1. Black Holes Mergers Frans Pretorius Princeton University XXIèmes Rencontres de Blois June 25, 2009 XXIèmes Rencontres de Blois June 25, 2009

2. Outline … Numerical relativity and black hole mergers: brief history and current state of the fieldNumerical relativity and black hole mergers: brief history and current state of the field Categorize the two-body problem in general relativity into two classes: rest mass dominated and kinetic energy dominated mergersCategorize the two-body problem in general relativity into two classes: rest mass dominated and kinetic energy dominated mergers rest mass dominated mergers: relevant to gravitational wave astronomyrest mass dominated mergers: relevant to gravitational wave astronomy phases: “Newtonian”, inspiral  quasi-circular inspiral, plunge/merger, ringdownphases: “Newtonian”, inspiral  quasi-circular inspiral, plunge/merger, ringdown brief overview of a couple of the more interesting results recently uncovered by numerical simulationsbrief overview of a couple of the more interesting results recently uncovered by numerical simulations –“simplicity” of the merger waveform –large recoil velocities

3. … Outline kinetic energy dominated mergers: relevant to super- Planck scale particle collisionskinetic energy dominated mergers: relevant to super- Planck scale particle collisions –though what do black hole collisions have to do with particle collisions? –black hole scattering problem: either a deflection or a merger, the latter will be followed by a ringdown phase –either case could be accompanied by copious amounts of gravitational wave emission –may contain regimes where the luminosity approaches the Planck luminosity, and the remnant black hole in a merger is near-extremal

4. Brief (and incomplete) history of the binary black hole problem in numerical relativity Hahn and Lindquist, Ann. Phys. 19, 304 (1964) First simulation of “wormhole” initial dataHahn and Lindquist, Ann. Phys. 19, 304 (1964) First simulation of “wormhole” initial data L. Smarr, PhD Thesis (1977) : First head-on collision simulationL. Smarr, PhD Thesis (1977) : First head-on collision simulation P. Anninos, D. Hobill, E.Seidel, L. Smarr, W. Suen PRL 71, 2851 (1993) : Improved simulation of head-on collisionP. Anninos, D. Hobill, E.Seidel, L. Smarr, W. Suen PRL 71, 2851 (1993) : Improved simulation of head-on collision B. Bruegmann Int. J. Mod. Phys. D8, 85 (1999) : First grazing collision of two black holesB. Bruegmann Int. J. Mod. Phys. D8, 85 (1999) : First grazing collision of two black holes mid 90’s-early 2000: Binary Black Hole Grand Challenge Alliancemid 90’s-early 2000: Binary Black Hole Grand Challenge Alliance –Cornell,PSU,Syracuse,UT Austin,U Pitt, UIUC,UNC, Wash. U, NWU … head-on collisions, grazing collisions, cauchy-characteristic matching, singularity excision B. Bruegmann, W. Tichy, N. Jansen PRL 92, 211101 (2004) : First full orbit of a quasi-circular binaryB. Bruegmann, W. Tichy, N. Jansen PRL 92, 211101 (2004) : First full orbit of a quasi-circular binary FP, PRL 95, 121101 (2005) : First “complete” numerical solution of a non head-on merger event: orbit, coalescence, ringdown and gravitational wave extraction, using generalized harmonic coordinatesFP, PRL 95, 121101 (2005) : First “complete” numerical solution of a non head-on merger event: orbit, coalescence, ringdown and gravitational wave extraction, using generalized harmonic coordinates M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006) ; J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006) obtained similar stable evolution using the “BSSN” formalismM. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006) ; J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006) obtained similar stable evolution using the “BSSN” formalism –note that to go from “a to b” here has required a tremendous amount of research in understanding the mathematical structure of the field equations, stable discretization schemes, dealing with geometric singularities inside black holes, computational algorithms, initial data, extracting useful physical information from simulations, etc.

