Aspects of Magnetized Black Holes1bodri.elte.hu/seminar/siahaan_20171018.pdf · +2am 2 r4 cos2 43 2 6 + 2a r2 3 3cos2 2cos a4 1 + cos4 +amr r 2+ a r2 3 + 6cos 2 cos 4 a2 1 6cos 3cos

Motivations and goals Black Holes CFT and AdS/CFT Kerr/CFT Destroying black holes Summary

Aspects of Magnetized Black Holes1

Haryanto M. Siahaan2

[email protected]

MTA Lendület Holographic QFT Group, Wigner Research Centre for Physics,Konkoly-Thege Miklós u. 29-33, 1121 Budapest , Hungary

andCenter for Theoretical Physics,

Physics Department, Parahyangan Catholic University,Jalan Ciumbuleuit 94, Bandung 40141, Indonesia

1Phys.Rev. D96 (2017) no.2, 024016 and Class.Quant.Grav. 33 (2016) no.15, 1550132Funded by The Tempus Public Foundation.

Haryanto M. Siahaan Aspects of Magnetized Black Holes




Outline

1 Black HolesBlack holes and Einstein-Maxwell systemErnst magnetizationErnst magnetizationMagnetized Kerr

2 CFT and AdS/CFTCFT in D and 2 dimensionsAdS/CFT correspondence

3 Kerr/CFTExtremal Kerr/CFT“c” for NHEKFrolov-Throne temperature and EntropyNon-extremal Kerr/CFTMagnetized Kerr/CFTFrolov-Throne temperature and Entropy of mag. black holes

4 Destroying black holesClassical test particle in black hole backgroundDestroying asymp. flat. charged rotating black holesMagnetic field: shield or not ?



Questions:

Does Kerr/CFT conjecture still work for (strongly) magnetized black hole?

Does strong magnetic field around a black hole give contribution to form a nakedsingularity?



Black holes and Einstein-Maxwell system

E.o.M. : Rµν = 2FµαFαν − 12 gµνF 2 where the Maxwell system is source free, i.e.

∇µFµν = 0.

Asymptotically flat solution: Kerr-Newman.

For an axial and stationary spacetime, Ernst formalism is powerful.

The line element and vector field:

ds2 = −f (dφ+ ωdt)2 + e2µ(

dρ2 + dz2)

+ f−1ρ2dt2

andA = At dt + Aφdφ

Ernst eqtns. (E + E + 2ΦΦ

)∇2E = 2

(∇E + 2Φ∇Φ

)• ∇E(

E + E + 2ΦΦ)∇2Φ = 2

(∇E + 2Φ∇Φ

)• ∇Φ

whereE = f − ΦΦ + iϕ , Φ = Aφ + i At

ϕ,ρ = −ρ−1f 2ω,z + i(ΦΦ,ρ − ΦΦ,ρ

), ϕ,z = ρ−1f 2ω,ρ + i

(ΦΦ,z − ΦΦ,z

)At,ρ = ρ−1f

(At,z − ωAφ,z

), At,z = −ρ−1f

(At,ρ − ωAφ,ρ

)The operator ∇ is gradient operator in (flat) cylindrical coord.



Black holes and Einstein-Maxwell system

E.o.M. : Rµν = 2FµαFαν − 12 gµνF 2 where the Maxwell system is source free, i.e.

∇µFµν = 0.

Asymptotically flat solution: Kerr-Newman.

For an axial and stationary spacetime, Ernst formalism is powerful.

The line element and vector field:

ds2 = −f (dφ+ ωdt)2 + e2µ(

dρ2 + dz2)

+ f−1ρ2dt2

andA = At dt + Aφdφ

Ernst eqtns. (E + E + 2ΦΦ

)∇2E = 2

(∇E + 2Φ∇Φ

)• ∇E(

E + E + 2ΦΦ)∇2Φ = 2

(∇E + 2Φ∇Φ

)• ∇Φ

whereE = f − ΦΦ + iϕ , Φ = Aφ + i At

ϕ,ρ = −ρ−1f 2ω,z + i(ΦΦ,ρ − ΦΦ,ρ

), ϕ,z = ρ−1f 2ω,ρ + i

(ΦΦ,z − ΦΦ,z

)At,ρ = ρ−1f

(At,z − ωAφ,z

), At,z = −ρ−1f

(At,ρ − ωAφ,ρ

)The operator ∇ is gradient operator in (flat) cylindrical coord.



