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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
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
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
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
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)−→+∞
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 3 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 4 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
The Regge-Wheeler equation
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`
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 5 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
The Regge-Wheeler equation
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 6 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
The Regge-Wheeler equation
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 7 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
The Regge-Wheeler equation
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
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 8 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 9 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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 →+∞.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 10 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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
Massive scalar field
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 11 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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
Massive scalar field
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
Massive scalar field
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
Massive scalar field
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 12 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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
Einstein gravityH=2,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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 13 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 14 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
A semiclassical analysis
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 15 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
A semiclassical analysis
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 16 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
The Regge-Wheeler equationIntrinsic giant ringingsResultsA semiclassical analysis
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 17 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 18 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
Results : Extrinsic giant ringings
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)
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 19 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
Results : Extrinsic giant ringings
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))
Massive scalar field
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
Einstein gravityH=2,n=0L
0 20 40 60 80 100
-0.1
0
0.15
Massive scalar field
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
Einstein gravityH=2,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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 20 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 21 / 23
IntroductionResonant behavior of the quasinormal excitation factors
Extrinsic giant ringingsConclusion and perspectives
ConclusionPerspectives
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.
Resonant excitation of black holes by massive bosonic fields and giant ringings SW-8 May 2014 22 / 23