ADMtoNP.tex

filename=“ADMtoNP.tex” 2003-0311 HShinkaihshinkai@postman.riken.go.jp

The published version of Gunnarsen-Shinkai-Maeda [1] has several typo in the process of printing.

This note is for confirming the notations in the article, including one more variations in the signature in the metric (sin (11)).

The Newman-Penrose formulation [2, 3] has many advantages, especially for treating gravitational wave dynamics.

• Natural framework for calculations in radiative space-time.

• Variables have geometrical meanings.

• Practical advantages in treating Petrov type-D space-time.

• Closely related with spinor formalism.

Newman-Penrose’s variables are based on real-valued null vectorsl,nand complex conjugate null vectors m,m, which satisfy

l·n=l^an_a=l_an^a= 1, m·m=m^am_a=m_am^a=−1, and else 0. (1) This set of null basis (l^a, n^a, m^a, m^a) have relations with orthogonal tetrad basis (t^a, x^a, y^a, z^a) as

l^a=o^Ao^A = (1/√

2)(t^a+z^a), m^a=o^Aι^A = (1/√

2)(x^a−iy^a), (2) n^a=ι^Aι^A = (1/√

2)(t^a−z^a), m^a=ι^Ao^A = (1/√

2)(x^a+iy^a), (3) where I also put spinor basis expressions (o^A, ι^A) of those.

Metric g_ab will be recovered by

g_ab =ηijeⁱ_ae^j_b = 2l_(an_b)−2m_(am_b)=tat_b−xax_b−yay_b−zaz_b (4) whereη_ij =η^ij = diag

0 1

1 0

,

0 −1

−1 0

. The Weyl curvature Cabcd is defined as

C_abcd =R_abcd−g_a[cR_d]b+g_b[cR_d]a−1

3Rg_a[cg_d]b. (5)

The 10 components of Weyl curvature are expressed by the following 5 complex scalars [4];

Ψ0 ≡ ψABCDo^Ao^Bo^Co^D =Cabcdl^am^bl^cm^d, n^a-directed transverse component,{4,0} (6) Ψ₁ ≡ ψ_ABCDo^Ao^Bo^Cι^D =C_abcdl^an^bl^cm^d, n^a-directed longitudinal component{2,0} (7) Ψ₂ ≡ ψ_ABCDo^Ao^Bι^Cι^D =C_abcdl^am^bm¯^cn^d, ‘Coulomb’ component,{0,0} (8) Ψ₃ ≡ ψ_ABCDo^Aι^Bι^Cι^D =C_abcdl^an^bm¯^cn^d, l^a-directed longitudinal component,{−2,0} (9) Ψ4 ≡ ψABCDι^Aι^Bι^Cι^D =C_abcdn^am¯^bn^cm¯^d, l^a-directed transverse component,{−4,0} (10) where{p, q} indicatesspin- and boost-weighted type and prime-operation will be defined later.

Gunnarsen-Shinkai-Maeda [1] derived a transformation formula of Weyl scalar Ψi from ADM variables (γ_ij, K_ij), motivated by an application to interpret numerically generated space-time. Here,

1

we consider vacuum space-time. Let (M, ηab) be real, 4-dimensional Lorentz vector space with volume formε_abcd;ε_abcdε^abcd=−4!. Let (t^a, x^a, y^a, z^a) be orthonormal basis of (M, η_ab), and define t^ata= +s (s=±1), ε_abc=ε^abcdt_d, (11) where the tensor fieldε_abc=ε_[abc]satisfiesε_abcε^abc= 3! We formulate our equations in the signatures both (+,−,−,−) and (−,+,+,+) by choosing s = 1 or −1, respectively¹, because the former notation is common in working with the spinors.

First, we define the Weyl curvature C_abcd by (5) and decompose those into its electric and a magnetic components,

E_ab≡ − C_ambnt^mtⁿ, B_ab ≡ −^∗C_ambntⁿt^m, (12) where ^∗C_abcd= ¹₂ε_ab^mnC_mncd is a dual of the Weyl tensor. These decomposed elementsE_ab andB_ab are also presented by the 3-metric γ_ab and the extrinsic curvatureK_ab as [6]

E_ab = R_ab−K_a^mK_bm+KK_ab− 2

3Λγ_ab, (13)

Bab = ε_a^mnDmKnb. (14)

This is why we emphasize that our inputs are ‘3+1’ elements. It follows from two constraint equations that the fields Eab, Bab are both trace-free and symmetric. We can reconstruct the Weyl curvature completely from E_ab and B_ab by