5. Current state of the field Two quite different, stable methods of integrating the Einstein field equations for this problemTwo quite different, stable methods of integrating the Einstein field equations for this problem –generalized harmonic with constraint damping, F.Pretorius, PRL 95, 121101 (2005) Caltech/CornellCaltech/Cornell LSU/LUI/BYULSU/LUI/BYU PITT/AEI/LSU-CCTPITT/AEI/LSU-CCT PrincetonPrinceton UMDUMD –BSSN with “moving punctures” M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006) Jena/FAUJena/FAU LSU/AEI/UNAMLSU/AEI/UNAM NASA GoddardNASA Goddard PennstatePennstate RITRIT UIUC/BowdoinUIUC/Bowdoin U. SperhakeU. Sperhake U.Tokyo/UWMU.Tokyo/UWM

6. Motivation to study astrophysical mergers: gravitational wave observatories! LIGO Hanford LIGO Livingston ground based laser interferometers LIGO/VIRGO/GEO/TAMA space-based laser interferometer (hopefully with get funded for a 20?? Launch) LISA ALLEGRO/NAUTILUS/AURIGA/… resonant bar detectors ALLEGRO AURIGA Pulsar timing network, CMB anisotropy The Crab nebula … a supernovae remnant harboring a pulsar Segment of the CMB from WMAP

7. source frequency (Hz) source frequency (Hz) source “strength” source “strength” 10 4 10 -12 10 -8 10 -4 1 relics from the big bang, inflation relics from the big bang, inflation exotic physics in the early universe: phase transitions, cosmic strings, domain walls, … exotic physics in the early universe: phase transitions, cosmic strings, domain walls, … 1-10 M ๏ BH/BH mergers 1-10 M ๏ BH/BH mergers NS/BH mergers NS/NS mergers pulsars, supernovae EMR inspiral EMR inspiral NS binaries WD binaries 10 2 -10 6 M ๏ BH/BH mergers 10 2 -10 6 M ๏ BH/BH mergers >10 6 M ๏ BH/BH mergers >10 6 M ๏ BH/BH mergers CMB anisotropy Pulsar timing LISALIGO/… Bar detectors Overview of expected gravitational wave sources

8. Anatomy of a Rest Mass Dominated Merger In the conventional scenario of a black hole merger in the universe, one can break down the evolution into 4 stages: Newtonian, inspiral, plunge/merger and ringdownIn the conventional scenario of a black hole merger in the universe, one can break down the evolution into 4 stages: Newtonian, inspiral, plunge/merger and ringdown NewtonianNewtonian –in isolation, radiation reaction will cause two black holes of mass M in a circular orbit with initial separation R to merge within a time t m relative to the Hubble time t H –label the phase of the orbit Newtonian when the separation is such that the binary will take longer than the age of the universe to merge, for then to be of relevance to gravitational wave detection, other “Newtonian” processes need to operate, e.g. dynamical friction, n-body encounters, gas-drag, etc. For e.g., two solar mass black holes need to be within 1 million Schwarzschild radii ~ 3 million kmtwo solar mass black holes need to be within 1 million Schwarzschild radii ~ 3 million km two 10 9 solar mass black holes need to be within 6 thousand Schwarzschild radii ~ 1 parsectwo 10 9 solar mass black holes need to be within 6 thousand Schwarzschild radii ~ 1 parsec

9. Anatomy of a Rest Mass Dominated Merger inspiral  quasi-circular inspiral (QSI)inspiral  quasi-circular inspiral (QSI) –In the inspiral phase, energy loss through gravitational wave emission is the dominate mechanism forcing the black holes closer together –to get an idea for the dominant timescale during inspiral, for equal mass, circular binaries the Keplarian orbital frequency offers a good approximation until very close to merger the dominant gravitational wave frequency is twice thisthe dominant gravitational wave frequency is twice this –Post-Newtonian techniques provide an accurate description of certain aspects of the process until remarkably close to merger –if the initial pericenter of the orbit is sufficiently large, the orbit will loose its eccentricity long before merger [Peters & Matthews, Phys.Rev. 131 (1963)] and become quasi-circular

