Newman-Penrose formalism in higher dimensions

Transcription

1 Newman-Penrose formalism in higher dimensions V. Pravda various parts in collaboration with: A. Coley, R. Milson, M. Ortaggio and A. Pravdová

2 Introduction - algebraic classification in four dimensions Weyl tensor in a complex null tetrad Petrov classification Algebraic classification in higher dimensions General method Weyl tensor Static and stationary spacetimes NP formalism in higher dimensions Ricci identities and their consequences Bianchi identities and their consequences Conclusions

3 The complex null tetrad in 4D and Weyl tensor In 4D the complex null tetrad - null vectors (l, n, m, m) with the only non-vanishing contractions being l a n a = 1, m a m a = 1 is usually used. It can be also constructed from the spinor basis (o A, ι A ) by l a o A ōȧ, n a ι A ῑȧ, m a o A ῑȧ, m a ι A ōȧ. Weyl tensor is fully described by its five complex frame components Ψ 0... Ψ 4. These can be expressed in terms of Weyl tensor C abcd or Weyl spinor Ψ ABCD as Ψ 0 = C abcd l a m b l c m d = Ψ ABCD o A o B o C o D Ψ 1 = C abcd l a n b l c m d = Ψ ABCD o A o B o C ι D Ψ 2 = C abcd l a m b m c n d = Ψ ABCD o A o B ι C ι D Ψ 3 = C abcd l a n b m c n d = Ψ ABCD o A ι B ι C ι D Ψ 4 = C abcd m a n b m c n d = Ψ ABCD ι A ι B ι C ι D.

4 Petrov classification of the Weyl tensor In general the Weyl spinor Ψ ABCD = α (A β B γ C δ D). Various classes correspond to multiplicities of principal null directions. Type I alg. general t. (1, 1, 1, 1) 4 different PNDs, Ψ 0 = 0, l b l c l [e C a]bc[d l f] = 0 Type II alg. special t. (2, 1, 1), Ψ 0 = Ψ 1 = 0, l b l c C abc[d l e] = 0 Type III alg. special t. (3, 1), Ψ 0 = Ψ 1 = Ψ 2 = 0, l c C abc[d l e] = 0 Type N alg. special t. (4), Ψ 0 = Ψ 1 = Ψ 2 = Ψ 3 = 0, l c C abcd = 0 Type O alg. special t., C abcd = 0

5 I II D III N O Penrose digram Peeling theorem for an isolated matter distribution: C abcd = [N] abcd r + [III] abcd r 2 + [II] abcd r 3 + [I] abcd r 4 + O(r 5 ).

6 Einstein s equations can be considerably simplified for algebraically special spacetimes. One of the reasons (in the vacuum case) is Goldberg-Sachs theorem: A vacuum metric is algebraically special if and only if it contains a shearfree geodetic null congruence l κ = σ = 0 Ψ 0 = Ψ 1 = 0. In fact, most of the known exact vacuum solutions in 4D are algebraically special: Kerr-Newmann, C-metric, Kerr-Schild metrics, Kundt metrics (e.g. pp-wave), Robinson-Trautman metrics etc.

7 Algebraic classification of tensors in n dimensions [Coley, Milson, Pravda, Pravdová, CQG 21, L35-L42, 2004] [Milson, Coley, Pravda, Pravdová, Int. J. Geom. Meth. Mod. Phys. 2, 41, 2005] Let us work in the frame m (0) = l, m (1) = n, m (i), i, j, k = 2... n 1, with two null vectors n, l l a l a = n a n a = 0, l a n a = 1, a = 0... n 1 and n 2 spacelike vectors m (i), m a (i) m (j)a = δ ij, i, j, k = 2... n 1. The metric has the form g ab = 2l (a n b) + δ ij m (i) a m (j) b.

8 The group of ortochronous Lorentz transformations is generated by null rotations spins and boosts ˆl = l + z i m (i) 1 2 zi z i n, ˆn = n, ˆm (i) = m (i) z i n, ˆl = l, ˆn = n, ˆm (i) = X i j m(j), ˆl = λl, ˆn = λ 1 n, ˆm (i) = m (i). A quantity q has a boost weight b if it transforms under a boost according to ˆq = λ b q. Boost order of a tensor T is the maximum boost weight of its frame components.

