excitation of black holes - umr 7332cosmo/sw_2014/ppt/ouldelhadj.pdf · introduction resonant...

IntroductionResonant behavior of the quasinormal excitation factors

Extrinsic giant ringingsConclusion and perspectives

Excitation of black holes :Resonant behavior induced by massive bosonic fields and giant ringings

Mohamed OULD EL HADJ ∗

UMR CNRS 6134 SPE, Faculté des Sciences, Université de CorseProjet : COMPA

Équipe physique théorique

SW8 – HOT TOPICS IN MODERN COSMOLOGY

CARGÈSE, CORSICA (FRANCE)

12-17 May 2014

∗. In collaboration with Antoine FOLACCI and Yves DÉCANINI : [arXiv:1401.0321] & [arXiv:1402.2481]Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 1 / 23

http://arxiv.org/abs/1401.0321

http://arxiv.org/abs/1402.2481



Introduction

Ultralight bosonic fields are important ingredients of the fundamental theories beyond :I The standard model of elementary particles.I The standard model of cosmology based on Einstein’s general relativity.

The particles associated with these fields are so light and are so weakly coupled with the visiblesector particles that they have escaped detection so far.

It is rather exciting to realize that ultralight bosonic fields interacting with black holes (BHs) couldlead to “macroscopic” effects.

In this context, in our recent work dealing with massive bosonic fields in the Schwarzschildspacetime, we have highlighted a new and unexpected effect in BH physics :I Around particular values of the mass, the excitation factors of the long-lived quasinormal

modes (QNMs) have a strong resonant behavior.I This effect has an immediate fascinating consequence : it induces giant and slowly decaying

ringings when the Schwarzschild BH is excited by an external perturbation.

Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 2 / 23



Introduction

Throughout this presentation :I We display our numerical results by using the dimensionless coupling constant α= 2Mµ/m2

p

(here M,µ and m2p denote respectively the mass of BH, the rest mass of the field and Planck

mass).I We adopt units such that ~= c = G = 1.I We consider the exterior of the Schwarzschild BH of mass M defined by the metric

ds2 =−(1−2M/r)dt2 +(1−2M/r)−1dr2 + r2dσ22 (1)

where dσ22 denotes the metric on the unit 2-sphere S2.

I The tortoise coordinate :r∗(r) = r +2M ln[r/(2M)−1] (2)

I For r −→ 2M (black hole horizon), r∗(r)−→−∞

I For r −→+∞ (spatial infinity), r∗(r)−→+∞




The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis

The Regge-Wheeler equation

In the Schwarzschild spacetime, the time-dependent Regge-Wheeler equation[− ∂2

∂t2 +∂2

∂r2∗−V`(r)

]φ`(t, r) = 0 (3)

with the effective potential V`(r) given by

V`(r) =

(1− 2M

r

)(µ2 +

A(`)r2 +β

2Mr3

)(4)

governs the partial amplitudes φ`(t, r) ofi. the modes of the massive scalar field (` ∈ N, A(`) = `(`+1) and β = 1)ii. the odd-parity modes of the Proca field (` ∈ N∗, A(`) = `(`+1) and β = 0)iii. the even-parity `= 0 mode of the Proca field (A(`) = 2 and β =−3)iv. the odd-parity `= 1 mode of the Fierz-Pauli field (Massive gravity)(A(`) = 6 and β =−8)v. the odd-parity modes of Einstein gravity (` ∈ N−0,1,A(`) = `(`+1) and β =−3)






The retarded Green function Gret` (t; r , r ′) associated with the partial amplitude φ`(t, r)

is a solution of [− ∂2

∂t2 +∂2

∂r2∗−V`(r)

]Gret` (t; r , r ′) =−δ(t)δ(r∗− r ′∗) (5)

I It satisfying the condition Gret` (t; r , r ′) = 0 for t ≤ 0

I It can be written as

Gret` (t; r , r ′) =−

∫ +∞+ic

−∞+ic

dω

2π

φinω`(r<)φ

upω`(r>)

W`(ω)e−iωt (6)

where c > 0, r< = min(r , r ′), r> = max(r , r ′) and with W`(ω) denoting the Wronskian ofthe functions φin

ω` and φupω`






φinω` and φ

upω` are linearly independent solutions of the Regge-Wheeler equation

d2φω`

dr2∗

+[ω

2−V`(r)]

φω` = 0. (7)

