Numerical comparison between maximal and geodesic slice in Schwarzschild black hole

1. Numerical comparison between maximal and geodesic slice in Schwarzschild black hole 2006. 04. 20 ~21 Hyung Won Lee , Kyung Yee Kim Inje Institute for Mathematical Sciences School of Computer Aided Science, Inje University

2. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

3. Solving Einstein’s Equations Numerically To learn radiated gravitational energy at infinity To understand dynamical processes near singularity What is Numerical Relativity? Introduction

4. Introduction Component to need the numerical construction

5. Introduction • Schwarzschild Black hole: • Vacuum, Spherically symmetric • Stationary and Static solution • Time reversal symmetry • Invariant on time translation • Asymptotically flat solution • Geometric mass:

6. Introduction Schwarzschild black hole line element: Transformation to isotropic coordinate

7. Introduction Isotropic coordinate • Invariant isometry transformation: • Allowing us to apply the isometry boundary condition. • Identical under mapping from the outside of throat to the inside of throat w.r.t. . • The form of map J which identifies two asymptotically flat sheet through the throat (isometry boundary condition):

8. : shift vector Introduction The 3+1 ADM (Arnowitt, Deser and Misner) Formalism : Lapse function To determine the proper time measured by an observer falling normal to the slice:

9. C B A Introduction The 3+1 ADM (Arnowitt, Deser and Misner) Formalism

10. Introduction The 3+1 ADM (Arnowitt, Deser and Misner) Formalism The Extrinsic Curvature • To describe the curvature of 3D slice in the 4D space-time in which it is embedded. • To be defined via the Lie derivative of the 3-metric with respect to the future-pointing unit normal vector to the 3-surface.

11. Introduction The 3+1 ADM (Arnowitt, Deser and Misner) Formalism The Einstein Equation: • Vacuum Einstein Equation • Hamilton Constraint Equation: • Momentum Constraint Equation: • Evolution Equation :

12. Introduction The 3+1 ADM (Arnowitt, Deser and Misner) Formalism Constraint equation Evolution equation Definition of

13. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

14. C B A The Gauge Conditions

15. The Gauge Condition :Geodesic slice geodesic slice : radially infalling free particle t( coordinate time) r =0 r =2m r =r0

16. t=0 r =0 r =2m r =2m at crash t>0 r =2m t>0 t=0 0<r <2m r =2m at crash r =0 The Gauge Condition : Geodesic Slice geodesic slice:

17. t= The Gauge Condition : Maximal Slice • Maximal slice: t=0 r =2m r =0 r =2m t= t>0 r =2m t>0 t=0 0<r <2m r =2m r =3/2m

18. The Gauge Condition : Maximal Slice • Condition:

19. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

20. To be determined 3-metric and to satisfying the constraint equation. • Using York’s conformal decomposition method • Insert 3-metric and extrinsic curvature to determine in Hamilton constraint equation : The Initial Value problem

21. The Initial Value problem Metric: Spherical symmetry line of element 3- metric: York’s conformal decomposition method

22. The Initial Value problem four dynamic variable two gauge variable

23. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

24. The Boundary Condition The Inner boundary condition: • Black hole ‘s throat • Inner- most surface is locally an areal minimum. • For Schwarzschild black hole, the throat is the event horizon. • Isometry boundary condition. • Invariant :

25. The Boundary Condition The outer boundary condition: • far away from the black hole • Asymptotically flat at infinite region. The Inner boundary condition The Outer boundary condition

26. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

27. The Numerical Algorithm Finite Difference method:

28. The Numerical Algorithm: Evolution Equation: • ½ time slice estimation of :

29. The Numerical Algorithm Grid:

30. The Numerical Algorithm: Evolution Equation for geodesic slice:

31. The Numerical Algorithm: To apply the finite difference method

32. The Numerical Algorithm: To apply the finite difference method

33. The Numerical Algorithm:

34. The Numerical Algorithm:

35. The Numerical Algorithm: Runge - Kutta method, shooting method

36. The Numerical Algorithm:

37. Contents • Introduction • Gauge Conditions* Geodesic slice * Maximal slice • The Initial Value Problem • The Boundary condition • The Finite difference Scheme • Results

38. Result Isotropic gauge: Static solution Code test result: static solution

39. Result T=0 T= 400(4M)

40. Result: geodesic slice The result Crashed around time:

41. Result Horizon [figure] metric a T = 100, 150, 200, 250

42. Result [Figure] Solution of Schwarzschild black hole

43. Result Schwarzchild Black hole Simulation Result Fig. a-variation with time

44. Result Schwarzchild Black hole Simulation Result Fig. b-variation with time

45. Result Schwarzchild Black hole Simulation Result Fig. -variation with time

46. Result Schwarzchild Black hole Simulation Result Fig. - variation with time

47. Result Schwarzchild Black hole Simulation Result

48. Result: Maximal slice

49. Result: Maximal slice [0M,5M,10M,20M,30M,40M,50M,60M,80M,90M] [gr-qc0307036 Fig9.Result]

50. Result: Maximal slice [0M,5M,10M,20M,30M,40M,50M,60M,80M,90M]