Ernst magnetization

Melvin universe

ds2 =

(1 +

B2ρ2

4

)2 (−dt2 + dρ2 + dz2

)+

(1 +

B2ρ2

4

)−2

ρ2dφ2

with

Fρφ =

(1 +

B2ρ2

4

)−1

ρB

This solution satisfies the source free and Einstein-Maxwell equations.

FUN FACT:



Ernst magnetization

Melvin universe

ds2 =

(1 +

B2ρ2

4

)2 (−dt2 + dρ2 + dz2

)+

(1 +

B2ρ2

4

)−2

ρ2dφ2

with

Fρφ =

(1 +

B2ρ2

4

)−1

ρB

This solution satisfies the source free and Einstein-Maxwell equations.FUN FACT:



Ernst magnetization

Harrison transformation to a known stationary and axially symmetric spacetime inEinstein-Maxwell theory3:

E ′ =EΛ, Φ′ =

1Λ

(Φ−

BE2

),

where Λ = 1 + BΦ− B2E4 .

The transformation leaves the Einstein-Maxwell eqtns. to be invariant, and sourcefree condition still holds.The components of magnetized line element to the old one:

f ′ = |Λ|2 f , ∇ω′ = |Λ|2∇ω + ρf−1 (Λ∗∇Λ− Λ∇Λ∗) .

Recall Kerr spacetime

ds2 = Σ

(−

∆

Ξdt2 +

dr2

∆+ dθ2

)+

Ξ sin2 θ

Σ(dφ− ωdt)2 ,

whereΞ =

(r2 + a2

)2−∆a2 sin2 θ , Σ = r2 + a2 cos2 θ ,

and∆ = r2 + a2 − 2mr , ωΞ = 2mra .

3F. J. Ernst. 1976. J.Math.Phys.,17,54; F. J. Ernst and W. J. Wild. 1976. J.Math.Phys.,17,182.



Magnetized Kerr

ds2 = Σ |Λ|2(−

∆

Ξdt2 +

dr2

∆+ dθ2

)+

Ξ sin2 θ

Σ |Λ|2(|Λ0|2 dφ− ω′dt

)2,

where ω′ = 16mra+ωB(r,θ)B4

8Ξ, ωB (r , θ) = 4a3m3r

(3 + cos4 θ

)+2am2

(r4((

cos2 θ − 3)2 − 6

)+ 2a2r2 (3− 3 cos2 θ − 2 cos4 θ

)− a4 (1 + cos4 θ

))+amr

(r2 + a2) (r2 (3 + 6 cos2 θ − cos4 θ

)− a2 (1− 6 cos2 θ − 3 cos4 θ

)).

Re Λ = 1 +B2

4

((r2 + a2

)sin2 θ +

2a2mr sin4 θ

Σ

),

Im Λ = −B2 cos θ

4

(2am

(2 + sin2 θ

)+

2a3m sin4 θ

Σ

).

A =(Φ0 − ω′Φ3

)dt + Φ3dφ

Φ0 = −a

8Ξ

4a4m2 + 2a4m r − 24a2m3r − 24a2m2r2 − 4a2mr3 − 12m2r4 − 6mr5

−∆(

12rm(

r2 + a2)

cos2 θ +(

2mr3 + a2(

4m2 − 6mr))

cos4 θ)

,

Φ3 =1

8Σ |Λ|2

4ΞB sin2 θ + B4

Σ(

r2 + a2)2

sin4 θ + 4a2mr(

r2 + a2)

sin6 θ

+4a2m2(

r2(

2 + sin2 θ)

cos2 θ + a2(

1 + cos2 θ)2)

,



CFT in D and 2 dimensions

CFT in D dimensions

CFT is invariant under translations, rotations, dilations, and special conformaltransformations.

The conformal invariance of CFT rules the two and three point funtions, e.g. thetwo point function

〈φ1 (x1)φ2 (x2)〉 =C12

|x1 − x2|h1+h2.



