![Page 1: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/1.jpg)
Solucion numerica de las ecuaciones de Einstein:
Choques de agujeros negrosJose Antonio Gonzalez
IFM-UMSNH
25-Abril-2008
ENOAN 2008
![Page 2: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/2.jpg)
• Introduction– Binary black hole problem
• Some ingredients– 3+1 decomposition
– Formulation of the equations
– Initial data
– Gauge
– Mesh refinement
– Boundary conditions
– Excision
– Diagnostic tools
• Applications• Conclusions
OverviewOverview
![Page 3: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/3.jpg)
The big pictureThe big picture
Model GR (numerical relativity) PN Perturbation theory Non-GR?
External Physics Astrophysics Fundamental physics Cosmology
DetectorsPhysical System
describes
observe
test implications
Help d
etec
t
![Page 4: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/4.jpg)
Numerical relativityNumerical relativity
-Two 10 solar mass black holes-Frequency ~ 100Hz-Distort the 4km mirror spacingby about 10^-18 m
![Page 5: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/5.jpg)
3+1 decomposition3+1 decomposition
GR: “Space and time exist together as Spacetime’’
Numerical relativity: reverse this process!
ADM 3+1 decomposition Arnowitt, Deser, Misner (1962)York (1979)Choquet-Bruhat, York (1980)
3-metric ijlapse
shift
i
lapse, shift Gauge
Einstein equations 6 Evolution equations
4 Constraints
Constraints preserved under evolution!
![Page 6: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/6.jpg)
ADM equationsADM equations
Evolution equations
ijijt KL 2)(
]2[)( KKKKRDDKL ijjm
imijjiijt
Constraints
02 ijijKKKR
0 KDKD iijj
Evolution
Solve constraints initially Evolve data Reconstruct spacetime Extract physics
![Page 7: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/7.jpg)
Formulation of the equationsFormulation of the equations
ADM: unsuccessful; weakly hyperbolic!
BSSN (most popular) Shibata, Nakamura ‘95Baumgarte, Shapiro ‘99
Balance laws: Bona, Massó (H-code)
Many more:
Sarbach et.al.‘02;Gundlach, Martin-Garcia
Split degrees of freedom (similar to initial data split) Hyperbolicity
Generalized harmonic formulation
Harmonic gauge well-posed Wave equations for BBH-breakthrough
Choquet-Bruhat ‘62
Pretorius ‘05g
ADM-like family:Harmonic family:Control of constraints:
KST, NOR,… Z4 LSU, Caltech, Gundlach
Garfinkle ‘04
![Page 8: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/8.jpg)
The BSSN formulationThe BSSN formulation
![Page 9: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/9.jpg)
Initial dataInitial data
Two difficulties: Constraints, realistic data
York-Lichnerowicz split ijij ~4 KAK ijijij 3
1
Conformal transverse traceless Physical transverse traceless Thin sandwich
York, LichnerowiczO’Murchadha, YorkWilson, Mathews; York
Conformal flatness:
Spurious radiation does not seem problematic, but alternatives studied
Generalized analytic solutions:
Time symmetric, -holes:
Spin, linear momenta:Punctures
Brill-Lindquist, Misner (1960s)Bowen, York (1980)Brandt, Brügmann (1997)
Isotropic Schwarzschild
N
Excision data: Isolated Horizon condition on excision surface
Meudon group; Cook, Pfeiffer; Ansorg
Quasi-circularity: Effective potential method PN fit helical killing vector
![Page 10: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/10.jpg)
GaugeGauge
Specific problem in GR: Coordinates constructed during evolutions
Highly non-trivial: Prescribe to avoid coordinate singularities
Einstein equations say nothing about , i, i
Maximal slicing, min.distortion shiftSmarr, York ‘78
Driver conditionsBalakrishnaet.al.’96
1+log, -driverAEI
~
Moving puncturesUTB, Goddard ‘06
Bona-Massó familyBona, Massó ‘95
Harmonic coordsChoquet-Bruhat‘62
Generalized harmonicGarfinkle ‘04Pretorius ‘05
Study singularity avoidanceAlcubierre ‘03
Analytic studies
gauge sources relation to i ,
Drive to stationarity
specialcase
specialcase
![Page 11: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/11.jpg)
Mesh-refinement, boundariesMesh-refinement, boundaries
3 length scales: BHWave lengthWave zone M
M
M
100
10
1
Choptuik ’93 AMR, Critical phenomena
Stretch coordinates: Fish-eye Lazarus, AEI, UTB
FMR, Moving boxes: Berger-Oliger BAM Brügmann’96 Carpet Schnetter et.al.’03
AMR: Steer resolution via scalar Paramesh: MacNeice et.al.’00, Goddard modified Berger-Oliger: Pretorius, Choptuik ’05 SAMRAI
Refinement boundaries: reflections, stability Lehner, Liebling, Reula ‘05
![Page 12: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/12.jpg)
Outer boundary conditionsOuter boundary conditions
Problems: Well-posedness of equations? Constraint violations?
