Upload
others
View
9
Download
0
Embed Size (px)
Citation preview
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
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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?
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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)
Harald P. Pfeiffer, Caltech
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)
Harald P. Pfeiffer, Caltech
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)
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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 !!
Harald P. Pfeiffer, Caltech
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!!
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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-
Harald P. Pfeiffer, Caltech
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!
Harald P. Pfeiffer, Caltech
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!
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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...)
Harald P. Pfeiffer, Caltech
19Quasi-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
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)
Harald P. Pfeiffer, Caltech
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.
Harald P. Pfeiffer, Caltech
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.
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
23
Binary black hole solutions (corotating, K = 0)
Lapse positive through horizon
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
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?
Harald P. Pfeiffer, Caltech
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)
Harald P. Pfeiffer, Caltech
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
Harald P. Pfeiffer, Caltech
28
How good is good enough?
Just some thoughts and conjectures — further opinions welcome!!
Harald P. Pfeiffer, Caltech
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.
Harald P. Pfeiffer, Caltech
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!)
Harald P. Pfeiffer, Caltech
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.
Harald P. Pfeiffer, Caltech
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, Caltech