CFT in D and 2 dimensions

CFT in 2 dimensions

The fields transform from z → z′ as

φ(z′, z′

)− φ (z, z) =

∞∑n=−∞

(εn ln + εn ln

)φ (z, z) ,

where ln = −zn+1∂z and ln = −zn+1∂z .

The generators obey

[ln, lm] = (n −m) ln+m ,[ln, lm

]= (n −m) ln+m ,

[ln, lm

]= 0 . (3.1)

Globally defined only for n = ±1, 0, and the algebra is SL(2,C)

[l±1, l0] = ∓l±1 , [l+, l−] = 2l0 .

Central extension of (3.1): [Lm, Ln] = (m − n) Lm+n + c12

(m3 −m

)δm+n,0.

Entropy formula in CFT2

SCardy =π2

3(cLTL + cRTR) .



AdS/CFT correspondence

Gravity theory in the bulk is dual to a CFT with no gravity living on the boundary.(Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231-252.)

Witten’s prescription (Witten, Adv.Theor.Math.Phys. 2 (1998) 253-291.)⟨exp

∫φ0O

⟩CFT

= Zgrav (φ0) .

An AdS/CFT test using scalar fields

⟨O(~x)O(~x ′)⟩∼

δ2Zgrav (φ0)

δφ0(~x)δφ0

(~x ′) ∣∣∣∣∣φ0=0

∼1∣∣~x − ~x ′∣∣2D .



AdS/CFT correspondence

Gravity theory in the bulk is dual to a CFT with no gravity living on the boundary.(Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231-252.)

Witten’s prescription (Witten, Adv.Theor.Math.Phys. 2 (1998) 253-291.)⟨exp

∫φ0O

⟩CFT

= Zgrav (φ0) .

An AdS/CFT test using scalar fields

⟨O(~x)O(~x ′)⟩∼

δ2Zgrav (φ0)

δφ0(~x)δφ0

(~x ′) ∣∣∣∣∣φ0=0

∼1∣∣~x − ~x ′∣∣2D .



Extremal Kerr/CFT

NHEK r → λr + r0 (near horizon extreme Kerr) 4 (Strominger et al, PRD80 (2009))

ds2 = 2GJΩ2

(−(1 + r2)dτ2 +

dr2

1 + r2+ dθ2 + Λ2(dϕ+ rdτ)2

).

The Killing vectors5 of NHEK;

J0 = 2∂τ , J1 =2r sin τ√

1 + r2∂τ − 2 cos τ

√1 + r2∂r +

2 sin τ√1 + r2

∂ϕ ,

J2 = −2r cos τ√

1 + r2∂τ − 2 sin τ

√1 + r2∂r −

2 cos τ√1 + r2

∂ϕ , JU(1) = −∂ϕ

The commutation between J ’s : SL(2,R) ×U(1) symmetry[J2, J0

]= 2J2 ,

[J1, J0

]= −2J1 ,

[J1, J2

]= 2J0

4 Ω2 ≡ 1+cos2 θ2 , Λ ≡ 2 sin θ

1+cos2 θ.

5∇µζν +∇νζµ = 0



“c” for NHEK

Brown and Henneaux 1986’ discovery6: The algebra of the canonical generatorsof symmetry of AdS3 spacetime7 with a set of boundary conditions adpoted atspatial infinity turns out to be just the Virasoro algebra (CFT2) with c = 3l/2.

Boundary condition for NHEK:

Lζgµν = hµν ∼

O(r2) O (1) O

(r−1) O

(r−2)

O (1) O (1) O(r−1) O

(r−1)

O(r−1) O

(r−1) O

(r−1) O

(r−2)

O(r−2) O

(r−1) O

(r−2) O

(r−3)

.

where accordingly ζµ∂µ = −rε′ (φ) ∂r + ε (φ) ∂φ.Conserved charges and central term:

Qζ = −1

16π

(d2x

)µν

(∇µζν −∇νζµ)→ Qζ =

∫V

dQζ =

∮∂V

Qζ

Qζ ,Qξ

= Q[ζ,ξ] −

∮∂V

kgζ

(gµν ,Lξgµν

)kgζ (gµν , hµν) =

√|g|

64πεµναβkµνζ dxα ∧ dxβ

kµνζ = ζν∇µh − ζν∇ρhµρ +h2∇νζµ − hνρ∇ρζµ + ζρ∇νhµρ − (µ→ ν)

The central charge for NHEK: c = 12J.

