Simulating merging black holes with spins beyond the Bowen … · 2011-05-31 · Simulating merging black holes with spins beyond the Bowen-York limit Geoffrey Lovelace (Cornell)

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

http://www.ligo.caltech.edu

http://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


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



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



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


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


6000 6200 6400 6600 6800


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

Documents

Simulating merging black holes with spins beyond the Bowen … · 2011-05-31 · Simulating merging black holes with spins beyond the Bowen-York limit Geoffrey Lovelace (Cornell)