9 Proposition: Let l, n, m (i) and ˆl, ˆn, ˆm (i) be two null-frames with l and ˆl scalar multiples of each other. Then, the boost order of a given tensor is the same relative to both frames. Thus boost order of a tensor depends (only) on the choice of a null direction - b T (l). Let T be a fixed tensor and let b max denote the maximum value of b(k) taken over all null vectors k. We will say that a null vector k is aligned with the tensor T whenever b(k) < b max and we will call integer b max b(k) 1 order of alignment.

10 Classification of the Weyl tensor in higher dimensions Operation { } : w {a x b y c z d} 1 2 (w [a x b] y [c z d] + w [c x d] y [a z b] ). Decomposition of the Weyl tensor in its frame components: C abcd = + boost weight 2 type G {}}{ 4C 0i0j n {a m (i) b n cm (j) d } 1, I {}}{ 8C 010i n {a l b n c m (i) d } + 4C 0ijk n {a m (i) b m(j) c m (k) d } +4C 0101 n {a l b n c l d } + 4C 01ij n {a l b m (i) c m (j) d } +8C 0i1j n {a m (i) b l cm (j) d } + C ijkl m (i) + + {a m(j) b m(k) c m (l) d } 1, III {}}{ 8C 101i l {a n b l c m (i) d } + 4C 1ijk l {a m (i) 2, N {}}{ 4C 1i1j l {a m (i) b l c m(j) d }. b m(j) c m (k) d } 0, II,D

11 For the Weyl tensor b max = 2. A generic Weyl tensor for n 5 does not possess any aligned null direction. All possible algebraical types are given in the table. Alignment type classification of the Weyl tensor in 4D reproduces the Petrov classification. n > 4 dimensions 4 dimensions Weyl type alignment type Petrov type G G I (1) I i (1,1) I II (2) II i (2,1) II D (2,2) D III (3) III i (3,1) III N (4) N

12 l b l c l [e C a]bc[d l f] = 0 = l is WAND, type I l b l c C abc[d l e] = 0 = l is WAND, type II l c C abc[d l e] = 0 = l is WAND, type III l c C abcd = 0 = l is WAND, type N Note that there is in general no equivalence in these relations in higher dimensions. For example it can be shown that the most general Weyl tensor satisfying l c C abcd = 0 has form C ijkl m (i) {a m(j) b m(k) c m (l) d } +4C 1ijk l {a m (i) b m(j) c m (k) d } +4C 1i1j l {a m (i) b l cm (j) d }. The curvature invariant of this Weyl tensor is C abcd C abcd = C ijkl C ijkl = Σ(C ijkl ) 2 which is non-zero as long as C ijkl has a non-vanishing component. Necessary and sufficient conditions in terms of l for various types are given in [Ortaggio, arxiv: ].

13 Compact notation Bost weight zero components C 0101, C 01ij, C 0i1j, C ijkl. Φ ij C 0i1j [(n 2) (n 2) real matrix] Φ S ij - symmetric part, ΦA ij - antisymmetric part, Φ - trace. From the symmetries and the tracelessness of the Weyl tensor C 01ij = 2C 0[i 1 j] = 2Φ A ij, C 0(i 1 j) = Φ S ij = 1 2 C ikjk, C 0101 = 1 2 C ijij = Φ. B. w. zero compts are determined by m(m 1) 2 components of Φ A ij and m2 (m 2 1) 12 independent components of C ijkl, where m = n 2. Negative b.w. components: Ψ i C 101i, Ψ ijk 1 2 C 1kij, Ψ ij 1 2 C 1i1j.

