Moving punctures that are neither moving, nor punctures Mark Hannam Friedrich-Schiller Universit¨ at, Jena, Germany From Geometry to Numerics workshop Paris November 20-24 2006 Ongoing work following ugmann, ´ Hannam, Husa, Pollney, Br¨ O Murchadha, gr-qc/0606099 Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 1 / 13
From geometry to numerics... and on to astrophysics! Full black-hole binary evolutions (inspiral, merger, ringdown) are now routine Recent results from the Jena group: 0.0075 � Re � r Ψ 4 � � M � 1 � l � 2,m � 2 � coarse � mid �� cf4 0.005 cf4 � 1.34473 0.0025 mid � high 0 � 0.0025 � 0.005 � 0.0075 0 50 100 150 200 250 300 Time � M � Figure: 4th-order convergence and BAM / LEAN comparison. (gr-qc/0610128) Merger time error of 0.2% for r 0 = 3 . 257 M . 0.5% for r 0 = 4 M . No phase shift applied! High-resolution runs take less than 48 hours on LRZ altix cluster. Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 2 / 13
From geometry to numerics, and on to astrophysics! Largest parameter study to date of binary merger evolutions Nonspinning unequal-mass binaries with mass ratios of 1:1 to 1:4 300 250 200 v (km/s) 150 100 Baker, et al Campanelli Damour and Gopakumar Herrmann, et al 50 Sopuerta, et al 0 0.15 0.2 0.25 η Kick velocity vs reduced mass ratio η = m 1 m 2 / ( m 1 + m 2 ) 2 . (gr-qc/0610154) Maximum recoil velocity of 175 . 2 ± 11 km s − 1 . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 3 / 13
From geometry to numerics, and on to astrophysics! Largest parameter study to date of binary merger evolutions Nonspinning unequal-mass binaries with mass ratios of 1:1 to 1:4 300 250 200 v (km/s) 150 100 Baker, et al Campanelli Damour and Gopakumar Herrmann, et al 50 Sopuerta, et al 0 0.15 0.2 0.25 η Kick velocity vs reduced mass ratio η = m 1 m 2 / ( m 1 + m 2 ) 2 . (gr-qc/0610154) Maximum recoil velocity of 175 . 2 ± 11 km s − 1 . Now we can Fully explore the physics of BBH mergers Provide waveforms to data analysts Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 3 / 13
Back to geometry How to deal with black-hole singularities in a numerical code “Excision”: Chop them out! (Pretorius, Caltech) “Punctures”: avoid them. (UTB, Goddard, Penn State, Jena (x2)) Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 4 / 13
Back to geometry How to deal with black-hole singularities in a numerical code “Excision”: Chop them out! (Pretorius, Caltech) “Punctures”: avoid them. (UTB, Goddard, Penn State, Jena (x2)) The “moving punctures” method is easy to implement and popular But... Are punctures a crude and dirty way to solve the problem? Or are they a simple and elegant solution? Attempt to explain how punctures evolve by looking at a Schwarzschild black hole. Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 4 / 13
Puncture initial data Schwarzschild in isotropic coordinates: � 2 � � 4 � 1 − M � 1 + M dt 2 + dr 2 + r 2 d Ω 2 � ds 2 2 r = . − 1 + M 2 r 2 r ψ 2 r . = R R extends from ∞ to 2 M (at r = M / 2), and back to ∞ (at r = 0). Slice connects two asymptotically flat ends; avoids the singularity 5 4 Schwarzschild R 3 2 1 0.5 1 1.5 2 2.5 3 3.5 4 Coordinate r Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 5 / 13
Puncture initial data Schwarzschild in isotropic coordinates: � 2 � � 4 � 1 − M � 1 + M dt 2 + dr 2 + r 2 d Ω 2 � ds 2 2 r = . − 1 + M 2 r 2 r ψ 2 r . R = R extends from ∞ to 2 M (at r = M / 2), and back to ∞ (at r = 0). Slice connects two asymptotically flat ends; avoids the singularity Initial data for a dynamical evolution: ψ = 1 + M ˜ γ ij = δ ij , 2 r ˜ K = 0 , A ij = 0 β i = 0 . α = 1 , With this choice of lapse and shift, there will be nontrivial evolution. Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 5 / 13
“Fixed puncture” evolutions The conformal factor diverges at the puncture. Assume that we keep the wormhole topology during evolution, and write it as � 1 + M � ψ = f , 2 r and φ = ln f = 0 initially. Then evolve φ = ln f . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 6 / 13
