simulating merging black holes with spins beyond the bowen ......simulating merging black holes with...
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Simulating merging black holes with spins beyond the Bowen-York limit
Geoffrey Lovelace (Cornell)in collaboration with
Mark A. Scheel & Béla Szilágyi (Caltech)see also arXiv:1010.2771
May 21, 2011
Outline• Black hole spin• Motivation
–Astrophysical–Physical
• Challenges –Initial data–Evolution
• Results–Black-hole eccentricity, spin–Gravitational waveforms–Horizon vorticity–Extremality & cosmic censorship
SS
(c)
• Dimensionless spin
–Geometrized units:–Stationary black hole:
• Naked singularity if
• Spacetime geometry–Nonspherical horizon–Spacetime rotates: gyroscopes
precess (“frame dragging”)
Horizon vorticity
Black hole spin
Horizon shape,
intrinsic scalar curvature,
tendicity
Horizonembedding
diagram
• Astrophysical motivation –Black holes could have spins
• Accretion models e.g. χ ~ 0.998: neglect magnetohydrodynamics - K. S. Thorne (1974) χ ~ 0.95: include magnetohydrodynamics - S. L. Shapiro (2005)
• Observational evidence (uncertain)e.g. microquasar GRS 1915+105 χ > 0.98 - J.E. McClintock et al (2006) χ ~ 0.7 - M. Middleton et al (2006)see also J. Blum et al (2009) & refs. therein
–Predict gravitational waveforms from binary black holes (BBHs)• Calibrate analytic waveform models
–Stellar-mass (ground detectors)–Supermassive (space detectors)
–Final holeʼs mass, spin, recoil
Image courtesy wikipedia
Introduction
Image courtesy www.ligo.caltech.edu
• Physical motivation–Explore general relativity
at its most extreme• Nonlinear behavior of
strongly-warped spacetimeFor details, see R. Owen et al (2011), cf. M. Scheelʼs talk this morning
–E.g. vorticity (“twisting”) and tendicity (“stretching/squeezing”)on and above horizon
–Extremality & cosmic censorship• Stationary holes: naked singularity if• Superextremal dynamical black holes?
Introduction
• Must satisfy constraints–Conformally flat:
• E.g. Bowen-York data: solve analyticallyJ. M. Bowen (1979), J. M. Bowen & J. W. York, Jr. (1980)
• Canʼt make equilibrium spinning black holesGarat & Price (2000)
• Binaries: Spins L. Rezzolla et al (2008) P. Marronetti et al (2008) S. Dain, C. O. Lousto, & Y. Zlochower (2008) M. Hannam et al (2010)
Initial dataMaxwell Einstein
χ < 0.93
• Must satisfy constraints–Superposed-Kerr-Schild
GL, R. Owen, H. P. Pfeiffer, & T. Chu (2008)
• Must solve all (coupled) constraint equations
Initial dataMaxwell Einstein
Metric of a boosted, spinning hole
Conformally flat except near horizons
0 100 200 300 400 500 600t / M
0.948
0.9485
0.949
0.9495
0.95
χz
Low resolutionMedium resolutionHigh resolution
Spin of individual horizon
Initial data
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1Spin / (Mass)2
-0.1 -0.10 0
0.1 0.10.2 0.20.3 0.30.4 0.40.5 0.50.6 0.60.7 0.70.8 0.80.9 0.9
1 11.1 1.1
E rot /
E rot,e
xtre
mal
Rotational energy / rotational energy if extremal
SKS initial data(this talk):first BBH mergersbeyond B.Y. limit
Not accessible withBowen-York (B.Y.)initial data
Accessible with Bowen-York data
• For high accuracy: spectral methods–Require more resolution
near mergeruse spectral adaptive mesh refinementFor details, cf. talk yesterday by B. Szilágyi• Adjust resolution as needed
–Need smooth, finite solution • Excise singularity• Grid coords. comove
with holes• Map between grid &
asymptotically inertial coords.
