1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,

Transcription

1 1 Gourgoulhon BSSN BSSN ϕ = 1 6 ( D i β i αk) (1) γ ij = 2αĀij 2 3 D k β k γ ij (2) K = e 4ϕ ( Di Di α + 2 D i ϕ D i α ) + α ] [4π(E + S) + ĀijĀij + K2 3 (3) Ā ij = 2 3Āij D k β k 2αĀikĀk j + αāijk +e 4ϕ [ 2α D i Dj ϕ + 4α D i ϕ D j ϕ + 4 D (i α D j) ϕ D i Dj α + α( R ij 8πS ij ) ] TF (4) constraint equations Λi = γ jk ˆDj ˆDk β i Γi Dj β j γik Dk Dj β j 2Ājk (δ i j k α 6αδ i j k ϕ α Γ i jk) 4 3 α γij j K 16πα γ ij S j (5) 2 3 K2 ĀijĀij + e 4ϕ ( R 8 D i ϕ D i ϕ 8 D 2 ϕ) 16πE = 0 (6) D j A ij + 6Āij D j ϕ 2 3 D i K 8πS i = 0 (7) det( γ ij ) = ˆγ (8) γ ij Ā ij = 0 (9) Λ i + ˆD j γ ij = 0 (10) Ricci R ij = 1 2 γkl ˆDk ˆDl γ ij + γ k(i ˆDj) Λk + Γ k Γ (ij)k + γ kl (2 Γ m k(i Γ j)ml + Γ m ik Γ mjl ) (11) γ ij ϕ D i ˆγ ˆD i K Āij R ij Γ i E t L β γ ij () Ricci T µν n µ n ν L β β Lie TF γ ij trace 0 Einstein Einstein 1

2 1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t, x i ) AC t( / t) BC (x i x i )( / x i ) = tβ i ( / x i ) tt / t t = tαn + tβi t x i (13) g tt = t t = α2 n n + 2αβ i n x i + βi β j x i x j (14) n n = 1 n ( / x i ) = 0 ( / x i ) ( / x j ) Σ t γ ij tx i g tt = α 2 + β i β j γ ij (15) x i x j g ti = t x i = αn g µν = g ij = x i + βj x j x i = βj γ ji (16) x i x j = γ ij (17) ( ) α 2 + β i β j γ ij γ ij β j γ ji β i γ ij (18) ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) (19) g µν = ( 1 α 2 β j α 2 β i α 2 γ ij βj β i α 2 ) (20) γ ij γ ij n µ Σ t v µ n µ v µ = 0 n µ 0 n µ = (a, 0, 0, 0) n µ = ( a/α 2, aβ i /α 2 ) n µ n µ = 1 a 2 /α 2 = 1 a = ±α n µ 0 t a = α ( ) 1 n µ = α, βi α (21) n µ = ( α, 0) (22) 2

3 γ µ ν = δ µ ν + n µ n ν γ µ νγ ν ρ = (δ µ ν + n µ n ν )(δ ν ρ + n ν n ρ ) (23) = δ µ ρ + n µ n ρ + n µ n ρ + n µ n ν n ν n ρ (24) = δ µ ρ + n µ n ρ = γ µ ρ (25) γ µ νn ν = δ µ νn ν + n µ n ν n ν (26) = n µ n µ = 0 (27) γ µ νn µ = δ µ νn µ + n µ n ν n µ (28) = n ν n ν = 0 (29) n 0 µ = µ 1 µ p ν = ν 1 ν q i = i 1 i p j = j 1 j q T i j T µ ν = T i jγ µ iγ j ν γ µ i = γ µ 1 i1 γ µ p ip γ j ν T γ ij γ µν γ µν = g µν + n µ n ν (30) * 1 Σ t ( ) γ ij (intrinsic curvature) (extrinsic curvature) K (u, v) K(u, v) = u v n (31) Σ t u v n = u µ v ν ν n µ (32) = v ν ν (u µ n µ ) n µ v ν ν u µ (33) = n µ u ν ν v µ + n µ u ν ν v µ n µ v ν ν u µ (34) µ = 0 0 ( ) u µ n µ *1 u v ι : Σ M ι : T p (Σ) T p (M) γu = ι ū ū v γ g ι g ι : T p (M) T p (Σ)γ(u, v) = g(γu, γv) = g(ι ū, ι v) = ι g(ū, v) = γ(ū, v) γ(u, v) = γ(ū, v) γ 3

4 0 n µ = α µ t n µ u ν ν v µ n µ v ν ν u µ µ tu ν ν v µ µ tv ν ν u µ (35) = u ν ( ν ( µ tv µ ) v µ ν µ t) v ν ( ν ( µ tu µ ) u µ ν µ t) (36) = u ν v µ ( ν µ t µ ν t) (37) = 0 (38) µ t u(v) µ Γ i jk u v n = n µ u ν ν v µ (39) = u ν ν (n µ v µ ) + u ν v µ ν n µ (40) = u ν v µ ν n µ (41) v µ n µ K K u γ µ νu ν = (u + (n u)n) µ µ = 0 0 K K(u, v) = K(u + (n u)n, v + (n v)n) (42) = (u µ + n ν u ν n µ )(v σ + n ρ v ρ n σ ) σ n µ (43) = u µ v σ σ n µ n ν u ν n µ v σ σ n µ u µ n ρ v ρ n σ σ n µ n ν u ν n µ n ρ v ρ n σ σ n µ (44) µ n µ ν n µ = ν (n µ n µ ) n µ ν n µ = n µ ν n µ = 0 K(u, v) = u µ v σ σ n µ u µ v ρ n ρ n σ σ n µ (45) K µν = ν n µ n ν n σ σ n µ µ n ν = K νµ n µ n σ σ n ν (46) γ µ σγ ν ρk µν = γ µ σγ ν ρ ν n µ (47) K µν = γ i µγ j ν K ij (48) γ µ σγ ν ρk µν = γ µ σγ ν ργ i µγ j ν K ij = γ i σγ j ρ K ij = K σρ (49) K µν = γ σ µγ ρ ν ρ n σ (50) n µ = α µ t n σ σ n µ = n σ σ ( α µ t) (51) = n σ σ α µ t αn σ σ µ t = n σ σ α µ t αn σ µ σ t (52) = 1 α nσ σ αn µ αn σ µ ( 1 ) α n σ (53) = 1 α nσ σ αn µ + n σ µ n σ 1 α nσ µ αn σ (54) = α 1 ( µ α + n µ n σ σ α) (55) = α 1 (δ σ µ + n σ n µ ) σ α (56) = α 1 γ σ µ σ α (57) 4