10. Anatomy of a Rest Mass Dominated Merger plunge/mergerplunge/merger –this is the time in the merger when the two event horizons coalesce into one we know the two black holes must merge into one if cosmic censorship holds (and no indications of a failure yet in any merger simulations)we know the two black holes must merge into one if cosmic censorship holds (and no indications of a failure yet in any merger simulations) –full numerical solution of the field equations are required to solve for the geometry of spacetime in this stage –full numerical solution of the field equations are required to solve for the geometry of spacetime in this stage Only within the last 3 years, following a couple of breakthroughs, has numerical relativity been able to complete the picture by filling in the details of the final, non-perturbative phase of the mergerOnly within the last 3 years, following a couple of breakthroughs, has numerical relativity been able to complete the picture by filling in the details of the final, non-perturbative phase of the merger At present two known stable formulations of the field equations, generalized harmonic [FP, PRL 95, 121101 (2005)], and BSSN with moving punctures [M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)]At present two known stable formulations of the field equations, generalized harmonic [FP, PRL 95, 121101 (2005)], and BSSN with moving punctures [M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)] –in all cases studied to date, this stage is exceedingly short, leaving its imprint in on the order of 1-2 gravitational wave cycles, at roughly twice the final orbital frequency

11. Anatomy of a Rest Mass Dominated Merger ringdownringdown –in the final phase of the merger, the remnant black hole “looses all its hair”, settling down to a Kerr black hole –one possible definition for when plunge/merger ends and ringdown begins, is when the spacetime can adequately be described as a Kerr black hole perturbed by a set of quasi-normal modes (QNM) –the ringdown portion of the waveform will be dominated by the fundamental harmonic of the quadrupole QNM, with characteristic frequency and decay time [Echeverria, PRD 34, 384 (1986)]: j=a/M f, the Kerr spin parameter of the black hole

12. Sample evolution --- Cook-Pfeiffer Quasi-circular initial data This animation shows the lapse function in the orbital plane. The lapse function represents the relative time dilation between a hypothetical observer at the given location on the grid, and an observer situated very far from the system --- the redder the color, the slower local clocks are running relative to clocks at infinity If this were in “real-time” it would correspond to the merger of two ~5000 solar mass black holes Initial black holes are close to non-spinning Schwarzschild black holes; final black hole is a Kerr a black hole with spin parameter ~0.7, and ~4% of the total initial rest-mass of the system is emitted in gravitational waves A. Buonanno, G.B. Cook and F.P.; Phys.Rev.D75:124018,2007

13. Gravitational waves from the simulation A depiction of the gravitational waves emitted in the orbital plane of the binary. Shown is the real component of the Newman Penrose scalar  , which in the wave zone is proportional to the second time derivative of the usual plus- polarization The plus-component of the wave from the same simulation, measured on the axis normal to the orbital plane

14. What does the merger wave represent? Scale the system to two 10 solar mass (~ 2x10 31 kg) BHsScale the system to two 10 solar mass (~ 2x10 31 kg) BHs –radius of each black hole in the binary is ~ 30km –radius of final black hole is ~ 60km –distance from the final black hole where the wave was measured ~ 1500km –frequency of the wave ~ 200Hz (early inspiral) - 800Hz (ring-down)

