View
0
Download
0
Category
Preview:
Citation preview
1
Quasi-equilibrium binary black hole initial data
Harald P. Pfeiffer
California Institute of Technology
Collaborators: Greg Cook, Larry Kidder, Mark Scheel,
Saul Teukolsky, James York
Numerical Relativity 2005, Goddard Space Flight Center, Nov 4, 2005
Outline:
1. Formalism & Numerics
2. Non-uniqueness in conformal thin sandwich
3. Properties of the constructed ID sets
4. Public initial data repository
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
2
Formalism & Numerics
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
3Quasi-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,bBBH ID with inner boundary conditions
basically right, but various deficiencies
• Cook & HP, 2002, 2003, 2004 (especially Cook & Pfeiffer, PRD 70, 104106, 2004)General quasi-equilibrium method with isolated horizon BCs
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
4
Quasi-equilibrium method (the easy pieces)
• Time-independence in corotating frame
⇒ vanishing time derivatives
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
4
Quasi-equilibrium method (the easy pieces)
• Time-independence in corotating frame
⇒ vanishing time derivatives
• Extented conformal thin sandwich formalism
∂tg̃ij = 0 = ∂tK∇̃2ψ −
1
8R̃ψ −
1
12K
2ψ
4+
1
8ÃijÃ
ij=0
∇̃j
ψ6
2NLβij
!−
2
3ψ
6∇̃iK−∇̃j
ψ6
2Nũij
!=0
∇̃2`Nψ´−Nψ
„1
8R̃+
5
12K
2ψ
4+
7
8ÃijÃ
ij«
=
−ψ5(∂t − βk∂k)K
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
4
Quasi-equilibrium method (the easy pieces)
• Time-independence in corotating frame
⇒ vanishing time derivatives
• Extented conformal thin sandwich formalism
∂tg̃ij = 0 = ∂tK∇̃2ψ −
1
8R̃ψ −
1
12K
2ψ
4+
1
8ÃijÃ
ij=0
∇̃j
ψ6
2NLβij
!−
2
3ψ
6∇̃iK−∇̃j
ψ6
2Nũij
!=0
∇̃2`Nψ´−Nψ
„1
8R̃+
5
12K
2ψ
4+
7
8ÃijÃ
ij«
=
−ψ5(∂t − βk∂k)K• Boundary conditions at infinity
ψ = 1
βi= (~Ωorbital × ~r)
i
N = 1
• New contribution: inner boundary conditions (next slides)
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
5Quasi-equilibrium excision boundary conditions
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppp ppppp ppppp ppppp ppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppp pppppp pppppp pppppppppppp pppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppp pppppppppp ppppp ppppp ppppp pppppppppp ppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppp pppppp pppppp pppppp pppppp pppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
Σ
kµ
S
• Excise topological spheres S
• Require1. S be apparent horizons2. The AH’s remain stationary in evolution
3. Shear of kµ vanishes (isolated horizon)
⇒ Lkθ = 0 ⇒ AH moves along kµ and MAH initially constant
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
5Quasi-equilibrium excision boundary conditions
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppp ppppp ppppp ppppp ppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppp pppppp pppppp pppppppppppp pppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppp pppppppppp ppppp ppppp ppppp pppppppppp ppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppp pppppp pppppp pppppp pppppp pppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
Σ
kµ
S
• Excise topological spheres S
• Require1. S be apparent horizons2. The AH’s remain stationary in evolution
3. Shear of kµ vanishes (isolated horizon)
⇒ Lkθ = 0 ⇒ AH moves along kµ and MAH initially constant
• Rewrite in conformal variables ⇒ BC’s on ψ and βi
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
5Quasi-equilibrium excision boundary conditions
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppp ppppp ppppp ppppp ppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppp pppppp pppppp pppppppppppp pppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppp pppppppppp ppppp ppppp ppppp pppppppppp ppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppp pppppp pppppp pppppp pppppp pppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
Σ
kµ
S
• Excise topological spheres S
• Require1. S be apparent horizons2. The AH’s remain stationary in evolution
3. Shear of kµ vanishes (isolated horizon)
⇒ Lkθ = 0 ⇒ AH moves along kµ and MAH initially constant
• Rewrite in conformal variables ⇒ BC’s on ψ and βi
• General spin possible (→ Greg Cook’s talk)
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
5Quasi-equilibrium excision boundary conditions
