1. 1 Ulrich Sperhake Friedrich-Schiller Universität Jena Numerical simulations of high-energy collisions Of black hole CENTRA IST Seminar Lisbon, 29 th February 2008 Work supported by the SFB TR7 Work with E.Berti, V.Cardoso, J.A.Gonzalez, F.Pretorius

2. 2 Overview Introduction Results Summary A brief history of BH simulations Results following the recent breakthrough Black holes in physics Ingredients of numerical relativity

3. 3 1. Black holes in physics

4. 4 Black Holes predicted by GR valuable insight into theory Black holes predicted by Einstein’s theory of relativity Vacuum solutions with a singularity For a long time: mathematical curiosity Term “Black hole” by John A. Wheeler 1960s but no real objects in the universe That picture has changed dramatically!

5. 5 How to characterize a black hole? Consider light cones Outgoing, ingoing light Calculate surface area of outgoing light Expansion:=Rate of change of that area Apparent horizon:= Outermost surface with zero expansion “Light cones tip over” due to curvature

6. 6 Black Holes in astrophysics Structure formation in the early universe Black holes are important in astrophysics Structure of galaxies Starbursts Black holes found at centers of galaxies

7. 7 Black Holes in astrophysics Black holes are important in astrophysics Gravitational lenses Important sources of electromagnetic radiation

8. 8 Fundamental physics of black holes Allow for unprecedented tests of fundamental physics Strongest sources of Gravitational Waves (GWs) Test alternative theories of gravity No-hair theorem of GR Production in Accelerators

9. 9 Gravitational wave physics Accelerating bodies produce GWs Weber 1960s Bar detector Claimed detection probably not real GWs displace particles GW observatories: GEO600, LIGO, TAMA, VIRGO Bar detectors

10. 10 The big picture Model GR (NR) PN Perturbation theory Alternative Theories? External Physics Astrophysics Fundamental Physics Cosmology Detectors Physical system describes observe test Provide info Help detection

11. 11 2. General relativity

12. 12 The framework: General Relativity Curvature generates acceleration Description of geometry Metric Connection Riemann Tensor “geodesic deviation” No “force” !!

13. 13 The metric defines everything Christoffel connection Covariant derivative Riemann Tensor Geodesic deviation Parallel transport …

14. 14 How to get the metric? The metric must obey the Einstein Equations Ricci-Tensor, Einstein-Tensor, Matter tensor “Trace-reversed’’ Ricci Einstein Equations Solutions: Easy! Take metric Calculate Use that as matter tensor Physically meaningful solutions: Difficult! “Matter”

15. 15 The Einstein equations in vacuum Field equations: Second order PDEs for the metric components Analytic solutions: Minkowski, Schwarzschild, Kerr, Robertson-Walker,… Numerical methods necessary for general scenarios! System of equations extremely complex: Pile of paper! “Spacetime tells matter how to move, matter tells spacetime how to curve” Invariant under coordinate (gauge) transformations

16. 16 3. The basics of numerical relativity

17. 17 A list of tasks Target: Predict time evolution of BBH in GR Einstein equations: Cast as evolution system Choose specific formulation Discretize for Computer Choose coordinate conditions: Gauge Fix technical aspects: Mesh-refinement / spectral domains Excision Parallelization Find large computer Construct realistic initial data Start evolution and wait… Extract physics from the data Gourgoulhon gr-qc/0703035

18. 18 3.1. The Einstein equations

19. 19 3+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 lapse shift lapse, shift Gauge Einstein equations 6 Evolution equations 4 Constraints Constraints preserved under evolution!

20. 20 The ADM equations Time projection Mixed projection Spatial projection Hamiltonian constraint Momentum constraints Evolution equations

21. 21 The structure of the ADM equations Constraints: They do not contain time derivatives They must be satisfied on each slice The Bianchi identities propagate the constraints: If they are satisfied initially, they are always satisfied Evolution equations: Commonly written as first order system Gauge: Equations say nothing about lapse and shift !

