Black hole dynamicsin generic spacetimes
Helvi Witek,V. Cardoso, L. Gualtieri, C. Herdeiro,A. Nerozzi, U. Sperhake, M. Zilhao
CENTRA / IST,Lisbon, Portugal
Phys. Rev. D 81, 084052 (2010), Phys. Rev. D 82, 104014 (2010),Phys. Rev. D 83, 044017 (2011), Phys. Rev. D 84, 084039 (2011),
Phys. Rev. D 82, 104037 (2010),work in progress
Molecule workshop:“Recent advances in numerical and analytical methods for black hole dynamics”
YITP, Kyoto, 28 March, 2012
H. Witek (CENTRA / IST) 1 / 26
Outline
1 Collisions of black holes in higher dimensional spacetimes
2 Black holes in a box
3 Conclusions and Outlook
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Black hole collisions
in higher dimensional spacetimes
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High Energy Collision of ParticlesConsider particle collsions with E = 2γm0c2 > MPl
Hoop - Conjecture (Thorne ’72)⇒ BH formation, if circumference ofparticle < 2πrS
Collisions of shock waves(Penrose ’74, Eardley & Giddings ’02)⇒ BH formation if b ≤ rS
numerical evidence in ultra relativisticcollision of boson stars⇒ BH formation if boost γc ≥ 2.9
Low Lorentz boost, γ = 1
Large Lorentz boost, γ = 4
Choptuik & Pretorius ’10
⇒ black hole formation in high energy collisions of particles
higher dimensional theories of gravity ⇒ TeV gravity scenarios
signatures of black hole production in high energy collision of particles
at the Large Hadron Colliderin ultra-high relativistic Cosmic rays interactions with the atmosphere
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Life cycle of Mini Black Holes
1 Formation
lower bound on BH mass from area theorem (Yoshino & Nambu ’02)
2 Balding phase: end state is Myers-Perry black hole
3 Spindown phase: loss of angular momentum and mass
4 Schwarzschild phase: decay via Hawking radiation
5 Planck phase: M ∼ MPl
Goal: more precise understanding of black hole formation⇒ compute energy and angular momentum loss in gravitational radiationToy model: black hole collisions in higher dimensions
H. Witek (CENTRA / IST) 5 / 26
Numerical Relativity
in D > 4 Dimensions
Yoshino & Shibata, Phys. Rev. D80, 2009,Shibata & Yoshino, Phys. Rev. D81, 2010
Okawa, Nakao & Shibata, Phys. Rev. 83, 2011
Lehner & Pretorius, Phys. Rev. Lett. 105, 2010
Sorkin & Choptuik, GRG 42, 2010; Sorkin, Phys. Rev. D81, 2010
Zilhao et al., Phys. Rev. D81, 2010, Witek et al, Phys. Rev. D82, 2010,Witek et al, Phys. Rev. D83, 2011, Zilhao et al., Phys. Rev. D84, 2011.
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Numerical Relativity in D Dimensions
consider highly symmetric problems
dimensional reduction by isometry on a (D-4)-sphere
general metric element
ds2 = gµνdxµdxν + λ(xµ)dΩD−4
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Numerical Relativity in D Dimensions
D dimensional vacuum Einstein equations GAB = RAB − 12 gAB R = 0 imply
(4)Tµν =D − 4
16πλ
[∇µ∇νλ−
1
2λ∂µλ∂νλ− (D − 5)gµν +
D − 5
4λgµν∇αλ∇αλ
]∇µ∇µλ = 2(D − 5)− D − 6
2λ∇µλ∇µλ
4D Einstein equations coupled to scalar field
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Formulation of EEs as Cauchy Problem in D > 4
3+1 split of spacetime (4)M = R +(3) Σ (Arnowitt, Deser, Misner ’62)ds2 = gµνdxµdxν =
(−α2 + βkβ
k)
dt2 + 2βidx idt + γijdx idx j
3+1 split of 4D Einstein equations with source terms=⇒ Formulation as initial value problem with constraints (York 1979)
Evolution Equations
∂tγij = −2αKij + Lβγij∂tKij = −DiDjα + α
((3)Rij − 2KilK
lj + KKij
)+ LβKij
−αD − 4
2λ
(DiDjλ− 2KijKλ −
1
2λ∂iλ∂jλ
)∂tλ = −2αKλ + Lβλ
∂tKλ = −1
2∂ lα∂lλ+ α
((D − 5) + KKλ +
D − 6
λK 2λ
−D − 6
4λ∂ lλ∂lλ−
1
2D lDlλ
)+ LβKλ
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Wave Extraction in D > 4
Generalization of Regge-Wheeler-Zerilli formalism by Kodama & Ishibashi ’03
Master function
