµ)

Γ(

12 − iκ+ iµ

)Γ(

12 + iκ+ iµ

)+c2(2B)

12

+iκ−iµ Γ(1− iκ+ iκ)Γ(2iµ)

Γ(

12 + iκ+ iµ

)Γ(

12 − iκ+ iµ

)



Flux formula (radial direction)

D =

∫dθdϕ i

√−ggrr(ΦDrΦ

∗ − Φ∗DrΦ)

Relevant fluxes:

D(in)B = −2(r20 + a2)µ

∣∣∣c(in)B

∣∣∣2 SD

(in)H = −2B(r20 + a2)(κ− κ)

∣∣∣c(in)H

∣∣∣2 SD

(out)B = 2(r20 + a2)µ

∣∣∣c(out)B

∣∣∣2 SD

(out)H = 2B(r20 + a2)(κ− κ)

∣∣∣c(out)H

∣∣∣2 Sangular part (irrelevant): S = 2π

∫dθ sin θΘΘ∗



Outer boundary condition: c(in)B = 0

c1 = −c2(2B)i(κ−κ) Γ(1− iκ+ iκ)Γ(12 + iκ− iµ)Γ(1

2 − iκ− iµ)

Γ(12 + iκ− iµ)Γ(1

2 − iκ− iµ)Γ(1− iκ+ iκ)

parameter c(out)B :

c(out)B = −c2(2B)

12

+iκ−iµ sinh(2πµ) sinh(πκ− πκ)

cosh(πκ− πµ) cosh(πκ+ πµ)

× Γ(1− iκ+ iκ)Γ(2iµ)

Γ(12 − iκ+ iµ)Γ(1

2 + iκ+ iµ)



Bogoliubov coefficients:

|α|2 =Dincident

Dreflected=

∣∣∣D(out)H

∣∣∣∣∣∣D(in)H

∣∣∣ =cosh(πκ− πµ) cosh(πκ+ πµ)

cosh(πκ+ πµ) cosh(πκ− πµ)

|β|2 =Dtransmitted

Dreflected=

∣∣∣D(out)B

∣∣∣∣∣∣D(in)H

∣∣∣ =sinh(2πµ) sinh(πκ− πκ)

cosh(πκ+ πµ) cosh(πκ− πµ)

Absorption cross section:

σabs =Dtransmitted

Dincident=

∣∣∣D(out)B

∣∣∣∣∣∣D(out)H

∣∣∣ =sinh(2πµ) sinh(πκ− πκ)

cosh(πκ− πµ) cosh(πκ+ πµ)


Thermal Interpretation

Mean number of produced pairs: extremal limit κ→∞

N =

(e−2π(κ−µ) − e−2π(κ+µ)

1 + e−2π(κ+µ)

)(1− e−2π(κ−κ)

1 + e−2π(κ−µ)

)Temperature for Schwinger effect:

TKN =m

2π(κ− µ)= TU +

√T 2U +

R8π2

effective mass

m = m

√1 +

λ+ 1/4

m2(r20 + a2)

=

√m2 − λ+ 1/4

2R

Unruh temperature and AdS curvature:

TU =κ

2πm(r20 + a2)

, R = − 2

r20 + a2



Another “temperature”: (R → 0⇒ TKN = 2TU , TKN = 0)

TKN =m

2π(κ+ µ)= TU −

√T 2U +

R8π2

Mean number of produced pairs: (AdS2)Rindler2

N = emTKN

(e− mTKN − e

− mTKN

1 + e− mTKN

)e− mTKN

(1− e

− ω−qΦH−pΦH−nΩHTH

)1 + e

− ω−qΦH−pΦH−nΩHTH e

− mTKN

ω = εω (frequency in “original” coordinates), TH (Hawkingtemperature), ΦH , ΦH (chemical potentials), ΩH (angularvelocity)



TH =B

2π, ΩH = − 2ar0B

r20 + a2

ΦH =Q(Q2 + P 2)B

r20 + a2

, ΦH =P (Q2 + P 2)B

r20 + a2

Hamilton-Jacobi action: R(r) = eiS(r) (WKB)

S(r) =

∫dr

r2 −B2

√(ω − κr)2 − m2(r2

0 + a2)(r2 −B2)