14 Static spacetimes l is a WAND l b l c l [e C a]bc[d l f] = 0. Assume l = (l t, l A ), A = 1... n 1. Static metric and therefore also the above equation is invariant under t = t. Therefore, in these new coordinates ñ = (l t, l A ) is also a WAND. In the original coordinates n = ( l t, l A ). Thus existence of a WAND l implies existence of a distinct WAND n with same order of alignment. Weyl types compatible with this property are types G, I i and D (or, trivially, O). Proposition: All static spacetimes in arbitrary dimension are of Weyl types G, I i or D, unless conformally flat. [n=4, vacuum: discussed already in Petrov, Einstein spaces, 1961] explicit examples: - static charged black ring - type G - static vacuum black ring - type I i - Schwarzschild-Tangherlini black hole - type D

15 Stationary spacetimes 5D Myers-Perry black hole ds 2 = ρ 2 4 dx2 + ρ 2 dθ 2 dt 2 + (x + a 2 ) sin 2 θdφ 2 + (x + b 2 ) cos 2 θdψ 2 + r 0 2 ρ 2 (dt + a sin2 θdφ + b cos 2 θdψ) 2, is invariant under t = t, φ = φ, ψ = ψ. WANDs l, n = (x + a2 )(x + b 2 ) [ t a x + a 2 φ b ] x + b 2 ψ ±2 x x. Thus in a stationary spacetime with appropriate reflection symmetry one can construct a second WAND n from the original one l by using the reflection symmetry. The only issue is that for some special cases one could in principle obtain n = l. Note, however, that expansion of both WANDs is identical l a ;a = n a ;a while the expansion of l is equal to l a ;a. Thus

16 Proposition: In arbitrary dimension, all stationary spacetimes with non-vanishing divergence scalar ( expansion ) and invariant under appropriate reflection symmetry are of Weyl types G, I i or D, unless conformally flat. Explicit examples of stationary spacetimes subject to the above Proposition are expanding stationary spacetimes with n 2 commuting Killing vector fields (e.g. apart from MP also black rings, doubly spinning black ring, black saturn, black di-rings, black strings etc.).

17 NP formalism in HD - Ricci rotation coefficients reminder: our frame is m (0) = n, m (1) = l, m (i), i = 2... n 1 Ricci rotation coefficients L ab, N ab and i M ab are defined by l a;b = L cd m (c) a m (d) b, n a;b = N cd m (c) a m (d) b, m (i) where a, b = 0... n 1. For example l a;b = L 11 l a l b +L 10 l a n b +L 1i l a m (i) b +L i1m (i) a a;b = M i cd m (c) a l b +L i0 m (i) a n b +L ij m (i) m (d) b, a m (j) b. Let us decompose L ij into its tracefree symmetric part σ ij (shear), its trace θ (expansion) and its antisymmetric part A ij (twist) L ij = σ ij + θδ ij + A ij, σ ij L (ij) n 2 1 kkδ ij, θ n 2 1 kk, A ij L [ij].

18 Optical scalars l is geodetic iff L i0 = 0. Other scalar quantities (apart from expansion) out of l a;b : shear and twist σ 2 σ 2 ii = σ ijσ ji, ω 2 A 2 ii = A ija ji. If l is affinely parametrized, i.e. take the form L 10 = 0, the optical scalars σ 2 = l (a;b) l (a;b) 1 n 2 ( l a ;a ) 2, θ = 1 n 2 la ;a, ω2 = l [a;b] l a;b.

19 Comparison with standard 4D notation For n = 4, L ab, N ab and M i ab are equivalent to the 12 complex NP spin coefficients. With the notation 2m = m (2) im (3) : 2κ 2l a;b m a l b = L 20 il 30, 2ρ 2l a;b m a m b = L 22 + L iL [23], 2σ 2l a;b m a m b = L 22 L 33 2iL (23), 2τ 2l a;b m a n b = L 21 il 31, 2ν 2na;b m a n b = N 21 + in 31, 2µ 2n a;b m a m b = N 22 + N 33 2iN [23], 2λ 2n a;b m a m b = N 22 N iN (23), 2π 2na;b m a l b = N 20 + in 30, ε + ε l a;b n a l b = L 10, ε ε m a;b m a l b = i 2 M 30, γ + γ n a;b l a n b = L 11, γ γ m a;b m a n b = i 2 M 31, 2(β ᾱ) 2ma;b m a m b = 2 M 33 + i 2 M 32, 2(β + ᾱ) 2l a;b n a m b = L 12 il 13.