Φ,t = (D − 2)r (D−4)/2 2rF,t − F rt
k2 − D + 2 + (D−2)(D−1)2
rD−3S
rD−3
, k = l(l + D − 3)
Energy flux & radiated energy
dEl
dt=
(D − 3)k2(k2 − D + 2)
32π(D − 2)(Φl
,t)2 , E =
∞∑l=2
∫ ∞−∞
dtdEl
dt
Momentum flux & recoil velocity
dP i
dt=
∫S∞
dΩd2E
dtdΩni , vrecoil =
∣∣∣∣∫ ∞−∞
dtdP
dt
∣∣∣∣H. Witek (CENTRA / IST) 10 / 26
Numerical Setup
use Sperhake’s extended Lean code (Sperhake ’07, Zilhao et al ’10)
3+1 Einstein equations with scalar fieldBaumgarte-Shapiro-Shibata-Nakamura formulation with moving puncturesdynamical variables: χ, γij , K , Aij , Γi , ζ, Kζ
modified puncture gauge
∂tα = βk∂kα− 2α(K + (D − 4)Kζ)
∂tβi = βk∂kβ
i − ηββ i + ηΓΓi + ηλD − 4
2ζγ ij∂jζ
measure lengths in terms of rS with
rD−3S =
16π
(D − 2)ASD−2 M
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Equal mass head-on in D = 4, 5, 6
Brill-Lindquist type initial data
ψ = 1 + rD−3S,1 /4rD−3
1 + rD−3S,2 /4rD−3
2
Key (technical) issues:
modification of gaugeconditionsmodification of formulation
increase in E/M with D⇒ qualitative agreement withPP calculations(Berti et al, 2010)
D rS ω(l = 2) E/M(%)4 0.7473− i 0.1779 0.0555 0.9477− i 0.2561 0.0896 1.140− i 0.304 0.104
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Unequal mass head-on in D = 5
consider mass ratios q = rD−3S,1 /rD−3
S,2 = 1, 1/2, 1/3, 1/4
Modes of Φ,t
0
0.04
Φ,t
l=2 /
r S
1/2
q = 1q = 1:2q = 1:4
0
0.005
Φ,t
l=3 /
rS
1/2
-10 0 10 20 30∆t / r
S
0
0.0009
Φ,t
l=4 /
rS
1/2
E/M ∼ η2 (M.Lemos ’10, MSc thesis)
fitting function EMη2 = 0.0164− 0.0336η2
within < 1% agreement with point particle calculation (Berti et al, 2010)
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Initial data for boosted BHs in D > 4
construct initial data by solving the constraints
assumption: γab = ψ4
D−3 δab , K = 0 , Kab = ψ−2Aab
constraint equations
∂aAab = 0 , 4ψ +D − 3
4(D − 2)ψ−
3D−5D−3 AabAab = 0 , with 4 ≡ ∂a∂a
analytic ansatz for Aab → generalization of Bowen-York type initial data
elliptic equation for ψ → puncture method (Brandt & Brugmann ’97)
ψ =1 +∑i
rD−3S(i) /4rD−3
(i) + u
⇒ Hamiltonian constraint becomes
4u +D − 3
4(D − 2)AabAabψ
− 3D−5D−3 = 0
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Head-on of boosted BHs (preliminary results)
evolution of puncture with z/rS = ±30.185 with P/rD−3S = 0.4
l = 2 mode of Φ,t
0 50 100 150 200 250t / r
S
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Φ, t
l=2
D = 5
D = 6
PP calculations (Berti et al, 2010)
Issues:
dependence of radiated energy on D and boost
long-term stable evolutions for larger boosts
adjustment of (numerical) gauge
requirement of very high resolution in wavezone for reasonable accuracy
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Black hole binaries
in a box
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AdS / CFT correspondence (Maldacena ’97)
Anti-de Sitter spacetime⇒ spacetime with negative cosmologicalconstant
duality betweentheory with gravity on AdSd × X
mconformal field theory on conformalboundary
consider scalar field propagation in SAdSbackground
d2
dr2∗ψ + (ω2 − V )ψ = 0
potential V −→∞ as r −→∞⇒ view AdS as boxed spacetime
⇒ toy model: BH evolution in a box Berti, Cardoso & Starinets ’09
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Black Hole stability
superradiant scattering of rotating BH (Penrose ’69, Christodoulou ’70,Misner ’72)
impinging wave amplified as it scatters off a BH if ω < mΩH
extraction of energy and angular momentum of the BH by superradiant modes
“black hole bomb” (Press & Teukolsky ’72)
consider Kerr BH surrounded by a mirrorsubsequent amplification of superradiant modes
stability of Kerr-AdS BHs (Hawking & Reall ’99, Cardoso et al. ’04)
AdS infinity behaves as box ⇒ amplification of superradiant instabilities?large Kerr-AdS BH: stablesmall Kerr-AdS BH: unstable
