Harald P. Pfeiﬀer

1

Binary black hole initial data

Harald P. Pfeiffer

California Institute of Technology

Collaborators: Greg Cook (Wake Forest), Larry Kidder (Cornell),

Mark Scheel (Caltech), Saul Teukolsky (Cornell), James York (Cornell)

Relativistic Astrophysics Workshop, IPAM, UCLA, May 4, 2005

Harald P. Pfeiffer, Caltech

2Black hole evolutions are important

See talks by

Joan Centrella

Frans Pretorius

Luis Lehner

Jerome Novak

Oscar Reula

Mark Scheel

Lee Lindblom

Matt Choptuik

Tvsi Piran

Saul Teukolsky

evolutions need initial data


3Initial data for GR

• Initial data consists of metric gij and extrinsic curvature Kij

on one hypersurface Σ.

• It must satisfy the constraints

R +K2 −KijK

ij= 0

∇j

“Kij − gijK

”= 0

• Challenges:

1. How to solve the constraints

2. How to choose gij and Kij


4Outline of talk

1. Construction of any initial data – Conformal method

2. Some surprising properties – Non-uniqueness

3. Construction of BBH initial data – Quasi-equilibrium method

4. Discussion & Thoughts – How good is good enough?


5Conformal method

Problem: Find solutions gij, Kij

of the constraint equations

R +K2 −KijK

ij= 0

∇j

“Kij − gijK

”= 0

Strategy: Split gij and Kij into smaller pieces, some

freely specifiable, the rest completely determined.

Specifically, gij = ψ4gij

Choose free data ⇒ Solve elliptic equations ⇒ Assemble gij, Kij


6Numerical method: Spectral elliptic solver

(HP, Kidder, Scheel, Teukolsky 2003)

Expand solution in basis-functions & solve for expansion-coefficients

Smooth solutions ⇒ exponential convergence

• Superior accuracy: Numerical errors physical effects

• Superior efficiency: Large parameter studies

• Domain decomposition: Multiple length-scales

30 45 60 75 90N

10-8

10-6

10-4

10-2

||H||2||M||2|δEADM||δMK||δJz|

Binary black hole ID (Cook & HP, 2004) Bowen-York ID (HP, Kidder et al, 2003)


7Standard and extended conformal thin sandwich

Standard CTS Extended CTS

Free data gij, ∂tgij, K, N Free data gij, ∂tgij, K, ∂tK

Four elliptic equationsFive elliptic equations

∇2ψ −

1

8Rψ −

1

12K

2ψ

4+

1

8AijA

ij= 0

∇j„

1

2NLβij

«−

2

3ψ

6∇iK−∇j„

1

2Nuij

«= 0

∇2ψ −

1

8Rψ −

1

12K

2ψ

4+

1

8AijA

ij=0

∇j„

1

2NLβij

«−

2

3ψ

6∇iK−∇j„

1

2Nuij

«=0

∇2`Nψ

7´−Nψ7

„1

8R+

5

12K

2ψ

4+

7

8AijA

ij«

=

−ψ5(∂t − βk∂k)K

+ Quite a few existence and uniqueness – Mathematical terra incognita

results (especially for K = 0)


8Some results on the standard system

Mathematics:

1. asymptotically flat

2. no inner boundaries

3. maximal slice K = 0

→ Yamabe constant Y[gij]:

Y[gij] > 0 ⇔ existence & uniqueness

(Cantor 1977, Murray & Cantor 1981,

Maxwell 2005)

Free data based on “Teukolsky wave”

ingoing, M=0, odd parity, centered at r=20

(HP, Kidder, Scheel, Shoemaker 2005)

gij = δij + Ahij

uij = A ∂thij

K = 0, N = 1

0 0.5 1 1.5 2A~

1

10

100

1.85 1.90 1.95 2.00A~

0.00

0.05

0.10

0.15

max(ψ)

1 / max(ψ)