BCs that satisfy constraints and/or well-posedness
Friedrich, Nagy ‘99 Calabrese, Lehner, Tiglio ‘02 Frittelli, Goméz ‘04 Sarbach, Tiglio ‘04 Kidder et.al.‘05, Lindblom et.al.‘06
Tested with success in BBH simulation: Lindblom et.al.‘06
Conformal, null-formulation: Untested in BBH simulations
Compactification in 3+1 Pretorius ‘05
Push boundaries “far out”, use Sommerfeld condition Used successfully by most groups; accuracy limits?
Multi-patch approach: Efficiency AEI (Cactus): Thornburg et.al.: excision, Char.Code LSU (below), Austin (below), Cornell-Caltech (below)
![Page 13: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/13.jpg)
Black hole excisionBlack hole excision
Cosmic censorship: Causal disconnection of region inside AH
Unruh ’84 cited in Thornburg ‘87
Grand Challenge: Causal differencing
“Simple Excision” Alcubierre, Brügmann ‘01
Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius
combined with Dual coordinate frame Caltech-Cornell
Mathematical properties: Wealth of literature
![Page 14: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/14.jpg)
Diagnostic toolsDiagnostic tools
A computer just gives numbers! These are gauge dependent! Convert to physical information…
ADM mass, momentum Arnowitt, Deser, Misner ‘62
Bondi mass, News function (Characteristic approach)
Gravitational Waves Zerilli-Moncrief formalism Newman-Penrose scalar hih tttt4
Black hole quantities: mass, momentum, spin, area,…
Apparent Horizon Alcubierre, Gundlach (Cactus) Schnetter ‘03 Thornburg ‘03 (AHFinderDirect) Pretorius
Event horizon Diener ‘03 Isolated, Dynamic Hor. Ashtekar, Krishnan ’03
Ashtekar et.al. Dreyer et.al. ’02
![Page 15: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/15.jpg)
2004
2007
How far we are?How far we are?
![Page 16: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/16.jpg)
Spinning holes: The orbital hang-upSpinning holes: The orbital hang-up
Spins alligned with inspiral delayed, largerL
radrad JE ,
Spins anti-alligned with inspiral fast smallerL
radrad JE ,
No extreme Kerr holes produced
![Page 17: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/17.jpg)
Gravitational recoilGravitational recoil
Anisotropic emission of GW carries away linear momentum recoil of remaining system
Merger of galaxies
Inspiral and merger of black holes Recoil of merged hole Displacement, Ejection?