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BBHs in a Box - Setup
mimic AdS-BH spacetime by BHs in a box
impose reflecting b.c. via∂tKij = 0 and ∂tγij = 0 at boundary
wave extraction: Newman Penrose scalarsΨ4 (outgoing) and Ψ0 (ingoing)
initial data: non-spinning, equal mass BHs
head-on collision ⇒ non-spinning final BHquasi-circular inspiral ⇒ spinning final BH witha/M = 0.69
Movie
rOuter
rOuterBoundary
DR
rijk
G0
1
G2
2G3
1
G1
1
G2
1 G3
2
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BBHs in a Box - Waveforms
<(Ψ422) and <(Ψ0
22)
0 100 200 300 400
t / M
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
rM R
e[ψ
0,4 2
2]
rM Re[ψ0
22(t / M - 10)]
rM Re[ψ4
22(t / M)]
spectrum of first outgoing pulse
0 0.2 0.4 0.6 0.8 1
Mω
0
0.1
0.2
0.3
0.4
(dE
/dω
) /
M2
Mωc = 0.4
spectrum of Ψ422 after merger and before interaction with boundary
assume quasi Kerr BH⇒ critical frequency for superradiance MωC = mΩ = 0.4
signal contains frequencies within and above superradiance regime
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BBHs in a Box - Convergence Test
<(Ψ422)
0 100 200 300 400
t / M
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
∆ (
rM R
e[ψ
4 22])
rM Re[ψ4
22,hc
- ψ4
22,hm
]
1.47 rM Re[ψ4
22,hm
- ψ4
22,hf
]
1.26 rM Re[ψ4
22,hm
- ψ4
22,hf
]
<(Ψ022)
0 50 100 150 200 250 300 350
t / M
-0.004
-0.002
0
0.002
0.004
∆(r
M R
e[ψ
0 22])
rM Re[ψ0
22,hc
- ψ0
22,hm
]
1.47 rM Re[ψ0
22,hm
- ψ0
22,hf
]
1.26 rM Re[ψ0
22,hm
- ψ0
22,hf
]
simulations at 3 different resolutions:hc = 1/48M, hm = 1/52M, hf = 1/56M
4th order accurate in merger signal
2nd order convergence in 1st & 2nd reflection
loosing convergence afterwards ⇒ consider first 3 cycles
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BBHs in a Box - Area and SpinRelative horizon mass and spin
0 50 100 150 200
∆t / M
0
0.005
0.01
0.015
M / M
0 -
1
M / M0 - 1
Mirr
/ Mirr,0
- 1
0.63
0.64
0.65
0.66
J
J
successive increase in horizon area, mass and angular momentumduring interaction between BH and gravitational radiation
absorption of ∼ 15% of GW energy per cycle
increase of ∼ 5% in angular momentum in first cycle
no indication of superradiant amplification
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BBHs in a Box
IDEA:complementary phenomena
high frequency modes:absorbtion of mass and angular momentum fromradiation
mlow frequency modes:amplification of superradiant modeswith ω < mΩH
frequencies in superradiant andabsorption regime⇒ at transition point between stable andsuperradiant regime?
ToDo:
improvement of bcs ⇒ long-term evolution
model highly spinning BHs Press & Teukolsky ’74
H. Witek (CENTRA / IST) 23 / 26
Conclusions I
Evolutions of BH head-on collisions in D = 5, 6 dimensions
good agreement between PP and numerical results for unequal mass binaries
collisions from rest: increase in radiated energy with increasing dimension
setup of initial data for boosted BH solving the constraints
evolutions of collisions with small boost parameter
Issues & ToDo list:
dependence of radiated energy on D and boost
go beyond D = 6
long-term stable evolutions for large boosts
adjustment of gauge conditions
modification of formulation
H. Witek (CENTRA / IST) 24 / 26
Conclusions II
mimic AdS-BH spacetimes by BHs in a box
“BH bomb” like setup
monitor interaction between Kerr BH and gravitational radiation
evidence for absorption of radiation by the BH
Issues & ToDo list:
no clear evidence for superradiant amplification
systems becomes numerically unstable after few reflections
improvement of boundary conditions⇒ long-term modelling of system
increasing amplification rate with increasing spin⇒ evolve highly spinning BHs
H. Witek (CENTRA / IST) 25 / 26
Arigato!
http://blackholes.ist.utl.pt
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