6Comm. Math. Phys. 104, No 2 (1986), 207-226.7S =

∫d3x√|g|(

R − 2l−2)



“c” for NHEK

Brown and Henneaux 1986’ discovery6: The algebra of the canonical generatorsof symmetry of AdS3 spacetime7 with a set of boundary conditions adpoted atspatial infinity turns out to be just the Virasoro algebra (CFT2) with c = 3l/2.Boundary condition for NHEK:

Lζgµν = hµν ∼

O(r2) O (1) O

(r−1) O

(r−2)

O (1) O (1) O(r−1) O

(r−1)

O(r−1) O

(r−1) O

(r−1) O

(r−2)

O(r−2) O

(r−1) O

(r−2) O

(r−3)

.

where accordingly ζµ∂µ = −rε′ (φ) ∂r + ε (φ) ∂φ.Conserved charges and central term:

Qζ = −1

16π

(d2x

)µν

(∇µζν −∇νζµ)→ Qζ =

∫V

dQζ =

∮∂V

Qζ

Qζ ,Qξ

= Q[ζ,ξ] −

∮∂V

kgζ

(gµν ,Lξgµν

)kgζ (gµν , hµν) =

√|g|

64πεµναβkµνζ dxα ∧ dxβ

kµνζ = ζν∇µh − ζν∇ρhµρ +h2∇νζµ − hνρ∇ρζµ + ζρ∇νhµρ − (µ→ ν)

The central charge for NHEK: c = 12J.6Comm. Math. Phys. 104, No 2 (1986), 207-226.7S =

∫d3x√|g|(

R − 2l−2)



Frolov-Throne temperature and Entropy

The spacetime outside of the Schwarzschild black hole is populated by quantumfields in the thermal state (Hartle-Hawking state) weighted by a Boltzmann factore−ω/TH .

For Kerr black holes, the corresponding thermal state (Frolov-Throne state) isweighted by the Boltzmann factor e−(ω−mΩH )/TH .

In the limits TH → 0 and ω → mΩH , e−(ω−mΩH )/TH → e−(m/TL) whereTL = 1/2π (Frolov-Thorne temp.).

Cardy formula and Bekenstein-Hawking entropy

SCardy =π2

3cT = SBH

The conformal symmetry which NHEK exhibits and the matching of entropy hintthe duality of extremal Kerr black hole and CFT2.

However, the specific dual CFT2 is unknown yet (OPEN PROBLEM).

Suppose this Kerr/CFT true, it only works for rotating black hole. How aboutSchwarzschild ?



Frolov-Throne temperature and Entropy

The spacetime outside of the Schwarzschild black hole is populated by quantumfields in the thermal state (Hartle-Hawking state) weighted by a Boltzmann factore−ω/TH .

For Kerr black holes, the corresponding thermal state (Frolov-Throne state) isweighted by the Boltzmann factor e−(ω−mΩH )/TH .

In the limits TH → 0 and ω → mΩH , e−(ω−mΩH )/TH → e−(m/TL) whereTL = 1/2π (Frolov-Thorne temp.).


SCardy =π2

3cT = SBH

The conformal symmetry which NHEK exhibits and the matching of entropy hintthe duality of extremal Kerr black hole and CFT2.

However, the specific dual CFT2 is unknown yet (OPEN PROBLEM).

Suppose this Kerr/CFT true, it only works for rotating black hole. How aboutSchwarzschild ?



Non-extremal Kerr/CFT

The near horizon limit of a non-extremal Kerr black hole is Rindler spacetime.The conformal structure is “hidden” in the low energy limit, Mω 1, of scalarwave Φ = e−iωt+imφR (r) S (θ) equation in the near region rω 1,[∂r (∆∂r ) +

(2Mr+ω − am)2

(r − r+)(r+ − r−)−

(2Mr−ω − am)2

(r − r−)(r+ − r−)

]R(r) = l(l +1) R(r) . (4.2)

The l.h.s. of (4.2) can be rewritten as a squared Casimir of SL(2,R)L × SL(2,R)Rgroup (Strominger et al, Phys.Rev. D82 (2010) 024008.),

ω+ =

√r − r+

r − r−exp(2πTRφ+ 2nR t) , ω− =

√r − r+

r − r−exp(2πTLφ+ 2nLt)

y =

√r+ − r−r − r−

exp(π(TR + TL)φ+ (nR + nL)t)

H2 = −H20 + (H1H−1 + H−1H1) , H2 = −H2

0 +(H1H−1 + H−1H1

).