9Extended system (HP & York, gr-qc/0504142)

gij = δij + Ahij

uij = A ∂thij

K = 0, ∂tK = 0

0 0.1 0.2 0.3A~

1.00

1.03

1.06

1.09

10-4 10-3 10-2

A~c,5-A~

0.003

0.01

0.03

ψcr

it- max

(ψ)

max(ψ)

ψ finite as A → Ac

Parabolic behaviorψ ≈ ψc − const.(δA)1/2


10A more comprehensive look

0.03 0.1 0.3 1.0A~

0.00

0.20

0.40

0.60

0.80

1.00

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)

max(β)

Equatorial plane, A = 0.3


11A second branch

• So far u−(A) = uc − vcpδA , u =

“ψ, β

i, N

”• Two branches??

u±(A) = uc ± vcpδA

• Problem: With “simple” initial guess, elliptic solver converges always to u−;

need good guess to converge to u+.

du−(A)

dA=

1

2pδA

vc

⇒ u+(A) ≈ u−(A) + 4 δAdu−(A)

dA

• Take two numeric solutions u− of five coupled 3-D nonlinear elliptic equations,

and finite-difference them !!


12Constructing the upper branch u+

0.03 0.1 0.3 1.0A~

0.03

0.1

0.3

1.0

3.0

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)max(β)

0.03 0.1 0.3 1.0A~

0.03

0.1

0.3

1.0

3.0

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)max(β)

0.03 0.1 0.3 1.0A~

0.03

0.1

0.3

1.0

3.0

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)max(β)

0.03 0.1 0.3 1.0A~

0.03

0.1

0.3

1.0

3.0

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)max(β)

0.03 0.1 0.3 1.0A~

0.03

0.1

0.3

1.0

3.0

1.08

1.10

1.12

0.56

0.64

0.72

0.296 0.300 0.3040.14

0.16

0.18

max(ψ)

min(N)min(N)

max(ψ)

max(β)max(β)

• Parabolic close to Ac

• u+ and u− meet at Ac

• u+ deviates strongly from

Minkowski at small A

• Two solutions forarbitrarily small A!!


13Energy & Apparent horizon

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

standard

syste

m

u-

u+

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

standard

syste

m

u-

u+apparenthorizon

Apparent horizons for A = 0.02

-30 -20 -10 0 10 20 30x-30

-20

-10

0

10

20

30

z

ψ=1.8

ψ=2.0

ψ=2.2


14Unique solutions, nevertheless?

• Physics is determined by gij, Kij. For example gij = ψ4gij = ψ4δij + ψ4A hij

• Physical amplitude of perturbation is A = ψ4A• Question: For given physical amplitude A, how many solutions exist?

0.1 0.3 1.0Amax=(max ψ)4A~

0.03

0.1

0.3

1.0

3.0

1.10

1.12

0.56

0.60

0.64

0.42 0.44 0.46

0.16

0.17

0.18

max(ψ)

min(N) min(N)

max(ψ)

max(β)

max(β)

0.1 1 10Amax=(max ψ)4A~

0.1

1

10

100ADM energy

standard system

u+

u-


15Lessons from journey into mathematical wonderland

• Life is more interesting than expected (HP & York, gr-qc/0504142).

• Similar issues possible in any system containing the extended

conformal thin sandwich equations (→ Jerome Novak’s talk?).

• Non-unique data sets are very different from BBH initial data. “Just solving”

works, so let’s go!


16Astrophysically realistic BBH initial data

Question: How to choose free data and boundaryconditions for a binary black hole in a circular orbit?

Traditional approach (conformally flat Bowen-York) has many problems:

ISCO wrong, BHs plunge too quickly, ...

Need better method!


