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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 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 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 (1)...a ( n ) [a1...an ] a (1)...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 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 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 , [ab] 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 a(xa) , function”, as : 1 g ab g ab t t - a b a b 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 ab gab + nanb ab ac bd ab a b g g g g cd g + n n 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 a b db + n nb a b a a The normal-projection operator is b -n na db - 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 d bwc + a wddc 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 a Kab - (anb) , n : unit normal to S extrinsic curvature provides information on how much the normal direction changes and hence, how S is curved parallel the antisymmetric part vanish transported b d n -K d x due to Frobenius‟s theorem : a ab a “For unit normal n to a slice a b n (x ) a b b n[abnc] 0 ” n (x +d x ) b a x 0 3(n n[abnc] ) - [bnc] b b S x +d 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 dadb 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 Ln 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 Da Dbwc a ( bwc ) d d abwc + (bwd )(a c ) + (d wc )(a b ) d d d abwc - 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 (abnc -banc ) where the right hand side is calculated as abnc a (-Kbc - nbac ) -Da Kbc- acanb -Da Kbc + ac Kab Then we obtain the Codazzi relation 4 d Rabc nd Db Kac - 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 (abnc - banc ) b n [-a (Kbc + nbac ) + b (Kac + naac )] b b b [Kbcan + 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 b b b b b b Ln Kac (n bKac + Kabcn + Kcban ) n bKac - KabKc - Kcb Ka 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 Kacbn + n Kbcan -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 ab n cnb 2n [cnb] -2n [ca b] -n (bca - cba) c c n nbc lna +db c lna Db lna where we have used the fact that is
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