**S** Brill Wave S1 S2 P1 P2 S2 S1 P2 P1 Black Hole Initial Data for Evolution • Distorted Black Holes: “Brill Wave plus Black Hole” (NCSA model) • ds2 = y4 (e2q(dr2+r2dq2)+ r2sin2qdf2) • q(r,q,f) --> true 3D distorted BH • Good model for late stage of BH merger • Binary IVP: Multiple Wormhole Model • Misner 2BH: equal mass, time symmetric, colliding BHs, variations • Cook et al fully general merger data • Binary IVP: Multiple “Punctures” • Brandt-Brügmann general merger data • Now evolving... • More coming: Meudon/AEI/Jena Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Why are Black Holes so Difficult?** Singularity Horizon • Fundamental Problem with Black Holes • Stretching of hypersurfaces leads to pathologies in metric functions • Why not cut singularity away if surrounded by horizon??? t=150 Crash! t=100 grr t=50 t=0 Collapsing Star Albert-Einstein-Institut www.aei-potsdam.mpg.de

**A Little Lesson in Black Hole Physics** • Black Hole Perturbation Theory • What happens when a BH is perturbed? • How can this be used in numerical work? • Going beyond linear theory through numerical relativity... Albert-Einstein-Institut www.aei-potsdam.mpg.de

**V(r)** y + V(r) y = 0 What happens when a black hole is perturbed? Horizon • Take existing black hole (Schwarzschild solution) • Analytic solution known since 1916! • Perturb by tossing pebble into it • Is it stable? • Perturb metric tensor gmu ds2 = -(1-2M/r) dt2 + dr2/(1-2M/r) + r2 (dq2 + sin2qdf2) by adding ehmn(Ylm), expand to first order in e... • Discover (after 30 years study!) … • Scattering off potential barrier • Find resonance frequencies of this potential barrier • y ~ exp(iw(t-x)) • w resonances are complex: get damped sinusoids • Frequency depends on mass and spin of BH, that’s all! • In all theoretical studies, these modes are excited • Collapse of matter to a BH, Distorted, Colliding BH’s... pebble Black hole Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Black Hole Normal Modes** Ringing modes: damped sinusoid ~eiwt • Measure this, can determine mass and spin of BH that created it • Questions, Questions: what happens when we go beyond linear theory? Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Merger Phase** Technique Time Scale Post N Colliding BH Roadmap:A patchwork to success Post Newt inspiral t ~ 500+ M Need Post-N BH Initial data gap Standard Numrel: Exciting new results, but will break down t ~ 100 M Final Plunge t ~ 30 M Ringdown Excision: making progress Perturbative Time Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Start with ADM Evolution Equations(used almost exclusively** for last 30 years…) The 3+1 evolution equations used in numerical relativity are normally written as: “Arnowitt-Deser-Misner” (or simply “ADM”) evolution equations. These equations are highly non-unique: add arbitrary multiples of constraints, obtain new evolution equations that are just as valid. New equations: same physical solutions, but very different mathematical properties, different solutions away from constraint hypersurface (“off-shell”). Highly computationally intensive: want larger computers than exist, must perform many, many developmental tests: very slow! Albert-Einstein-Institut www.aei-potsdam.mpg.de

**First attempt to go beyond axisymmetric, non-orbital data** Brandt-Brügmann Puncture data Unequal mass, spin, orbital J Maximal slicing, no shift Nested grids to get boundary away from holes Merger of AH S2 S1 P2 P1 First True 3D Grazing Black Hole Collision Bernd Brügmann 97 Single outer AH appearsas holes merge MTSs inside M2 M1 ADM evolution to about t=7M but then crash! Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Very important new thing: The BSSN Formulation** All recent simulations use the Baumgarte-Shapiro (BS) Shibata-Nakamura (SN) reformulation of the EEs. This BSSN formulation is based on a conformal decomposition of the standard ADM equations: A crucial ingredient of the BSSN formulation is the introduction of the conformal connection functions as independent variables: The BSSN formulation has been found to be extremely stable when compared to standard ADM in a large variety of cases, both with and without matter, so it is our preferred choice. Will apply to NS, too!! Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Return to Brügmann Grazing Collision…Alcubierre, Allen,** Benger, Brügmann,Lanfermann, Nerger, ES, Takahashi • BSSN • Spent 2 years understanding BSSN (Alcubierre…) • Focused on pure waves (demanding, but w/o singularity complications) • Then applied to BH problem • Big Boost! 5x as far! • Finally doing real (toy?) physics with 3D numerical relativity… • Next: NS-NS Could never do this before 1999 Albert-Einstein-Institut www.aei-potsdam.mpg.de