22. 22 The ADM equations as an initial value problem Entwicklungsgleichungen

23. 23 Alternatives to the ADM equations Unfortunately the ADM equations do not work in NR !! Weak hyperbolicity: Nearby initial data can diverge From each other super-exponentially Many alternative formulations have been suggested Two successful families so far ADM based formulations: BSSN Generalized harmonic formulations Shibata & Nakamura ’95, Baumgarte & Shapiro ‘99 Choque-Bruhat ’62, Garfinkle ’04, Pretorius ‘05

24. 24 The BSSN equations

25. 25 3.2. Gauge choices

26. 26 The gauge freedom Remember: Einstein equations say nothing about Any choice of lapse and shift gives a solution This represents the coordinate freedom of GR Physics do not depend on So why bother? Avoid coordinate singularities! Stop the code from running into the physical singularity No full-proof recipe, but Singularity avoiding slicing Use shift to avoid coordinate stretching

27. 27 3.3. Initial data

28. 28 Initial data problem Two problems: Constraints, realistic data York-Lichnerowicz split Rearrange degrees of freedom Conformal transverse traceless Physical transverse traceless Thin sandwich York, Lichnerowicz Conformal flatness: Kerr is NOT conformally flat! Non-physical GWs: problematic for high energy collisions! Wilson, Mathews; York O’Murchadha, York

29. 29 2 families of initial data Generalized analytic solutions: Time-symmetric, -holes: Spin, Momenta: Punctures Brill-Lindquist, Misner (1960s) Bowen, York (1980) Brandt, Brügmann (1997) Isotropic Schwarzschild: Excision Data: IH boundary conditions on excision surface Meudon group; Cook, Pfeiffer; Ansorg Quasi-circular: Effective potential PN parameters helical Killing Vektor

30. 30 4. Extracting physics

31. 31 Basic assumptions Extracting physics in NR is non-trivial !! We assume that the ADM variables Lapse Shift 3-metric Extrinsic curvature are given on each hypersurface Even when using other formulations, the ADM variables are straightforward to calculate Newtonian quantities are not always well-defined !!

32. 32 Global quantities ADM mass: Global energy of the spacetime Total angular momentum of the spacetime By construction all of these are time-independent !!

33. 33 Local quantities Often impossible to define !! Isolated horizon framework Calculate apparent horizon Ashtekar and coworkers irreducible mass, momenta associated with horizon Total BH mass Christodoulou Binding energy of a binary:

34. 34 Gravitational waves Most important result: Emitted gravitational waves (GWs) Newman-Penrose scalar GWs allow us to measure Radiated energy Radiated momenta Angular dependence of radiation Predicted strain Complex 2 free functions

35. 35 Angular dependence Waves are normally extracted at fixed radius Decompose angular dependence Spin-weighted spherical harmonics Modes are viewed from the source frame !!

36. 36 5. Status of black hole simulations

37. 37 A very brief history Pioneers: Hahn and Lindquist 1960s, Eppley and Smarr 1970s Problems: Computer resources Instabilities: Gauge, Equations Breakthrough: Pretorius 2005, Brownsville, NASA Goddard 2005 AMR Now about 10 codes GHG Moving puncture

38. 38 The BBH breakthrough Simplest configuration GWs circularize orbit quasi-circular initial data Pretorius ‘05 Initial data: scalar field Radiated energy 25 50 75 100 4.7 3.2 2.7 2.3 Eccentricity BBH breakthrough

39. 39 Astrophysical binaries: basics Free parameters NR can only simulate the last orbits 1: Total mass (merely a scaling factor) 6: Spin 1: Mass ratio 1: Eccentricity; GWs circularize orbit 4 stages of BBH merger: Newtonian inspiral (not GW driven) Post-Newtonian (PN) inspiral phase (GW driven) merger (Numerical Relativity) Ring-down (Perturbation Theory)