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppp ppppp ppppp ppppp ppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppp pppppp pppppp pppppppppppp pppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppp pppppppppp ppppp ppppp ppppp pppppppppp ppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppp pppppp pppppp pppppp pppppp pppppppppppppppppppppppppppp
pppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp
Σ
kµ
S
• Excise topological spheres S
• Require1. S be apparent horizons2. The AH’s remain stationary in evolution
3. Shear of kµ vanishes (isolated horizon)
⇒ Lkθ = 0 ⇒ AH moves along kµ and MAH initially constant
• Rewrite in conformal variables ⇒ BC’s on ψ and βi
• General spin possible (→ Greg Cook’s talk)
• One still must specify...1. Conformal metric g̃ij2. Shape of excision surfaces S3. Mean curvature K
4. Lapse boundary condition
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
6Spectral elliptic solver (HP, Kidder, Scheel & Teukolsky, 2003)
Expand solution in basis-functions & solve for expansion-coefficients
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
6Spectral elliptic solver (HP, Kidder, Scheel & Teukolsky, 2003)
Expand solution in basis-functions & solve for expansion-coefficients
Smooth solutions ⇒ exponential convergence
30 45 60 75 90N
10-8
10-6
10-4
10-2
||H||2||M||2|δEADM||δMK||δJz|
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
6Spectral 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
30 45 60 75 90N
10-8
10-6
10-4
10-2
||H||2||M||2|δEADM||δMK||δJz|
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
6Spectral 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
30 45 60 75 90N
10-8
10-6
10-4
10-2
||H||2||M||2|δEADM||δMK||δJz|
HP, Kidder, Scheel, Teukolsky 2003
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
6Spectral 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: Nontrivial topologies & Multiple length-scales
30 45 60 75 90N
10-8
10-6
10-4
10-2
||H||2||M||2|δEADM||δMK||δJz|
HP, Kidder, Scheel, Teukolsky 2003
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
7
Non-uniqueness
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
8Extended conformal thin sandwich equations
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Exten
ded CT
S
g̃ij = δij + Ãhij
∂tg̃ij = ÃḣijK = ∂tK = 0
(perturbed flat space w/o inner b’dries)
HP & York, 2005
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
8Extended conformal thin sandwich equations
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Exten
ded CT
S
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Extented CTS, too!
Exten
ded CT
S
g̃ij = δij + Ãhij
∂tg̃ij = ÃḣijK = ∂tK = 0
(perturbed flat space w/o inner b’dries)
HP & York, 2005
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
8Extended conformal thin sandwich equations
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Exten
ded CT
S
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Extented CTS, too!
Exten
ded CT
S
0.01 0.1 1A~
0.01
0.1
1
10
100
ADM energy
Standa
rd CT
S
Extented CTS, too!
Exten
ded CT
S
apparenthorizon
g̃ij = δij + Ãhij
∂tg̃ij = ÃḣijK = ∂tK = 0
(perturbed flat space w/o inner b’dries)
Apparent horizons exist for small Ã!
HP & York, 2005
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
9
Properties of QE-ID sets
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
10Corotating BBH solutions
Arbitrary choices: Conformal flatness, S=sphere. Gauge choices: K = 0, ∂n(Nψ) = 0.
30 45 60 75 90N
10-8
10-6
10-4
10-2
||H||2||M||2|δEADM||δMK||δJz|
Exponential convergence Lapse positive through horizon
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
11Sequences of quasi-circular orbits & ISCO
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
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)
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
12Towards evolving these ID
• ISCO and other diagnostics very promising
• But, ID only up to AH, whereas evolution codes excise inside AH
• Extrapolate data inward to 0.75rAH
• Constraints violated for r < rAH
0.6 0.8 1 21.4r
10-10
10-8
10-6
10-4
10-2Hamiltonian Constraint
Low resolution "Lev0"
high resolution "Lev5"Extrapolate
• The next slides highlight aspects ofevolution which are relevant to ID
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
13Evolution with fixed gauge – horizon motion
Same data as in Mark Scheel’s talk – separation 10.