**good** even even even y22 y22 y22 QNM fit (lowrez) QNM fit (medrez) better best QNM fit (highrez) Gauge-Invariant waveform extraction:Extract physics, predict rotation and mass of final BH • QNM’s seen at late times • Good fit for two lowest a/M = 0.73 Kerr QN modes • Fit improves with resolution • Fit poor if use different a/M values! • Energy radiated in all modes of order 1% MADM • Energy Estimate: • Mir≈ 3.0 • MAH ≈ 3.3 • Compare • MADM = 3.22 • Have done parameter studies • Low, med, high spin cases t/M Convergence of waveform t/M Fit to Kerr QNM’s Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Crash!** Computationaldomain singularity horizon Excised region Going Further: Black Hole Excision(Alcubierre, Brügmann, Pollney, Takahashi, ES) • Two Ingredients • Excise a region inside the AH. • (Superluminal) Shift vector! • Hard! • Spherical topology, ill adapted to cartesian coords • Large shifts can cause instabilities: causal differencing • Moving across grid?? • Everything from here forward unpublished so far… • Unruh, ‘84, ES, Suen, ‘92, Alcubierre, Brügmann, ‘00, many others Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Shift Conditions(Alcubierre, et al…)** New shift condition obtained by making the conformal connection functions to be time-independent: “Gamma-freezing” shift condition closely related to “minimal distortion” shift condition, but related to fundamental variables in BSSN! Obtain hyperbolic conditions by making second time derivatives of the shift proportional to the elliptic operator contained in the above condition. VERY important: will apply to NS, too! Albert-Einstein-Institut www.aei-potsdam.mpg.de

**2D run, no excision, no shift, high resolution** Crash! 3D run, no excision, no shift Crash! 3D run, excision+shift Schwarzschild: Horizon MassMADM=2, x = 0.2, 1283 grid points • Very robust • No fine tuning! • Remember: started with • a = 1 • bi = 0 • It just works naturally… • This is very new in the community… Albert-Einstein-Institut www.aei-potsdam.mpg.de

**How about rotating,distorted BH?: Waveforms (4)Mimics** final stages of BBH Coalescence MADM=7.54, a/M=0.62, x = 0.5, 1923 points S • No fine tuning! • It just works… very naturally as gauges drive system static… • Shift has min. distortion character: like Boyer-Lindquist for Kerr so keeps coords OK • Even waves are fine… • Lots of other examples… Remarkable agreement between 2D and 3D! Crash! 2D runs crashes! Albert-Einstein-Institut www.aei-potsdam.mpg.de

** Grazing results ** in press… New, low res test simulation w/shift!! Grazing Collision with shift and excision!!Starting to put it together • Could never do this before… • Only excise final BH • Just a test, only 1953 • Horizons great • Waveforms, too • Runs “forever” now • Will be able to do 5123,-7683 • Plenty of parameter tuning to do • How far can this go? • Move into ISCO and beyond… MAH Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Preliminary Result: Plunge from the ISCO using new shift,** excision • grid: • 195x195x100 • h = 0.083 • excision of final BH Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Lazarus** Full non- Linear??? The Evolution of the Evolution of 3D Black Holes t(M) Distort Grazing Pre-ISCO?? Evolution times for robust, accurate evolutions, with solid horizon physics, waveforms Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Conclusions Main message: unusual progress, will benefit** entire Network • I have waited many years to see such results • 1D broke through barrier in 1987 • 2D in 1993-5 • 3D got stuck until last year… • Why is all this happening now? • We have a great team! (Critical mass, talent, focus, collaboration) • Developed much better theoretical understanding/practical experience. • We developed great collaborative technology (otherwise not possible.) • We have much better computers (need even more access) • 3+1 codes have smashed through the t = 30M barrier • Shift to cure grid-stretching • Evolutions orders of magnitude better than last year • All this applies to NS-NS and other problems! You get this for free! Albert-Einstein-Institut www.aei-potsdam.mpg.de

**Hot off the Presses: Most Recent Waveform Result** Albert-Einstein-Institut www.aei-potsdam.mpg.de