Residue contributions of the contour integrate at threesimple poles: r = ±B and r =∞

S+ = − κ+ κ

2, S− =

κ− κ2

, S∞ = µ



Contour for Schwinger effect in AdS space

Na = ei(−2πi)(S−+S++S∞) = e−2π(κ−µ) = e− mTKN

Nb = ei(−2πi)(S−+S+−S∞) = e−2π(κ+µ) = e− mTKN



Contour for Schwinger effect in Rindler space

Na = ei(−2πi)(S−−S+−S∞) = e−2π(κ+µ)

Nb = ei(−2πi)(S−−S++S∞) = e−2π(κ−µ)


Dual CFT Description

Absorption cross section (pair production)

σabs =sinh(2πµ)

π2sinh(πκ−πκ)

∣∣∣∣Γ(1

2+i(µ−κ)

)∣∣∣∣2 ∣∣∣∣Γ(1

2+i(µ+κ)

)∣∣∣∣2

from 2-point correlator in 2D CFT

σabs ∼ T 2hL−1L T 2hR−1

R sinh

(ωL2TL

+ωR2TR

) ∣∣∣∣Γ(hL + iωL

2πTL

)∣∣∣∣2×∣∣∣∣Γ(hR + i

ωR2πTR

)∣∣∣∣2

Conformal weight: hL = hR = 12 + iµ (complex)



Threefold CFT descriptions for Kerr-Newman black holesCMC, Huang, Sun, Wu, Zou, PRD 82 (2010) 066004 [arXiv:1006.4097 [hep-th]]

J-picture: cJL = cJR = 12J (central charges) temperatures

T JL =r2+ + r2

− + 2a2

4πa(r+ + r−), T JR =

r+ − r−4πa

⇒ T JL ∼r20 + a2

4πar0, T JR ∼

B

2πa

Q-picture: cQL =cQR= 6Q(Q2+P 2)` (central charges) temperatures

TQL =(r2

++r2−+2a2)`

4πQ(r+r−−a2), TQR =

(r2+−r2

−)`

4πQ(r+r−−a2)⇒ TQL ∼

(r20 +a2)`

2πQ(Q2+P 2), TQR ∼

r0B`

πQ(Q2+P 2)

P -picture: cPL =cPR= 6P (Q2+P 2)` (central charges) temperatures

TPL =(r2

++r2−+2a2)`

4πP (r+r−−a2), TPR =

(r2+−r2

−)`

4πP (r+r−−a2)⇒ TPL ∼

(r20 +a2)`

2πP (Q2+P 2), TPR ∼

r0B`

πP (Q2+P 2)

geometrical meaning of `: radius of embedded extra circle



CFT entropy: (for all three pictures)

SCFT =π2

3(cLTL + cRTR) ∼ π(r2

0 + a2 + 2r0B)

Black hole entropy and temperature

SBH = π(r2+ + a2) ⇒ SBH ∼ π(r2

0 + a2 + 2r0B)

TH =r+ − r−

4π(r2+ + a2)

⇒ TH ∼B

2π

Identification via first law of thermodynamics(δSBH = δSCFT)

δM − ΩHδJ − ΦHδQ− ΦHδP

TH=ωLTL

+ωRTR



angular velocity and chemical potential (at r = B)

ΩH =2ar0r20 + a2

B, ΦH = − Q3B

r20 + a2

variation of parameters: δM = ω, δJ = n, δQ = q, δP = pCFT “frequencies”:

J-picture(δJ = n, δQ = 0, δP = 0) : ωJR =ω

a, ωJL = n

Q-picture(δJ = 0, δQ = q, δP = 0) : ωQR =2r0`ω

Q(Q2 + P 2), ωQL = −q`

P -picture(δJ = 0, δQ = 0, δP = p) : ωPR =2r0`ω

P (Q2 + P 2), ωPL = −p`

for all three picturesωL2TL

= −πκ, ωR2TR

= πκ


Summary

Spontaneous pair production of scalar dyons in nearextremal dyonic KN black holes: exact Bogoliubovcoefficients

There is a remarkable thermal interpretation.

The pair production (unstable mode) is holographicallydual to an operator with complex conformal weight.

pair production of scalar dyons in kerr-newman black holes · kerr-newman black holes chiang-mei...

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