sign in sign up
<<
Home , ADM formalism, BSSN formalism, Numerical relativity

Lecture note on 3+1 formalism of

Masaru Shibata (Yukawa Institute, Kyoto U)

2010/06/09 1 Basis equations for Numerical Relativity G GT 8 4  Einstein equation c  T 0    u  0    u Yll Q Matter fields:    Fj  4 Next time if any  []F  0 f p f  p  S   p 2 Others in Numerical Relativity

• Imposing gauge conditions • Extracting gravitational waves • Finding black holes (finding ) • Adaptive mesh refinement

3 Contents 1. Structure of Einstein‟s equation (briefly) 2. 3+1 formalism of Einstein‟s equation 3. BSSN formalism 4. Gauge conditions 5. Initial value problem 6. Implementation of finite differencing 7. Extracting gravitational waves 8. Adaptive mesh refinement 9. Apparent horizon finder 4 Solution of Einstein’s equation for dynamical phenomena • We have to solve Einstein‟s equations as an initial value problem 4 • Einstein‟s equations, G = 8Gc T , are equations for space and time  Space and time coordinates appear in a mixed way; “time coordinate” does not always have the property of time. E.g., Schwarzschild coordinates; t is time for r > 2M, but is not for r < 2M  Special formalism is necessary to follow

dynamics of a variety of 5 Schwarzschild

1 2 222GM  2  GM  2 2 2 2 2 ds  c   dt  1 2  dr  r d  sin  d   r   rc  t is a radial t = const coordinate

Coordinate t is the time singularity coordinate.

Time has to be always “time”

in numerical relativity 6 Several formulations 1. 3+1 (N+1) formalism 2. Formulation based on a special (harmonic) coordinates (e.g., Pretorius formalism; also often used in Post-Newtonian theory) 3. Hyperbolic formalism (e.g., Kidder-Scheel-Teukolsky formalism) • Others …..

In this lecture, I focus on the first one 7 Einstein’s equation = hyperbolic equations

1 G G : G R   g  R  8 T   4 R : Ricci tensor 2 c         R         

1   g   g    g    g  2  g G     g g  g   g  g    g t         2 t  : Pseudo tensor of Landau-Lifshitz Og    de-Donder Gauge:  gg  0    Wave equation    2  G   g g     g g  O  g      8 Einstein’s equation is similar to scalar wave equation

2 ① t    Post-Newtonian way

t ②  3+1 way t   or ,   Kij , gij

t i   i Hyperbolic way ③  i ti   Similar: However, spacetime is not

a priori given in 9 Section I: 3+1 (ADM) formalism Concept 1. Foliate spacetime by spacelike surfaces 2. “Choose” time coordinates for a direction of time at each location 3. Follow dynamics of spacelike surfaces

Spacelike Time axes hypersurfaces

10 3+1 decomposition (N+1) Time 4 Evolve gij & Kij 3 Choose time axes at each point Space i t + t  nn = -1 2 Draw timelike t unit normal wrt n t spacelike surface, and choose lapse

t g ij, K ij  1 Choose a spacelike surface Kij~g ij : Talk later & give space-metric 11 Definition of variables g  (g , ,  k) gij = space metric  ij Kij= extrinsic curvature Dynamics  = lapse function  i = shift vector Gauge  Lie derivative g g  n n : g i n  0 1 Kn g g    L g ij i j 2 n ij 1   tg ij DD i  j  j  i  2  n t ; n   0 12 Line element i Time  t t i Space dx t + t

t n t

2 2 i i j j ds  dt  gij  dx   dt dx   dt 13 Structure of variables i 1  n,  , n   , 0 ; cf. t   1, 0  ij : lapse function,  :shift vector; i g ij  2  k ,  1/22 , i /  ggki,   ij i j 2 * ,g ij * ,g   /  k ,  0 ,0 Cf. g  ki, g    ij * ,g ij * ,g j j j KKij , ij  0 ,0 KK ,   ij * ,KKij  * , 14 Geometric meaning of Kij If space-like hyper-surface is curved, n is not parallel-transported. un0  n n n u

ii u Kij g j u  n i  0

Kij denotes the “curved” degree of chosen spatial hyper-surfaces 15 Next step is to rewrite

Einstein’s equation by gij & Kij  As a first step, it is necessary to define

associated with gij

Dig jk  0 Required property ijk T lmn : A spacetime tensor ijk h''''''' i j k l m n i j k Define DTTh lmng h g i ' g j ' g k ' g l g m g n h ' l ' m ' n ' Then, projection i'''' j k i jk'' Dig jk g i g j g k  i''' g j k  g i g j g k i'''''()g j k n j n k =gi''' g j g k n n  n  n  0 i j k j'''''' i k k i j OK 16 3+1 definition of Kij

 1   Kij g i g j  n    g i g j    n     n   2       ggi jn   g i  n i n g j   n   n   n   0   in j  n i n  n j

1  Kij    i n j   j n i  n   n i n j  2   1     gj  in  g i   j n  n   g ij  2 1    ggj  in  i  j nng ij  2 1   tg ij DD i  j  j  i  2 17 Usuful, often used relations

 K   n  n  n   n    n  n D ln  Note : n n  D  ln : acceleration

We will use it for many times

18 3+1 decomposition of Einstein’s equation

    G n n 8 T  n n : Hamiltonian constraint     G ngkk 8  T  n g : Momentum constraint     GTgi g j 8   g i g j : Evolution equation

• First two eqs = constraint eqs ・・ no second derivative of spatial metric • Last one = evolution eqs ・・ hyperbolic eqs of spatial metric • No time derivative for  &  k

19 Similar to Maxwell’s eqs

i ieE 4 i Constraint eqs i B 0 iii t E   B  4 j Evolution eqs i i t BE    

Step 1: Give an initial condition which satisfies constraints. Step 2: Solve evolution equations.

20 All eqs have to be written by (g, K) • Method: Derive Gauss-Codacci equations Start from definition of 3D Riemann tensor (3) l DDDDRDi j j i k  ijk  l :  l  3 vector Using defintion of 3D covariant derivative,

a b c l m DDijk g ijkabclm g g  g g    a b c a b m l gi g j g k  a  b  c  g i g j g k  a g b l m  a l c m gi g j g k  a g c  l  m  a b c a m c gi g j g k  a  b  c n  a  m g k K ij  K ik K j  c 21 where we used

b b b b b b cacaag   g  n n   ca n n  n ac  n  n  ca n b b b  na K c  n c D ln  n K ac  n c D a ln  c a b b  ggi j cg a  nK ij l m l m m a gjn l  m   g j  m  l n   m K j  n  a  0

Then, DiDDD j j ik gives (3)R l = ga g b g cRKKKK l   l  l ijk l i j k abc l l j ik l i jk Gauss (3) a b c d Codacci  RRKKKKijkl g i g j g k g l abcd il jk ik jl eq. ()N a b c d N 1 Note: RRijklg i g j g k g l abcdKKKK il jk ik jl 22 Gauss-Codacci equations I, continued (3) a b c d RRKKKijklg i g j g k g l abcd  il jk  ikK jl Multiplying g ik

(3) b d a c k Rjlg jg l R bd  R abcd n n  K jk K l  KK jl Multiplying g jl again This will be used later

(3)ac il 2 R R 2 Rad n n  K il K  K a c a c a c Here, R 2 Rad n n 2 Gabb n nn 16Tna (3) 2 il ac  RKKKil 16Tab n n Hamiltonian constraint 1 component 23 • Derive Gauss-Codacci equations II Start from

k abk c abk dc e Dij D ng ijcab g g  D n  g ijcabed g g  g g  n  a b k c a b k d c gijcab g g  nn  g ijc g g  ab g   d a b k c e gi g j g c  a g e  bn a b k c k d c b k e gi g j g c  a  bn  g c K ij n  d n  g j K i n e  b n a b k c k gi g j g c  a  bnD   ln  Kij d c c e where nd n  D ln , n e  b n  0

24 • Derive Gauss-Codacci equations II

k a b k c k Di D j ng i g j g c  a  b n   Dln   K ij (i,k)  i a b c i Di D j ng c g j  a  b n   Dln   K ij   D D nig a g b   n c  D i ln  K (j,k)  j i j c a b  ij i i a b c  Di D j n  D j D i n gg c j  a  b   b  a  n a b c d ggc jRn abd iii a b c d b d Dij n  K, D n  KRj, g c g j abdnn  R bd g j i b d b d  DKDKi j j  R bdgj n  8 T bd g j n Momentum constraint 3 components 25 Derive evolution equation Start from contracted Gauss-Codacci eq. I

(3) b dac i RRKKKKjlgg j l bd Rnabcd n   il j  jl a c 2  Rabcd n n contains tg ij or  tK ij d Let us calculate Rabcd n  a  b   b  a n c

a  bnK c   a  bc  nD b c ln 

 abK c Kab n a D b ln Dc ln

nDb a c ln

Remember: bnc   K bc  n bD c ln 26 d From a  b   b  a n c  R abc n d  a c b d a c b gj g lR abcd n n g j g l n  a K bc   b K ac 

DDDDj ln l ln  j l ln  a c b a b gj g ln  a K bc  g j K bl  a n a b b k  gjK bl K a  n a D ln    K kl K j a c b a c bb gj g ln b K ac  g j g lL n K ac  K ab  cn Kbc a n  k Ln KKKjl 2 jk l a c b d k 1  gj g lR abcd n n Ln K jlKKDD jk l  j l  27 Thus, the G-C equation becomes (3)bd  1 RRKDDjlg j g l bd L n jl   j l  k  2KKKKkl j jl Here, L 1 k k k niK j  tKDKKD ij   k ij  ik j   KD jki (3) k  tKRDDKK ij  ij  i j     ij  2K ik K j  k k k b d  DKKDKDk ij ik j   jk i   g ig j Rbd

b d b d g ij Note: gi g jRTT bd  8  g ig j bd  2  Evolution equations = 6 components 28 Summary of 3+ 1 formalism

(3)ij 2  R Kij K  K 16 T n n i  Constraints Di K j D j K  8g T n j (3) l KRKKKKDDij ij  ij 2 ilj  ij l l l KDKDDKil j  jl i    l ij Evolution

 1     8T gi g j  ( g  n n ) g ij 2 gij 2 KDD ij  i  j  j  i i ,  Gauge condition 29 Constrained system • ADM equations seem to have too many components: Constraints seem to be

redundant equations, because gij & Kij are determined by solving evolution equations

• Constraints are guaranteed to be satisfied if evolution equations are solved correctly  No inconsistency

30 Evolution of constraints

AGT 8  i i i i H0 n n   Hig  n   H i g  n   H ij g  g 

HH0 0, i 0 H & M Constraints

Hij  0 Evolution eqs.  A 0  l i i ij t   l H00   KH 2 H i D    D i H   H ij K l kk t   l H i   H0 D i    KH i  H k  ,i DH k i  If constrains are zero at t = 0 and evolution equations are satisfied for any t, constraints are always satisfied. 31 Nature of standard 3+1 formalism • Evolution equations are wave equations of 6 components, but it is not simple one: Many additional terms even in linear order 1 K ~  g ij2 ij 1 (3) kl Rij~ g ij  g giklj,, g jkli ij g kl  2 

 ggij   ij  cf . Maxwell's equation   AA        0 32 Linearized equations Linearized Einstein equations with  =1 & i  0 gij ijhh ij; | ij | 1

Evolution eq. : hij  h ij  h ik,,, kj  h jk ki  h kk ij

Constraint H : hhii  ik, ki  0

M : hhij, i  ii, j  0

This causes a problem 33 Stability analysis Decomposition: TT hij A ij  C,(,) ij  2 B i j  h ij TT AC, : scalar , Bi : vector , hij :tensor TTTT definition: Bii,, 0, hij j  h ii  0

 Trace hii  3 A   C ,

Divergence hAij, j , i  ()CB,ii  substitute  hij   h ij  h ik,,, kj  h jk ki  h kk ij

hii  h ik,,, ki  0, h ij i  h ii j  0 34 3+1 Equations

H: A 0 Constraints :  M: t  2AB, i   i   0 TTTT Wave hhij ij  equation B(,)ij  0 Evolution eqs. :  AA  Strange CA forms

35 Solutions I TT 1 Equations for hij =Wave equations  True degree of GWs: No problem 2 Constraint (H) : A  0 & Evolution equation for A = wave eq.  Violated constraint will propagate away. i 3 Constraint (M) :  2A,ii   B  F x  i Numerical integration For A 0, BiB F x  i of zero is zero i Evolution equation for Bi gives B i F B  x  0 & BxF xi t F i Perhaps no problem iB B    i   36 Solutions II C is not constrainted by constraints, but determined by CAAA ,   If constraint is violated and A  0 initially,

flm ()() r t  g r  t AY  lm lm, r f()() r t  g r  t CYC  Ct   lm r 1 2 Constraint violation is serious in this case

Small error in A results in serious error in C 37 To summarize • In the original 3+1 (N+1) formalism, if constraints are violated even slightly, the error increases with time even in a nearly flat spacetime with no limit • Namely, it is unsuitable for numerical relativity • Source:

hhij  ij  hhik,,, kj jk kih kk ij

First, realized by T. Nakamura (1987) 38 Section II: BSSN formalism Essence • Need reformulate of 3+1 formalism • At least, in the linear level, constraint violation mode must not appear Define new variables Fh  i ij, j Evolution eqs.  hii for Fi and  ? and rewrite as

hijh ij Fi,, j F j, i   ij 39 Reformulation using constraint equations

Momentum constraint: hhij,, j jji 0

 Fii ,  0 : Evolution eq for Fi

Trace of hhij ij FFi,,, j j i ij

   2   2Fii,

Hamiltonian constraint: Fii,  0

  0    0  Fi  0

 hhij ij No problem !! 40 Reformulation increasing the variables and using constraints appears to be robust

• Similar definition of new variables

Fi &  is possible in the non- linear case

First, derived by T. Nakamura (1987) Subsequently modified by Shibata (1995), Baumgarate and Shapiro (1998)

41 Original BSSN formalism (Shibata-Nakamura 1995) First of all, write the line element

2 2i 2 i 4 i j ds   i   dt 2g i dx dt  e ij dx dx

Here, detg ij   1. ( corresponds to  .)

As conjugates for gij and , define

4 1 ij Aij= e K ij g ij K and KKK trace ij g ij 3 Up to here, we increase 2 variables (, K) and two constraints, detgg 1 and A 0  ij ij ij 42 Then, the equations are 2 ( l  ) g   2 A  g  l  g  l  g  l t l ij ij il , j jl , i3 ij , l

ll1 ()t   l    K   , l  6

l 4411    ()t   lA ij   e R ij  g ij R   e  D i D j   g ij    33    2 +KAAAAA 2 l    l    l   l A  ij il j il j jl i 3 ,l ij 1  8eT4 g  g  g  g Used  i j ij 3 H-constraint

l ij 1 2    (t   l )K   A ij A  K     4  T  n n  g  3 Not sufficient in this stage ! 43 Note • The linear analysis for the simple conformally-decomposed formalism shows “System is even more unstable” • An exponentially growing mode appear

hhij ij  hhik,, kj jk ki TT hij  C,ij  Bij,   h ij  CC 

44 Linear analysis shows that the problems come from Rij  RRRRij ij ij: ij is Ricci tensor of g ij

 k RDDDDDDij  2 i j  2 g ij k   4 i  j  k  4gijDD k   scalar part 1 kl Rij g gij,, kl  gik jl g jk, il  2  kl l k Already g ,k l, ij  jk  il rewritten

See appendix by  45 Write formally gij ijf ij kl g gijkl,,,  g ikjl  g jkil  kl Linear  flat,g ijg ik jl g jk ,il  kl nonlinear f gij,, kl  g ik jl  g jk , il  kl  Define Fi  g ik, l as in the linear case kl  flatgij   g ik , jl  g jk , il 

=flatg ij FF i , j  j , i 

46 Next step: Derive equations for the new variables using constraints

• As in the linear case, the equation for Fi should be derived from momentum constraint Momentum constraint: 2 D e6 Ai  e 6  D K8 J e 6  i j 3 j j

ii2 or DADDADKJi j 6  i   i  j   j  8  j     3

l l l2 l Here, 2Aij  (  t    l ) g ij  g il  , j  g jl  , i  g ij  , l 3 47 ik Thus, a term Digg t jk  appears and ik ik l l Dig t g jk  g  i  t g jk   ij  t g kl   ik t g jl  ik ik l l t Fj f i  tg jk  g  ij  t g kl ik  t g jl 

Namely, momentum constraint can be regarded as the evolution equation for Fi

Other terms with tg jk are rewritten using 2 ( l  ) g   2 A  g  l  g  l  g  l t l ij ij il , j jl , i3 ij , l 48 Equation for Fi

k t   k F i

jk jk12 jk k  2f Aij, k  g , k A ij  A g jk , i  6  , k A i  K , i 23 jk jl k  2 ,kA ij   , l g ij , k

k k2 k jl gik  , j  g jk  , i  g ij  , k   16  J i 3 ,l

Note no nonlinear term of hij , A ij ; h ijg ij ij

49 Summary of BSSN formalism • Definition of 3 additional variables

(5 components) (Fi, K, ) is essential. • Conformal transformation is not essential at all: Only with conformal transformation, the resulting formalism does not work  “Conformal formalism” is misunderstanding (stupid) naming. • The increase of variables results in the increase of new constraints: Now, 17 equations with 9 constraint equations. 50 Alternative (Baumgarte-Shapiro ,1998)

i ij jk Define = g , j instead of F i   g ij,k .

(In the linear level, both reduce to hij, j .)

k i i jk2 ij ij t  k  26   jkAKA  g,, j   j 3 2 1  j i   i j  g jk i  gik j ,jj3 , , jk 3 , jk ij  16g J j

1 kl k k Slightly simpler Rij g g ij, kl  g ik  , j  g jk  , i  2

1 kl kl k l k g ,j g ik , l  g jk , l g , i  g ij , k    ik  jl 2 51 Nine components of constraints Hamiltonian constraint (1) (3)ij 2  R Kij K  K  16 T n n Momentum constraint (3) i  Di K j D j K   8g T n j

Tracefree condition for Aij (1) ij Aijg  0

Determinant=1 for g ij (1)

det g ij  1 Auxiliary variable (3) i ij Figg ij, j or    , j 52 Puncture-BSSN (Campanelli et al. 2005) Define   eW42 (or  e ) instead of  to follow a spacetime. Schwarzschild spacetime in the isotropic coordinates: 24 221 M / 2 r 22  M  2 ds   dt  1    dx  dy  dz  1 M / 2r   2r 

M r0   ln 1    2r  4 M Define  1  : regular everywhere  2r   BH spacetime can be numerically followed with no special technique 53 New conformal factor  e42 or W e

ll1 ()t   l    K   , l  6

llW  ()t   lWK     , l  3 No divergence 4   Rij 4 k  2e  Di D j  g ij   2 D i  D j   2 g ij D k  D  4  k   Rij WD i D j Wg ij W W2 D k WDW No irregular term even for BH spacetime 54 Other prescriptions: Special gauge Linearized Einstein equations with  1 & i 0 gij  ij h ij; | h ij | 1,   1  a / 2; | a | 1

Evolution eq. : hij h ij  h ik,,, kj  hjk ki  h kk ij

a,,ij i j   j,i

Constraint H : hhii ik, ki  0

M : hhij,,i  iij 0

55 Harmonic gauge: a  i  0   ,i  g 0   ih ij, j 0

hik,,,,,,,, kj  h jk ki  h kk ij  a ij  i j  j i  h kk ij  a ij  hij  h ij  h kk,, ij  a ij   Constraint (H) a hij, ji  a   h kk 2  th kk  a   h kk  a Only wave equations appear  No problem

This shows an evidence why Pretorius formulation works56 Extension to N+1 case (N > 3)

• ADM equations are unchanged • BSSN formalism is slightly modified because the dimension is different (Yoshino-Shibata 09) • In the following, spacetime dimension is denoted by D = N+1

57 2 2k 2 k 1 i j For ds   k  dt  2  k dx dt   g ij dx dx 2 ( l  ) g   2 A  g  l  g  l  g  l t l ij ij il , j jl , iD 1 ij , l

ll2 ()t   l   K   , l  D 1

l 11    ()t   lARRDD ij   ij  g ij     i j   g ij    DD11    2 +KAAAAAA 2 l    l    l   l  ij il j il j jl iD 1 , l ij

 1   8T gi g j g g ij D 1 2 l ij K 8    (t   l )K   A ij A      T  D  3 n n  g  DD12 58 l i i jkDD21 ij ij t   l  2   jkAKA  g,, j   j D 12 23D   j i   i  j  g jk  i  g ik  j ,jDD11 , j , jk , jk ij  16g J j  RRRij ij ij

 D  31k D  3 RDDij i i  gijDDDD k  2  i  j  22 4

D 1 k  2 gijDD k   4

Robust for any dimension (at least up to 7D,

Shibata & Yoshino, „10) 59 Section III: Gauge conditions

Any time slice and any time axis can be chosen in numerical relativity Normal direction Time Shift axis

Lapse 60 Slice: Required properties － Black hole singularities and coordinate singularities have to be avoided

t Singularity

Or, effectively excised

61 Well-known good slice Maximal slice:  KKSKT0      g ij , ij , 

Estbrook et al. PRD 1973

Drawback: Solving elliptic-type equation is computationally expensive 62 Spatial gauge; required property

Frame dragging coordinates distort Coordinate shear has to be suppressed.

63 Minimal distortion gauge (Smarr-York, PRD1978) : Minimize global distortion defined as I dV ( g )(  g ) gik gjl g 1/ 2  t ij t kl

I k 1/ 2 i  0 D gg t ik   0  1  g k D DRk   k jk3 j k jk

1/ 2 k l l k2 kl m +gjlDDDDS k ln g    g m   j 3 Very physical & beautiful.

But, elliptic eqs. & expensive 64 Puncture gauge; =0 at puncture

l tl        K;

 2 for 4D & 1< 2 for N-D D 1 kl gk V2  F   t  F  t2D  2  l t l or

l kD 132 kDV4, 1 k tl   VBB  2D  2 4 l k l k k t   l BB   t   l   22 VV 1 for 4D and 1 is preferable for N-D  ~1/M (Alcubierre, Bruegman 03, and others)  BH 65 Properties • Slice: Asymptotically approaches to maximal slice-like slice;

slice freezes (,t = 0, and thus, K  const) • Shift: Similar to minimal distortion gauge !! • Computational costs are very small • Horizon sucking does not matter because it happens inside horizon and hyperbolic gauge is suitable for this “effective excision” (Bruegmann et al., 2006)

66 Section IV: Initial value problem; How to impose constraints

(3)ij 2  R Kij K  K 16 T n n  16  H i  Di K j D j K  88 T n g j   J j Only 4 components equations for 12 component of g & K

These are not hyperbolic equations  Write in elliptic equations (Method of O‟Murchadha-York, 1973) 67 Hamiltonian constraint

4 gij  g ij

(3) 18  RR 4    See Appendix  (3) ij 2 RKKK ij16 H 5 5 ij 2   RKK 2 H   ij  K  88 Elliptic equation for a given set of

gij , Ki j , H 

68 Momentum constraint 1 g 4 g, KAK   g ij ij ij ij3 ij jj66 2 DKDKDADKDj i i  j  i  i  i g jk  0 3 Then, set 2 g6ADWDWDWj j  j  j k  K TT j i i i3 i k i i TT j TT i D KK i  0  i  12  WDDW jj  RWD  6 K  8J 6 i3 i j i j 3 ii

Vector elliptic equation of Wi

for a given set of gij , , KJ , j  69 Section V: Numerical implementation • Discretize all variables;

g (x, y, z)  g i, j, k (i, j, k = 1—n) • And then, perform finite difference; e.g., 2nd order finite difference 1 xg i, j , k  g i 1, j , k  g i 1, j , k  2 4th order finite difference 1  xijkg,,   g ijk 2,,  g ijk  2,, 8 g ijk  1,,  g ijk  1,,  12  70 Basic equations i SAg gij,,, ij   gg     ij ij i     SK ,,,        SAKg,,,,,, ii   AA    A ij ij ij   ij   KK    SAKg,,,,   k K ij ij tk        iii i     SAK gij,,,,,  ij  ,           SK  ,  ii     SBi ii      BB     ii SAKBg ij,,,,,,  ij    71 Method for space finite difference • Metric = smooth, no shock (different from fluid)  Every variable can be expand as the Taylor series

234   g x    g   g'  g ''  g '''  g '''' ij0 ij ij2! ij 3! ij 4! ij 42 ' '' th gij x0 22   g ij   g ij  g ij 4 order 2! scheme 834 16 gg''' '''' 3!ij 4! ij

4 derivatives  5 values 72 Need 1st and 2nd derivatives 1  xg ab   g ab, j 2, k , l  g ab , j 2, k , l  12   8 gg  ab, j 1, k , l ab , j 1. k . l  1 2  xg ab   g ab, j 2, k , l  g ab , j 2, k , l  122    16gab,1,, j k l  g ab ,1,, j k l  30 g ab ,,, j k l 

Write all the derivatives using these rules: th Currently, 4 -order is most popular 73 Einstein’s equation is complicated

2 1  xyab g    g abijk,2,2,   g abijk ,2,2,   8 g abijk ,1,2,    g abijk ,1,2,    12 

+gabi,2,2, jk   g abi ,2,2,  jk   8 g abijk ,1,2,    g abijk ,1,2,   

+88gabi,2,1, jk   g abi ,2,1,  jk   g abijk ,1,1,    g abijk ,1,1,   

 8 gabi,2,1, jk   g abi ,2,1,  jk   8 g abijk ,1,1,    g abijk ,1,1,   ]

Many points are used

74 Advection term: Upwind scheme I Consider the simplest wave equation:

txf  c  f  0  f  F x  ct Simple Finite difference

n1 nct n n fj f j  f j11  f j  x  unstable !! j von Neumann stability analysis n n n ujjexp ikx  exp ikj  x Cf. Numerical ct Recipe    1  isin k x  1 (Press et al.) x 75 Advection term: Upwind scheme II

Simplest stable method = First-order upwind scheme

 nct n n  fj f j  f j1  c  0 n1  x f j    nct n n fj f j1  f j  c  0  x von Neumann stability analysis   1

j j

c>0 c<760 Advection term: Upwind scheme III l th Advection term: gl ij 4 -order scheme  Need to use upwind scheme For  l 0, 1 g'   g x 3   6 g x  2   18 g  x   ij12  ij0 ij 0 ij 0   10ggij xx00  3 ij    For  l 0, 1 g'   g x 3 6 g x  2 18 g  x  ij12  ij0 ij 0 ij 0  10ggxx  3    ij 00 ij   77 Time evolution: Standard one • Use Runge-Kutta method • Simple 2nd order method  Unstable • 3rd or 4th order (popular)  Stable

tQFQ a a   k10 tF Qa  t   k2  tF  Qa  t 0  k 1 /2   k3  tF Qa  t 0  k 2 /2   k4  tF Qa  t 0  k 3

k122 k 2  k 3  k 4 5 Qaa t00  t  Q t   O  t  6 78 Boundary conditions

• obeys wave equations Outgoing boundary condition

Fa  t r/ c 2 Qa  O1/ r  r  : characteristic angular frequency Usually, only the leading order is taken into account. Location of boundary should be r 1

79 Numerical implementation of outgoing boundary condition t r t Determine by interpolation r

rQaa F t r/ c

 rQa  t r / c

r   r Qa  t   t  ( r   r ) / c 80 Section: Extracting gravitational waves by complex Weyl scalar Using a complex Weyl scalar   g  4  Rg n m n m

Rg : 4D Riemann tensor n, l  , m  , m   : Null tetrad 1 r ,    h ih 4 2    h  ih   2 dt ' dt ''   t ''  tt' 4 81 Weyl scalar for gravitational waves I

n, l  , m  , m   : Null tetrad      n n l l   m m  0,  n l   1  m m 

g  n  l   l n   m  m   m  m  1 n : Ingoing Null  n  r   2 m : Complex orthnormal to n  & l  n : Unit timelike normal as before  r : Unit radial vector, r n  0

82 Weyl scalar for gravitational waves II 1   (2(4)R n m  n g m   (4) R n  m  r g m  4 2 g g (4)   g   Rg r m r m ) (4)   g  k i j i j Rg n m n mRKKKKEij i jk ij m m ij m m (4)   g  i j k i j k Rnmrmg DKDKj ik k ij  mrmBijk mrm (4)   g  i j k l Rg r m r m  Rijkl K ikK jl  Kil K jk  r m rm cf, derivation of Gauss-Codacci eqs. Note in 3D, R RRRRRijklg ik jl  g il jk  g jk il  g jl ik  g ik g jl  g il g jk  2 83 Weyl scalar for gravitational waves III

RijklRKKKK ijkl  ik jl  il jk l RikRKKKKE ik  ik  il k  ik , 2 ji ERKKK ij 0 (vacuum, Hamiltonian constr.) E  Rijkl g ikE jl  g ilEEE jk  g jk il  g jl ik g ik g jl gil g jk  2 i j k l j l & then Rijklr m r m E jl m m j l j l j l where gjlm m 0, g jl r m  0, g jl r r  1 Finally, i k i j k 4  Eik m m  B ijk m r m  Quite simple 84 Section: Adaptive Mesh Refinement (AMR)

L   l ~ 4GM/c2 Why is it required ? • More than 2 different length scales • For binary, gravitational wavelength and radius of compact star  Resolve stars while extracting waves 85 Typical length scales Gm Radius of black hole R  c2 Gm Radius of R  5~8 c2 GM Angular velocity of binary   , M  m  m r3 12 c Gravitational wavelength    3/ 2 r3  r   GM    cR10022    GM10 GM / c  c  GM r is orbital radius  ~5 2 c 86 AMR grid in numerical relativity • Prepare N=5—10 levels of different resolutions and domain size:

Domain Dl, Size Ll, Grid spacing l where l=1—N

• For the simple case, we set 2Ll= Ll-1 , 2l=l-1 (same grid number for l=1—N) 2 and L1 > , N << Gm/c N-1 N-2 Ratio=2:1 N

87 AMR time step • For the finer grid spacing, the time step is

smaller: 2tl=tl-1

7 5 6 Refinement boundary 4 2 1 3

For each evolution, th 4 -order Runge-Kutta method is used88 Key: Interpolation and treatment of buffer zone • Grid in each domain is composed of main and buffer zones

c

Buffer Main Buffer

Main : Solve equations with no prescription Buffer: Interpolation is necessary 89 A method for interpolation: Example I • Before the first Runge-Kutta step  Quantity in the buffer zone is determined by Lagrange interpolation, e.g., 5th-order interpolation (using 6 points)

c Main

3 For 3-dimension, 6 =216 grid points are used90 A method for interpolation II • The Runge-Kutta evolution step • Inner 3 buffer points are solved in the same manner as in main region • 4th one: The same as above but for advection term; mixture of 2nd and 4th order upwind schemes • 5th and 6th one: Interpolation in space and time: Time = 2nd-order interpolation

c

Bruegmann et al., Yamamoto et al., PRD2008 91 AMR time interpolation • Concept of time interpolation

3 1 2

Runge-Kutta middle steps

92 Moving domain • Small-size domains (finer domains) which cover compact objects move with them • Large-size domains (coarser domains) do not move (fixed grid) • Finer domains are moved when the center of black hole or neutron star moves dxi For black holes, solve  i dt k tk     0 at black hole center,  =0    in the puncture gauge For neutron stars, find the location of max density93 Section: Finding a black hole

• Popular method = Finding apparent horizon • Apparent horizon = surface for which expansion of outgoing null ray is zero • Important property (Hawking-Ellis, 1973): If an apparent horizon exists for a globally hyperbolic spacetime, horizon of a black hole always exists  Apparent horizon is formed = A black hole is formed

94 Basic equation for AH I  • Define outgoing null vector r l n  r  n : Time like unit normal  r: Radial unit normal, r n  0. r  is perpendicular to AH sphere. ab , : Unit normal on sphere

 g  n n   r  r   a  a   b  b

95 Basic equation for AH II • Expansion = change rate of area r      a a  b b  l    g r r  n   r   k i j  K  Dk r  r r K ij Outside AH Apparent horizon:  0 r k i j  Dk r  r r K ij  K  0

Inside AH96 Basic equation for AH III • AH=two sphere, r=h(, )

ri  C1,   h ,   h i C is determined by rri  1

kk1  Dkk r    g r  g

221 C  h  cot   h 2   h  .... sin     0 2D elliptic type equation

97 Basic equation for AH IV • Boundary condition , 0  , 

 Periodic

     0 0 0  2 

Well defined  Standard solver is OK 98 Appendix: Conformal decomposition I 4 gij  g ij, DD i g jk  0, i g jk  0

i1 il jk g  jkl g   kjl g   ljk g  2

1 il g j g kl   k g jl   l g jk  2

2 il + g gkl j   g jl  kg  jk  l  

i 2 i i i = jk  kDDD j   j k  g jk    =i Ci jk jk 99 Conformal decomposition II i i i l i l R jk ijk  jik  jkil   jlik i i i l i l i jk  jik  jkil   jlik i i i l i l  iCCCCCC jk   jik  jkil  jlik i l i l i l i l  jkCCCC il   jl ik  jk  il  jlik  i i i l il R jk DCDCi jk j ik CCC jk il  jlC ik  i i i l i l RDCDCCCCCjk ijk  jik  jkil  jlik

1 l 2  6DDDDj k  2 g jk l     2 DDj k  g jk    100  Methods for code validation

• Confirm to reproduce exact solutions: -- Black hole solutions (check area etc) -- Propagation of linear gravitational waves • Check violation of constraints is small: Monitor Hamiltonian and momentum constraints • Check convergence: If 4th-order scheme is used, quantities th 4 converges at 4 order; Q = Q0 + Q4 x

101 Some technical issues I • Imposing some constraints within O(4):

detggij  1 and A ij ij 0 1/3 g det g g  ij ij ij  1/3 1 AAAdetgg Tr  ij ij ij ij ij    3 1/ 6 WW det g   ij   KKATr   ij 

g ij , K ij are unchanged 102 Some technical issues II • Kreiss-Oliger-type dissipation > O(4):

Q Q   x6 Q (6)  is a constant of O 0.1 Q(6) is the sum of sixth derivatives

Purpose is to suppress high-frequency numerical noise

103