I When Im(ω)> 0, φinω` is uniquely defined by its ingoing behavior at the event horizon r = 2M

(i.e., for r∗→−∞)φ

inω`(r) ∼

r∗→−∞e−iωr∗ (8a)

and, at spatial infinity r →+∞ (i.e., for r∗→+∞), it has an asymptotic behavior of the form

φinω`(r) ∼

r∗→+∞

[ω

p(ω)

]1/2

×(

A(−)` (ω)e−i[p(ω)r∗+[Mµ2/p(ω)] ln(r/M)]

+A(+)` (ω)e+i[p(ω)r∗+[Mµ2/p(ω)] ln(r/M)]

)(8b)






I Similarly, φupω` is uniquely defined by its outgoing behavior at spatial infinity

φupω`(r) ∼

r∗→+∞

[ω

p(ω)

]1/2

e+i[p(ω)r∗+[Mµ2/p(ω)] ln(r/M)] (9a)

and, at the horizon, it has an asymptotic behavior of the form

φupω`(r) ∼

r∗→−∞B(−)` (ω)e−iωr∗ +B(+)

` (ω)e+iωr∗ . (9b)

In Eqs. (8) and (9), p(ω)=(ω2−µ2

)1/2denotes the “wave number" while A(−)

` (ω), A(+)` (ω),

B(−)` (ω) and B(+)

` (ω) are complex amplitudes.

By evaluating the Wronskian W`(ω) at r∗→−∞ and r∗→+∞, we obtain

W`(ω) = 2iωA(−)` (ω) = 2iωB(+)

` (ω). (10)






If the Wronskian W`(ω) vanishes, the functions φinω` and φ

upω` are linearly dependent and propa-

gate inward at the horizon and outward at spatial infinity, a behavior which defines the QNMs

-50 0 50 100 150 200r*

0.05

0.10

0.15

0.20VlHr*L

e-i Ω r*

Ω P@ΩD AH+LHΩLe+ i B P HΩ L r*+J M Μ2

P HΩL N ln I r

MMF

ΦinΩ

Ω P@ΩD AH-LHΩLe- i B P HΩ L r*+J M Μ2

P HΩL N ln I r

MMF

P@ΩD= Ω2 - Μ2

-40 -20 0 20 40 60 80 100r*

0.05

0.10

0.15

0.20VlHr*L

BH+LHΩLe+i Ω r*

BH-LHΩL e- i Ω r*

ΦupΩ

Ω P@ΩD e+ i B P HΩ L r*+K M Μ2

P HΩL O ln I r

MMF

P@ΩD= Ω2 - Μ2





Intrinsic giant ringings

By Cauchy’s theorem, we can extract from the retarded Green function (6) a residueseries over the quasinormal frequencies ω`n lying in the fourth quadrant of the complexω plane.

We then obtain the contribution describing the BH ringing. It is given by

GretQNM` (t; r , r ′) = ∑

nGretQNM`n (t; r , r ′) (11)

with

GretQNM`n (t; r , r ′) = 2Re

[B`nφ`n(r)φ`n(r

′)

×e−i[ω`n t−p(ω`n)r∗−p(ω`n)r′∗−[Mµ2/p(ω`n)] ln(rr

′/M2)]]. (12)





Intrinsic giant ringings

Here

B`n =

12p(ω)

A(+)` (ω)

dA(−)` (ω)

dω

ω=ω`n

(13)

denotes the excitation factor corresponding to the complex frequency ω`n.

i. In Eq. (12), the real part symbol Re has been introduced to take into account the symmetryof the quasinormal frequency spectrum with respect to the imaginary ω axis.

ii. The modes φ`n(r) are defined by

φ`n(r)≡ φinω`n`

(r)/[

[ω`n/p(ω`n)]1/2 A(+)

` (ω`n)

×ei[p(ω`n)r∗+[Mµ2/p(ω`n)] ln(r/M)]]

(14)

and are therefore normalized so that φ`n(r)∼ 1 as r →+∞.





