initial data for binary black holes: the conformal thin- sandwich puncture method mark d. hannam utb...
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Initial data for binary black holes: the conformal thin-
sandwich puncture method
Mark D. Hannam
UTB Relativity Group SeminarSeptember 26, 2003
Overview: the smallest picture possible
• We want to simulate a (realistic) binary black hole collision. To do that,
1. Rewrite Einstein’s equations as a Cauchy problem2. Set up initial data for two black holes in orbit3. Evolve the system.
• Problems: we can’t do (2) or (3) very well.• Partial solution: try to create good initial data
close to the interesting physics…• Describe two black holes in quasi-circular, quasi-
equilibrium orbit just before they plunge together.
)(2
1ijjiijtij N
K
))((222 dtdxdtdxdtNds jjiiij
Initial data: ijij K,
3,2,1,0,,82
1 TRgR Space and time are mixed…
ijjiijijt
ijij
lijl
ljilijl
l
jiljilijijijt
ililil
ijij
NK
SSGN
KKK
NKKKKRNK
jGKK
GKKKR
2
))(2(4
)2(
8)(
162 Initial value constraints
Evolutionequations
ij'
ijt
tt
tN
ti
What quantities are constrained?
12 independent components - 4 constraint equations
8 free quantities: 4 dynamical 4 gauge
Which are which?
Use a conformal decomposition…
ijij K,
Conformal thin-sandwich decomposition
NNiiijij
~~~ 64
KKAAKAK ijijijijij
~~
3
1 2
ijtijk
kijijjiij
ijijij
uL
uLN
A
~~,~~
3
2~~~
,~~~
2
1~
KKKSGAANN
jNuN
NKNNL
GAAKR
tllij
ij
iijj
ij
ijiL
ijij
~
12
5)2(2
~~
8
7)
~()
~(
~
~16~
~1~~~~
3
4~ln
~~~
2~~~~
52548772
106
578
1258
1
8
12
From ... Kt
CTS: the essentials
• Free data:
• Solve for:
• Construct:
K
K
tijt
ij
~
~
Ni ~,,
ijij K,
“Easy” examples• Schwarzschild (single stationary black hole):
• Brill-Lindquist (multiple stationary black holes)
ed...)undetermin are (,~21
~
0~
0,~21
0,0,0~,~
7i
N
i i
i
iji
N
i i
i
tijtijij
cr
cN
Ar
m
KKf
0~
0,~21
~,~2
1
0,0,0~,~
7
iji
tijtijij
Ar
MN
r
M
KKf
Orbits in the CTS decomposition
• In a corotating reference frame, the black holes will be almost stationary.
• Choose
• These choices are physically motivated– Free data choices in old decompositions were
made for convenience
0,0~ Ku tij
CTS solutions
• Gourgoulhon, Grandclément, and Bonazzola (GGB), 2001.– Solved with– Excised regions containing singularities– Employed boundary conditions on excised
surfaces(…there were inconsistencies here)
• I want to avoid inner boundary conditions
Puncture method.
0,~ Kfijij
CTS-puncture approachRecall Brill-Lindquist solution:
Extend to .
The shift has no singular part– What corresponds to black holes with Pi and Si ?
vr
cN
N
i i
i ~21
~ 7
ur
mN
i i
i ~21
7~N
(what are ci?)
KKAAvr
cv l
lijij
N
i i
i
~
12
5~~
8
7~2
1~ 52582
Hamiltonian constraint:
Constant-K equation:
Solve for v
2572
12
1~~
8
1~KAAu ij
ij
Regular if
3
6
~~
~~~~
rK
rAA ijij
Solve for u
Issues: Slicing choices(for one black hole)
• Two principle choices:
• Schwarzschild
– But: on some surface…
• Estabrook (N = 1).
but
– This is a “dynamical” slicing!– The stationary Schwarzschild black hole will APPEAR
to have dynamics
• This isn’t necessarily fatal to the method
[Ref: MDH, C.R. Evans, G.B. Cook, T.W. Baumgarte, gr/qc-0306028]
mc
mc
0~ N ijij L
NA ~
~2
1~
mc 1
~ N 0~ ijt A
r
cN ~2
1~ 7
r
mN ~2
1~ 7
r
mN ~2
1~ 7
Issue #2: The shift vectorConditions at the puncture?
• The analytic, singular part of the conformal factor gave us a black hole solution, without the need for inner boundary conditions
• There is no known analytic part of the shift for a black hole with non-zero Pi and Si, and the puncture form of the lapse.
• We need to impose suitable conditions at the puncture.
• Methods to date do not give convergent results…
Future work
• Convert code to Cactus, where much greater resolution is possible – Maybe the momentum constraint solver will
converge.
• Construct data with an everywhere positive lapse
• Examine the level of stationarity of quasi-circular orbits (located by, for example, the effective potential method)– Maybe the “Estabrook” lapse choice is Ok.