H1 = i∂+, H0 = i(ω+∂+ +12

y∂y ), H−1 = i((ω+)2∂+ + ω+y∂y − y2∂−)

H1 = i∂−, H0 = i(ω−∂− +12

y∂y ), H−1 = i((ω−)2∂− + ω−y∂y − y2∂+)

With TL/R =r++/−r−

4πa and the assumption the “c” is smoothly unchanged, the

Cardy formula gives SCardy = π2

3 (cLTL + cRTR) = 2πMr+ = SBH .



Magnetized Kerr/CFT

In the NH of extremal mag. Kerr (NHEMK), the metric is

ds2 = Σ+ |Λ|2+

(−(1 + r2) dt2 + dr2

(1+r2)+ dθ2

)+

A+ sin2 θ|Λ0|2+Σ+|Λ|2+

(dφ+

(1−B4m4

)|Λ0|2+

rdt

)2

Σ+ = m2 (1 + cos2 θ), A+ = 4m2

|Λ|2+ =

(1+B2m2

)2+(

1−B2m2)2

cos2 θ

1+cos2 θ, |Λ0|2+ = |Λ|2+

∣∣∣θ=0

and the vector field is Aµdxµ = −PQL(θ)rdt + L(θ)dφ.

In Einstein-Maxwell theory, the charges come from the diffeomorphisms Lζgµνand LζAµ.

As r →∞, NHEMK behaves just like NHEK, so the same bound. conds. can beused.

For Aµ, we can impose Aµ (r →∞) ∼[O (r) ,O

(r−1) ,O (1) ,O

(r−2)].

Central charges: cg = 12m2 (1− B2m2) and cA = 0.

Recall: [Lm, Ln] = (m − n) Lm+n + c12

(m3 −m

)δm+n,0.



Frolov-Throne temperature and Entropy of mag. black holes

We can use the same Frolov-Throne state, i.e. weighted by the Boltzmann factore−(ω−mΩH )/TH .

In the limits TH → 0 and ω → mΩH , e−(ω−mΩH )/TH → e−(m/TL) whereTL = 1+B4M4

2π(1−B4M4). (TL can be negative for strong B)


SCardy =π2

312m2

(1− B2m2

) 1 + B4M4

2π(1− B4M4

) = SBH

It seems the strong magnetization breaks the hidden conformal symmetry, due tothe non-separability of wave equation for a scalar perturbation, i.e. magnetizedKerr metric does not possess the Killing-Yano tensor.



Classical test particle in black hole background

We can go back the the asymptotically flat black holes in Einstein-Maxwell theory,i.e. Kerr-Newman black hole.

Horizons: r± = M ±√

M2 − a2 − Q2, where a = JM−1.

Extremal state: M2 = a2 + Q2.

Dynamics of classical test particle around this black hole is dictated by thegeodesic equation:

duµ

ds+ Γµαβuαuβ =

qm

Fµνuν

Effectively, the e.o.m. can be obtained from

L =12

mgαβ xαxβ + qAµxµ

Two conserved quantities:

E = −∂L∂ t

= −m(

gtt t + gtφφ)− qAt , L =

∂L∂φ

= m(

gtφ t + gφφφ)

+ qAφ .

Naked singularity formation condition (for Kerr-Newman):

(i) . (M + E)2 < (Q + q)2 +

(|J|+ |L|M + E

)2and (ii) . Veff = −

r2

2< 0 ∀r > r+



Destroying asymp. flat. charged rotating black holes

For the limits of static or neutral black holes, i.e. RN or Kerr black holes, theseblack holes can be overspun or overcharged from its extremal state. V. E. Hubeny,

PhysRevD.59.064013 (1999); T. Jacobson and T. P. Sotiriou, PhysRevLett.103.141101 (2009)

However, when these black holes are near extremal, a classical test particle maydestroy their horizon.

The last observation holds for a rotating and charged black hole.

Even for an extremal rotating and charged black hole, a test particle can destroyits horizon. Gao and Zhang, PhysRevD.87.044028 (2013); HMS, PhysRevD.93.064028 (2016).

However, there are pros and cons related to this results. Some authors show thatif self-force effect is considered, such destruction can never happen. Ref. eg.P. Zimmerman, I. Vega, E. Poisson and R. Haas, PhysRevD.87.041501 (2012).



Destroying asymp. flat. charged rotating black holes

Example: rotating charged black hole in string theory with horizons

r± = M − b ±√

(M − b)2 − a2 where b = Q2/2M.

Extremal condition: 2 (M + E)2 = 2 |J + L|+ (Q + q)2.Near-extreme parameter: δ = M − b − a. After capturing a test particle:

δM + 2ME + E2 < Qq + L + q2

2 . → Constraint for the maximum energy.The condition for minimum energy is obtained from the conserved quantity Eexpression evaluated at r = r+.



Magnetic field: shield or not ?

What is mass, angular momentum, and charged in magnetized spacetime?M. Astorino, G. Compère, R. Oliveri and N. Vandevoorde, PhysRevD.94.024019 (2016)

M =

[M2 + 2JQB +

(2J2 +

32

M2Q2 − Q4)

B2 + JQ(

2M2 −32

Q2)

B3

+

(J2M2 −

12

J2Q2 +1

16M2Q4

)B4] 1

2.

J = J − Q3B −32

JQ2B2 −Q4

(8J2 + Q4

)B3 −

J16

(16J2 + 3Q4

)B4

Q = Q + 2JB −Q3B2

4Interesting relation: Christodoulou-Ruffini mass for Kerr-Newman black hole D.

Christodoulou and R. Ruffini, Phys. Rev. D4 (1971) 3552

M2 =S4π

+Q2

2+π(

Q4 + 4J2)

4S

Non-negative Entropy:

0 ≤ M4 − Q2M2 − J2 = M2

(M2 −

J2

M2− Q2

)(∣∣ΛRN,0∣∣2 +

∣∣ΛK ,0∣∣2 + 2JQB3

)Haryanto M. Siahaan Aspects of Magnetized Black Holes



We consider the extremal magnetized Kerr only : M2 = J.

Make sure that test particle’s geodesic exists on the equatorial plane(perpendicular to the rotational and magnetic field axis).

Cosmic censor violation (constraint for Emax )

(M + E

)2<

(Q + q

)2+

√(Q + q

)2+ 4

(J + L

)2

2

Minimum energy: Emin =L(

1−B4M4)

+4qM4B3(

1+2B4M4)

2M(1+B4M4).




However, such particle cannot reach the horizon.

Illustration



Conclusions and future works

The extremal Kerr/CFT conjecture holds for a Kerr black hole immersed in astrong magnetic field.

The hidden conformal symmetry for magnetized Kerr black hole does not exist.

An extremal Kerr black hole immersed by strong magnetic field cannot bedestroyed by a classical test particle.

Can test particle destroy a near extremal magnetized Kerr black hole ?

Can one obtain a magnetized black hole system in 4d Einstein-Maxwell-Λ theory ?

Non-unitary CFT2 dual⇔ “non-unitary” physics of extremal magnetized Kerr blackhole ?



Conclusions and future works

The extremal Kerr/CFT conjecture holds for a Kerr black hole immersed in astrong magnetic field.

The hidden conformal symmetry for magnetized Kerr black hole does not exist.

An extremal Kerr black hole immersed by strong magnetic field cannot bedestroyed by a classical test particle.

Can test particle destroy a near extremal magnetized Kerr black hole ?

Can one obtain a magnetized black hole system in 4d Einstein-Maxwell-Λ theory ?

Non-unitary CFT2 dual⇔ “non-unitary” physics of extremal magnetized Kerr blackhole ?



Acknowledgements

My host: Dr. Zoltan Bajnok.

The Tempus Public Foundation.


Documents

Aspects of Magnetized Black Holes1bodri.elte.hu/seminar/siahaan_20171018.pdf · +2am 2 r4 cos2 43 2 6 + 2a r2 3 3cos2 2cos a4 1 + cos4 +amr r 2+ a r2 3 + 6cos 2 cos 4 a2 1 6cos 3cos