“Fixed puncture” evolutions The conformal factor diverges at the puncture. Assume that we keep the wormhole topology during evolution, and write it as � 1 + M � ψ = f , 2 r and φ = ln f = 0 initially. Then evolve φ = ln f . Sometimes works for single black holes, head-on collisions, orbiting binaries Always needs a lot of fine-tuning of gauge parameters. By definition, the puncture is always under-resolved. Gauge parameters chosen such that β i = O ( r 3 ) at the punctures. ⇒ even for binaries, punctures are fixed on the grid. The evolution does not find a stationary slice. (Reimann and Br¨ ugmann, ’04) Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 6 / 13
Example: “Fixed puncture” evolution of Schwarzschild Evolve using initial data of Schwarzschild in isotropic coordinates (from earlier slide) α = 1 and β i = 0 initially BSSN “fixed puncture” reformulation of the 3+1 evolution equations ˜ Γ-driver shift evolution 1+log slicing, ∂ t α = − 2 α K ⇒ For a stationary solution, ∂ t α = 0 ⇒ K = 0, maximal slicing. Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 7 / 13
Example: “Fixed puncture” evolution of Schwarzschild Evolve using initial data of Schwarzschild in isotropic coordinates (from earlier slide) α = 1 and β i = 0 initially BSSN “fixed puncture” reformulation of the 3+1 evolution equations ˜ Γ-driver shift evolution 1+log slicing, ∂ t α = − 2 α K ⇒ For a stationary solution, ∂ t α = 0 ⇒ K = 0, maximal slicing. Look at value of Tr ( K ) on the (outer) horizon R = 2 M : 0.02 0.015 0.01 0.005 Tr � K � 0 � 0.005 � 0.01 � 0.015 0 5 10 15 20 25 30 35 Time � M � Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 7 / 13
“Moving punctures” Now llet (Goddard) 1 + m � � φ = ln ψ = ln , 2 r or (UTB) χ = ψ − 4 , and evolve φ or χ . (Don’t assume anything about ψ .) Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 8 / 13
“Moving punctures” Now llet (Goddard) 1 + m � � φ = ln ψ = ln , 2 r or (UTB) χ = ψ − 4 , and evolve φ or χ . (Don’t assume anything about ψ .) BINARY BLACK HOLE PROBLEM SOLVED! Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 8 / 13
“Moving punctures” Now llet (Goddard) 1 + m � � φ = ln ψ = ln , 2 r or (UTB) χ = ψ − 4 , and evolve φ or χ . (Don’t assume anything about ψ .) BINARY BLACK HOLE PROBLEM SOLVED! The “moving punctures” package: BSSN (with φ or χ variables) Singularity-avoiding slicing (maximal, 1+log, ...) ˜ Γ-freezing shift evolution “Puncture” initial data Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 8 / 13
“Moving puncture” evolution of Schwarzschild Using “maximal” 1+log slicing, ∂ t α = − 2 α K , and “˜ Γ-driver” shift evolution. 0.15 0.1 Tr � K � 0.05 0 � 0.05 20 40 60 80 100 Time � M � Reaches stationary (maximal) slice in about 40 M . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 9 / 13
“Moving puncture” evolution of Schwarzschild Using “maximal” 1+log slicing, ∂ t α = − 2 α K , and “˜ Γ-driver” shift evolution. 35 0.15 30 25 0.1 Schwarzschild R 20 Tr � K � 0.05 15 0 10 5 � 0.05 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 Time � M � Coordinate r Reaches stationary (maximal) slice in about 40 M . Evolution of Schwarzschild R ( r ): T = 0. Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 9 / 13
“Moving puncture” evolution of Schwarzschild Using “maximal” 1+log slicing, ∂ t α = − 2 α K , and “˜ Γ-driver” shift evolution. 35 0.15 30 25 0.1 Schwarzschild R 20 Tr � K � 0.05 15 0 10 5 � 0.05 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 Time � M � Coordinate r Reaches stationary (maximal) slice in about 40 M . Evolution of Schwarzschild R ( r ): T = 1 M . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 9 / 13
“Moving puncture” evolution of Schwarzschild Using “maximal” 1+log slicing, ∂ t α = − 2 α K , and “˜ Γ-driver” shift evolution. 35 0.15 30 25 0.1 Schwarzschild R 20 Tr � K � 0.05 15 0 10 5 � 0.05 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 Time � M � Coordinate r Reaches stationary (maximal) slice in about 40 M . Evolution of Schwarzschild R ( r ): T = 2 M . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 9 / 13
“Moving puncture” evolution of Schwarzschild Using “maximal” 1+log slicing, ∂ t α = − 2 α K , and “˜ Γ-driver” shift evolution. 35 0.15 30 25 0.1 Schwarzschild R 20 Tr � K � 0.05 15 0 10 5 � 0.05 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 Time � M � Coordinate r Reaches stationary (maximal) slice in about 40 M . Evolution of Schwarzschild R ( r ): T = 3 M . Mark Hannam (FSU Jena) Moving Punctures Paris, November 23 2006 9 / 13
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