15. What does the merger wave represent? fractional oscillatory “distortion” in space induced by the wave transverse to the direction of propagation has a maximum amplitude  L/L ~ 3x10 -3fractional oscillatory “distortion” in space induced by the wave transverse to the direction of propagation has a maximum amplitude  L/L ~ 3x10 -3 a 2m tall person will get stretched/squeezed by ~ 6 mm as the wave passesa 2m tall person will get stretched/squeezed by ~ 6 mm as the wave passes LIGO’s arm length would change by ~ 12m. Wave amplitude decays like 1/distance from source; e.g. at 10Mpc the change in arms ~ 5x10 -17 m (1/20 the radius of a proton, which is well within the ballpark of what LIGO is trying to measure!!)LIGO’s arm length would change by ~ 12m. Wave amplitude decays like 1/distance from source; e.g. at 10Mpc the change in arms ~ 5x10 -17 m (1/20 the radius of a proton, which is well within the ballpark of what LIGO is trying to measure!!) despite the seemingly small amplitude for the wave, the energy it carries is enormous — around 10 30 kg c 2 ~ 10 47 J ~ 10 54 ergsdespite the seemingly small amplitude for the wave, the energy it carries is enormous — around 10 30 kg c 2 ~ 10 47 J ~ 10 54 ergs peak luminosity is about 1/100 th the Planck luminosity of 10 59 ergs/s !!peak luminosity is about 1/100 th the Planck luminosity of 10 59 ergs/s !! luminosity of the sun ~ 10 33 ergs/s, a bright supernova or milky-way type galaxy ~ 10 42 ergs/sluminosity of the sun ~ 10 33 ergs/s, a bright supernova or milky-way type galaxy ~ 10 42 ergs/s if all the energy reaching LIGO from the 10Mpc event could directly be converted to sound waves, it would have an intensity level of ~ 80dBif all the energy reaching LIGO from the 10Mpc event could directly be converted to sound waves, it would have an intensity level of ~ 80dB

16. Highlights of recent results: simplicity of merger waveform the “non-linear” phase of the merger is surprisingly shortthe “non-linear” phase of the merger is surprisingly short great boon for data analysis, as this suggests an efficient LIGO template bank could be compiled by stitching together quick-to-calculate perturbative waveforms, guided by a handful of numerical waveformsgreat boon for data analysis, as this suggests an efficient LIGO template bank could be compiled by stitching together quick-to-calculate perturbative waveforms, guided by a handful of numerical waveforms to-date, some of the best examples employ the Effective One Body (EOB) approach [A. Buonanno and T. Damour, et al., Phys.Rev.D59:084006,1999]. Example here is for quasi-circular inspiral of non-spinning BH’s [A. Buonanno et al., Phys.Rev.D76:104049,2007]to-date, some of the best examples employ the Effective One Body (EOB) approach [A. Buonanno and T. Damour, et al., Phys.Rev.D59:084006,1999]. Example here is for quasi-circular inspiral of non-spinning BH’s [A. Buonanno et al., Phys.Rev.D76:104049,2007] –EOB inspiral connected to 3 leading QNMs –added “pseudo” 4PN term to EOB model, with coefficient determined by a best-fit match to a set of numerical results –used simulation results for final spin and black hole mass to fix the QNM frequencies and decay constants 4:1 mass ratio example

17. Highlights of recent results : large recoil velocities significant recoil can be imparted to the remnant black hole due to asymmetric beaming of radiation during the merger, up to 4000km/s in some casessignificant recoil can be imparted to the remnant black hole due to asymmetric beaming of radiation during the merger, up to 4000km/s in some cases –Herrmann et al., gr-qc/0701143; Koppitz et al., gr-qc/0701163; Campanelli et al. gr- qc/0701164 & gr-qc/0702133, Gonzalez et al, arXiv:gr-qc/0702052, Tichy & Marronetti, arXiv:gr-qc/0703075v1 there are far reaching consequences to this, some that could be detected via electromagnetic observations, in particular for supermassive black hole mergersthere are far reaching consequences to this, some that could be detected via electromagnetic observations, in particular for supermassive black hole mergers –offset or double galactic nuclei, displaced active galactic nuclei, wiggling jets, enlarged cores, lopsided cores, x-ray afterglows, feedback trails, off- center flares from tidally disrupted stars, hypervelocity stars, a population of galaxies without supermassive black holes, etc. Merritt et al., ApJ. 607 (2004) L9-L12; Milosavljevic & Phinney, ApJ 622, L93(2005); Gualandris & Merrit arXiv:0708.0771 &, arXiv:0708.3083; Lippai et al. arXiv:0801.0739; Kornreich & Lovelace, arXiv:0802.2058; Devecchi et al. arXiv:0805.2609; Komossa & Merrit arXiv:0807.0223 & arXiv:0811.1037; Fujita arXiv:0808.1726 & arXiv:0810.1520 –a 2650km/s recoiling black hole could explain the emission line spectra from quasar SDSSJ092712.65+294344. 0 [S. Komossa et al., ApJ.678:L81,2008]

18. Kinetic energy dominated mergers: the black hole scattering problem consider v~c. In general two, distinct end-states possibleconsider v~c. In general two, distinct end-states possible for b<b* one black hole, after a collisionfor b<b* one black hole, after a collision for b>b* two isolated black holes, after a deflectionfor b>b* two isolated black holes, after a deflection because there are two distinct end-states, there must be some kind of threshold behavior approaching the critical impact parameter b*because there are two distinct end-states, there must be some kind of threshold behavior approaching the critical impact parameter b* b m 1,v 1 m 2,v 2

19. Motivation: Black hole formation at the LHC and in the atmosphere? large extra dimension scenarios [ N. Arkani-Hamed, S. Dimopoulos & G.R. Dvali, PLB429:263-272; L. Randall & R. Sundrum, PRL.83:3370-3373 ] suggest the true Planck scale can be very different from what then would be an effective 4-dimensional Planck scale of 10 19 GeV calculated from the fundamental constants measured on our 4-D Branelarge extra dimension scenarios [ N. Arkani-Hamed, S. Dimopoulos & G.R. Dvali, PLB429:263-272; L. Randall & R. Sundrum, PRL.83:3370-3373 ] suggest the true Planck scale can be very different from what then would be an effective 4-dimensional Planck scale of 10 19 GeV calculated from the fundamental constants measured on our 4-D Brane In the TeV range is a “natural” choice to solve the hierarchy problemIn the TeV range is a “natural” choice to solve the hierarchy problem Implications of this are that super-TeV particle collisions would probe the quantum gravity regimeImplications of this are that super-TeV particle collisions would probe the quantum gravity regime –collisions sufficiently above the Planck scale are expected to be dominated by the gravitational interaction, and arguments suggest that black hole formation will be the most likely result of the two-particle scattering event [Banks & Fishler hep-th/9906038, Dimopoulos & Landsberg PRL 87 161602 (2001), Feng & Shapere, PRL 88 021303 (2002), …] current experiments rule out a Planck scale of <~ 1TeVcurrent experiments rule out a Planck scale of <~ 1TeV The LHC should reach center-of-mass energies of ~ 10 TeVThe LHC should reach center-of-mass energies of ~ 10 TeV cosmic rays can have even higher energies than this, and so in both cases black hole formation could be expectedcosmic rays can have even higher energies than this, and so in both cases black hole formation could be expected these black holes will be small and decay rapidly via Hawking radiation, which is the most promising route to detectionthese black holes will be small and decay rapidly via Hawking radiation, which is the most promising route to detection if a lot of gravitation radiation is produced during the collision this could show up as a missing energy signal at the LHCif a lot of gravitation radiation is produced during the collision this could show up as a missing energy signal at the LHC ATLAS experiment at the LHC One of the water tanks at the Pierre Auger Observatory

20. But do super-Planck scale particle collisions form black holes? The argument that the ultra-relativistic collision of two particles should form a black hole is purely classical, and is essentially based on Thorne’s hoop conjectureThe argument that the ultra-relativistic collision of two particles should form a black hole is purely classical, and is essentially based on Thorne’s hoop conjecture –(4D) if an amount of matter/energy E is compacted to within a sphere of radius R=2GE/c 4 corresponding to the Schwarzschild radius of a black hole of mass M=E/c 2, a black hole will form –applied to the head-on collision of two “particles” each with rest mass m, characteristic size W, and center-of-mass frame Lorentz gamma factors , this says a black hole will form if the Schwarzschild radius corresponding to the total energy E=2m  is greater that W –the quantum physics comes in when we say that the particle’s size is given by its de Broglie wavelength W = hc/E, from which one gets the Planck energy E p =(hc 5 /G) 1/2

21. Evidence to support this From the classical perspective, evidence to support this would be solutions to the field equations demonstrating that weakly self-gravitating objects, when boosted toward each other with large velocities so that the net mass of the space time (in the center of mass frame) is dominated by the kinetic energy, generically form a black hole when the interaction occurs within a region smaller than the Schwarzschild radius of the spacetimeFrom the classical perspective, evidence to support this would be solutions to the field equations demonstrating that weakly self-gravitating objects, when boosted toward each other with large velocities so that the net mass of the space time (in the center of mass frame) is dominated by the kinetic energy, generically form a black hole when the interaction occurs within a region smaller than the Schwarzschild radius of the spacetime –generic: the outcome would have to be independent of the particular details of the structure and non-gravitational interactions between the particles, if the classical picture is to have any bearing on the problem –interaction: the non-linear interaction of the gravitational kinetic energy of the boosted particles will be key in determining what happens consider the trivial counter-examples to the hoop conjecture applied to a single particle boosted to ultra-relativistic velocities, or a white hole “explosion”consider the trivial counter-examples to the hoop conjecture applied to a single particle boosted to ultra-relativistic velocities, or a white hole “explosion” What is oft quoted as evidence comes from studies of the infinite boost limit of BH collisions [Penrose 1974, Eardley & Giddings PRD 66, 044011 (2002)]What is oft quoted as evidence comes from studies of the infinite boost limit of BH collisions [Penrose 1974, Eardley & Giddings PRD 66, 044011 (2002)] –however, it is not obvious that this describes large-but-finite collisions: it is a singular limit, the gravitational field changes character from Petrov type D (Coloumb-like) to type N (pure gravitational wave) and is nowhere a good approximation to a boosted massive particle geometry, the spacetime is no longer asymptotically flat, … etc

22. High speed soliton collision simulations Try to test this hypothesis by colliding self-gravitating solitons, boson stars in this case (work with M.W.Choptuik)Try to test this hypothesis by colliding self-gravitating solitons, boson stars in this case (work with M.W.Choptuik) The follow use initial conditions where the Thorne estimates says The follow use initial conditions where the Thorne estimates says  –get BH formation at  already!  scalar field magnitude, 

23. High Speed Black Hole Collisions If these soliton collisions are confirming the expected generic behavior of high energy particle collisions, this implies we can use any model of a particle to study the classical gravitational signatures of super-Planck scale collisions, including black holes!If these soliton collisions are confirming the expected generic behavior of high energy particle collisions, this implies we can use any model of a particle to study the classical gravitational signatures of super-Planck scale collisions, including black holes! Will describe some on-going studies of such scenarios with U. Sperhake, V. Cardose, E. Berti, J.A Gonzalez, T. Hinderer & N. Yunes (see also M. Shibata, H. Okawa & T. Yamamoto, PRD 78:101501, 2008)Will describe some on-going studies of such scenarios with U. Sperhake, V. Cardose, E. Berti, J.A Gonzalez, T. Hinderer & N. Yunes (see also M. Shibata, H. Okawa & T. Yamamoto, PRD 78:101501, 2008) However, with application to the LHC in mind, such a project can only be pursued in some approximate mannerHowever, with application to the LHC in mind, such a project can only be pursued in some approximate manner –unknown Planck-scale physics –unknown structure of the extra dimensions –4D simulations “barely” feasible … generic 10(11?) dimensional simulations impossible in foreseeable future

24. Aside : the Planck Luminosity regime The Planck Luminosity L p isThe Planck Luminosity L p is –notice that Planck’s constant does not enter, hence this is a regime that is ostensibly described by classical general relativity –L p may be a limiting luminosity for “reasonable” processes, and trying to reach it may reveal some of the more interesting aspects of the theory would super-Planck luminosities be associated with violations of cosmic censorship?would super-Planck luminosities be associated with violations of cosmic censorship?

25. The Planck Luminosity regime Suppose a single black hole is formed during the collision, and a fraction e of the total energy of the system 2  mc 2 (equal mass) is released as gravitational waves. Estimate that the shortest timescale on which the energy can be released is the light-crossing time of the remnant black hole ~2R/c = 4G(1-e)M/c 3. This givesSuppose a single black hole is formed during the collision, and a fraction e of the total energy of the system 2  mc 2 (equal mass) is released as gravitational waves. Estimate that the shortest timescale on which the energy can be released is the light-crossing time of the remnant black hole ~2R/c = 4G(1-e)M/c 3. This gives Implies that a very efficient (>80%) prompt energy release mechanism will result in super-Planck luminosity, howeverImplies that a very efficient (>80%) prompt energy release mechanism will result in super-Planck luminosity, however –we know in the low-speed regime e is small (fraction of a percent) –if the limiting solution is a collision of Aichelburg-Sexl shock waves, Penrose found trapped surfaces at the moment of collision implying e<29%; seems to be consistent with simulation results (next slides) –expect to get much larger emission rates in grazing collisions

26.  collision example   (M=  M 0 ) Re[  4 ]

27.  collision example   (M=  M 0 ) Re[  4 ]

28.  collision example   (M=  M 0 ) Re[  4 ]

29.  collision example   (M=  M 0 ) Re[  4 ]

30.  collision example   (M=  M 0 ) Re[  4 ]

31.  collision example   (M=  M 0 ) Re[  4 ]

32.  collision example   (M=  M 0 ) Re[  4 ]

33.  collision example   (M=  M 0 ) Re[  4 ]

34.  collision example   (M=  M 0 ) Re[  4 ]

35.  collision example Re[  4 ]   (M=  M 0 )

36.  collision example roughly 8% of the total energy is emitted in gravitational waves during the collision … peak luminosity ~0.3% of L proughly 8% of the total energy is emitted in gravitational waves during the collision … peak luminosity ~0.3% of L p –about a factor of 6 less than the order-of-magnitude estimate (the relevant time scale seems to be given by the dominant quasi-normal mode frequency of the final black hole, the period of which is a few times larger that the light-crossing time of the black hole) spectrum is reasonably flat (left) below a cut-off given by the QNM frequencies of the black hole, a result predicted by the “zero-frequency-limit” (ZFL) approximationspectrum is reasonably flat (left) below a cut-off given by the QNM frequencies of the black hole, a result predicted by the “zero-frequency-limit” (ZFL) approximation extrapolation to infinite boost limit using a curve motivated by the ZFL (right, red line) suggests the result will be ~ 14 %, a bit less than 1/2 the Penrose boundextrapolation to infinite boost limit using a curve motivated by the ZFL (right, red line) suggests the result will be ~ 14 %, a bit less than 1/2 the Penrose bound

37. Conclusions We are learning a lot about the the nature of gravity through numerical solution of the field equationsWe are learning a lot about the the nature of gravity through numerical solution of the field equations– Black hole collisions are a fascinating probe of strong-field general relativity, though it is remarkable how “simple” some of the these highly non-linear, dynamical scenarios are turning out to beBlack hole collisions are a fascinating probe of strong-field general relativity, though it is remarkable how “simple” some of the these highly non-linear, dynamical scenarios are turning out to be –merger phase and waveforms from quasi-circular inspiral –in the ultra-relativistic (UR) limit that the leading order behavior of the dynamics of the collision is captured by the cut-and-paste Penrose construction Future workFuture work –barely scratched the surface of the UR limit – things are becoming very interesting in non head-on collisions : much higher luminosities, nearly extremely rotating BH’s are formed, onset of “zoom-whirl” behavior