Dual null formulation (and its Quasi-Spherical version)

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1 filename= dualnull.tex Hisaaki Shinkai Dual null formulation (and its Quasi-Spherical version This note is for actual coding of the double null formulation by Hayward [1, 2], expecially its quasi-spherical approximated version [3, 4]. Contents 1 Dual-null formulation (without conformal scaling metric Lie derivatives Geometrical quantities Full version Full set of Einstein equation Full set of Einstein equation HS notes: friendly expressions Quasi-spherical version Full set of Einstein equation Full set of Einstein equation 2 (quasi-spherical approximation HS notes: friendly expressions Dual-null formulation (with conformal scaling Introducing the conformal decomposition Full version Quasi-spherical version HS notes Numerical Scheme for Quasi-Spherical approx. space-time 8 4 Schwarzschild metric as an example 9 1

2 1 Dual-null formulation (without conformal scaling 1.1 metric ds 2 = h ij (dx i + p i dx + + m i dx (dx j + p j dx + + m j dx 2e f dx + dx (1 = h ij dx i dx j +2p l dx l dx + +2m l dx l dx + p l p l dx ml m l dx 2 +2(pl m l e f dx + dx (2 In the matrix form, and p 2 e f + p l m l p j g ab = e f + p l m l m 2 m j (3 p i m i h ij 0 e f e f m i g ab = e f 0 e f p i (4 e f m j e f p j h ij e f (p i m j + p j m i 1.2 Lie derivatives We use Lie derivatives L ± along the null normal vectors l ± = u ± s ± = e f g 1 (n, (5 that is a l + a l = u a + p a =(1, 0, 0, 0 (0, 0,p 1,p 2 =(1, 0, p 1, p 2 (6a = u a m a =(0, 1, 0, 0 (0, 0,m 1,m 2 =(0, 1, m 1, m 2. (6b The Lie derivative is defined in general as ξ S = ξ α α S, (7a ξ V µ = ξ α α V µ V α α ξ µ = ξ α α V µ V α α ξ µ, (7b ξ U µ = ξ α α U µ + U α µ ξ α, = ξ α α U µ + U α µ ξ α, (7c ξ T µν = ξ α α T µν T αν α ξ µ T µα α ξ ν = ξ α α T µν T αν α ξ µ T µα α ξ ν, (7d ξ W µν = ξ α α W µν + W αν µ ξ α + W µα ν ξ α = ξ α α W µν + W αν µ ξ α + W µα ν ξ α (7e therefore for scalar, vector and tensor quantities, L + S = l a + a S = x +S p k k S (8a L S = l a a S = x S m k k S (8b L + V a = x +V a p k k V a V k k p a (8c L V a = x V a m k k V a V k k m a (8d L + h ij = x +h ij p k k h ij +2h k(i j p k (8e L h ij = x h ij m k k h ij +2h k(i j m k (8f 2

3 1.3 Geometrical quantities From the original expressions The fields (θ ±,σ ±,ν ±,ω encode the extrinsic curvature of the dual-null foliation. These extrinsic fields are unique up to duality ± and diffeomorphisms which relabel the null hypersurfaces, i.e. dx ± e λ ± dx ± for functions λ ± (x ±. θ ± = L ± 1 (9a σ ± = L ± h θ ± h (9b ν ± = L ± f (9c ω = 1 2 ef h([l,l + ] (9d where is the Hodge operator of h ab. The functions θ ± are the expansions, the traceless bilinear forms σ ± are the shears, the 1-form ω is the twist, measuring the lack of integrability of the normal space, and the functions ν ± are the inaffinities, measuring the failure of the null normals to be affine. HS notes For more friendly expressions, these are 1 θ ± = L ± det h ij (10a det hij More concrete, σ ±ab = h c ah d bl ± h cd θ ± h ab (10b ν ± = L ± f (10c ω a = 1 2 ef h ab [l,l + ] b = 1 2 ef h ab (l c c l+ b l+ c c l b = 1 2 ef h ab (l c c l+ b l+ c c l b (10d θ + = θ = 1 [ det x + det h ij p k k det h ij ] hij 1 [ det x det h ij m k k det h ij ] hij (11a (11b σ +ab = h c ah d bl + h cd θ + h ab = h c ah d b[ x +h cd p k k h cd +2h k(c d p k ] θ + h ab (11c σ ab = h c ah d bl h cd θ h ab = h c ah d b[ x h cd m k k h cd +2h k(c d m k ] θ h ab (11d ν + = L + f = x +f p k k f (11e ν = L f = x f m k k f (11f ω a = 1 2 ef h ab ( x l+ b m k k l+ b x +l b + p k k l b (11g 1.4 Full version Full set of Einstein equation 1 Inverting the definitions of the momentum fields yields the propagation equations (L + s L s + = 2e f h 1 (ω+[s,s + ] (12a L ± h = θ ± h + σ ± (12b L ± f = ν ±. (12c 3

4 The full set of Einstein equations is obtained with the below Full set of Einstein equation 2 From Appendix B in [1] (with the current convention: L + θ + = ν + θ θ hac h bd ( σ +ab σ +cd (13a L + θ = θ + θ + e f 1 2 R + hab (ω a ω b 1 2 D ad b f D afd b f + ω a D b f D a ω b (13b ( L + ν = 1 2 θ +θ hac h bd σ +ab σ cd + e f 1 2 R + hab (3ω a ω b 1 4 D afd b f ω a D b f (13c L + σ ab = 1 2 (θ +σ ab θ σ +ab +h cd σ +c(a σ bd +2e f ( ω a ω b 1 2 D ad b f D afd b f + ω (a D b f D (a ω b and e f h cd ( ω c ω d 1 2 D cd d f D cfd d f + ω c D d f D c ω d h ab (13d L + ω a = θ + ω a (Dν + Dθ + θ + Df+ 1 2 hbc D c σ +ab (13e L θ = ν θ 1 2 θ2 1 4 hac h bd ( σ ab σ cd (14a L θ + = θ + θ + e f 1 2 R + hab (ω a ω b 1 2 D ad b f D afd b f ω a D b f + D a ω b (14b ( L ν + = 1 2 θ +θ hac h bd σ +ab σ cd + e f 1 2 R + hab (3ω a ω b 1 4 D afd b f + ω a D b f (14c L σ +ab = 1 2 (θ +σ ab θ σ +ab +h cd σ c(a σ +bd +2e f ( ω a ω b 1 2 D ad b f D afd b f ω (a D b f + D (a ω b e f h cd ( ω c ω d 1 2 D cd d f D cfd d f ω c D d f + D c ω d h ab (14d L ω a = θ ω a 1 2 (Dν Dθ θ Df 1 2 hbc D c σ ab (14e where D is the covariant derivative of h ab. There is no L + σ +ab, L σ ab, L + ν +, and L ν. In spherical symmetry, (σ ±,ω,dvanish, while (θ ±,ν ±,D ± generally do not HS notes: friendly expressions For the equations in ( h a b L + m b L p b = h a b ( x +m b p k k m b m k k p b x p b + m k k p b + p k k m b ( = h a b x +m b x p b =2e f h ab ω b + m c c p a p c c m a (15a h c ah d bl + h cd = h c ah d b( x +h cd p k k h cd +2h k(c d p k =θ + h ab + σ +ab (15b h c ah d bl h cd = h c ah d b( x h cd m k k h cd +2h k(c d m k =θ h ab + σ ab (15c L + f = x +f p k k f = ν + (15d L f = x f m k k f = ν (15e 4

5 1.5 Quasi-spherical version Full set of Einstein equation 1 Inverting the definitions of the momentum fields yields the propagation equations (L + s L s + = 2e f h 1 (ω+[s,s + ] (16a L ± h = θ ± h + σ ± (16b L ± f = ν ±. (16c The full set of Einstein equations is obtained with the below (17a-(17e. There is no L + σ +ab, L σ ab, L + ν +, and L ν Full set of Einstein equation 2 (quasi-spherical approximation After the truncations for quasi-spherical approximations, L ± θ ± = ν ± θ ± 1 2 θ2 ± (17a L ± θ = θ + θ e f r 2 (17b L ± ν = 1 2 θ +θ e f r 2 (17c L ± σ = ± 1 2 (θ +σ θ σ + (17d L ± ω = θ ± ω ± 1 2 (Dν ± Dθ ± θ ± Df (17e where D is the covariant derivative of h ab. (θ ±,ν ±,D ± generally do not. In spherical symmetry, (σ ±,ω,dvanish, while HS notes: friendly expressions For the equations in L ± θ ± = ± θ ± s k ± k θ ± = ν ± θ ± 1 2 θ2 ± that is L + θ + = + θ + p k k θ + = ν + θ θ2 + (18a L θ = θ m k k θ = ν θ 1 2 θ2 (18b L ± θ = ± θ s k ± k θ = θ + θ e f r 2 that is L + θ = + θ p k k θ = θ + θ e f r 2 (18c L θ + = θ + m k k θ + = θ + θ e f r 2 (18d L ± ν = ± ν s k ± k ν = 1 2 θ +θ e f r 2 that is L + ν = + ν p k k ν = 1 2 θ +θ e f r 2 (18e L ν + = ν + m k k ν + = 1 2 θ +θ e f r 2 (18f h c ah d bl ± σ cd = h c ah d b( x σ cd s k ± k σ cd +2σ k(c d s k ± = ± 1 2 (θ +σ ab θ σ +ab (18g h b al ± ω b = h b a( ± ω b s k ± k ω b + ω k b s k ± = θ ± ω a ± 1 2 (D aν ± D a θ ± θ ± D a f = θ ± ω a ± 1 2 ( aν ± a θ ± θ ± a f (18h 5

6 2 Dual-null formulation (with conformal scaling 2.1 Introducing the conformal decomposition It is also possible to integrate all the way from I to I + by a conformal transformation. We take the conformal decomposition of h ab = r 2 k ab, (19 such that D ± 1 =0 (D ± det k ab =0 (20 where is the Hodge operator of k, satisfying 1 = r 2. The shear equations, composed into a second-order equation for k, become k ab =0 (21 where is the quasi-spherical wave operator: ( φ = 2e f L (+ L φ +2r 1 L (+ rl φ ( = e f L + L φ + L L + φ +2r 1 L + rl φ +2r 1 L rl + φ In practice, one may use the conformal factor and the rescaled expansions and shears which are finite and generally non-zero at I. 2.2 Full version Not available here. 2.3 Quasi-spherical version (22 Ω=r 1 (23 ϑ ± = rθ ± (24a ς ± = r 1 σ ± (24b Of the dynamical fields and operators introduced above, (s ±,σ ±,ω,dvanish in spherical symmetry, while (h, f, θ ±,ν ±, ± generally do not. The quasi-spherical approximation consists of linearizing in (s ±,σ ±,ω,d, i.e. setting to zero any second-order terms in these quantities. This yields a greatly simplified truncation [3] of the full field equations, the first-order dual-null form of the vacuum Einstein system[2]. In particular, the truncated equations decouple into a three-level hierarchy, the last level being irrelevant to determining the gravitational waveforms. The remaining equations are the quasi-spherical equations ± Ω = 1 2 Ω2 ϑ ±, (25 ± f = ν ±, (26 ± ϑ ± = ν ± ϑ ± 1 4 Ω ς ± 2, }{{} (27 2nd order ± ϑ = Ω( 1 2 ϑ +ϑ + e f, (28 ± ν = Ω 2 ( 1 2 ϑ +ϑ + e f 1 4 ς +,ς, }{{} (29 2nd order 6

7 and the linearized equations ± k = Ως ±, (30 ± ς = Ω( ς + k 1 ς }{{} 1 2 ϑ ς ±. (31 1st but missing in below These are all ordinary differential equations; no transverse D derivatives occur. Thus we have an effectively two-dimensional system to be integrated independently at each angle of the sphere. The 2nd order terms are pointed out in [5]. The initial-data formulation is based on a spatial surface S orthogonal to l ± and the null hypersurfaces Σ ± generated from S by l ±, assumed future-pointing. The initial data for the above equations are (Ω,f,k,ϑ ± ons and (ς ±,ν ± onσ ±. We will take l + and l to be outgoing and ingoing respectively. For the quasi-spherical approximation to be valid near I ±,amodification of the integration scheme is suggested, such that initial data is given at spatial infinity i 0 rather than Σ. Specifically, one may fix (Ω,f,ϑ ±,k=(0, 0, ± 2,ɛati 0, where ɛ = dϑ dϑ + sin 2 ϑdϕ dϕ is the standard metric of a unit sphere. The first step would then be to integrate backwards along I as far as Σ, fixing ν + =0onΣ +. The remaining coordinate data is given by ν on the ingoing null hypersurface Σ, which is left free so that one may adapt the foliation of Σ to the surfaces which are most spherical. 2.4 HS notes For scalar and tensor quantities: ± S = L ± S = ( ± S s k ± k S= ± S s k ± k S ± W ij = L ± W ij = h a i h b jl ± W ab = h a i h b j( ± W ab s k ± k W ab +2W k(a b s k ± Therefore we can write the evolution equations as the following: ± Ω = ± Ω s k ± k Ω= 1 2 Ω2 ϑ ± that is + Ω=p k k Ω 1 2 Ω2 ϑ + Ω=m k k Ω 1 2 Ω2 ϑ (32a (32b ± f = ± f s k ± k f = ν ± that is + f = p k k f + ν + f = m k k f + ν ± ϑ ± = ± ϑ ± s k ± k ϑ ± = ν ± ϑ ± that is + ϑ + = p k k ϑ + ν + ϑ + ϑ = m k k ϑ ν ϑ ± ϑ = ± ϑ s k ± k ϑ Ω( 1 2 ϑ +ϑ + e f that is + ϑ = p k k ϑ Ω( 1 2 ϑ +ϑ + e f ϑ + = m k k ϑ + Ω( 1 2 ϑ +ϑ + e f (32c (32d (32e (32f (32g (32h ± ν = ± ν s k ± k ν Ω 2 ( 1 2 ϑ +ϑ + e f that is + ν = p k k ν Ω 2 ( 1 2 ϑ +ϑ + e f ν + = m k k ν + Ω 2 ( 1 2 ϑ +ϑ + e f (32i (32j 7

8 ± k ij = h a i h b j( ± k ab s k ± k k ab +2k k(a b s k ±=Ως ±ij that is h a i h b j + k ab = h a i h b j(p k k k ab 2k k(a b p k +Ως +ij h a i h b j k ab = h a i h b j(m k k k ab 2k k(a b m k +Ως ij (32k (32l ± ς ij = h a i h b j( ± ς ab s k ± k ς ab +2ς k(a b s k ±= 1 2 Ωϑ ς ±ij that is h a i h b j + ς ab = h a i h b j(p k k ς ab 2ς k(a b p k 1 2 Ωϑ ς +ij h a i h b j ς +ab = h a i h b j(m k k ς +ab 2ς +k(a b m k 1 2 Ωϑ +ς ij (32m (32n 3 Numerical Scheme for Quasi-Spherical approx. space-time The variables are (Ω,f,ϑ +,ϑ,ν +,ν,k ab,ς +ab,ς ab 1. Prepare initial data on Σ Set metric components (f,k ab onσ. Set extrinsic curvature components (ϑ ± onσ. 2. Prepare data on Σ Assume m a =0onΣ Set (ς,ν onσ. Integrate (Ω,f,ϑ ±,ν +,k ab,ς +ab by equations. Ω=m k k Ω 1 2 Ω2 ϑ f = m k k f + ν ϑ = m k k ϑ ν ϑ ϑ + = m k k ϑ + Ω( 1 2 ϑ +ϑ + e f ν + = m k k ν + Ω 2 ( 1 2 ϑ +ϑ + e f h a i h b j k ab = h a i h b j(m k k k ab 2k k(a b m k +Ως ij h a i h b j ς +ab = h a i h b j(m k k ς +ab 2ς +k(a b m k 1 2 Ωϑ +ς ij (33a (33b (33c (33d (33e (33f (33g 3. One step to go in U (from 0 to x + first time Assume p a =0inU. Set (ς +,ν + onσ + ( x +. Integrate (Ω,f,ϑ ±,ν,k ab,ς ab by + equations. + Ω=p k k Ω 1 2 Ω2 ϑ + + f = p k k f + ν + + ϑ + = p k k ϑ + ν + ϑ + + ϑ = p k k ϑ Ω( 1 2 ϑ +ϑ + e f + ν = p k k ν Ω 2 ( 1 2 ϑ +ϑ + e f h a i h b j + k ab = h a i h b j(p k k k ab 2k k(a b p k +Ως +ij h a i h b j + ς ab = h a i h b j(p k k ς ab 2ς k(a b p k 1 2 Ωϑ ς +ij (34a (34b (34c (34d (34e (34f (34g 8

9 Note that we do not integrate ν + and ς +, that is we do not have them except on x + =0 surface and on the Σ + surface. Therefore we have to use ν + (x + =0and ς + (x + =0in the above RHSs. 4. Integrate along l direction Assume m a =0inU. Set (ς +,ν + onσ + ( x +. Integrate (ν +,ς +ab by equations. ν + = m k k ν + Ω 2 ( 1 2 ϑ +ϑ + e f h a i h b j ς +ab = h a i h b j(m k k ς +ab 2ς +k(a b m k 1 2 Ωϑ +ς ij (35a (35b where we need interpolate ϑ ±, Ω,f,ς ij in ODE solver, since they are only given on the grid points. To get the accurate solution, we have better to integrate these variables (except ς. Ω=m k k Ω 1 2 Ω2 ϑ f = m k k f + ν ϑ = m k k ϑ ν ϑ ϑ + = m k k ϑ + Ω( 1 2 ϑ +ϑ + e f (36a (36b (36c (36d As for ς and ν,wehave to interpolate them from the grid points. In order to check the accuracy, we also integrate (k ab and compare it with the results of step 2. h a i h b j k ab = h a i h b j(m k k k ab 2k k(a b m k +Ως ij (37 5. check the accuracy If bad, then go back to the step 2. Integrate equations using the obtained ν + ( x + and ς + ( x + together with ν + (0 and ς + (0 with their linear interpolations. If good, then fix the values ν + ( x + and ς + ( x + and go to the step 2 for the integration from x + to 2 x +. 4 Schwarzschild metric as an example We are using the metric with the form ds 2 = Ω 2 k ij (dx i + p i dx + + m i dx (dx j + p j dx + + m j dx 2e f dx + dx (38 Schwarzschild metric is ds 2 = (1 2m r dt2 + dr2 1 2m r + r 2 (dθ 2 + sin 2 θdφ 2 (39a = (1 2m r [ dt2 + dr 2 ]+r 2 (dθ 2 + sin 2 θdφ 2 (39b = (1 2m r 2dudv + r2 (dθ 2 + sin 2 θdφ 2 (39c 9

10 where and r = r +2m ln( r 2m 1, dr dr =1 2m r (40 u = 1 2 (t r, v = 1 2 (t + r. (41 Immediately we obtain Ω = r 1 (42a k ij = diag(1, sin 2 θ (42b ( f = ln 1 2m (42c r m i = p i =0 (42d Take + = v, = u, then 2 θ ± = ± r (1 2m r that is ϑ ± = ± 2(1 2m r (43a ν ± = 2 m r 2 (43b ( 1 0 σ ±ab = θ ± 0 sin 2 that is θ ( 1 0 ς ±ab = Ωθ ± 0 sin 2 (43c θ References [1] S.A. Hayward, Ann. Inst. H. Poincaré 59, 399 (1993. [2] S.A. Hayward, Class. Quantum Grav. 10, 779 (1993. [3] S.A. Hayward, Phys. Rev. D63, (2000. [4] H. Shinkai and S.A. Hayward, Phys. Rev. D64, (2001. [5] S.A. Hayward, gr-qc/