Evolution
Figure courtesy Béla Szilágyi
3400 3450 3500 3550Time / Total mass
10-7
10-6
10-5
Cons
train
t vio
latio
n ou
tside
hor
izon
Low resolutionMedium resolutionHigh resolution
Anti-aligned, equal mass BBH (spin=0.95)
Incoming char. field
Excision with high spins• Excision surface
–Must be inside horizon–No boundary condition (BC)
• OK if all info goes down the hole: “only outgoing characteristic fields”
• Control excision surface shape, velocity
No BC needed
Need BC
E.g. wave equation
6200 6250 6300 6350Time / Total Mass
0
0.0002
0.0004
0.0006
0.0008Min char speed (char speed control)Min char speed (size control)
Only outgoingchar. fieldsIncoming char. fields
Outgoing char. field
-5 0 5x
-5
0
5
yResults
-5 0 5x
-5
0
5
y
-6000 -4000 -2000 0Time before merger / Total Mass
00.10.20.30.4
Am
plitu
de
χ = +0.97χ=-0.95
-0.4-0.2
00.20.4
Wav
efor
m
χ = -0.95
-0.4-0.2
00.20.4
Wav
efor
m χ = +0.97
Waveforms
0 1000 2000 3000Time / total mass
-0.95
-0.94
-0.93
Dim
ensio
nles
s spi
n
Low resolutionMedium resolutionHigh resolution
Spin of individual horizon
3500 4000Time / total mass
0.15
0.2
0.25
0.3
0.35
0.4
Dim
ensio
nles
s spi
n
Low res.Medium res.High res.
Spin of common horizon
0 1000 2000 3000Time / total mass
-0.95
-0.94
-0.93
Dim
ensio
nles
s spi
n
Low resolutionMedium resolutionHigh resolution
Spin of individual horizon
3500 4000Time / total mass
0.15
0.2
0.25
0.3
0.35
0.4
Dim
ensio
nles
s spi
n
Low res.Medium res.High res.
Spin of common horizonSpin vs. time
Campanelli, Lousto, and Zlochower (2006)Tichy and Marronetti (2008)Boyle and Kesden (2008)
Barausse and Rezzolla (2009)Rezzolla et al (2008)Buonanno, Kidder, and Lehner (2008)
0 1000 2000 3000 4000 5000 6000Time / Total Mass
0.94
0.95
0.96
0.97
0.98
0.99D
imen
sionl
ess s
pin
Low resolutionMedium resolution
6000 6200 6400 6600 6800
Low resolutionMedium resolution
Individual horizons
Common horizons
0 1000 2000 3000 4000 5000 6000Time / Total Mass
0.94
0.95
0.96
0.97
0.98
0.99D
imen
sionl
ess s
pin
Low resolutionMedium resolution
6000 6200 6400 6600 6800
Low resolutionMedium resolution
Individual horizons
Common horizons
Spin vs. time
Campanelli, Lousto, and Zlochower (2006)Tichy and Marronetti (2008)Boyle and Kesden (2008)
Barausse and Rezzolla (2009)Rezzolla et al (2008)Buonanno, Kidder, and Lehner (2008)
• Initial data
• Evolution
Extremality
0.37 0.38 0.39Ωr
0.80.850.90.9511.051.1
Spin
0.811.21.41.61.82
Mas
s or r
adiu
s
Horizon Excision surface
ξ
χ
Mirr
min radius
max radius
SKS, conformal spin 0.99
6410 6410.1 6410.2 6410.3 6410.4Time / Total Mass
0.8
0.85
0.9
0.95
1
χξζ
Horizon vorticity
Movie courtesy Dave Kotfis
Conclusion• Nearly extremal spins
–Astrophysical black holes may have spins ~ 1–Improved initial data: can simulate BBH with spins
beyond the Bowen-York limit of • This talk: spins up to
• Future work–More simulations
• Unequal masses, generic spin direction
• Even higher spins–Extract science from
high-spin simulations
χ = 0.97
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1Spin / (Mass)2
-0.1 -0.10 0
0.1 0.10.2 0.20.3 0.30.4 0.40.5 0.50.6 0.60.7 0.70.8 0.80.9 0.9
1 11.1 1.1
E rot /
E rot,e
xtre
mal
Rotational energy / rotational energy if extremal
SKS initial data(this talk):first BBH mergersbeyond B.Y. limit
Not accessible withBowen-York (B.Y.)initial data
Accessible with Bowen-York data