Results : Massive scalar field (`= 3,n = 0)

æææææææææææææææææææææææææææææææææææææææææ

æ

æ

æ

æ

0.0 0.5 1.0 1.5 2.0-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Re@2MΩ30D

Im@2

MΩ

30D

HaL

Massive scalar field

Α=0

Α=1.80

Α=2.20

Massive scalar fieldAbs@B30D

Re@B30D

Im@B30D

0.0 0.5 1.0 1.5 2.0

-2000

-1000

0

1000

2000

Α

B30

HbL


Α

= 1.79

0 20 40 60 80 100

-5000

0

5000

tH2ML

G3

0retQ

NMHt

,r,r

'L

HdL

Α

= 1.63

Α

= 2.03

0 20 40 60 80 100-50

-20

0

20

60

Α

= 0

0 20 40 60 80 100

-0.11

0

0.1Massive scalar field

Α

= 0

Α

= 1.63

Α

= 1.79

Α

= 2.03

0 20 40 60 80 10010-11

10-8

10-5

0.01

10

tH2ML

G30re

tQN

MHt

,r,r

'L

HeL

FIGURE: The (`= 3,n = 0) QNM of the massive scalar field. (a) The complex quasinormal frequency 2Mω30 forα = 0,0.05, . . . ,2.15,2.20. (b) The resonant behavior of the excitation factor B30.(d) and (e) Some intrinsic ringingscorresponding to values of the mass near and above the critical value α30. We compare them with the ringing corresponding tothe massless scalar field. The results are obtained from (12) with r = 50M and r ′ = 10M.





Results : Massive scalar field (`= 3,n = 1)

ææææææææææææææææææææææææææææææææææææææææææææææææææææ

0.0 0.5 1.0 1.5 2.0-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Re@2MΩ31D

Im@2

MΩ

31D

HaL


Α=0

Α=1.80

Α=2.55

Massive scalar fieldAbs@B31D

Re@B31D

Im@B31D

0.0 0.5 1.0 1.5 2.0 2.5-6

-4

-2

0

2

4

6

Α

B31

HbL


Α

= 1.50

Α

= 1.75

Α

= 2.40

0 20 40 60 80 100-10

-5

0

5

10

tH2ML

G31re

tQN

MHt

,r,r

'L

HdL

Α

= 0

0 20 40 60 80 100

-0.2

0

0.1

0.3


Α

= 0

Α

= 1.50

Α

= 1.75

Α

= 2.40

0 20 40 60 80 10010-28

10-23

10-18

10-13

10-8

0.001

tH2ML

G3

1retQ

NMHt

,r,r

'L

HeL

FIGURE: The (`= 3,n = 1) QNM of the massive scalar field. (a) The complex quasinormal frequency 2Mω31 forα = 0,0.05, . . . ,2.50,2.55. (b) The resonant behavior of the excitation factor B31.(d) and (e) Some intrinsic ringingscorresponding to values of the mass near and above the critical value α31. We compare them with the ringing corresponding tothe massless scalar field. The results are obtained from (12) with r = 50M and r ′ = 10M.





Results : Fierz-Pauli field (odd-parity sector) (`= 1,n = 0)

ææææææææ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0.0 0.2 0.4 0.6 0.8 1.0

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Re@2MΩ10D

Im@2

MΩ

10D

HaL

Fierz-Pauli field Hodd-parity sectorL

Α® 0

Α= 0.35

Α= 0.70

Α= 1.05

Fierz-Pauli field Hodd-parity sectorLAbs@B10D

Re@B10D

Im@B10D

0.0 0.2 0.4 0.6 0.8 1.0

-10

0

10

20

Α

B10

HbL

Fierz-PauliHodd-parity sectorLΑ= 0.89H=1,n=0L

Α= 0.97H=1,n=0L

Α= 1.05H=1,n=0L

0 20 40 60 80 100-60

-40

-20

0

20

40

tH2ML

GnretQ

NMHt

,r,r

'L

HdL

Einstein gravityH=2,n=0L

0 20 40 60 80 100-0.11

0

0.1

0.25

Fierz-Pauli fieldHodd-parity sectorLΑ

= 0.89H=1,n=0L

Α

= 0.97H=1,n=0L

Α

= 1.05H=1,n=0L


0 20 40 60 80 100

10-8

10-5

0.01

10

tH2ML

GnretQ

NMHt

,r,r

'L

HeL

FIGURE: The odd-parity (`= 1,n = 0) QNM of the Fierz-Pauli field. (a) The complex quasinormal frequency 2Mω10 forα = 0,0.05, . . . ,1.00,1.05. (b) The resonant behavior of the excitation factor B10.(d) and (e) Some intrinsic ringingscorresponding to values of the mass near and above the critical value α10. We compare them with the ringing corresponding tothe odd-parity (`= 2,n = 0) QNM of the massless spin-2 field. The results are obtained from (12) with r = 50M and r ′ = 10M.