5 torsion-free γ σ µ σ γ µν D µ T n D µ T α 1α 2 α p β1β 2 β q = γ α 1 µ1 γ α p µp γ ν 1 β1 γ ν q βq γ σ ρ σ T µ 1 µ p ν1 ν q (58) D µ n D µ Leibnitz γ µν 0 µ Leibnitz D ϵ (T α βs γ ϵ) = γ α µγ ν βγ γ ργ λ δγ σ ϵ σ (T µ νs ρ λ) (59) = γ α µγ ν βγ σ ϵ( σ T µ ν)γ γ ργ λ δs ρ λ +γ α µγ ν βt µ νγ γ ργ λ δγ σ ϵ σ S ρ λ (60) = (D ϵ T α β)s γ δ + T α β(d ϵ S γ δ) (61) µ Leibnitz D σ T µ µ = D σ (T µ µ) D σ T µ µ = γ µ αγ β βγ γ σ γ T α β (62) = γ β αγ γ σ γ T α β (63) = γ γ σ γ (γ β αt α β) γ γ σt α β(n β γ n α + n α γ n β ) (64) = γ γ σ γ (T µ µ) = D σ (T µ µ) (65) γ γ β α = n β γ n α + n α γ n β u µ n f u µ D µ f = u µ γ ν µ ν f = u ν ν f (66) ν 0 f D µ D ν f = γ λ µγ ρ ν λ (γ σ ρ σ f) (67) = γ λ µγ ρ ν( λ γ σ ρ σ f + γ σ ρ λ σ f) (68) = γ λ µγ ρ ν(n σ λ n ρ + n ρ λ n σ ) σ f + γ λ µγ ρ ν λ σ f (69) = γ λ µγ ρ ν λ n ρ n σ σ f + γ λ µγ ρ ν λ σ f (70) γ σ ρ = δ σ ρ + n σ n ρ γ n γ λ µγ ρ ν λ n ρ = K µν σ 0 µ ν f µ ν D µ D ν f D ν D µ f = γ λ µγ ρ ν λ (γ σ ρ σ f) γ λ νγ ρ µ λ (γ σ ρ σ f) = 0 0 5

6 0 γ µ αγ ν βγ ρ σ ρ γ µν = γ µ αγ ν βγ ρ σ(n µ ρ n ν + n ν ρ n µ ) (71) = γ ρ σ(γ µ αn µ γ ν β ρ n ν + γ µ αγ ν βn ν ρ n µ ) = 0 (72) σ g µν = 0 γ n (58) α 1 γ σ µ σ α = α 1 D µ α (73) n σ σ n µ = α 1 D µ α (74) K µν = ν n µ α 1 (D µ α)n ν (75) 1.2 Einstein Lie αn Lie * 2 L αn γ µ ν = αn σ σ γ µ ν γ σ ν σ (αn µ ) + γ µ σ ν (αn σ ) (79) = αn σ σ (n µ n ν ) γ σ νn µ σ α αγ σ ν σ n µ + γ µ σn σ ν α + αγ µ σ ν n σ (80) = αn σ σ (n µ n ν ) + γ σ ν(αk µ σ + n σ D µ α n µ σ α) γ µ σ(αk σ ν + n ν D σ α n σ ν α) (81) = n ν D µ α + n µ D ν α + αk µ ν n µ D ν α αk µ ν n ν D µ α = 0 (82) (75) (74) (58) γ n αn Lie (p, q) T µ ν ( µ ν ) γ µ σγ ρ νt σ ρ = T µ ν (83) L αn (γ µ σγ ρ νt σ ρ) = L αn (T µ ν) (84) L αn (γ µ σγ ρ νt σ ρ) = i γ µ 1 σ1 L αn γ µ i σi γ µ p σp γ ρ νt σ ρ + j γ µ σγ ρ1 ν 1 L αn γ ρj ν j γ ρq ν q T σ ρ +γ µ σγ ν ρl αn T µ ν (85) = γ µ σγ ν ρl αn T µ ν (86) γ µ σγ ν ρl αn T µ ν = L αn T µ ν (87) *2 tortion free L αn γ µ ν = αn σ σ γ µ ν γ σ ν σ (αn µ ) + γ µ σ ν (αn σ ) (76) = αn σ σ γ µ ν αn σ Γ µ σλγ λ ν + αn σ Γ λ σνγ µ λ γ σ ν σ (αn µ ) + γ σ νγ µ σλαn λ + γ µ σ ν (αn σ ) γ µ σγ σ νλαn λ (77) = αn σ σ γ µ ν γ σ ν σ (αn µ ) + γ µ σ ν (αn σ ) (78) 6

7 T αn Lie Lie := L αn = L t L β (88) L t (1, 1) T µ ν t δ µ L t T µ ν = δ σ 0 σ T µ ν T σ ν σ δ µ 0 + T µ σ ν δ σ 0 = t T µ ν (89) = t L β (90) γ µν γ µν = αn σ σ γ µν + γ σν µ (αn σ ) + γ µσ ν (αn σ ) (91) = αn σ σ (n µ n ν ) γ σν (αk σ µ + n µ D σ α n σ µ α) γ µσ (αk σ ν + n ν D σ α n σ ν α) (92) = n µ D ν α + n ν D µ α αk νµ n µ D ν α αk µν n ν D µ α = 2αK µν (93) Lie µ g σρ = 0 γ ij = 2αK ij (94) Einstein Einstein R µν 1 2 Rgµν = 8πT µν (95) ( R µν = 8π T µν 1 ) 2 T g µν (96) T = g µν T µν E := T µν n µ n ν S µν := T σρ γ σ µγ ρ ν S := γ ij S ij = g µν S µν *3 T = g µν T µν = γ µν T µν n µ n ν T µν = γ ρσ γ µ ργ ν σt µν E = γ ρσ S ρσ E = S E Einstein γ i σ γ j ρ *4 [ R.H.S. = 8π S ij 1 ] 2 (S E)γ ij (99) ()Gauss L.H.S. = γ i σ γ j ρ R σρ (100) *3 γ µν γ µν γ µν = g µν + n µ n ν (97) S µν T n g µν S µν = γ µν S µν ( ) γ µν 0 0 = 0 γ ij (98) γ µν S µν = γ ij S ij = S *4 δ i σ δ j ρ 7

8 D µ Riemann Q µ νσρ n v D σ D ρ v µ D ρ D σ v µ = Q µ νσρv ν (101) v σ ρ n v γ µ νv ν n D µ ( ν σ ρ ) n σ ρ D σ γ α σ ν v D σ D ρ γ µ νv ν D ρ D σ γ µ νv ν = Q µ νσρv ν (102) γ µ ν = δ µ ν + n µ n ν γ σ νn σ = 0 v µ n µ = 0 (50) D σ D ρ γ µ νv ν = γ α σγ β ργ µ γ α (γ δ βγ γ ν δ (γ ν σv σ ) (103) = γ α σγ β ργ µ γ(n δ α n β γ γ ν δ (γ ν σv σ ) + n ν α n γ γ δ β δ (γ ν σv σ ) + γ δ βγ γ ν α δ (γ ν σv σ )) (104) = γ α σγ β ργ µ γ(n δ α n β γ γ ν δ (γ ν σv σ ) γ ν σv σ α n γ γ δ β δ n ν + γ δ βγ γ ν α δ (γ ν σv σ )) (105) = K σρ γ µ νn δ δ (γ ν σv σ ) K µ σk ρν v ν + γ α σγ δ ργ µ ν α δ (γ ν σv σ ) (106) (58) γ µ ν = δ µ ν + n µ n ν γ σ νn σ = 0 n µ v µ = 0 (50) K Riemann µ ν (γ ρ σv σ ) ν µ (γ ρ σv σ ) = R ρ σµνγ σ λv λ (K σν K µ ρ K ρν K µ σ)γ ν λv λ + γ τ σγ φ ργ µ λr λ ντφγ ν θv θ = Q µ νσρv ν (107) K σν n γ τ σγ φ ργ µ λγ θ νr λ θτφv ν = Q µ νσρv ν + K ρν K µ σv ν K σν K µ ρv ν (108) v γ τ σγ φ ργ µ λγ θ νr λ θτφ = Q µ νσρ + K ρν K µ σ K σν K µ ρ (109) Gauss µ σ γ τ λγ φ ργ θ νr λ θτφ = (δ τ λ + n τ n λ )γ φ ργ θ νr λ θτφ (110) = γ φ ργ θ νr θφ + n τ n λ γ φ ργ θ νr λθτφ (111) = γ φ ργ θ νr θφ + n τ n λ γ φ ργ θ νr θλφτ (112) = γ φ ργ θ νr θφ + n τ n λ γ φ ργ θν R θ λφτ (113) Riemann Ricci γ σ µγ ρ νr σρ + γ µσ n ρ γ λ νn τ R σ ρλτ = Q µν + KK µν K µσ K σ ν (114) Q µν = Q σ µσν Gauss Riemann R γ µσ n ρ γ λ νn τ R σ ρλτ = γ µσ n τ γ λ ν( λ τ n σ τ λ n σ ) (115) 8

9 γ µσ n τ γ λ ν( λ τ n σ τ λ n σ ) = γ µσ n τ γ λ ν[ λ (K σ τ + n τ n ρ ρ n σ ) + τ (K σ λ + n λ n ρ ρ n σ )] (116) = γ µσ n τ γ λ ν[ λ (K σ τ + n τ α 1 D σ α) + τ (K σ λ + n λ α 1 D σ α)] (117) = γ µσ γ λ ν[k σ τ λ n τ + λ (α 1 D σ α) + n τ τ K σ λ +α 1 D λ αα 1 D σ α + n τ n λ τ (α 1 D σ α)] (118) = γ µσ γ λ νk σ τ ( K τ λ n λ α 1 D τ α) + D ν D µ (ln α) +γ µσ γ λ νn τ τ K σ λ + D ν ln αd µ ln α (119) = K µτ K τ ν + D ν D µ (ln α) +γ σ µγ λ νn τ τ K σλ + D ν ln αd µ ln α (120) (75) K µ σn σ = 0 n σ ν n σ = 0 (75) γ n µ g νρ = 0 γ µσ n ρ γ λ νn τ R σ ρλτ = K µσ K σ ν + α 1 D ν D µ α + γ σ µγ ρ νn λ λ K σρ (121) K µν = αn σ σ K µν + K σν µ (αn σ ) + K µσ ν (αn σ ) (122) γ µ λγ ν ρ (87) K µν γ µν K λρ = αγ µ λγ ν ρn σ σ K µν + γ µ λk σρ (n σ µ α + α µ n σ ) + γ ν ρk λσ (n σ ν α + α ν n σ ) (123) = αγ µ λγ ν ρn σ σ K µν + γ µ λk σρ α( K σ µ n µ n τ τ n σ ) + γ ν ρk λσ α( K σ ν n ν n τ τ n σ ) (124) = αγ µ λγ ν ρn σ σ K µν 2αK ρσ K σ λ (125) Riemann γ µσ n ρ γ λ νn τ R σ ρλτ = α 1 K µν + α 1 D µ D ν α + K µσ K σ ν (126) Gauss γ σ µγ ρ νr σρ = α 1 K µν α 1 D µ D ν α + Q µν + KK µν 2K µσ K σ ν (127) µ = i ν = j Einstein K ij = D i D j α + α(q ij + KK ij 2K ik K k j + 4π[(S E)γ ij 2S ij ]) (128) K iµ K µ j = K ik K k j Hamiltonian constraint Einstein R µν n µ n ν R = 8πT µν n µ n ν (129) Gauss γ trace γ µν γ σ µγ ρ νr σρ = (g µν + n µ n ν )R µν = R + R µν n µ n ν (130) γ µν γ µσ n ρ γ λ νn τ R σ ρλτ = n ρ n τ γ λ σr σ ρλτ (131) = n ρ n τ R ρτ + n ρ n τ n λ n σ R σ ρλτ (132) = R µν n µ n ν (133) 9

10 Riemann R + 2R µν n µ n ν = Q + K 2 K ij K ij (134) Hamiltonian constraint 2 Hamiltonian constraint Q + K 2 K ij K ij 16πE = 0 (135) momentum constraint n µ γ µν = 0 R σρ n σ γ ρ µ = 8πS µ (136) S µ = T σν n σ γ ν µ Codazzi Riemann ( µ ν ν µ )n σ = R σ ρµνn ρ (137) γ σ µγ ρ νγ λ τ R τ θσρn θ = γ σ µγ ρ νγ λ τ ( σ ρ n τ ρ σ n τ ) (138) γ σ µγ ρ νγ λ τ σ ρ n τ = γ σ µγ ρ νγ λ τ σ ( K τ ρ n ρ n α α n τ ) (139) = D µ K λ ν γ σ µγ ρ νγ λ τ ( σ n ρ n α α n τ + n ρ σ n α α n τ + n ρ n α σ α n τ ) (140) = D µ K λ ν + γ σ µγ ρ νγ λ τ (K ρσ n α α n τ + n σ n β β n ρ n α α n τ ) (141) = D µ K λ ν + K µν (n σ σ )n λ (142) µ ν γ σ µγ ρ νγ λ τ R τ θσρn θ = D µ K λ ν + D ν K λ µ (143) Codazzi λ µ Riemann γ σ µγ ρ νγ µ τ R τ θσρn θ = R θρ γ ρ νn θ + γ ρ νn θ n σ n τ R τ θσρ = R θρ γ ρ νn θ (144) γ ρ µn σ R σρ = D µ K D σ K σ µ (145) Codazzi momentum constraint D i K D j K j i + 8πS i = 0 (146) K σ µn σ = 0 K 0 µ = 0 Einstein R µν 1 2 Rg µν = 8πT µν (147) 10 g µν Bianchi ( ) γ ij K ij 12 Hamiltonian momentum constraint 4 ( ) 4 * *

11 1.3 Einstein Einstein γ ij = e 4ϕ γ ij (148) γ ˆγ det( γ ij ) = ˆγ (149) γ ij conformally related metric ϕ ( exponent) ˆγ ij conformally related metric γ ij γ ij γ ij = e 4ϕ γ ij (150) conformally related metric D i γ ij D i Christoffel Γ i jk Γ i jk C i jk = Γ i jk Γ i jk D i *6 D i v j = D i v j + C j ikv k (151) C i jk D i D i C i jk = C i kj D i γ ij 0 0 = D i γ jk = D i γ jk C l ijγ lk C l ikγ jl (152) ijk jki kij C i jk = 1 2 γil ( D j γ lk + D k γ jl D l γ jk ) (153) γ ik γ kj = δ i j D i γ ij (148) 0 = D i γ jk = D i (e 4ϕ γ jk ) = 4e 4ϕ γ jk Di ϕ + e 4ϕ Di γ jk (154) D i γ jk = 4γ jk Di ϕ (155) C C i jk = 2(δ i D j k ϕ + δ i D k j ϕ D i ϕ γ jk ) (156) γ il γ jk = γ il γ jk D i = γ ij Dj D j γ ij Ricci (102) Riemann D µ D i Riemann 3 R i jkl Q µ νρσ 3 R i jkl *6 n D µ 11

12 (102) n n µ Q µ νσρ = 0 Q 0 νσρ = 0 Ricci Q ij = Q σ iσj = Q k ikj (157) Q i jkl Q i jkl = dx i, D ek D el e j D el D ek e j (158) e j n D µ D D ei e j G µ ij *7 D ei e j = G µ ije µ (159) *8 e 0 µ 0 G 0 ij = dx 0, D ei e j n, D ei e j = 0 (160) µ = 0 0 n n D n D ei e j = G k ije k (161) dx i, D ek D el e j = dx i, D ek (G m lje m ) (162) = dx i, D ek (G m lj)e m + G m ljd ek e m (163) = dx i, k (G m lj)e m + G m ljg n kme n (164) = k (G i lj) + G m ljg i km (165) e k n D ek k Riemann Q i jkl = k G i lj l G i kj + G m ljg i km G m kjg i lm (166) γ µν D µ 0 * 9 0 = D i γ jk = i γ jk G l ijγ lk G l ikγ jl (167) γ µν γ ij ijk jki kij i γ jk + j γ ki k γ ij 2G l ijγ lk = 0 (168) *7 Γ µ νρ eν e ρ = Γ µ νρe µ *8 D n D n G 0 0 G 0 D e 0 Q e 0 *9 n v u e i (i = 1, 2, 3) D u v = D u (v j e j ) = (D u v j )e j + u i v j D ei (e j ) v j u n D uv = u i ( i v j )e j + u i v j G k ije k D i v j = i v j + G j ikv k Leibnitz n D i ω j = i ω j G k ijω k γ µν 12

13 G i jk = 1 2 γil ( j γ lk + k γ lj l γ jk ) (169) G i jk = Γ i jk D i Riemann 3 R i jkl = k Γi lj l Γi kj + Γ m lj Γ i km Γ m kj Γ i lm (170) Q i jkl = 3 R i jkl Q ij = 3 R ij Ricci conformally related metric Ricci D i Ricci Riemann v 3 R ij v j = D j Di v j D i Dj v j (171) D i D i 3 R ij v j = D j ( D i v j ) + C j D jk i v k C k D ji k v j D i ( D j v j ) (172) = D j ( D i v j + C j ikv k ) + C j D jk i v k + C j jkc k ilv l C k D ji k v j C k jic j klv l D i ( D j v j + C j jkv k ) (173) = D j Di v j + ( D j C j ik)v k + C j D ik j v k + C j D jk i v k + C j jkc k ilv l C k D ji k v j C k jic j klv l D i Dj v j ( D i C j jk)v k C j D jk i v k (174) = D j Di v j D i Dj v j + ( D j C j ik)v k C k jic j klv l + C j jkc k ilv l ( D i C j jk)v j (175) D i Ricci R ij 3 R ij = R ij + D l C l ij C k lic l kj + C l lkc k ij D i C k kj (176) C k ki = 2(3 D i ϕ + D i ϕ D i ϕ) = 6 D i ϕ (177) D l C l ij = 2 D i Dj ϕ + 2 D j Di ϕ 2 γ ij Dl Dl ϕ (178) C k lic l kj = 4(3 D i ϕ D j ϕ + D i ϕ D j ϕ D i ϕ D j ϕ D i ϕ D j ϕ + D i ϕ D j ϕ γ ij Dl ϕ D l ϕ D i ϕ D j ϕ γ ij Dk ϕ D k ϕ + D i ϕ D j ϕ) (179) = 20 D i ϕ D j ϕ 8 γ ij Dk ϕ D k ϕ (180) C l lkc k ij = 12 D i ϕ D j ϕ + 12 D j ϕ D i ϕ 12 γ ij Dk ϕ D k ϕ (181) D i C k kj = 6 D i Dj ϕ (182) 3 R ij = R ij 2 D i Dj ϕ 2 D k Dk ϕ γ ij + 4 D i ϕ D j ϕ 4 D k ϕ D k ϕ γ ij (183) trace 3 R(= γ ij 3 R ij ) = e 4ϕ ( R 8( D k Dk ϕ + D k ϕ D k ϕ)) (184) R = γ ij Rij (134) Q Q = γ µν Q µν = γ ij Q ij = γ ij 3 R ij = 3 R (185) 13

14 Hamiltonian constraint Q R trace trace traceless : A ij = K ij 1 3 Kγ ij (186) γ ij trace 0 traceless Ā ij = e 4ϕ A ij (187) γ trace 0 γ ij Ā ij = 0 (e 4ϕ γ ij ) = 2αA ij 2 3 Kγ ij (188) (e 4ϕ γ ij ) = 4e 4ϕ γ ij ϕ + e 4ϕ γ ij (189) γ ij γ ij γ ij = ln det( γ ij ) = ln ˆγ = L β ln ˆγ = L β ln det( γ ij ) (190) tˆγ = 0 L β ln det( γ ij ) = γ ij L β γ ij = γ ij (β k Dk γ ij + γ ik Dj β k + γ kj Di β k ) = 2 D i β i (191) * 10 : ϕ = 1 6 ( D i β i αk) (195) γ ij = e 4ϕ γ ij Āij traceless conformally related metric γ ij = 4 γ ij ϕ 2αĀij 2 3 K γ ij (196) γ ij = 2αĀij 2 3 D k β k γ ij (197) Āij := γ ik γ jl Ā kl 2αĀij 2 3 D k β k γ ij = γ ik γ jl γ ij (198) = γ ik ( ( γ jl γ ij ) γ ij γ jl ) (199) = γ ik (δ l i) δ k j γ jl (200) = γ kl (201) *10 β 0 L β γ ij = β k k γ ij + γ ik j β k + γ kj i β k (192) n D ( Lie ) β k D k γ ij + γ ik D j β k + γ kj D i β k (193) D (151) D β k Dk γ ij + γ ik Dj β k + γ kj Di β k (194) 14

15 γ ij = 2αĀij D k β k γ ij (202) trace K = (K ij γ ij ) = K ij γ ij + γ ij K ij = 2αK ij K ij + γ ij K ij (203) * 11 γ ij trace L.H.S. = γ ij K ij = K 2αK ij K ij (207) R.H.S. = D i D i α + α(q + K 2 2K ij K ij + 4π(3(S E) 2S)) (208) K = D i D i α + α(q + K 2 + 4π(S 3E)) (209) Hamiltonian constraint Q + K 2 = K ij K ij + 16πE K = D i D i α + α(4π(s + E) + K ij K ij ) (210) D i D i α = D i Di α = D i (γ ij Dj α) = D i (γ ij Dj α) + C i ijγ jk Dk α (211) = D i (e 4ϕ γ ij Dj α) + 6e 4ϕ γ jk Dj ϕ D k α (212) = 4e 4ϕ Di ϕ γ ij Dj α + e 4ϕ Di Di α + 6e 4ϕ γ jk Dj ϕ D k α = e 4ϕ Di Di α + 2e 4ϕ γ jk Dj ϕ D k α traceless ( K ij K ij = A ij + 1 ) 3 Kγ ij (A ij + 13 ) Kγij = A ij A ij K2 = ĀijĀij K2 (214) (213) K Āij = e 4ϕ A ij ] K = e 4ϕ ( D Di i α + 2 D i ϕ D i α) + α [4π(E + S) + ĀijĀij + K2 3 (215) traceless K ij = A ij Kγ ij K γ ij (216) = A ij Kγ ij 2 3 αkk ij (217) K = (K ij γ ij ) = K ij γ ij + γ ij K ij (218) *11 γ ik γ jl γ ij = γ ik (γ jl γ ij ) γ ik γ ij γ jl = γ kl (204) γ ik γ jl K ij = K kl (205) γ ij = 2αK ij (206) 15

16 A ij = K ij 1 3 γ ij K αkk ij (219) K Einstein A ij = D i D j α + α(q ij + KK ij 2K ik K k j + 4π((S E)γ ij 2S ij )) 1 3 γ ij( D i D i α + α(q + K 2 + 4π(S 3E))) αkk ij (220) ( = D i D j α 1 ) ( 3 γ ijd k D k α + α Q ij 1 ) ( 5 3 γ ijq + α 3 KK ij 2K ik K k j 1 ) 3 γ ijk 2 ( 8πα S ij 1 ) 3 γ ijs (221) traceless ( A ij + 1 ) 3 Kγ ij KK ij 2K ik K k j 1 3 γ ijk 2 = 5 3 K ( A ik Kγ ik ) ( A k j + 1 ) 3 Kδk j 1 3 γ ijk 2 (222) = 5 3 KA ij K2 γ ij 2 (A ik A kj + 13 A ijk + 13 KA ij + 19 ) K2 γ ij 1 3 K2 γ ij (223) = 1 3 KA ij 2A ik A k j (224) A ij = 1 3 αka ij 2αA ik A k j + ( D i D j α + αq ij 8παS ij ) TF (225) TF traceless (trace free) T ij TF T ij 1 3 T γ ij conformally related metric A ij = (e 4ϕ Ā ij ) = e 4ϕ Ā ij + 4e 4ϕ Ā ij ϕ = e 4ϕ Ā ij e4ϕ Ā ij ( D i β i αk) (226) D i D j α = D i Dj α = D i Dj α (227) = D i Dj α C k ij D k α (228) = D i Dj α 2( D j ϕ D i α + D i ϕ D j α γ ij Dk ϕ D k α) (229) ( D i D j α 1 ) 3 γ ijd k D k α = D i Dj α + 2( D j ϕ D i α + D i ϕ D j α γ Dk ij ϕ D k α) γ ij( D Di i α + 2 γ jk Dj ϕ D k α) (230) = ( D i Dj α) TF + 2( D j ϕ D i α) TF + 2( D i ϕ D j α) TF (231) = ( D i Dj α 4 D (j ϕ D i) α) TF (232) TF γ ij trace 0 Ricci Q ij (183) Q ij 1 3 Qγ ij = R ij 2 D i Dj ϕ 2 γ ij Dk Dk ϕ + 4 D i ϕ D j ϕ 4 γ ij Dk Dk ϕ 1 3 γ ij( R 8( D Dk k ϕ + D k ϕ D k ϕ)) = (233) R TF ij 2( D i Dj ϕ) TF + 4( D i ϕ D j ϕ) TF (234) 16

17 S ij 1 3 Sγ ij = S ij 1 3 S klγ kl γ ij = S ij 1 3 S kl γ kl γ ij = S TF ij (235) A ik A k j = γ kl A ik A lj = e 4ϕ γ kl e 4ϕ Ā ik e 4ϕ Ā lj = e 4ϕ Ā ik Ā k j (236) Ā ij = 2 3 D k β k Ā ij + αkāij 2αĀikA k j +e 4ϕ ( D i Dj α + 4 D (i α D j) ϕ + α( R ij 2 D i Dj ϕ + 4 D i ϕ D j ϕ 8πS ij )) TF (237) Ricci constraints Hamiltonian constraint momentum constraint 2 3 K2 ĀijĀij 16πE + e 4ϕ ( R 8 D k Dk ϕ 8 D k ϕ D k ϕ) = 0 (238) D i K = D i K = D i K (239) D j K j i = D j K j i = D j (A j i + 1 ) 3 Kδj i = D j A j i D i K (240) D j A j i = D j Ā j i + C j jlāl i C l jiāj l (241) = D j Ā j i + 6Āl D i l ϕ 2Āj lδ l D j i ϕ 2Āj D i j ϕ + 2Āij D j ϕ (242) = D j Ā j i + 6Āj D i j ϕ (243) 2 3 D i K γ ik Dj Ā jk 6 γ ik Ā jk Dj ϕ + 8πS i = 0 (244) 1.4 BSSN BSSN Ricci Q ij = 3 R ij R ij R ij Ricci ˆR ij (= 0) Christoffel ˆΓ i jk Γ i jk = Γ i jk ˆΓ i jk 0 = D k γ ij = ˆD k γ ij Γ l ki γ lj Γ l kj γ li (245) ijk jki Γ i jk = 1 2 γil ( ˆD j γ lk + ˆD k γ jl ˆD l γ jk ) (246) D i Ricci Christroffel Γ i jk R ij = k Γk ij i Γk kj + Γ l lk Γ k ij Γ k li Γ l kj (247) 17

18 R ij = k (ˆΓ k ij + Γ k ij) i (ˆΓ k kj + Γ k kj) +(ˆΓ l lk + Γ l lk)(ˆγ k ij + Γ k ij) (ˆΓ k li + Γ k li)(ˆγ l kj + Γ l kj) (248) = k ˆΓk ij iˆγk kj + ˆΓ l ˆΓ lk k ij ˆΓ k liˆγ l kj + k Γ k ij + ˆΓ l lk Γ k ij ˆΓ k li Γ l kj i Γ k kj + ˆΓ k ij Γ l lk ˆΓ l kj Γ k li + Γ l lk Γ k ij Γ k li Γ l kj (249) = ˆR ij + k Γ k ij + ˆΓ l lk Γ k ij ˆΓ k li Γ l kj ˆΓ l ki Γ k lj i Γ k kj ˆΓ l kj Γ k li + ˆΓ k ij Γ l lk + ˆΓ l ki Γ k lj + Γ l lk Γ k ij Γ k li Γ l kj (250) 0 ˆD i R ij = ˆD l Γ l ij ˆD i Γ k kj Γ k li Γ l kj + Γ l lk Γ k ij (251) Γ ijk = γ il Γ l jk Γ ijk = Γ ikj (252) ˆD i γ jk = Γ jki + Γ kji (253) ˆD k Γ k ij = 1 2 ˆD k ( γ kl ( ˆD i γ jl + ˆD j γ il ˆD l γ ij )) (254) = ˆD k γ kl Γ lij γkl ( ˆD k ˆDi γ jl + ˆD k ˆDj γ il ˆD k ˆDl γ ij ) (255) 0 = ˆD k (δ i j) = ˆD k ( γ il γ lj ) = γ il ˆDk γ il + γ il ˆDk γ il ˆD k γ ij = γ il γ jm ˆDk γ lm (256) = γ il γ jm ( Γ lmk + Γ mlk ) (257) = γ jm Γ i mk γ il Γ j lk (258) * 12 ˆD k Γ k ij = 1 2 γkl ( ˆD k ˆDi γ jl + ˆD k ˆDj γ il ˆD k ˆDl γ ij ) γ kl Γ m kl Γ mij Γ l ij Γ k kl (259) ˆD j Γ k ki = 1 2 ˆD i γ kl ˆDj γ kl γkl ˆDj ˆDi γ kl (260) = 1 2 ( γkm Γ l mi + γ lm Γ k mi)( Γ klj + Γ lkj ) γkl ˆDj ˆDi γ kl (261) = 1 2 γkl ˆDj ˆDi γ kl Γ k lj Γ l ki γ kl Γ m lj Γ mki (262) R ij = 1 2 γkl ( ˆD k ˆDi γ jl + ˆD k ˆDj γ il ˆD k ˆDl γ ij ˆD i ˆDj γ kl ) + γ kl ( Γ m lj Γ mki Γ m kl Γ mij ) (263) *12 ˆDk γ kl = γ lm Γ k mk γ km Γ l mk 18

19 Laplace Laplace BSSN Γ k := γ ij Γ k ij 0 = ˆD i (δ j k) = ˆD i ( γ jl γ lk ) = ( ˆD i γ jl ) γ lk + γ jl ˆDi γ lk Γ k := γ ij Γ k ij (264) = 1 2 γij γ kl ( ˆD i γ lj + ˆD j γ il ˆD l γ ij ) (265) = 1 2 γij γ lj ˆDi γ kl 1 2 γij γ li ˆDj γ kl γ kl ˆDl ln γ (266) = ˆD l γ kl γ kl ˆDl ln γ (267) 1 2 γkl ˆDk ˆDi γ jl = 1 2 ˆD k ( γ kl ˆDi γ jl ) 1 2 ˆD k γ lk ˆDi γ jl (268) = 1 2 ˆD k ( γ jl ˆDiˆγ kl ) Γk ˆDi γ jl γkl ( ˆD k ln γ) ˆD i γ jl (269) = 1 2 ˆD k γ jl ˆDi γ kl 1 2 γ jl ˆD i ˆDk γ kl Γk ˆDi γ jl γkl ( ˆD k ln γ) ˆD i γ jl (270) = 1 2 ( Γ jlk + Γ ljk )( γ km Γ l im + γ lm Γ k im) γ jl ˆD i ( Γ l + γ lm ˆDm ln γ) Γk ˆDi γ jl γkl ( ˆD k ln γ) ˆD i γ jl (271) = γ km Γ l mi Γ jlk γkm Γ l mi Γ lkj Γl mi Γ m lj 1 2 γ lj ˆD i Γ l ˆD i ˆDj ln γ Γl Γ jil Γl Γ lij (272) ˆD i i j 1 2 γkl ˆDi ˆDj γ kl = 1 2 ˆD i ( γ kl ˆDj γ kl ) ˆD i γ kl ˆDj γ kl (273) = ˆD i ˆDj ln γ 1 2 ( γkm Γ l im + γ lm Γ k im)( Γ klj + Γ lkj ) (274) = ˆD i ˆDj ln γ Γ l mi Γ m li γ km Γ l mi Γ lkj (275) 1 2 γkl ˆDk ˆDi γ jl γkl ˆDk ˆDj γ il 1 2 γkl ˆDi ˆDj γ kl = γ km Γ l m(i Γ j)lk + γ l(i ˆDi) Γ l + ˆD i ˆDj ln γ + Γ l Γ (ij)l + γ km Γ l km Γ lij (276) Λ k constraint Λ k Γ k = 0 Ricci R ij = 1 2 γkl ˆDk ˆDl γ ij + γ k(i ˆDj) Λk + Γ k Γ (ij)k + γ kl (2 Γ m k(i Γ j)ml + Γ m ik Γ mjl ) (277) Āij Λ k 19