40. 40 Astrophysical binaries: Results Recoil or kick Unequal mass and/or spin kick Equal-mass, non-spinning binaries of the total mass of the system Good agreement with PN predictions for GW emission GWs quadrupole dominated Certain spin configurations Enough to eject BHs from galaxies Observations such large kicks not generic Spin-flip Merger spin realignment X-shaped radio sources

41. 41 Zoom whirl orbits 1-parameter family of initial data: linear momentum Pretorius & Khurana ‘07 Fine-tune parameter ”Threshold of immediate merger” Analogue in gedodesics ! Reminiscent of ”Critical phenomena”

42. 42 6. High energy collision

43. 43 Motivation No known mechanism to accelerate BHs to Such collisions unlikely to occur in astrophysical scenarios But: Probe GR in the most dynamic range Near the speed of light, structure is lost Collisions might describe generic collisions Test cosmic censorship ”Threshold of immediate merger” Pretorius 2006 Indication of unexpected phenomena

44. 44 Setup of the problem Take two black holes Total rest mass Linear momentum Initial position For now: Equal-mass, head-on Target: GW energy Mode distribution of GW energy

45. 45 Analytic studies of a single boosted BH Schwarzschild boosted with Aichelburg-Sexl metric Planar shock wave travelling along Minkowskian everywhere else No event horizon! Analytically studied case:

46. 46 Analytic studies of a BH binary Superpose two Aichelburg-Sexl metrics Shock waves collide Penrose, Eardley & Giddings Future trapped surface! Max. gravitational radiation: Penrose Perturb superposed Aichelburg-Sexl metrics D’Eath & Payne ’90s First correction term reduces GWs to

47. 47 Analytic studies of a BH binary Instant Collision approximation Applied to superposed Aichelburg-Sexl metric Smarr; Adler & Zeks i) Flat at sufficiently low Energy spectrum ii) Radiation isotropic in the limit iii) Functional relation iv) Multipole structure Numerical simulations Grand Challenge ’90s Limited accuracy, mild boosts

48. 48 Numerical simulations LEAN code Sperhake ‘07 BSSN formulation Puncture initial data Brandt & Brügmann 1996 Based on the Cactus computational toolkit Mesh refinement: Carpet Schnetter ‘04 Elliptic solver: TwoPunctures Ansorg 2005 Numerically very challenging! Length scales: Horizon Lorentz-contracted “Pancake” Mergers extremely violent Substantial amounts of unphysical “junk” radiation

49. 49 Case example Boost initial separation Total radiated energy Mode energy Junk radiation

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60. 60 Modes

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62. 62 Modes

63. 63 Modes

64. 64 Modes

65. 65 Modes

66. 66 Energy spectrum Junk radiation affects spectrum for small separations

67. 67 Conclusions Introduction to Numerical Relativity NR has solved the binary problem Key results on kicks, spin-flips, PN, data analysis High energy collisions Penrose limit Numerical simulations very difficult (junk radiation!) Merger radiation for Todo: Improve accuracy, larger

68. 68 Gravitational recoil Anisotropic emission of GWs radiates momentum recoil of remaining system Leading order: Overlap of Mass-quadrupole with octopole/flux-quadrupole Bonnor & Rotenburg ’61, Peres ’62, Bekenstein ‘73 Merger of galaxies Merger of BHs Recoil BH kicked out?

69. 69 Gravitational recoil Ejection or displacement of BHs has repercussions on: Escape velocities Globular clusters dSph dE Giant galaxies Structure formation in the universe BH populations Growth history of Massive Black Holes IMBHs via ejection? Structure of galaxies Merrit et al ‘04

70. 70 Kicks of non-spinning black holes Parameter study Jena ‘07 Target: Maximal Kick Mass ratio: 150,000 CPU hours Maximal kick for Convergence 2 nd order Spin Simulations PSU ’07, Goddard ‘07

71. 71 Super Kicks Test Rochester-Prediction: Jena ‘07 Two models: Lean Bam Model I: Lean code Moving puncture method: version Not quasi-circular Resolutions: Model II: Safety check with independent code BAM

72. 72 Convergence Discretization error: