Introduction to 3+1 and BSSN Formalism

Introduction to 3+1 and BSSN formalism

Yuichiro Sekiguchi (YITP)

1 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

Solving the Setting ‘realistic’ or constraint ‘physically motivated’ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein’s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder )

BH excision

2 APCTP International school on NR and GW July 28-August 3, 2011 Scope of this lecture

BH excision

3 APCTP International school on NR and GW July 28-August 3, 2011 The Goal

 The main goal that we are aiming at:

“To Derive Einstein’s equations in BSSN formalism”

4 APCTP International school on NR and GW July 28-August 3, 2011 Notation and Convention

 The signature of the metric: ( - + + + )  (We will use the abstract index notation)

 e.g. Wald (1984); see also Penrose, R. and Rindler, W. Spinors and spacetime vol.1, Cambridge Univ. Press (1987)  Geometrical unit c=G=1  symmetric and anti-symmetric notations 1 1 T  T , T  sgn( )T (a1...an )  a ()...a ( n ) [a1...an ]  a ()...a ( n ) n!  n!  1 1 T  (T +T ), T  (T -T ) (ab) 2 ab ba [ab] 2 ab ba

5 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

BH excision

6 APCTP International school on NR and GW July 28-August 3, 2011 Solving Einstein’s equations on computers

 Einstein’s equations in full covariant form are a set of coupled partial differential equations  The solution, metric gab, is not a dynamical object and represents the full geometry of the spacetime just as the metric of a two-sphere does  To reveal the dynamical nature of Einstein’s equations, we must break the 4D covariance and exploit the special nature of time  One method is 3+1 decomposition in which spacetime manifold and its geometry (graviational fields) are divided into a sequence of ‘instants’ of time  Then, Einstein’s equations are posed as a Cauchy problem which can be solved numerically on computers

7 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of spacetime manifold

 Let us start to introduce foliation or slicing in the spacetime manifold M  Foliation {S } of M is a family of slices (spacelike hypersurfaces) which do not intersect each other and fill the whole of M  In a globally hyperbolic spacetime, each S is a Cauchy surface which is parameterized by a global time function, t , as St a  at  Foliation is characterized by the gradient one-form

a  at , [ab]  0

8 APCTP International school on NR and GW July 28-August 3, 2011 The lapse function

 The norm of a is related to a function called “lapse function”, a(xa) , as : ab ab 1 g ab  g atbt  - a 2

 As we shall see later, the lapse function characterize the proper time between the slices  Also let us introduce the normalized one-form : ab na  -a a , g nanb  -1

 the negative sign is introduced so that the direction of n corresponds to the direction to which t increases  na is the unit normal vector to S

9 APCTP International school on NR and GW July 28-August 3, 2011 The spatial metric of S : gab

 The spatial metric gab induced by gab onto S is defined by g  g + n n ab ab a b g ab  g acg bdg  g ab + nanb cd  Using this ‘induced’ metric, a tensor on M is decomposed into two parts: components tangent and normal to S  The tangent-projection operator is defined as a   a + nan b b b a b a a  The normal-projection operator is b  -n na  b - b  Then, projection of a tensor into S is defined by

a1...ar a1 ar b1 bs c1...cr  T b ...b  ...   ...  T d ...d 1 s c1 cr d1 ds 1 s

 It is easy to check c d  gab a b gcd  g ab

10 APCTP International school on NR and GW July 28-August 3, 2011 Covariant derivative associated with gab

 Covariant derivative acting on spatial tensors is defined by

a1...ar a1...ar DeT b1...bs   eT b1...bs

f a1 ar b1 bs c1...cr    ...   ...   T d ...d e c1 cr d1 ds f 1 s  The covariant derivative must satisfy the following conditions  It is a linear operator : (obviously holds from linearity of ∇)  Torsion free : DaDb f = DbDa f , (easy to check by direct calculation)  Compatible with the metric : Dc gab=0 , (easy to check also)  Leibnitz’s rule holds : c b c b d c b d c Da (v wc )  a b (v wc )  a v  d bwc + a wdc bv b d c c b d d c  a v (d +d )bwc + a wd (c +c )bv c d b c d d c  v Da wc + wd Dav + a (d v bwc + c wd bv ) c d c d d  v Da wc + wd Dav for d v  c wd  0

11 APCTP International school on NR and GW July 28-August 3, 2011 Intrinsic and extrinsic curvature for S  The Riemann tensor for the slice S is defined by d (Da Db - DbDa ) vc  vd R abc  An other curvature tensor, the extrinsic curvature for S is defined by K  -   n , na : unit normal to S ab (a b)  extrinsic curvature provides information on how much the normal direction changes and hence, how S is curved parallel  the antisymmetric part vanish transported b  n  -K  x due to Frobenius’s theorem : a ab a “For unit normal n to a slice a b n (x ) a b b n [ a  b n c ]  0 ” n (x + x )

b a x 0   3(n n[abnc] )  -  [bnc] b b S x + x 12 APCTP International school on NR and GW July 28-August 3, 2011 The other expressions of Kab

 First, note that c d c c d d anb  ab cnd  (a +a )(b +b )cnd

  anb - na  ab  - Kab - naab  where ab is the acceleration of nb which is purely spatial b a b a b n ab  n n anb  n a (nbn )  0  Because the extrinsic curvature is symmetric, we have 1 1 K  - n - n a  - L (g + n n )  - L g ab (a b) (a b) 2 n ab a b 2 n ab

 where L n is the Lie derivative with respect to n 1  Also, we simply have K  -   n  -  L g ab (a b) 2 n ab  Thus the extrinsic curvature is related to the “velocity” of the spatial metric gab

13 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of 4D Riemann tensor

 Geometry of a slice S is described by gab and Kab  gab and Kab represent the “instantaneous” gravitational fields in S  In order that the foliation {S} to “fits” the spacetime manifold, gab and Kab must satisfy certain conditions known as Gauss, Codazzi, and Ricci relations  They are related to 3+1 decomposition g 4 of Einstein’s equations ab Rabcd  These equations are obtained by K Rabcd ab taking the projections of the g ab 4D Riemann tensor 4 4 d  Rabcd ,  Rabcdn , 4 b d  Rabcdn n

14 APCTP International school on NR and GW July 28-August 3, 2011 Gauss relation: spatial projection to S

 Let us calculate the spatial Riemann tensor D D w    (  w ) a b c a b c d d   abwc +  (bwd )(a c ) +  (d wc )(a b ) d d d   abwc -  KacKb w -  Kabn d wc  where we used (also note that n vanishes if n is uncontracted) d d d d d d a b  na (bn ) + n (anb )  -nb (Ka + naa ) - n (Kab + naab )  Then we obtain the Gauss relation d 4 d d d (Da Db - Db Da )wc  Rabc wd   Rabc wd - KacKb wd + KbcKa wd 4  Rabcd  Rabcd + KacKbd - KadKbc  The contracted Gauss relations are 4 4 b d b  Rac+ Rabcdn n  Rac + KKac - KabKc 4 4 a b 2 ab R +2 Rabn n  R + K - KabK

15 APCTP International school on NR and GW July 28-August 3, 2011 Codazzi relation : mixed projection to S and n

 Next, let us consider the “mixed” projection 4 d  Rabc nd   (abnc -banc )  where the right hand side is calculated as

 abnc   a (-Kbc - nbac )  -Da Kbc-  acanb  -Da Kbc + ac Kab  Then we obtain the Codazzi relation 4 d  Rabc nd  DbKac - Da Kbc  The contracted Codazzi relation is 4 b b  Rabn  Da K - DbKa

16 APCTP International school on NR and GW July 28-August 3, 2011 Gauss and Codazzi relations

 Note that the Gauss and Codazzi relations depend only on the spatial metric gab, the extrinsic curvature Kab, and their spatial derivatives  This implies that the Gauss-Codazzi relations represent integrability conditions that gab and Kab must satisfy for any slice to be embedded in the spacetime manifold  The Gauss-Codazzi relations are directly associated with the constraint equations of Einstein’s equation

17 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (1)

 Let us start from the following equation 4 b d b  Rabcdn n   n (abnc -banc ) b   n [-a (Kbc + nbac ) + b (Kac + naac )] b b b   [Kbcan + aac + n b Kac + acaa + nan bac ] b b b b   [Kbc(-Ka - naa ) + aac + n b Kac + acaa + nan bac ] b b  - KbcKa + Daac +  n b Kac + acaa  The Lie derivative of Kac is  L K   (nb K + K  nb + K  nb )   nb K - K K b - K K b n ac b ac ab c cb a b ac ab c cb a  Then we obtain the Ricci relation 4 b d b  Rabcdn n   Ln Kac + KabKc + Daac + acaa

18 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (2)

 Note that the Lie derivative of Kab is purely spatial, as a a c a c a c a c n Ln Kab  n n c Kab + n Kacbn + n Kbcan  -Kaba + Kbca  0

 Thus the Ricci relation is 4 b d b  Rabcdn n  Ln Kac + KabKc + Daac + acaa

19 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (3)  The acceleration ab is related to the lapse function a, as c c c c a  n  n  2n  n  -2n  a   -n (  a -   a) b c b [c b] [c b] b c c b  ncn  ln a + c ln a  D lna b c b c b  where we have used the fact that  is closed one-form  Then the Ricci relation can be written as 4 b d b  Rabcdn n  Ln Kac + KabKc + Da Dc ln a + Da ln a Da ln a 1  L K + K K b + D D a n ac ab c a a c  Furthermore, using the contracted Gauss relation 4 4 b d b  Rac+ Rabcdn n  Rac + KKac - KabKc  we obtain 1 4R  - L K - D D a + R + KK - 2K K b ac n ac a a c ac ac ab c 20 APCTP International school on NR and GW July 28-August 3, 2011 a “Evolution vector” and a n

 What is the natural “evolution” vector ?  As stated before, the foliation is characterized by the closed one- form  a a  Dual vectors t to  will be the evolution vector : at =1 a a  One simple candidate is t =an

 Note that na is not the natural evolution vector because a c a c a a c c a a a a Ln b  n c b - b cn + c bn  n c (n nb ) + Kb - (Kb + nba )) a  n ab  0 a  This means that the Lie derivative with respect to n of a tensor tangent to S is NOT a tensor tangent to S a  On the other hand, L a n  b  0 and any tensor field tangent to S is Lie transported by ana to a tensor field tangent to S

21 APCTP International school on NR and GW July 28-August 3, 2011 The shift vector

 We have a degree of freedom to add any spatial vector, a a a called “shift vector”, b to an because ab = 0 a a a  Therefore the general evolution vector is : t =an +b  This freedom in the definition of the evolution time vector stems from the general covariance of Einstein’s equations

 It is convenient to rewrite the Ricci relation in terms of the Lie derivative of the evolution time vector, as

4 1 1 b  Rac  - (Lt - Lb )Kac - Da Dca + Rac + KKac - 2KabKc a a  where we have used c c c Lt Kab  LanKab + Lb Kab an c Kab + Kacb (an ) + Kbca (an ) + Lb Kab aLnKab + Lb Kab

22 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of Einstein’s equations (1) - Decomposition of Tab

 Now we proceed 3+1 decomposition of Einstein’s equations 1 G 4R - 4Rg  8T ab ab 2 ab ab using the Gauss, Codazzi, and Ricci relations  To do it, let us decompose the stress-energy tensor as

Tab  Enanb + 2P(anb) + Sab b  where E  n a n b T ab , P a  -  ( n T ab ), and S ab   T ab are the energy density, momentum density/momentum flux, and stress tensor of the source field measured by the Eulerian observer  the trace is T  S - E  We shall also use Einstein’s equations in the form of

4  1  Rab  8Tab - gabT   2 

23 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of Einstein’s equations (2) - Hamiltonian constraint

 We first project Einstein’s equation into the direction perpendicular to S to obtain 1 4R nanb + 4R  8E ab 2  For the left-hand-side, we use the contracted Gauss relation 4 4 a b 2 ab R +2 Rabn n  R + K - KabK  We finally obtain the Hamiltonian constraint 2 ab R + K - KabK 16E  This is a single elliptic equation which must be satisfied everywhere on the slice

24 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of Einstein’s equations (3) - Momentum constraint

 Similary, “mixed” projection of Einstein’s equations gives 4 b  Rabn  -8 Pa  Using the contracted Codazzi relation 4 b b  Rabn  Da K - DbKa  We reach the momentum constraint b DbKa - Da K  8 Pa  includes 3 elliptic equations

25 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of Einstein’s equations (4) - Evolution equations

 The evolution part of Einstein’s equations is given by the full projection onto S of Einstein’s equations :

4  1 1   1   Rab   8Sab + 2n(a Pb) - g ab(S - E) + nanb (S + E)  8Sab - g ab(S - E)  2 2   2   Using a version of the Ricci relation 1 1 4R  - (L - L )K - D D a + R + KK - 2K K b ac a t b ac a a c ac ac ab c  We obtain the evolution equation for Kab

c  1  (Lt - Lb )Kab  -Da Dba +aRab + KKab - 2KacKb ]-8a Sab - g ab(S - E)  2   The evolution equation for gab is given by an expression of Kab

(Lt -Lb )g ab  -2a Kab

26 APCTP International school on NR and GW July 28-August 3, 2011 Summary of 3+1 decomposition

 Einstein’s equations are 3+1 decomposed as follows Gauss rel. Hamiltonian constraint a b a b 2 ab G n n  8 T n n R + K - K K 16E ab ab ab Momentum constraint b b Codazzi rel. b

 Gabn  8 Tabn D K - D K  8 P b a a a Evolution Eq. of Kab Ricci rel. c  G  8  T (Lt - Lb )Kab  -Da Dba +aRab + KKab - 2KacKb ] ab ab - 4a (2S -g (S - E)) ab ab Definition of Kab Evolution Eq. of gab

Kab  -  (anb) (Lt -Lb )g ab  -2a Kab

 3+1 decomposition of the stress-energy tensor E  n n T , P  -  (nbT ), S  T Tab  Enanb + 2P(anb) + Sab a b ab a ab ab ab

27 APCTP International school on NR and GW July 28-August 3, 2011 Similarity with the Maxwell’s equations

ab a b b a abc ab a a a F n E - n E + Bc b F 4J  4  ( e n + J ) ab a b b a abc *F n B - n B - Ec ab a a a []aFF bc 0   a *  0 E na0, B n a , J n a 0

abJ a a na b F n a (4 ) DEa  4e Gauss’s law, No monopole Normal no time derivatives of E, B projection nF*0ab a ab DBa  0 Constraint equations

 F ab  (4 J a )  -L EDBJKEa  abc( a ) - 4 a a + a a b Spatial ( tb ) b c projection ab L a abc a  b *0F  (t -b ) BDEKB  - b() a c + a

Faraday’s law, Ampere’s law Evolution equations

28 APCTP International school on NR and GW July 28-August 3, 2011 Evolution of constraints

 It can be shown that the “evolution” equations for the Hamiltonian (CH) and Momentum (CM) constraints becomes k k ij (t - Lb )CH  -Dk (aCM ) -CM Dka +aK(2CH - F) +aK Fij i ij i j i k i (t - Lb )CM  -Dj (aF ) + 2aK jCM +aKCM +aD (F -CH ) + (F - 2H)D a

 Where Fij is the spatial projection : the evolution equation

 4  1  Fab    Rab -8Tab - Tgab    2   The evolution equations for the constraints show that the constraints are “preserved” or “satisfied” , if  They are satisfied initially (CH = CM = 0)  The evolution equation is solved correctly (Fab=0)

29 APCTP International school on NR and GW July 28-August 3, 2011 Coordinate-basis vectors  Let us choose the coordinate basis vectors a  First, we choose the evolution timelike vector t as the a a time-basis vector : t  (e0 ) a  The spatial basis vectors are chosen such that a (ei )  0 a  The spatial basis vectors are Lie transported along t : a b a b a a Lt (ei )  t b (ei ) -(ei ) bt [t, ei ]  0 a  ( e i ) remains purely spatial because a a a a Lt (a (ei ) )  (Lta )(ei ) - aLt (ei )  (Lta )(ei ) b b a b a  (t ba + bat )(ei )  2t [ba] (ei )  0 a  ( em ) constitute the commutable coordinate basis

 Then Lt  t  m m a We define the dual basis vectors by ( )a : ( )a (em )

30 APCTP International school on NR and GW July 28-August 3, 2011 Components of geometrical quantities (1)

 Now we have set the coordinate basis we proceed to calculate the components of geometrical quantities  Because the evolution time vector is the time-coordinate a m a a m basis we have t  t (em )  (e0 )  t  [ 1 0 0 0 ]  From the property of the spatial basis, we have a m ni  0,  0  a (ei )  mi  a ni  Then, 0th contravariant components of spatial tensors vanish  0 0   0 0  b m  [ 0 b i ], g m  , K m   ij   ij   0 g   0 K   From the definition of the time vector and normalization a a a a m - - i condition of n , we obtain t  an + b  n  [ a -a b ]

a n na  -1  nm [-a 0 0 0 ]

31 APCTP International school on NR and GW July 28-August 3, 2011

 The normalization of na gives Components of geometrical quantities (2)

 From the definition of spatial metric, we have -a -2 a -b i  g ab  g ab - nanb  g m   - i ij - i j  a b g -a b b  g  g - n n  g  g ab ab a b ij ij  We here note that from the spatial component of the following equation, we have m m m  m m  ik i g g  (g + n n )g   + n   g g kj   j  This means that the indices of spatial tensors can be lowered and raised by the spatial metric  Then, from the inverse of gab, we obtain 2 k

-a + bk b bi  2  2 i i j j gm    , ds  -a dt +g ij (dx + b dt)(dx + b dt)  bi g ij 

32 APCTP International school on NR and GW July 28-August 3, 2011 An intuitive interpretation

b idt a i i na t dx + b dt  The lapse function measures proper time between two adjacent slices St+dt

a i an dx  The shift vector gives relation of the spatial origin between slices St

2  2 i i j j ds  -a dt +g ij (dx + b dt)(dx + b dt)

33 APCTP International school on NR and GW July 28-August 3, 2011 Conformal decomposition

 The importance of the conformal decomposition in the time evolution problem was first noted by York (PRL 26, 1656 (1971); PRL 28, 1082 (1972))  He showed that the two degrees of freedom of the gravitational field are carried by the conformal equivalence classes of 3-metric, which are related each other by the conformal transformation : g  4g~ ij ij  In the initial data problems, the conformal decomposition is a powerful tool to solve the constraint equations, as studied by York and O’Murchadha (J. Math. Phys. 14, 456 (1973); PRD 10, 428 (1974)) (see for reviews , e.g., Cook, G.B., Living Rev. Rel. 3, 5 (2000); Pfeiffer, H. P. gr-qc/0412002)  In the following, we shall derive conformal decomposition of Einstein’s equations

34 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of Ricci tensor (1)

 The covariant derivative associated with the conformal metric is characterized by ~ ~ Dcg ab  0  The two covariant derivatives are related by (e.g. Wald ) i ~ i i l l i DkTj  DTj +CklTj -CkjTl i  where C jk is a tensor defined by difference of Christoffel symbols k k ~ k 1 kl ~ ~ ~ C ij   ij -  ij  g (D g + D g - D g ) 2 i lj j il l ij ~  2( k D ln + k D ln -g~ Dk ln ) i j j i ij  By a straightforward calculation, we can show (e.g Wald )

j j ~ ~ ~ ~ j ~ k ~ k l k k l j Rijv  (Dj Di - Di Dj )v  (Dj Di - Di Dj )v + (DkCij - DiCkj + ClkCij -Cil Ckj )v ~ j ~ k ~ k l k k l j  Rijv + (DkCij - DiCkj + ClkCij -Cil Ckj )v

35 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of Ricci tensor (2)

 Thus the Ricci tensor is decomposed into two parts, one which is the Ricci tensor associated with the conformal metric and one which contains the conformal factor   More explicitly one can show (see e.g. Wald (1984)) ~ ~ ~ ~ ~ ~ k Rij  Rij - 2Di D j ln - 2g ij Dk D ln ~ ~ ~ ~ + 4(D ln )(D ln ) - 4g~ (D ln )(Dk ln ) i j ij k ~   Rij + Rij  Then, the Ricci scalar is decomposed as - ~ ~ ~ k ~ ~ k R  [R -8(Dk D ln + (Dk ln )(D ln ))] - ~ - ~ ~k  R -8 Dk D 

36 APCTP International school on NR and GW July 28-August 3, 2011 Conformal decomposition of extrinsic curvature

 The first step is to decompose Kij into trace (K) and traceless (Aij) parts as

1 ij ij 1 ij Kij  Aij + g ij K, K  A + g K 3 3  Then, we perform the conformal decomposition of the traceless part as  ~ ij ik jl - ~ Aij  Aij , A  g g Akl  Aij

 Under these conformal decompositions of the spatial metric and the extrinsic curvature, let us consider the conformal decomposition of Einstein’s equation

37 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (0) – an additional constraint

 In the following, with BSSN reformulation in mind, we set the determinant of the conformal metric to be unity: ~ ~ g  det g ij 1  with this setting, the conformal factor becomes 1 ln  ln g 12  In the BSSN formulation, the conformal factor is defined by  = ln so that  = lng/12  In the case that we do not impose the above condition to the background conformal metric, the equations derived in the following are modified slightly (for this, see Gourgoulhon, E., gr-qc/0703035)

38 APCTP International school on NR and GW July 28-August 3, 2011

“Conformal” decomposition of the evolution equations (1) : the conformal factor

 Let us start from the evolution equation of the spatial metric g : ij (L -L )g  L g  -2a K t b ab an ab ab  Taking the trace of this equation, we have ab g Lang ab  -2a K

 Now we use an identity for any matrix A : det[exp A]  exp[tr A]  By setting gij = exp A and taking the Lie derivative, we obtain ij Lang  exp[tr (lng ij )] Lan (tr (lng ij ))  gg Lang ij  Now we can derive the evolution equation for the conformal factor : ij g Lang ij  Lan ln g 12Lan ln  -2a K 1 (L - L )ln  - aK t b 6

39 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (2) : the conformal metric

 Again, we start from the evolution equation for gij :

(Lt -Lb )g ab  Lang ab  -2a Kab  Substituting the decomposition of gij and Kij, we obtain

 ~ 3 ~  4 ~ 1  ~   (Lt - Lb )g ij + 4 g ij (Lt - Lb )  -2a Aij +  g ij K   3  ~ ~  ~ 1 ~  (Lt - Lb )g ij + 4g ij (Lt - Lb )ln  -2a Aij + g ij K   3   Now, we shall use the evolution equation for the conformal factor, and finally, we get ~ ~ (Lt -Lb )g ij  -2a Aij

40 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (3) : the inverse conformal metric

 For the later purpose, let us derive the evolution equation for the inverse of the conformal metric  It is easily obtained from the evolution equation for the conformal metric, as

~ik ~ jl ~ ~ij g g (Lt - Lb )g kl  -2a A ~ik j ~ ~ jl ~ij g [(Lt - Lb ) k -g kl (Lt - Lb )g ]  -2a A ~ij ~ij (Lt - Lb )g  2a A

41 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (4a) : the trace of the extrinsic curvature

 We start from the evolution equation for Kij : k LanKij  -Di Dja +aRij + KKij - 2Kik K j ]+ 4a (g ij (S - E) - 2Sij )

 We first simply take the trace of this equation L L ij i 2 ij anK - Kij ang  -Di D a +aR + K - 2Kij K ]+ 4a (3(S - E) - 2S)

 Here, let make use of the evolution equation for the ab ab inverse of the spatial metric, (Lt -Lb )g  2a K

then we obtain L K  -D Dia +aR + K 2 ]+ 4a (S -3E) an i  Finally, using the Hamiltonian constraint, we obtain i ij (Lt -Lb )K  -Di D a +aKij K + 4(E + S)]

2 ab R + K - KabK 16E

42 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (4b) : the trace of the extrinsic curvature

 For convenience, let us express the right-hand-side in terms of the conformal quantities, as well as give a suggestion how to evaluate the derivative term :

k 1 k -  jk - 2 ~ jk Dk D a  k ( g D a ) k ( g Dja ) k ( g  ja ) g

1 ~ ~ 1 K K ij  A Aij + K 2  A Aij + K 2 ij ij 3 ij 3

43 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (5a) : the traceless part of the extrinsic curvature

 We start from the Lie derivative of Kij : 1 1 L K  L A + g L K + KL g an ij an ij 3 ij an 3 an ij

 Substituting the following equations into this yields L K  -D D a +aR + KK - 2K K k ]+ 4a (g (S - E) - 2S ) an ij i j ij ij ik j ij ij i 2 L g  -2a K LanK  -Di D a +aR + K ]+ 4a (S -3E) an ij ij

TF TF TF 5 k 1 2  L an Aij  -(Di Dja) +a(Rij -8Sij ) +a KKij - 2Kik K j - K g ij  3 3   where TF denotes the trace free part : TijTF = Tij - (1/3)gij(trT)  The terms that involve K in the right-hand-side can be written as

5 k 1 2 1 k  1 ~ ~ ~k  KKij - 2Kik K j - K g ij  KAij - 2Aik Aj    KAij - 2Aik Aj  3 3 3 3 

44 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the evolution equations (5b) : the traceless part of the extrinsic curvature

 We further proceed to decompose the left-hand-side : ~ ~  ~ 2 ~  L 4 L L 4 L an Aij   an Aij + 4Aij an ln   an Aij - aKAij   3   Combining all of the result, we finally reach ~ -4 TF TF TF ~ ~ ~k (Lt -Lb )Aij  -(Di Dja) +a(Rij -8 Sij )+aKAij - 2Aik Aj 

 We here note that the second-order covariant derivative of the lapse function may be calculated as ~ k Di Dja  Di ja  Di ja -Cij ka ~ k ~ ~kl  i ja - ij ka- 22(i ln  j)a -g ijg k ln la

  NOTE: there is the same 2nd order derivative in Rij

45 APCTP International school on NR and GW July 28-August 3, 2011 “Conformal” decomposition of the constraint equations

 Let us turn now to consider the conformal decomposition of the constraint equations  Hamiltonian constraint 2 ab R + K - KabK 16 E ij ~ ~ij 2 Kij K  Aij A + K /3 - ~ - ~ ~k R  R -8 Dk D  ~ ~i  ~   ~ ~ij 1 2   Di D - R +  Aij A - K + 2 E  0    12   Momentum constraint ij ij i ab a a D j K  D j A + D K / 3 DbK - D K  8 P ij ~ ij i kj j ik D j A  D j A + C jk A + C jk A ~ ij ij ~  DA +10A D j ln ~ ~ij ~ij ~  ~i  i Dj A + 6A Dj ln - D K  8 P - ~ ~ij ~ij ~   D j A + 6A D j ln 

46 APCTP International school on NR and GW July 28-August 3, 2011 Summary of conformal decomposition

 With the conformal decomposition defined by 4 ~ 4 1 ~ ~ g  g Kij  Aij + g ij K g  det g 1 ij ij 3 ij  The 3+1 decomposition (ADM formulation) of Einstein’s equations becomes Constraint equations

~ ~i  ~   ~ ~ij 1 2   Di D - R +  Aij A - K + 2 E  0 Evolution equations    12  1 ~ ~ij ~ij ~  ~i  i (Lt - Lb )ln  - aK D A + 6A D ln - D K  8 P 6 j j  ~ ~ (Lt -Lb )g ij  -2a Aij

i ij (Lt -Lb )K  -Di D a +aKij K + 4(E + S)]

~ -4 TF TF TF ~ ~ ~k (Lt -Lb )Aij  -(Di Dja) +a(Rij -8 Sij )+aKAij - 2Aik Aj 

47 APCTP International school on NR and GW July 28-August 3, 2011 Lie derivatives of tensor density  A tensor density of weight w is a object which is a tensor w/2 w/ 2 times g : Tij  g Tij  One should be careful because the Lie derivative of a tensor density is different from that of a tensor, as  s r  L a1...as  b c a1...as - a1.. c ..as  b ai - a1...as  b c + w a1...as  b k b Tb1...br  cTb1...br Tb1...br c Tb1.. c ..br bi  Tb1...br k  i1 i1    L a1...as  + w a1...as  b k b Tb1...br w0 Tb1...br k

i w/ 2 i w/ 2 k i k i i k i w- Lb Tj  Lb (g Tj )  g b kTj -Tj k b +Tk  j b +Tj (w/ 2)g Lb g k w/ 2 i i k w/ 2 k i i k i w- ij  b k (g Tj ) -Tj b kg - Tj k b + Tk  j b +Tj (w/ 2)g gg Lb g ij  k i i (w-1)/ 2 k 1/ 2 k i i k i w k  b k Tj -Tj wg b kg - Tj k b + Tk  j b +Tj wg Dk b k i i (w-1)/ 2 k 1/ 2 k i i k i w - 1/ 2 k  b k Tj -Tj wg b kg - Tj k b + Tk  j b +Tj wg g k (g b ) k i k i i k i w k  b k Tj - Tj k b + Tk  j b + wTj g k b

48 APCTP International school on NR and GW July 28-August 3, 2011 Lie derivatives in conformal decomposition 1/12  The weight factor of the conformal factor  = g is 1/6  Thus the weight factor of the conformal metric and the conformal extrinsic curvature is -2/3, so that a  Note that the Lie derivative along t is equivalent to the partial derivative along the time direction  Thus 1 (L - L )ln  ( - b k )ln -  b k t b t k 6 k 2 (L - L )g~  ( - b k )g~ -g~  b k -g~  b k + g~  b k t b ij t k ij ik j jk i 3 ij k

~ ~ ~ ~ 2 ~ (L - L )A  ( - b k )A - A  b k - A  b k + A  b k t b ij t k ij ik j jk i 3 ij k

49 APCTP International school on NR and GW July 28-August 3, 2011 Evolution of constraints

 Where Fij is the spatial projection of the evolution equation

50 APCTP International school on NR and GW July 28-August 3, 2011 Numerical-relativity simulations based on the 3+1 decomposition is unstable !!

 It is known that simulations based on the 3+1 decomposition (ADM formulation), unfortunately crash in a rather short time  This crucial limitation may be captured in terms of notions of hyperbolicity (e.g. see textbook by Alcubierre (2008) )  Consider the following first-order system i tU + A iU  0  The system is called  Strongly Hyperbolic ,if a matrix representation of A has real eigenvalues and complete set of eigenvectors  Weakly Hyperbolic , if A has real eigenvalues but not a complete set of eigenvectors  The key property of strongly and weakly hyperbolic systems :  Strongly hyperbolic system is well-posed, and hence, the solution for the finite-time evolution is bounded  Weakly hyperbolic system is ill-posed and the solution can be unbounded 51 APCTP International school on NR and GW July 28-August 3, 2011 Numerical-relativity simulations based on the 3+1 decomposition is unstable !!

 It is known that the ADM formulation is only weakly hyperbolic ( Alcubierre (2008) )  Consequently, the ADM formulation is ill-posed and the numerical solution can be unbounded, leading to termination of the simulation  We need formulations for the Einstein’s equation which is (at least) strongly hyperbolic  Let us consider Maxwell’s equations in flat spacetime to capture what we should do to obtain a more stable system

m k  E i  4 A  (, A ) i i e i E  4 e j k i B    A  B  0 i ijk j j i t Ei  Di D Aj - D D j Ai - 4 ji j k  E    B - 4 j t i ijk i t Ai  -Ei - Di j k t Bi  - ijk E Note the similarity of these equations to those in the ADM formulation

52 APCTP International school on NR and GW July 28-August 3, 2011 Consideration in Maxwell’s equations

 First of all, let us note the similarity of the Maxwell’s equations with the ADM equations (for simplicity in vacuum) i (constraint eqs.) i E  0 (constraint eq.) j j 2L K  -2aR +  wave-like part t E i  Di D Aj - D D j Ai an ij ij  a g kl   g + g kl   g -g kl   g +  t A i  -Ei - Di ( l i kj j k il k l ij )

Lang ij  -2aKij “mixed-derivative” part  Second, the Maxwell’s equations are ‘almost’ wave equation 2 k j -t Ai + D Dk Ai - Di D Aj  Dit j  Recall that in the Coulomb gauge DjA =0, the longitudinal part (associated with divergence part) of the electric field E does not obey a wave equation but is described by a Poisson equation (see a standard textbook, e.g., Jakson)

53 APCTP International school on NR and GW July 28-August 3, 2011 Reformulating Maxwell’s equations (1) - Introducing auxiliary variables

 A simple but viable approach is to introduce independent auxiliary variables to the system  Let us introduce a new independent variable defined by k F  D Ak  The evolution equation for this is k i k t F  t D Ak  -D Ei - Dk D   Then, the Maxwell’s equations for the vector potential become a wave equation in the form : 2 k -t Ai + D Dk Ai  Dit + Di F

54 APCTP International school on NR and GW July 28-August 3, 2011 Reformulating Maxwell’s equations (2) - Imposing a better gauge

 A second approach is to impose a good gauge condition  In the Lorenz gauge, the Maxwell’s equations in the flat spacetime are wave equations m m A  0  Alternatively, by introducing a source function, one may “generalize” the Coulomb gauge condition so that Poisson- like equations do not appear k m D Ak  H(x )

j  Recall again, that in the Coulomb gauge DjA =0, the longitudinal part (associated with divergence part) of the electric field E is described by a Poisson-type equation

55 APCTP International school on NR and GW July 28-August 3, 2011 Reformulating Maxwell’s equations (3) - Using the constraint equations

 A third approach is to use the constraint equations  To see this, let us back to the example considered in “introducing auxiliary variables” i i Di E  4 e Di E  4 e k k t Ei  Di F - D Dk Ai - 4 ji t Ei  Di F - D Dk Ai - 4 ji

t Ai  -Ei - Di t Ai  -Ei - Di i k k t F  -D Ei - Dk D  t F  -4 e - Dk D   The constraint equation can be used to rewrite the evolution equation for the auxiliary variable  Seen as the first-order system, the hyperbolic properties of the two system is different: the hyperbolicity could be changed !  It is important and sometimes even crucial to use the constraint equations to change the hyperbolic properties of the system

56 APCTP International school on NR and GW July 28-August 3, 2011 Reformulating Einstein’s equations  The lessons learned from the Maxwell’s equations are  Introducing new, independent variables  BSSN ( Shibata & Nakamura PRD 52, 5428 (1995); Baumgarte & Shapiro PRD 59, 024007 (1999) )  see also Nakamura et al. Prog. Theor. Phys. Suppl. 90, 1 (1987)  Kidder-Scheel-Teukolsky ( Kidder et al. PRD 64, 064017 (2001) )  Bona-Masso ( Bona et al. PRD 56, 3405 (1997) )  Nagy-Ortiz-Reula ( Nagy et al. PRD 70, 044012 (2004) )  Choosing a better gauge  Generalized harmonic gauge ( Pretorius, CQG 22, 425 (2005) )  Z4 formalism ( Bona et al. PRD 67, 104005 (2003) )  Using the constraint equations to improve the hyperbolicity  adjusted ADM/BSSN ( Shinkai & Yoneda, gr-qc/0209111 )  BSSN outperforms ( Alcubierre (2008) ) !  Exact reason is not clear 57 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism (1)

 Let first analyze the conformal Ricci tensor ~k ~  By noting that 2  ik   i ln g   the conformal Ricci tensor is ~ ~ k ~ k ~ k ~l ~ k ~l Rij  k ij -  jik + ij kl - il kj Wij

1 ~kl ~ ~ ~  - g (  g -   g -   g )+ (terms with g g ) 2 k l ij i l jk j l ik ~ij ij ij  If we divide the conformal metric formally as g   + f , we have k ~ k k Wij  k g ij -(i g kj +  j g ki )+ (terms with f f )  Thus we can eliminate the “mixed derivative” terms by introducing new auxiliary variable ( Shibata & Nakamura (1995) ) jk ~ j ~ Fi   kg ij   g ij

k ~ Wij  k g ij -(i Fj +  j Fi )+ (terms with f f )

58 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism (2)

 Baumgarte and Shapiro introduced the slightly different auxiliary variables i  - g~ij j  In this case, the mixed-second-derivative terms are encompassed as

~ 1 ~kl ~ ~ k ~ k Rij  - (g klg ij -g ik j -g jki )+ (terms with g g ) 2  In linear regime, SN and BS are equivalent

59 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism (3)

 Finally let us consider the evolution equation for the auxiliary variables (giving only a rough sketch of derivation)  Let us start from the momentum constraint equation ~ ~ ~ ~  ~ D (g~ jk A ) + 6A g~ jk D ln - D K  8  P k ij ij k  i i

~ ~ij ~ij ~  ~i  i Dj A + 6A Dj ln - D K  8 P   Substituting the evolution equation for the conformal extrinsic curvature ~ ~ (Lt -Lb )g ij  -2a Aij

~ij ~ij (L -L )g  2a A t b i  We obtain the evolution equations for Fi and  , respectively  It can be seen from the above sketch of derivation, the evolution i equation for  is slightly simpler

60 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism (4)

 The explicit forms of the evolution equations are

k  jk ~ ~ ~ jk 1 ~ jk ~ ~ j 2  (t - b k )Fi  -16aPi + 2a  f k Aij + Aijkg - A ig jk + 6Ai  j ln - i K  2 3 

jk  ~ l ~  ~ l ~ l 2 ~ l  + - 2Aijka + (k b )(lg ij ) + k g il j b +g jli b - g ijl b    3 

k i i ~i ~ jk ~ij 2 ~ij  ~ij (t - b k )   -16aP + 2a jk A + 6A  j ln - g  j K - 2A  ja  3  2 1 + b j i -  j b i + i b j + g~ij  b k +g~ jk  b i j j 3 j 3 j k j k

61 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism : summary (1)

~ ~i  ~   ~ ~ij 1 2   Di D - R +  Aij A - K + 2 E  0    12  ~ ~ ~ ~  ~ D Aij + 6Aij D ln - Di K  8  Pi j j  1 1 ( - b k )ln  - aK +  b k t k 6 6 k k ~ ~ ~ k ~ k 2 ~ k ( - b  )g  -2a A +g  b +g  b - g  b t k ij ij ik j jk i 3 ij k Hamiltonian  - b k K  -D Dia +aK K ij + 4(E + S)] ( t k ) i ij constraint is used

k ~ -4 TF TF TF ~ ~ ~k (t - b k )Aij  - (Di Dja) +a(Rij -8 Sij )+aKAij - 2Aik Aj  ~ ~ 2 ~ + A  b k + A  b k - A  b k ik j jk i 3 ij k

62 APCTP International school on NR and GW July 28-August 3, 2011 BSSN formalism : summary (2)

k  jk ~ ~ ~ jk 1 ~ jk ~ ~ j 2  (t - b k )Fi  -16aPi + 2a  f k Aij + Aijkg - A ig jk + 6Ai  j ln - i K  2 3 

jk  ~ l ~  ~ l ~ l 2 ~ l  + - 2Aijka + (k b )(lg ij ) + k g il j b +g jli b - g ijl b    3  Momentum constraint is used

k i i ~i ~ jk ~ij 2 ~ij  ~ij (t - b k )  -16aP + 2a jk A + 6A  j ln - g  j K - 2A  ja  3  2 1 + b j i -  j b i + i b j + g~ij  b k +g~ jk  b i j j 3 j 3 j k j k

63 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

BH excision

64 APCTP International school on NR and GW July 28-August 3, 2011 Gauge conditions

 Associated directly with the general covariance in general relativity, there are degrees of freedom in choosing coordinates (gauge freedom)  Slicing condition is a prescription of choosing the lapse function  Shift condition is that of choosing the shift vector  Einstein’s equations say nothing about how the gauge conditions should be imposed  As we have seen in the reformulation of the ADM system, choosing “good” gauge conditions are very important to achieve stable and robust numerical simulations  An improper slicing conditions in a stellar-collapse problem will lead to appearance of (coordinate and physical) singularities  Also, the shift vector is important in resolving the frame dragging effect in simulations of e.g. compact binary merger

65 APCTP International school on NR and GW July 28-August 3, 2011 Preliminary – decomposition of covariant derivative of na – a  The covariant derivative of a timelike unit vector z can be decomposed as 1  z   + + h  - z  a b ab ab 3 ab a b

hab  gab + za zb , (induced metric)  Where, the deformation of the ab   [a zb], (twist) congruence of the timelike vector TF     z , (shear) is characterized by these tensors ab (a b)    zc , (expansion) c  a  zc z a , (acceleration) c a  For the unit normal vector to S, n we have  The expansion is –K 1  The shear is –Aab anb  -Aab - g abK - naab  The twist vanishes 3

66 APCTP International school on NR and GW July 28-August 3, 2011 Geodesic slicing a=1, bi=0

 In the geodesic slicing, the evolution equation of the trace ij of the extrinsic curvature is t K  Kij K + 4(E +3S)  For normal matter (which satisfies the strong energy condition), the right-hand-side is positive  Thus the expansion of time coordinate ( -K ) decreases monotonically in time 1/2  In terms of the volume element g , this means that the volume element goes to zero, as 1  ln g   g ij g  -aK + D b k  - K t 2 t ij k  This behavior results in a coordinate singularity  As can be seen in this example, how to impose a slicing condition is closely related to the trace of the extrinsic curvature

67 APCTP International school on NR and GW July 28-August 3, 2011 Maximal slicing  Because the decrease in time of the volume element results in a coordinate singularity, let us maximize the volume element  We take the volume of a 3D-domain S : V[S]  g d 3x S and consider a variation along the time vector t a ana + b a i  At the boundary of S, we set a=1, b =0

3 i 3 LtV[S]  d x-aK g + i ( g b ) - aK g d x S S  Thus if K=0 on a slice, the volume is extremal (maximal)  We shall demand that this maximal slicing condition holds for all slices and set 0  (L -L )K  -D Dia +aK K ij + 4(E + S)] t b i ij  The maximal slicing has a strong singularity avoidance property (E.g. Estabrook & Wahlquist PRD 7, 2814 (1973); Smarr & York, PRD 17, 1945/2529 (1978) )  However this is a elliptic equation and is computationally expensive

68 APCTP International school on NR and GW July 28-August 3, 2011 (K-driver)/(approximate maximal) condition  As a generalization of the maximal slicing condition, let us consider the following condition with a positive constant c  K  -cK t  This (elliptic) condition drives K back to zero even when K deviates from zero due to some error or insufficient convergence  Balakrishna et al. ( CQG 13, L135 (1996) ) and Shibata ( Prog. Theor. Phys. 101, 251 (1999) ) converted this equation into a parabolic one by adding a “time” derivative of the lapse : i ij i a  Di D a -aKij K + 4(E + S)]- b Di K + cK  If a certain degree of the “convergence” is achieved and the lapse relaxes to a “stationary state”, it suggests  This condition is called K-driver or approximate maximal slicing condition ta  -(t K + cK ),   t

69 APCTP International school on NR and GW July 28-August 3, 2011 Harmonic slicing

c a  The harmonic gauge condition  c  x  0 have played an important role in theoretical developments (Choquet-Bruhat’s textbook)  Existence and uniqueness of the solution of the Cauchy problem of Einstein’s equations (somewhat similar to Lorenz gauge in EM)  The harmonic slicing condition is defined by c m c t  0  m ( - gg ) 0  Note that - g  a g  The harmonic slicing condition can be written as k  (t -k b ) a  -a K  This is an evolution equation  It is known that the harmonic slicing condition has some singularity avoidance property, although weaker than that of the maximal slicing ( e.g. Cook & Scheel PRD 56, 4775 (1997), Alcubierre’s textbook )

70 APCTP International school on NR and GW July 28-August 3, 2011 Generalized harmonic slicing

 Bona et al. ( PRL 75, 600 (1995) ) generalized the harmonic slicing condition to

k  (t -k b ) a  -a f (a)K

 This family of slicing includes the geodesic slicing ( f=0 ), the harmonic slicing ( f=1 ), and formally the maximal slicing ( f= )  The choice f(a)=2/a , which is called 1+log slicing , has stronger singularity avoidance properties than the harmonic slicing ( Anninos et al. PRD 52, 2059 (1995) )  The 1+log slicing has been widely used and has proven to be a successful and robust slicing condition

71 APCTP International school on NR and GW July 28-August 3, 2011 Minimal distortion (shift) condition

 Smarr and York (PRD 17, 1945/2529 (1978)) proposed a well motivated shift condition called the minimal distortion condition  As seen in the preliminary, the “distortion” part of the congruence is contained in the shear tensor I  S Sab g dx3  They define a distortion functional by  ab and take a variation in terms of the shift  here the distortion tensor is defined by 1 TF TF TF Sab   Ltg ab ~ -Kab   (anb) 2 ab  The resulting shift condition is DaS  0  Beautiful and physical but vector elliptic equations (computationally expensive)

c a c b b ab   ab  Dc D b + Da Dcb + Rabb  D 2aAab 2A Dba +a g Db K +16Pa    

72 APCTP International school on NR and GW July 28-August 3, 2011 -Freezing and approximate minimal condition

 With some calculations, one can show that the minimal distortion condition is written as ~ D j (  g~ )  0  The conformal factor is coupled ! t ij  Modifications of the minimal distortion condition are proposed by Nakamura et al. (Prog. Theor. Phys. Suppl. 128, 183 (1997)) and Shibata (Prog. Theor. Phys. 101, 1199 (1999))  E.g., Nakamura et al. proposed instead to solve the decoupled pseudo- ~ minimal distortion condition : D j ( g~ )  0 t ij  Alcubierre and Brugmann (PRD 63, 104006 (2001)) proposed an approximate minimal distortion condition called Gamma- Freezing : ~ D ( g~ij )   i  0 j t t  Anyway, these conditions are elliptic-type !

73 APCTP International school on NR and GW July 28-August 3, 2011 -Driver condition

 Alcubierre and Brugmann (PRD 63, 104006 (2001)) converted the - freezing elliptic condition into a parabolic one by adding a time derivative of the shift (somewhat similar to the K-driver)  i t b  kt  Alcubierre et al. (PRD 67, 084023 (2003)) and others (Lindblom & Scheel PRD 67, 124005 (2003); Bona et al. PRD 72, 104009 (2005)) extended the - freezing condition to hyperbolic conditions i i t b  kB  There are several alternative conditions  Bi   i -Bi t t  Shibata (ApJ 595, 992 (2003)) proposed a hyperbolic shift condition i ~ij t b  g (Fi + tstept Fi ), tstep : time-step used in simulation  To date, the above two families of shift conditions are known to be robust

74 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

BH excision

75 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Energy Conservation Equation (1)

 First, substitute the 3+1 decomposition of Tab to obtain b b b b b 0  bTa  b (En na + P na + Pan + Sa )  n nb E + Ea - KEn - Pb K + n  Pb + nb P - KP +  S b a b a a ba a b b a a b a  Then, let us project it onto normal direction to S. Noting that Pa, Kab, and ab is purely spatial, we obtain b a a b a b - n bE + KE -a P + n n bPa + n bSa  0 a  Because n Sab=0, we have a b b a b a a ab n bSa  -Sa bn  Sa (Kb + nba )  S Kab  Similarly, nanb P  -P nb na  -P ab b a a b b a  The divergence term of P is

a b a b b a a a Da P a bP  (a + nan )bP  a P - Paa

76 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Energy Conservation Equation (2)

 Combining altogether, we reach the energy conservation equation

b b b ab n b E + Db P + 2P ab - KE - KabS  0 k b ab b (t - b Dk )E +aDb P - KE - KabS ]+ 2P Dba  0  E + D (aPk - Eb k ) + E(D b k -aK) -aK S ab + Pb D a  0 t k k ab b  where we have used b - - k n bE  LnE a (Lt -Lb )E a (t - b Dk )E

ab  Db lna  The last equation will be used to derive the conservative forms of the energy equation

77 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Momentum Conservation Equation (1)

 To this turn, let us project the equation onto S to obtain Ea - PbK + c nb P - KP + c  S b  0 a ba a b c a a b c b  The spacetime-divergence term of S c can be replaced by the spatial-divergence by b d e b e d d b e b d DbSc  b c d Se  c (b + nbn )d Se  c bSe - Sc ad  The projection term with the covariant derivative of Pc is c b -1 c b -1 c d a n bPc a a (an )bPc a a (LanPc - Pdc (an ))  Note that (an)-Lie derivative of any spatial tensor is spatial, and

a a a a a a b (an )  n ba +a bn  n ba -a(Kb + nba )  so that

c b - b a n bPc a LanPa + KabP

78 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Momentum Conservation Equation (2)

 Combining altogether, we obtain the momentum conservation equation : b b (Lt - Lb )Pa +a[DbSa + Sa ab - KPa + Eaa ]  0 L b b (t - b )Pa +a[DbSa - KPa ]+ Sa Dba + EDaa  0 c b c b ( t - b Dc )Pa +aDbSa + (Dcb -aK)Pa + Sa Dba + EDaa  0  P + D (aS c - b c P ) + (D b c -aK)P - P D b c + ED a  0 t a c a c c a c a a  Where we have expressed the Lie derivative by spatial covariant derivative  The last equation will be used in conservative reformulation a  NOTE: In York (1979), because he used P instead of Pa, a extra term appear in the equation.

79 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Conservative Formulation (1)

 Now we will show the energy and momentum conservation equations can be recast to conservative form c c c ab b t E + Dc (aP - Eb ) + (Dcb -aK)E -aKabS + P Dba  0  P + D (aS c - b c P ) + (D b c -aK)P - P D b c + ED a  0 t a c a a c a c a a  First, by taking the trace of evolution eq. of gab, we get

ab a 1 ij 1 g (tg ab - Dabb - Dbba )  -2aK  Dab -aK  g tg ij  t g 2 g  Second, note that for any rank-(1,1) spatial tensor,

k k k j j k k j j k DkTi  kTi + jkTi - ikTj  kTi + ( j ln g )Ti - ikTj

1 k j k  k ( g Ti )-  ikTj g

80 APCTP International school on NR and GW July 28-August 3, 2011 ab 3+1 decomposition of  a T  0 - Conservative Formulation (2)

 Using the equations derived the above, we can finally reach the conservative forms of the energy and momentum equations

 g E +  g (aPk - Eb k )  g (aK S ij - Pk D a) t ( ) k ( ) ij k k k k k j j  t ( g Pi )+ k ( g (aSi - b Pi )) g Pk Di b - EDia +  ij (aSk - b Pk )

 For the perfect fluid, for instance, these equations may be solved by high resolution shock capturing schemes

81 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

BH excision

82 APCTP International school on NR and GW July 28-August 3, 2011 Locating the apparent horizon (1)

 Apparent horizon (e.g. Wald (1984)): the apparent horizon is the boundary of the (total) trapped region  Trapped region: the trapped region is collections of points where the expansion of the null geodesics is negative or zero  Thus, to locate the apparent horizon, we must calculate the expansion of the null geodesics and determine the points where the expansion vanishes

 Recall that the expansion is related to the trace of the extrinsic curvature : K  expansion  So that let us first define the extrinsic curvature of a null surface N generated by an outgoing null vector on a slice S:

83 APCTP International school on NR and GW July 28-August 3, 2011 Locating the apparent horizon (2)

 Let S to be an intersection of the slice S and the null surface N a  We denote the unit normal of S in S, as s a a  Then the outgoing ( k ) and ingoing ( l ) null vectors on S are

a 1 a a a 1 a a a a k  (n + s ), l  (n - s )  Using k and l , the metric 2 2 on S induced by gab is given by

ab  gab + kalb + kbla  g + n n - s s ab a b a b  Thus we can define the projection operator to S :

a a a a Pb  b + n nb - s sb

84 APCTP International school on NR and GW July 28-August 3, 2011 Locating the apparent horizon (3)

 Using the projection operator, the extrinsic curvature for N is defined by c d kab  -Pa Pb (ckd )

a  Because k is the outgoing null vector on S, the 2D-surface S a is the apparent horizon if tr[k] = k a = k = 0 a  This condition can be written in terms of s as k i j Dk s - K + Kij s s  0

k  This is a single equation for the three unknown “functions” s ! a  However, the condition that S is closed 2-sphere and that s is a unit normal vector bring two additional relation to sk  For detail, see ( e.g. Bowen, J. M. & York, J. W., PRD 21, 2047 (1980); Gundlach, C. PRD 57, 863 (1998) )

85 APCTP International school on NR and GW July 28-August 3, 2011 Energy and Momentums

86 APCTP International school on NR and GW July 28-August 3, 2011 Canonical formulation (1)

 The Lagrangian density of gravitational field in General Relativity is (e.g. Wald (1984)) 4 LG  - g R  Because the 4D Ricci scalar is written as 4 a b 4 a b R  2(Gabn n - Rabn n ) ab 2  a g (R + KabK - K ) + ( Divergence terms )  Noting that the extrinsic curvature is 1 K  (g - D b b - D b a ) ab 2a ab a b ab  The conjugate momentum  is defined by

L  ab  G  g (K ab - Kg ab) gab

87 APCTP International school on NR and GW July 28-August 3, 2011 Canonical Formulation (2)

 Now we obtain the Hamiltonian density as ab H G   gab -LG

a  ab 1   - ab   g  -g R +  ab -   - 2bb Da (g  ) + ( Divergence terms ) g  2    The Hamiltonian is defined by 3 HG  H Gdx   The constraint equations are derived by taking the variations with respect to the lapse and the shift, respectively, as

- ab 1 -  CH  -R +g  ab - g   0 : Hamiltonian constraint 2 b -1/ 2 ab C M  Da (g  ) 0 : Momentum constraint where we have dropped the surface term

88 APCTP International school on NR and GW July 28-August 3, 2011 Canonical Formulation (3)

 The evolution equations are derived by taking the variations with respect to the canonical variables (e.g. Wald (1984)) :

H  1  g  G  2ag -1/ 2  - g  + 2D b  B ab ab  ab ab  (a b) ab   2  H  1  1  1   1   ab G  ab -1/ 2 ab cd 2 -1/ 2 ac b ab    - ag R - Rg  + ag g   cd -   - 2ag   c -   g ab  2  2  2   2   a b ab c  -1/ 2 c ab c(a b) ab + g (D D a -g Dc D a )+ g Dc (g b  )- 2 Dcb  A

 again, we here dropped the divergence terms

89 APCTP International school on NR and GW July 28-August 3, 2011 Energy for Asymptotically Flat spacetime (1)

 Let us consider the energy of gravitational field in the asymptotically flat spacetime  Although there is no unique definition of ‘local’ gravitational energy in General Relativity, we can consider the total energy in the asymptotically flat spacetime  Asymptotically flat spacetime represent ideally isolated spacetime, and hence, there will be the conserved energy  A simple consideration based on the Hamiltonian density, b H G  g aCH - 2bbCM  + ( Divergence terms ) may lead to a conclusion that the energy of any spacetime is zero when the constraint equations are satisfied !  This “contradiction” stems from the wrong treatment of the divergence (surface) terms (which we have dropped)

90 APCTP International school on NR and GW July 28-August 3, 2011 Energy for Asymptotically Flat spacetime (2)  The boundary conditions to be imposed are not fixed ones Q|boundary = 0 where Q denotes relevant geometrical variables, but the “asymptotic flatness” : -1 i -1 -1 ij -2 a 1+O(r ), b  O(r ), g ij -ij  O(r ),   O(r )  Keeping the divergence terms, the variation of the Hamiltonian now becomes (Regge & Teitelboim, Ann. Phys 88. 286 (1974)) H  - M ijkl aD (g ) - (D a)g d G   k ij k ij  l - 2b  kl + (2b k jl - b l jk )g d  k jk  l  where we have assumed that the constraint equations and the evolution equations are satisfied, dl is the volume element of the ijkl boundary sphere and M is defined as 1 M ijkl  g g ikg jl +g ilg jk - 2g ijg kl  2 91 APCTP International school on NR and GW July 28-August 3, 2011 Energy for Asymptotically Flat spacetime (3)

 Under the boundary conditions of the asymptotic flatness, the non-zero contribution of the surface terms is, ijkl ij kl - M D (g ) d  - g g g  g - g d  k ij l  ( j ik k ij ) l  Thus, we define the Hamiltonian of the asymptotically flat spacetime as asympt.flat HG  HG +16 EG[g ij ]

1 ij kl E [g ]  g g g ( g -  g ) d G ij 16  j ik k ij l

 Then, the energy of the gravitational fields is not zero but E[gij]  The overall factor is determined by the requirement that the energy of an asymptotically flat spacetime is M

 2M   2Mxi x j  ds2  - 1- dt 2 +  + dxidx j    ij 3   r   r  92 APCTP International school on NR and GW July 28-August 3, 2011 Momentums for Asymptotically Flat spacetime (1)

 The action for the region V (t1

 Making use of g ij  - L  g ij  - ( D i  j + D j  i ) , we obtain

t 2 d 3 ij ij  SG  dt d x- Di (2  j )+ 2i Dj  0 t  1 dt  Note that the second term in the integrand vanished thanks to the momentum constraint 93 APCTP International school on NR and GW July 28-August 3, 2011 Momentums for Asymptotically Flat spacetime (2)

 Finally the variation of the action is reduced to

t2 kl SG   d l (- 2 k )  0  t1  Because the Killing vector approaches at the boundary (spacelike infinity) to a constant translation vector field ta , we have 1 t P k (t ) - P k (t )  0, P k [g ]  - d  kl k  G 2 G 1  G ij  l 8 k  This equation means that PG represent the total linear momentum  Similarly, the generator of the rotational Lie transport approaches j k  ijk j x (j is a constant vector field,  is the totally anti-symetric tensor) , we may define the total angular momentum by 1 j k LG (t ) - LG (t ) 0, LG[g ]  d   jl xk k 2 k 1 k ij 8  l ijk

94 APCTP International school on NR and GW July 28-August 3, 2011 Energy and Momentums : summary

 To summarize, we define the energy, the linear momentum, and the angular momentum in the asymptotically flat spacetime by

1 ij kl E [g ]  g g g ( g -  g ) d G ij 16  j ik k ij l

k 1 kl PG [g ij ]  - d l 8 

G 1 jl k Lk [g ij ]  d lijk x 8 

 A number of examples of the actual calculation will be found in a textbook (Baumgarte & Shapiro (2010))

95 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity

BH excision

96 APCTP International school on NR and GW July 28-August 3, 2011 References

 General Reference on General Relativity  Wald R. M., General Relativiy, Chicago University Press (1984)

 The classics on the basis of Numerical Relativity

 Arnowitt R., Deser S., & Misner C. W., The Dynamics of General Relativity, in Gravitation, ed. L. Witten, Wiley (1962), (Gen. Rel. Grav. 40, 1997 (2008))

 York J. W., Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation, ed. L. Smarr, Cambridge University Press (1979)

97 APCTP International school on NR and GW July 28-August 3, 2011 References

 Recent Textbooks on Numerical Relativity  Baumgarte T. & Shapiro S. L., Numerical Relativity. Cambridge University Press (2010)  Results of Numerical Relativity Simulations  Alcubierre M., Intrduction to 3+1 Numerical Relativity, Oxford University Press (2008)  Boundary Conditions  Gourgoulhon E., 3+1 Formalism and Bases of Numerical Relativity, to be published in Lecture Notes in Physics, (gr-qc/0703035)  Einstein’s equations as the Cauchy Problem  Choquet-Bruhat Y., General Relativity and Einstein Equations, Oxford University Press (2009)

98 APCTP International school on NR and GW July 28-August 3, 2011 References

 Locating the Horizons  Thornburg, J. , Event and Apparent Horizon Finders for 3+1 Numerical Relativity, Living Rev. Relativ. 10, 3 (2007)  ADM mass, and Some useful calculations  Poisson E. , A Relativist’s Toolkit, Cambridge University Press (2004)  Initial Data Problems  Cook, G. B. , Initial Data for Numerical Relativity, Living Rev. Relativ. 3, 5 (2000)  Pfeiffer, H. P. , Initial value problem in numerical relativity, gr-qc/0412002

99 APCTP International school on NR and GW July 28-August 3, 2011 References

 General Relativistic Numerical Hydrodynamics  Font, J. A. , Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity, Living Rev. Relativ. 11, 7 (2008)  Marti, J. M. & Muller, E. , Numerical Hydrodynamics in Special Relativity, Living Rev. Relativ. 6, 7(2003)  Toro, E. F. , Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction , Springer (2009)  Extracting Gravitational Waves  Reisswig, C., Ott, C. D., Sperhake, & U. & Schnetter, E. , Gravitational wave extraction in simulations of rotating stellar core collapse, Phys. Rev. D 83, 064008 (2011)

100 APCTP International school on NR and GW July 28-August 3, 2011