A semiclassical analysis

It is well known that the weakly damped QNMs of BHs which are associated with masslessfields can be interpreted in terms of waves trapped close to the so-called photon sphere with“radius" rc = 3M and we have for the corresponding impact parameter bc = 3

√3M.

For massive fields, the corresponding geometrical parameters are :

i. The critical radius rc(ω) parameter is given by

rc(ω) = 2M

(3+(1+8v2(ω)

)1/2

1+(1+8v2(ω))1/2

). (15)

Here v(ω) denotes the particle speed at large distances from the BH which can be expres-sed in term of the particle momentum p(ω) by

v(ω) =

√1− µ2

ω2 = p(ω)/ω. (16)






ii. The critical impact bc(ω) parameter is given by

bc(ω) =M√

2v2(ω)

[8v4(ω)+20v2(ω)−1 +

(1+8v2(ω)

)3/2]1/2

. (17)

Semiclassically, the QNMs of massive fields can be written in the form

φ±ω`(r) = exp

[±ip(ω)

∫ r∗(

1+2Mbc(ω)

2/rc(ω)2

r ′

)1/2(1− rc(ω)

r ′

)dr ′∗

]v±

ω`(r) (18)

which permits us to obtain analytically the quasinormal frequencies and their excitation factors.






The derivation of the quasinormal excitation factors B`n can be realized by using the standardWKB techniques as well as the usual matching procedures. After a tedious calculation, weobtain

B`n =i(`+1/2)−1√

8πn!

(−216i(`+1/2)

ξ

)n+1/2

exp[2ip(ω`n)ζc(ω`n)] (19)

where ξ = 7+4√

3 and with ζc(ω) given by

ζc (ω)2M =

b2c (ω)

2r2c (ω)− rc (ω)

2M

√1+ 2Mb2

c (ω)

r3c (ω)

−(

b2c (ω)

2r2c (ω)− rc (ω)

2M +1)

ln

[b2c (ω)

2r2c (ω)

+ rc (ω)2M + rc (ω)

2M

√1+ 2Mb2

c (ω)

r3c (ω)

]+(

rc (ω)2M −1

)√1+ b2

c (ω)

r2c (ω)

ln

[(rc (ω)

2M −1)(

b2c (ω)

2r2c (ω)

+1+

√1+ b2

c (ω)

r2c (ω)

)]−(

rc (ω)2M −1

)√1+ b2

c (ω)

r2c (ω)

ln

[b2c (ω)

2r2c (ω)

(rc (ω)

2M +1)+ rc (ω)

2M + rc (ω)2M

√(1+ 2Mb2

c (ω)

r3c (ω)

)(1+ b2

c (ω)

r2c (ω)

)]+ln2. (20)





Exact and asymptotic behaviors

Massive scalar field (`= 3,n = 0) Proca field odd-parity (`= 2,n = 0)

Massive scalar fieldExact

Asymptotic

0.0 0.5 1.0 1.5 2.0-2000

-1000

0

1000

2000

Α

Re@B

30D

0.0 0.2 0.4 0.6 0.8 1.0-0.2

-0.1

0

0.1

0.2

Proca field Hodd-parity sectorLExact

Asymptotic

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-10

0

10

20

Α

Re@B

20D

0.0 0.2 0.4 0.6 0.8

0

0.1

0.2

Massive scalar fieldExact

Asymptotic

0.0 0.5 1.0 1.5 2.0

-2000

-1000

0

1000

2000

Α

Im@B

30D

0.0 0.2 0.4 0.6 0.8 1.0-0.2

-0.1

0

0.1

0.2

FIGURE: The quasinormal excitation factor of the(`= 3,n = 0) QNM of the scalar field. Exact and asymptoticbehaviors.

Proca field Hodd-parity sectorLExact

Asymptotic

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-10

0

10

20

Α

Im@B

20D

0.0 0.2 0.4 0.6 0.8-0.2

-0.1

0

0.1

0.2

FIGURE: The quasinormal excitation factor of the(`= 3,n = 1) QNM of the scalar field. Exact and asymptoticbehaviors.