Astrophysical relevanceAstrophysical relevance
BH inspiral kick possible ejection of BH from host
Escape velocities: globular clusters dSph dE large galaxies Merritt et al.’04
km/s 30km/s 10020 km/s 300100
km/s 1000
![Page 18: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/18.jpg)
Non-spinning binariesNon-spinning binaries
Emerging picture: Kicks unlikely to exceed a few km/s 100
Numerical relativity allows accurate estimates Campanelli ’05 Herrmann et al.’06 Baker et al.’06
Close limit calculations Sopuerta et al.’06 a,b Upper and lower bounds
Including eccentricity increases kick
for small eccentricities
)1(kick ev
EOB approximation: account for deviations from Kepler law Damour & Gopakumar ‘06
![Page 19: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/19.jpg)
Non-spinning binariesNon-spinning binaries
Systematic parameter study Gonzalez et al.’06
Moving puncture method BAM code Nested boxes, resolutions Extract calculate linear momentum Vary mass ratio:
150,000 CPU hours
/40mh s4
16.0...25.0
,4:1...1:1
q
Higher order PN Blanchet et al.’05
![Page 20: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/20.jpg)
Non-spinning binaries: Maximal kickNon-spinning binaries: Maximal kick
km/s 117.175 005.0195.0 Maximal kick: at
![Page 21: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/21.jpg)
Recoil of spinning binariesRecoil of spinning binaries
Kidder ’95: PN study including recoil of spinning holes
= “unequal mass” + “spin(-orbit)”
Penn State ‘07: Spin-orbit term larger
extrapolated:
8.0,...,2.0m
a
km/s 475v
AEI ’07:
extrapolated:
6.01 m
a6.0,...,0.02
m
a
km/s 440v
![Page 22: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/22.jpg)
Recoil of spinning binariesRecoil of spinning binaries
UTB-Rochester
maximum predicted:km/s 454v
km/s 1300v
NASA Goddard:
km/s 52
km/s 3015
long
trans
v
v Spin effect
Unequal-mass effect
PN predictions remarkably robustFitting formulas
![Page 23: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/23.jpg)
Discretization error: km/s 43v
Trajectories:
Getting even larger kicksGetting even larger kicks
![Page 24: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/24.jpg)
Dependence on Extraction radiusDependence on Extraction radius
Error fall-off: km/s 120v
![Page 25: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/25.jpg)
Reducing eccentricityReducing eccentricity
![Page 26: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/26.jpg)
Data analysis and PN comparisonsData analysis and PN comparisons
Thick red line NR waveformsDashed black ‘best matched’ 3.5 PN waveformsThin green Hybrid waveforms
Since it is expensive to generate an entire physical bank of templates using numerical simulations, it is better to construct a phenomenological bank –unequal mass, non spinning black holes-
![Page 27: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/27.jpg)
EccentricityEccentricity
![Page 28: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/28.jpg)
IMRI’s: Motivation• Stellar mass black holes M~1-10 Msun
• Intermediate mass bh’s M~102-4 Msun
• Supermassive bh’s M~106-9 Msun
Why IMRIs and EMRIs are interesting?•Astrophysics•Data Analysis and gravitational waves detection:
Gravitational waves emited during the merger of stellar-mass black holes into a IMBHs will lie in the frequencies of Advanced LIGO (Brown et al. 2007)
•Tests of General Relativity•Comparison with PN and perturbation theory
![Page 29: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/29.jpg)
•Numerical simulations are expensive
•How many orbits are required?
Data analysis 10? 100?
•How far we need to go in mass ratios?
Compare with PN!
1:100? 1:1000???
Hopefully not!
![Page 30: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/30.jpg)
Mass ratio 1:10
–Resolution:
[η]= 1/M•Problems:
• M1 = 0.25 , M2 = 2.5 , M = M1+M2
• D = 19.25 = 7M
• q = M1/M2 = 10 , η = q/(1+q)2 = 0.0826
Parameters:
–Gauge:
![Page 31: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/31.jpg)
![Page 32: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/32.jpg)
![Page 33: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/33.jpg)
Kick
Fitchett (MNRAS 203 1049,1983) Gonzalez et al. (PRL 98 091101, 2007)
V~62 km/s
![Page 34: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/34.jpg)
Radiated energy ΔE/M=0.580192 η2
ΔE/M~0.004018
Berti et al. (2007)
![Page 35: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/35.jpg)
Final spin
Damour and Nagar (2007)
aaF/MF~0.2602
![Page 36: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/36.jpg)
ERAD = 0.011001
l=2 75.62%
l=3 16.36%
l=4 4.96%
l=5 1.74%
Energy distributionEnergy distribution
![Page 37: Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008](https://reader035.vdocuments.site/reader035/viewer/2022062518/56649ecd5503460f94bda2e4/html5/thumbnails/37.jpg)
Conclusions• After a lot of work and effort….it seems to
work!
• It is over? No way!– It is necessary to improve accuracy– Now it is possible to do physics –the original
purpose of everything-– Data analysis– Parameter estimations
• Matter