C_abcd = 4t_[aE_b][ct_d]+ 2ε_ab^mB_m[ct_d]+ 2ε_cd^mB_m[at_b]+ε_ab^mε_cdⁿEmn. (15) The next step is to choose a unit vector field ˆz^aon Σ, and to decomposeE_ab, B_ab into components along and perpendicular to ˆz^a. We set

e = Eabzˆ^azˆ^b, (16)

e_a = E_bczˆ^b(δ_a^c+sˆz_azˆ^c), (17) e_ab = E_cd(δ_a^c+sˆz_azˆ^c)(δ_b^d+sˆz_bzˆ^d) +1

2es_ab, (18)

b = B_abzˆ^azˆ^b, (19)

b_a = B_bczˆ^b(δ_a^c+sˆz_azˆ^c), (20) b_ab = B_cd(δ_a^c+sˆzazˆ^c)(δ_b^d+sˆz_bzˆ^d) +1

2bs_ab, (21)

wheres_ab=γ_ab−zˆ_azˆ_b. We note that E_ab, B_ab is again reconstructed from (16)-(21)

E_ab = eˆz_azˆ_b+ 2e_(azˆ_b)+e_ab−(1/2)s_abe. (22) B_ab = bˆzazˆ_b+ 2b_(azˆ_b)+b_ab−(1/2)s_abb. (23) Such decompositions will be useful to discuss the effects of curvatures on the transversal plane to the ˆz^adirection.

We put a rotation operator on the plane spanned by ˆx_a and ˆy_a as,

J_a^b≡ε_a^bcdzˆ_ct_d. (24)

It is easy to check this mapping preserves s_ab, and is also easy to check J_a^cJ_c^b = −(δ_a^b +sˆzazˆ^b), which shows us J_a^b has a complex structure, i.e., J_a^b lets us define complex multiples of vectors

1That is, the metric isηab=s(tatb−xaxb−yayb−zazb).

2

x^a∈Pz, according to the formula (m+in)x^a=mx^a+nJ_b^ax^b. In short,J_a^b expresses a rotation by 90 degrees in the plane orthogonal to ˆz^a.

By substituting (15) and (2, 3) into (6)-(10), we get Ψ_i using (??) and (24):

Ψ0 = −(e_ab+sJ_a^cb_bc)m^am^b, (25) Ψ1 = −(s/√

2) (ea+sJ_a^cbc)m^a, (26)

Ψ₂ = −(1/2) (e+ib), (27)

Ψ₃ = −(s/√

2)(e_a−sJ_a^cb_c) ¯m^a, (28) Ψ₄ = −(e_ab−sJ_a^cb_bc) ¯m^am¯^b. (29) This relation has been applied to many groups’ numerical codes, and helps their simulation’s physical understandings. Weyl scalars are also useful for evaluating Riemann (Kretchman) invariant as

CabcdC^abcd = Ψ4Ψ0−4Ψ1Ψ3+ 3Ψ²₂. (30) Note that

RabcdR^abcd =CabcdC^abcd+ 2RabR^ab−(1/3)R². (31) I hope this note helps you.

References

[1] L. Gunnarsen, H. Shinkai, and K. Maeda, Class. Quant. Grav.12 (1995) 133

[2] E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962);ibid. 4, 998 (1963) errata.

[3] R. Penrose and W. Rindler, Spinors and Space-time, vol.1 and 2, (Cambridge University Press, 1984, 1986)

S. Chandrasekhar,The Mathematical Theory of Black Holes, (Oxford Univ. Press, 1992).

J. Stewart,Advanced General Relativity, (Cambridge Univ. Press, 1990).

D. Kramer, H. Stephani, M. MacCallum, E. Herlt,Exact solutions of Einstein’s field equations, (Cambridge University Press, 1980).

[4] R.K. Sachs, Proc. Roy. Soc. London, A264, 309 (1961); ibidA270, 103 (1962).

P. Szekeres, J. Math. Phys. 6, 1387 (1965).

[5] R. Gerosh, A. Held and R. Penrose, J. Math. Phys. 14, 874 (1973).

[6] L. Smarr, Ann. N.Y. Acad. of Sci.302, 569 (1977).

Hisaaki SHINKAI

____________________________________________________________

E-mail: hshinkai@postman.riken.go.jp

URL: http://atlas.riken.go.jp/~shinkai/

address: Computational Science Division

Institute of Physical & Chemical Research (RIKEN) Hirosawa 2-1, Wako, Saitama, 351-0198 Japan

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