0 1 2 3 4 5t/MAH
-0.001
-0.0005
0
0.0005
0.001
min(rAH) - rinitial
max(rAH) - rinitial
Initially at rest, no transient
N and βi are excellent initial gauge
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
13Evolution with fixed gauge – horizon motion
Same data as in Mark Scheel’s talk – separation 10.
0 1 2 3 4 5t/MAH
-0.001
-0.0005
0
0.0005
0.001
min(rAH) - rinitial
max(rAH) - rinitial
Initially at rest, no transient
0 5 10 15 20 25t/MAH
-0.04
-0.02
0
0.02
0.04
0.06
min(rAH ) - r
initial
max(r A
H) -
r initial
Horizon crosses excision bdry
On longer time-scales, AH deforms
N and βi are excellent initial gauge
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
14Apparent horizon mass
0 10 20 30 40 50t / MAH
1
1.00002
1.00004
1.00006
MAH / MAH(t=0)(3 different resolutions)
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
15Not all is well – Tidal distortions
Tidal distortions not captured correctly with current choices for g̃ij and S— Work in progress —
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
16
Public ID repository
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
17Initial data repository
• http://www.tapir.caltech.edu/~harald/PublicID• Equal mass BBHs in corotation• Two choices for Lapse-BC – Eq. (59a) or (59b) from Cook&HP, 2004
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
http://www.tapir.caltech.edu/~harald/PublicID
17Initial data repository
• http://www.tapir.caltech.edu/~harald/PublicID• Equal mass BBHs in corotation• Two choices for Lapse-BC – Eq. (59a) or (59b) from Cook&HP, 2004
Concentrate on Lapse-BC (59a) for uniformity
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
http://www.tapir.caltech.edu/~harald/PublicID
17Initial data repository
• http://www.tapir.caltech.edu/~harald/PublicID• Equal mass BBHs in corotation• Two choices for Lapse-BC – Eq. (59a) or (59b) from Cook&HP, 2004
Concentrate on Lapse-BC (59a) for uniformity
10 20 30 40coordinate separation d
1
235
10
10 20 30 40
0.01
0.02
0.03
0.05
0.1
mΩ
orbits un
til ISCO
(1.5PN)
PretoriusBruegmann et al 2004
Available separations
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
http://www.tapir.caltech.edu/~harald/PublicID
18Using the public QE-BBH initial data
http://www.tapir.caltech.edu/~harald/PublicID
The web-site contains:
• Data sets, containing gij, Kij, N, βi in Cartesian components
• Library to interpolate the data to any desired point (x, y, z)(as long as it is inside the covered computational domain)
• Example executable and example data-set(Schwarzschild in Kerr-Schild coordinates)
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
http://www.tapir.caltech.edu/~harald/PublicID
19Summary
• Framework for BBH initial data in a kinematical setting (helical Killing vector)
• Advantages:1. Agreement with PN
2. N > 0, AH initially constant, MAH exceedingly constant
• Tidal distortions not yet captured
• Data sets publicly availablehttp://www.tapir.caltech.edu/~harald/PublicID
1. Compute waveforms!
2. Compare and validate evolution codes on the same initial data
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
http://www.tapir.caltech.edu/~harald/PublicID
20Contents of a data set
1. The data in several resolutions (Lev2, ... Lev5), each in its own subdirectory
2. The file Convergence listing errors for each resolution:
#....N Nor-Linf Nor-L2 Ham-Linf Ham-L2 Mom-Linf Mom-L232.184 0.2280 0.03185 0.0339 0.00202 0.00552 0.000217
21Interpolation Library – suggestions welcome!
• Library libSpECLibraryID.a (compiled with gcc 3.4.3 on RHE 9)
• Header file PublicID.hpp:#include
void ReadData(const double Omega); // import from disk
void InterpolateData(const vector& x,const vector& y,const vector& z,vector& gxx, ... , vector& gzz,vector& Kxx, ... , vector& Kzz,vector& Betax, ... , vector& Betaz,vector& N);
void ReleaseData(); // free memory
• Test-executable InterpolateExample.cpp:
g++ InterpolateExample.cpp libSpECLibraryID.a -lblas
H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005
Recommended