20 γ = ˆγ Λ k = ˆD l γ kl γ kl ˆDl ln γ (278) = ˆD l γ kl γ kl ˆDl ln ˆγ (279) ( = ˆD l γ kl 1 2 γklˆγ ij ˆDlˆγ ij ) (280) = ˆD l γ kl (281) (202) ˆD j divergence γ ij = t γ ij β k ˆDk γ ij + γ kj ˆDk β i + γ ik ˆDk β j (282) ˆD j t γ ij = t ˆDj γ ij = t Λi = ( ˆD j β k ) ˆD k γ ij + β k ˆDk ˆDj γ ij ( ˆD j γ kj ) ˆD k β i γ kj ˆDj ˆDk β i ( ˆD j γ ik ) ˆD k β j (283) γ ik ˆDk ˆDj β j + 2( ˆD j α)āij + 2α ˆD j Ā ij ( ˆD j ˆDk β k ) γ ij ˆD k β k ˆDj γ ij (284) = 2Āij ˆDj α + 2α ˆD j Ā ij + Λ k ˆDk β i β k ˆDk Λi γ kj ˆDj ˆDk β i 1 3 γik ˆDk ˆDj β j 2 3 ˆD k β k Λi (285) Λ i t = 2α ˆD j Ā ij 2Āij ˆDj α + β k ˆDk Λi Λ k ˆDk β i Λ i ˆDk β k + γ jk ˆDj ˆDk β i γij ˆDj ˆDk β k (286) β k ˆDk Λi Λ k ˆDk β i = L Λi β det( γ ij ) = ˆγ ˆD k β k = k β k + ˆΓ k kjβ j = k β k j ln ˆγβ j = k β k j ln det( γ kl )β j = k β k + Γ k kjβ j = D k β k ˆD j D j j D j Ā ij = ˆD j Ā ij + Γ i jkākj + ( Γ j jk ˆΓ j jk)āik = ˆD j Ā ij + Γ i jkākj (287) momentum constraint D j Ā ij = 2 3 γki Dk K 6Āji Dj ϕ + 8π γ ik S k (288) Λi = γ jk ˆDj ˆDk β i Γi Dj β j γik Dk Dj β j 2Ājk (δ i j k α 6αδ i j k ϕ α Γ i jk) 4 3 α γij j K 16πα γ ij S j (289) BSSN γ ij K ij = 12 Hamiltonian 1 momentum 3 constraint 8 γ ij K ij 8 constraint 4 Conformal transvers-traceless method 20

21 traceless extrinsic curvature A ij à ij = e 10ϕ A ij = e 6ϕ Ā ij (290) Ãij γ ij transverse longitudinal à ij = ( LX) ij + Ãij TT (291) Ãij TT transverse traceless γ ij à ij TT = 0 (292) transverse D j à ij TT = 0 (293) LX longitudinal Killing L X ( LX) ij := D i X j + D j X i 2 3 D k X k γ ij (294) γ ij ( LX) ij = 0 Ãij divergence D j à ij = D j ( LX) ij = D j Dj X i + D j Di X j 2 3 D i Dk X k (295) = D j Dj X i + D i Dj X j + R i jx j 2 3 D i Dk X k (296) = D j Dj X i D i Dj X j + R i jx j (297) D i Dj X k D j Di X k = R k lijx l D j Di X j D i Dj X j = R i jx j Laplacian L X i Hamiltonian constraint à e ϕ Dk Dk ϕ + e ϕ Dk ϕ D k ϕ 1 8 Re ϕ + ÃijÃij e 7ϕ 1 12 K2 e 5ϕ + 2πEe 5ϕ = 0 (298) momentum constraint D j à jk 2 3 e6ϕ Di K γ ik Dj à jk + 8πe 6ϕ S i = 0 (299) L X i 2 3 e6ϕ Di K = 8πe 6ϕ γ ik S k (300) ϕ X i γ ij Ãij TT K constraints ϕ X i constraint conformal transverse-traceless method * 13 Conformal thin-sandwich method *13 ( ) γ ij (6) K ij (6) 12 ϕ(1) γ ij (5) K(1) à ij TT (2) Xi (3) 12 ϕ X i Hamiltonian constraint momentum constraint 8 γ ij (5) K(1) Ãij TT (2) α βi 4 β i 3 γ ij = 2αĀij 2 3 D k β k γ ij γ ij α 1 K = D i D i α + α( 3 R + K 2 + 4π(S 3E)) K (α) (β i ) γ ij 2 Āij 2 4 γ ij K 8 21

22 Conformal transverse-traceless method γ K γ t γ γ ij = 2αĀij D k β k γ ij (301) L β γ ij = β k Dk γ ij γ ik Dk β j γ kj Dk β i = γ ik Dk β j γ kj Dk β i = ( Lβ) ij 2 3 γij Dk β k (302) Ā ij = 1 ( t γ ij + ( Lβ) ij) (303) 2α Āij = e 6ϕ à ij conformal lapse ᾱ := e 6ϕ α momentum constraint à ij = 1 ( t γ ij + ( Lβ) ij) (304) 2ᾱ 2 3 e6ϕ Di K γ ik Dj à jk + 8πe 6ϕ S i = 0 (305) ( ) ( ) e6ϕ Di K γ ik Dj α t γ ij + γ ik Dj 1 α ( Lβ) ij + 16πe 6ϕ S i = 0 (306) D j ( 1 α ( Lβ) ij ) + D j ( 1 α t γ ij ) 4 3 e6ϕ Di K 16πe 6ϕ γ ij S j = 0 (307) t γ ij K S j ϕ shift vector β ϕ Hamiltonian constraint conformal transverse-traceless method γ ij t γ ij K α E S i ϕ β i constraint conformal thin-sandwich method (conformal) lapse t K α D Di i α + 2 D i ϕ D i α = e ϕ ( D Di i (αe ϕ ) α D Di i e ϕ ) (308) K K = e 5ϕ ( D i Di (αe ϕ ) α D i Di e ϕ ) + α ] [4π(E + S) + ĀijĀij + K2 3 (309) αe ϕ = αe 7ϕ Hamiltonian constraint ( 1 D Di i ( αe 7ϕ ) ( αe 7ϕ ) 8 R K2 e 4ϕ + 8ÃijÃij 7 ) e 8ϕ + 2π(E + 2S)e 4ϕ +( t K β i Di K)e 5ϕ = 0 (310) γ ij t γ ij K t K E S S i ϕ α β extended conformal thin-sandwich method 1.6 BSSN Lapse shift Lapse shift (geodesic slicing maximal slicing harmonic slicing 1 + log slicing) (normal coordinates minimal distortion Gamma freezing Gamma driver) 22

23 geodesic slicing α α = 1 (311) geodesic slicing α = 1 β i = 0 t K = 4π(E + S) + K ij K ij 0 (312) K K = γ µν µ n ν = g µν µ n ν n µ n ν µ n ν = g µν µ n ν = µ n µ (313) K n * 14 β = 0 n t maximal slicing maximal slicing K = 0 (323) Lapse K = 0 K D i D i α = α [ 4π(E + S) + K ij K ij] (324) *14 u µ divergence affine parameter timelike unit vector u µ affine parameter affine parameter u µ normal vector affine parameter affine parameter deviation vector ξ µ deviation vector L u ξ = 0 u unit vector t + t ξ t ξ tu Lie u µ µ ξ ν = ξ µ µ u ν t + t u ξ u ξ(t + t) = u ξ(t) + t t (u ξ) (314) = u ξ(t) + tu µ µ (u ξ) (315) = u ξ(t) + tu νu µ µξ ν (316) = u ξ(t) + tu ν ξ µ µ u ν (317) = u ξ(t) (318) unit vector u µ ν u µ = 0 u ξ = 0 ξ u leaf foliation lapse shift lapse 1 shift 0 ( t) γ(t)d 3 x t + t ( ) γ(t) γ(t + t)d 3 t x = + 2γ tγ d 3 x (319) lapse 1 shift 0 tγ ij = 2K ij trace γ ij t γ ij = 1 γ tγ = 2K (320) ( ) γ(t + t)d 3 x = γ(t) K t d 3 x (321) ( ) = γ(t) + µu µ t d 3 x (322) timelike unit vector divergence K 23

24 K = 0 µ n µ = 0 (325) harmonic slicing harmonic slicing µ µ t = 0 (326) µ µ t = 1 g µ ( gg µν ν t ) (327) = 1 ( µ gg µν δ 0 ) ν g (328) = 1 g µ ( gg µ0 ) (329) µ ( gg µ0 ) = 0 (330) g = α γ * 15 ( γ t α t ( γ α ) + ( ) γ γ α x i α βi = α 2 γ = α 2 γ = α 2 ) + ( ) γ x i α βi = 0 (331) γ α γ βi α2 x i + α α βi x i α γ γ t t 1 γ α t ( α t ( α t α βi x i α γ γ t β i x i + βi γ α + α γ β i γ x i x i (332) ) + α βi x i + α ) γ x i ( γβ i ) (333) (334) α t [ α 1 βi x i α ] γ γ D i β i = 0 (335) t γ ij = 2αK ij (336) trace γ ij γ ij = γ ij t γ ij γ ij L β γ ij (337) = t ln γ γ ij (β k D k γ ij + γ ik D j β k + γ kj D i β k (338) = 2 t ln γ 2D k β k (339) = 2αK (340) α = α 2 K (341) *15 g 00 γ gg 00 g 00 = α 2 g = α 2 γ 24

25 harmonic slicing α β = 0 harmonic condition C(x i ) t ( ) γ = 0 (342) α α = C(x i ) γ (343) γ α 1 + log slicing harmonic slicing α = Kα 2 f(α) f(α) = 2/α 1 + log slicing α α = 2αK (344) t α L β α = t ln γ 2D i β i (345) β = 0 t α = t ln γ (346) α = 1 + log γ (347) harmonic slicing γ γ normal coordinates β i = 0 (348) normal coordinates coordinate shear minimal distortion Q ij γ ij traceless * 16 Q ij := γ ij t 1 3 γkl γ kl t γ ij (350) traceless 5 shift 3 (A ij )transverse longitudinal X i D j (LX) ij = D j Q ij (351) *16 γ γ t γ(t + t) γ(t) = t 2 1 γ tγ = t 2 γij t γ ij (349) tγ ij γ ij trace tγ ij traceless 25

26 (LX) ij = D i X j + D j X i 2 3 D kx k γ ij (352) X i Q TT ij = Q ij (LX) ij (353) transverse Q TT ij γ ij minimal distortion X = 0 D j Q ij = 0 Q ij = 2αK ij + L β γ ij 1 3 γ ij( 2αK + 2D k β k ) = 2αA ij + (Lβ) ij (354) minimal distortion 2αD j A ij 2A ij D j α + D j (Lβ) ij = 0 (355) momentum constraint D i K D j (A j i δj ik ) + 8πS i = 0 (356) D j A ji 2 3 Di K 8πS i = 0 (357) 4 3 αdi K 2A ij D j α 16παS i + D j (Lβ) ij = 0 (358) Laplacian D j (Lβ) ij = D j D j β i Di D j β j + R i jβ j (359) D j D j β i Di D j β j + R i jβ j = 16παS i αdi K + 2A ij D j α (360) minimal distortion β coordinate shear Gamma freezing conformally related metric γ ij = e 4ϕ γ ij (361) det( γ ij ) = ˆγ det( γ ij ) = e 12ϕ det(γ ij ) (362) ln det(γ ij ) = 12ϕ + ln ˆγ (363) γ ij t γ ij = t ln det(γ ij ) = 12 t ϕ + t ln ˆγ = 12 t ϕ (364) conformally related metric Q ij = t γ ij 4γ ij t ϕ = t (e 4ϕ γ ij ) ( t e 4ϕ ) γ ij = e 4ϕ t γ ij (365) 26

27 D j Q ij = 0 ˆD j ( t γ ij ) = 0 (366) Gamma freezing minimal distortion 0 = ˆD j ( t γ ij ) = t ( ˆD j γ ij ) = t Λi (367) Λ i constraint t Γ k = 0 (368) Delta Gamma Gamma freezing β (289) D j ˆD j * 17 Λi t = 0 Gamma driver k t β i = k t Γ i (369) β Gamma driver t 2 β i = k t Γ i (η t ln k) t β i (370) Gamma driver t β i = kb i (371) t B i = t Γ i ηb i (372) k = 3/4 puncture gauge 1 + log slicing Gamma driver puncture gauge *17 Dj β j = ˆD j β j 27