17Quasi-equilibrium method

Basic idea:Approx. time-independence in corotating frame

Approx. helical Killing vector

(both concepts essentially equivalent,

both useful depending on context)

History:

• Wilson & Matthews 1985: Binary neutron stars

• Gourgoulhon, Grandclement & Bonazzola, 2002a,b

BBH ID with inner boundary conditions

(basically right, but various details deficient)

• Cook & HP, 2002, 2003, 2004

General quasi-equilibrium method with isolated horizon BCs


18

Quasi-equilibrium method (the easy pieces)

• Time-independence in corotating frame

⇒ natural choice: vanishing time derivatives

• Extented conformal thin sandwich formalism

1. ∂tgij = 0 = ∂tK

2. gij and K still undetermined

• Boundary conditions at infinity from asymptotic flatness & corotation:

ψ = 1

βi= (~Ωorbital × ~r)

i

N = 1

• New contribution: inner boundary conditions (next slide...)


19Quasi-equilibrium excision boundary conditions

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Σ

si

kµ

S

kµ

– outward pointing null normal to Ssi

– (conformal) spatial normal to S

• Excise topological sphere(s) S

• Require

1. S be apparent horizon(s)

2. When evolved, the coordinate locations of the AH’s remain stationary

3. The shear σµν of kµ vanishes

• Item 3 is an isolated horizon condition. It implies for the expansion θ

Lkθ = −1

2θ

2 − σµνσµν = 0 on S

⇒ AH moves along kµ, and its area is constant (initially)


20Quasi-equilibrium excision boundary conditions cont’d

• Rewrite in variables of conformal thin sandwich

∂rψ = . . . on S (1)

βi

= ψ2Ns

i+ β

i‖ on S (2)

Boundary conditions for ψ and βi

• Rotating black holes“Vanishing shear” restricts βi‖. Freedom to specify spin of BH remains.

• Lapse boundary condition not fixed by IH (also Jaramillo et al, 2004).

• Determine orbital frequency by EADM = MK.


21Numerical solutions: Single black holes I

• gij, K, S not determined. and lapse-BC.

• For now arbitrary choices: Conformal flatness, S=sphere

• Do not use knowledge of single BH solutions – use single BHs to test method

Spherical symmetry

1. w.l.o.g. conformally flat

2. Try different choices for K and lapse boundary condition

3. any spherically symmetric K and any spherically symmetric lapse-BC yield:

– exact slice through Schwarzschild

– totally vanishing time-derivatives ∂tgij = ∂tKij = 0

4. Full success: Recover Schwarzschild independent of arbitrary choices.


22Numerical solutions: Single black holes II

• Spinning/Boosted black holesCompute quantities that vanish for Kerr:

Erad ≡qE2

ADM− P 2

ADM−

qM2

irr + J2ADM

/(4M2irr) (3)

∆Ω ≡ Ωr −JADM/E

3ADM

2 + 2q

1− J2ADM

/E4ADM

, ∆v ≡ v −PADM

EADM(4)

0.01 0.04 0.1 0.2Mirr Ωr

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Erad / EADM∆Ω EADM|∂tlnψ|

ISCO orbital frequency

0.04 0.1 0.4 1v10-7

10-6

10-5

10-4

10-3

10-2

10-1

Erad / EADM∆v|∂tlnψ|

ISCO velocities


23

Binary black hole solutions (corotating, K = 0)

Lapse positive through horizon


24

Sequences of quasi-circular orbits (corotating, K = 0)

Three different lapse boundary conditions

Compare to GGB and post-Newtonian results

3.6 4 4.4 4.8J/µm

-0.06

-0.05

-0.04

-0.03

-0.02

E b/µ

CO: MS - d(αψ)/dr = 0CO: MS - αψ = 1/2CO: MS - d(αψ)/dr = (αψ)/2r

3.39 3.42

-0.065

-0.06

3.6 4 4.4 4.8J/µm

-0.06

-0.05

-0.04

-0.03

-0.02

E b/µ

CO: MS - d(αψ)/dr = (αψ)/2rCO: HKV - GGBCO: EOB - 3PNCO: EOB - 2PNCO: EOB - 1PN

3.4 3.6

-0.06

-0.05

No difference – solution robust

Excellent agreement


25Testing the 2nd law

Normalize sequences such that dEADM = Ω0 dJADM

Irreducible mass along these sequences

0.50000

0.50005

Mirr

CO: MS - d(αψ)/dr = 0CO: MS - αψ = 1/2CO: MS - d(αψ)/dr = (αψ)/2r

0.03 0.06 0.09 0.12mΩ0

0.498

0.499

0.500

Mirr

IR: MS - d(αψ)/dr = 0IR: MS - αψ = 1/2IR: MS - d(αψ)/dr = (αψ)/2r

⇐ corotating sequences

(three different lapse BC’s)

Mirr slightly increasing during inspiral – ok

⇐ irrotational sequences

(three different lapse BC’s)

Mirr decreasing during inspiral

– Normalization of sequences wrong?