Results : Extrinsic giant ringings

Extrinsic giant ringings

We consider that the BH perturbation is generated by an initial value problem with(nonlocalized) Gaussian initial data. More precisely, we assume that the partial ampli-tude φ`(t, r) solution of (3) is given, at t = 0, by

φ`(t = 0, r) = φ0 exp

[− a2

(2M)2 [r∗(r)− r∗(r0)]2]

(21)

and, moreover, satisfies ∂t φ`(t = 0, r) = 0.

By Green’s theorem and using (3) and (5), we can show that

φ`(t, r) =∫

∂t Gret` (t; r , r ′)φ`(t = 0, r ′)dr ′∗ (22)

describes the time evolution of φ`(t, r) for t > 0.





Extrinsic giant ringings

We can now insert (6) into (22) and deform again the contour of integration in thecomplex ω plane in order to capture the contribution of the QNMs.We have

φQNM` (t, r) = ∑

nφ

QNM`n (t, r) (23)

with

φQNM`n (t, r) = 2Re

[iω`nC`n ×e−i[ω`n t−p(ω`n)r∗−[Mµ2/p(ω`n)] ln(r/M)]

]. (24)

Here C`n denotes the excitation coefficient of the (`,n) QNM. It is defined from thecorresponding excitation factor B`n. We have

C`n = B`n

∫φ`(t = 0, r ′)φin

ω`n`(r ′)√

ω`n/p(ω`n)A(+)` (ω`n)

dr ′∗. (25)





Results : Massive scalar field and Fierz-Pauli field

The results are obtained from (24) with r = 50M and by using (21) with φ0 = 1, a = 1 and r0 = 10M.

Massive scalar field (`= 3,n = 0) Fierz-Pauli field (odd-parity (`= 1,n = 0))


Α= 1.79

0 20 40 60 80 100-20

-10

0

10

20

tH2ML

Φ30

QN

MHt

,rL

HaL

Α= 1.63

Α= 2.03

0 20 40 60 80 100-3

-2

0

2

3

Α= 0

0 20 40 60 80 100-0.2

0

0.1

0.25

Fierz-Pauli field Hodd-parity sectorLΑ= 0.89 H=1,n=0L

Α= 0.97 H=1,n=0L

Α= 1.05 H=1,n=0L

0 20 40 60 80 100

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

tH2ML

Φ

nQN

MHt

,rL

HaL


0 20 40 60 80 100

-0.1

0

0.15


Α= 0

Α= 1.63

Α= 1.79

Α= 2.03

0 20 40 60 80 100

10-10

10-8

10-6

10-4

0.01

1

tH2ML

Φ30

QN

MHt

,rL

HbL

FIGURE: (a) and (b) Some extrinsic ringings corresponding tovalues of the mass near and above the critical value α20 andcomparison with the ringing associated with the masslessscalar field.

Fierz-Pauli field Hodd-parity sectorLΑ= 0.89 H=1,n=0L

Α= 0.97 H=1,n=0L

Α= 1.05 H=1,n=0L


0 20 40 60 80 100

10-10

10-8

10-6

10-4

0.01

1

tH2MLΦ

nQN

MHt

,rL

HbL

FIGURE: (a) and (b) Some extrinsic ringings corresponding tovalues of the mass near and above the critical value α10 andcomparison with the ringing associated with the odd-parity(`= 2,n = 0) QNM of the massless spin-2 field.




ConclusionPerspectives

Conclusion

By considering massive bosonic field theories in Schwarzschild spacetime, we have pointed outa new effect in BH physics :I The excitation factors of the long-lived QNMs have a strong resonant behavior (observed

numerically and confirmed semiclassically).

This has an amazing consequence :I The existence of giant and slowly decaying ringings in rather large domains of the mass

parameter. (We have also checked that the resonant effect and the associated giant ringingphenomenon still exist when the source of the BH perturbation is described by an initialvalue problem with Gaussian initial data.)

In the framework of massive gravity, even if the graviton is an ultralight particle, when it interactswith a supermassive BH, values of the coupling constant α leading to giant ringings can beeasily reached and used to test the various massive gravity theories or their absence could allowus to impose strong constraints on the graviton mass and to support, in a new way, Einstein’sgeneral relativity.





Perspectives

It would be interesting to study more realistic perturbations than the distortion descri-bed by an initial value problem, e.g :I The excitation of BH by a particle falling radially or plunging.I To extend our study to the Kerr BH.





Merci de votre attention.


excitation of black holes - umr 7332cosmo/sw_2014/ppt/ouldelhadj.pdf · introduction resonant...

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