– Remaining free data insufficient?

– Rigidity theorem?


26ISCO location

Caution: ISCO is not a sharp,

well-defined concept! Anyway...

Color: Corotating BH’s

Grey: Irrotational BH’s

0.08 0.12 0.16mΩ0

-0.022

-0.02

-0.018

-0.016

-0.014

E b / m GGB ’02

Cook&Pfeiffer ’04 (3 data points)

2,3 PN (standard)1,2,3 PN (EOB)

Cook ’94

Cook&Pfeiffer ’04 (3 data points)

PN (EOB)

PN (standard)

• Excellent agreement between NR and PN

• Superior to Bowen-York w/ effective potential (cf. Cook 1994)


27Qualitity of todays QE-BBH initial data

Contains two black holes of course

ISCO agrees with PN predictions yes

ID-solve results in lapse & shift yes

BHs at rest early in evolution yes

Time-derivatives small ψ/∂tψ ∼ 50Torbit

50Torbit Tinspiral at large separation!

Spins incorporated exactly? no

Tidal distortion of BHs correct? no

Satisfies testmass limit? no

Correct r (slightly negative)? no

Contains wavetrain of earlier insipral? no

⇒ superior to Bowen-York, but room for further improvement


28

How good is good enough?

Just some thoughts and conjectures — further opinions welcome!!


29How good is good enough?

Answer depends on application and desired accuracy.

Short term – improving evolution codes:

• Testing evolution codesID doesn’t matter.

(But smaller initial transient may well be advantageous for dynamic gauge conditions)

• Qualitative features of plunge wave-forms (→ Frans Pretorius’ talk)

My feeling is that ID won’t matter (must be demonstrated, though!).

• The first evolution of two orbits and plungeID must contain BHs in circular-ish orbits. I believe this is the case (no proof).

Such evolutions must quantify initial transient caused by initial data.


30Long term – gravitational wave science

1. If ID contains significant initial transients, then one must wait until they are beyond

wave-extraction radius Rextract

Twait ≈ 2Rextract ≈ 100M ∼ 105. . . 10

6CPU-hours

Expensive. But let’s assume we’re willing to pay...

2. Initial transient changes coordinatesystem, masses, spins, orbital elements.

Accumulated phase-error very sensitive to (e.g.) eccentricity ε.

How to measure ε in a dynamical spacetime in an unknown coordinatesystem?

This matters for

• Testing PN-results

• Parameter extractionMay be limited by our knowledge of evolution-parameters.

• Coherent template PN-inspiral+numerical evolution+ringdownMost stringent test of strong field GR.

Total phase error must be (much) smaller than QN-period. (LISA SMBH removal!)


31Summary

• Non-unique solutions exist (and may bite us)

• Quasi-equilibrium initial data vast improvement over Bowen-York...

• ... and contains handles for next round of improvements (gij,S)

• Further improvements essential, especially when evolutions mature

• QE-initial data publicly available soon.


32Summary of QE method

• Framework for BBH initial data in a kinematical setting (helical Killing vector)

• Explicitly displays the remaining choices gij, K,S, Lapse-BC

• Close in spirit to GGB, but greatly improved:

1. Constraints are satisfied

2. Incorporates isolated horizon boundary conditions

3. General spins possible

4. Retains freedom to choose any gij, K,S.

5. Lapse is positive on horizon

• Agrees very well with PN (even with simple choices)

• Data sets will be publicly available soon.


Harald P. Pfeiﬀer

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