20 Ricci identities [Ortaggio, Pravda, Pravdová, CQG 24, 1657 (2007)] Contractions of v a;bc v a;cb = R sabc v s with various combinations of the frame vectors (and with v a = l a, n a and m a ) lead to (i) a set of first order differential equations. Covariant derivatives along the frame vectors: D l a a, n a a, δ i m a (i) a. An example of a Ricci identity (Sachs equation) DL ij δ j L i0 = L 10 L ij L i0 (2L 1j + N j0 ) L i1 L j0 + 2L k[0 k M i j] L ik (L kj + k M j0) C 0i0j 1 n 2 R 00δ ij In a frame parallely propagated along a geodetic congruence l DL ij = L ik L kj C 0i0j 1 n 2 R 00δ ij

21 For type I (or more special) with R 00 = 0 (e.g. vacuum) DL ij = L ik L kj, matrix form: DL = L 2. A direct consequence: Proposition: In an n > 4 dimensional spacetime, n odd, for a geodetic WAND A ij 0 σ ij 0, i.e., a twisting geodetic WAND must also be shearing. A counterexample to a HD Goldberg-Sachs theorem (algebraically special vacuum spacetimes are shearfree in n = 4). E.g. n=5 Myers-Perry black holes. If L is invertible we obtain linear ODEs [Ortaggio, Pravda, Pravdová, arxiv: (2009)] DL 1 = I.

22 Bianchi identities Contractions of R ab{cd;e} = R abcd;e + R abde;c + R abec;d = 0 with various combinations of the frame vectors - set of first order PDEs. An example: For vacuum type N spacetimes the Weyl tensor has the form C abcd = 8Ψ ij l {a m (i) b l cm (j) d }, where Ψ ij = 1 2 C 1i1j is symmetric and traceless.

23 Bianchi identities for vacuum type N read DΨ ij = k 2Ψ k(i M j)0 Ψ ik L kj 2Ψ ij L 10, δ [k Ψ j]i = s Ψ is M [jk] + M s i[j Ψ k]s + Ψ i[j L k]1 + 2L 1[j Ψ k]i, 0 = L k[i Ψ j]k, 0 = Ψ i[j L k]0, 0 = L k[j Ψ m]i + L i[m Ψ j]k, 0 = Ψ i{k A jm}. These equations have two important consequences The multiple WAND l is geodetic (L i0 = 0), Rank of L ij = rank of Ψ ij = 2. We can see that here geodetic part of Goldberg-Sachs holds while shearfree part does not. Note, however, that shear of the only non-vanishing two-block in L ij is zero. This in fact holds in much more general context.

24 Geodeticity of WANDs For geodetic WAND in vacuum, Bianchi equations for the leading boost weight terms of the Weyl tensor containig operator D (determining r-dependence) decouple. This leads to considerable simplifications. There are however explicit examples of vacuum spacetimes with non-geodetic multiple WANDs. For n 7 see [Pravda, Pravdová, Ortaggio, CQG 24, 4407 (2007)] For n 5 [Godazgar, Reall, arxiv: (2009)], [Durkee, arxiv: (2009)] Nevertheless, all these examples are somewhat exceptional. In fact one can prove that:

25 Proposition: In arbitrary dimension, multiple WAND of type III or N vacuum spacetime is always geodetic. Proposition: In arbitrary dimension, multiple WANDs of type II and D Einstein spacetimes are geodetic if at least one of the following conditions is satisfied: i) Φ A ij is non-vanishing ; ii) for all eigenvalues of Φ S ij : p (i) Φ. Thus multiple WAND in a generic vacuum type II and D Einstein spacetime is geodetic.

26 r-dependence for various algebraical types [Pravdová, Pravda, CQG 25, (2008)] Special case with all non-vanishing eigenvalues of L ij being equal and A ij = 0. r is affine parameter along geodetic multiple WAND Indices p, q,... - non-vanishing eigenvalues of L ij. Indices v, w,... - vanishing eigenvalues of L ij. type N: ψ pq 1 r, the rest vanishes; type III: ψ p 1 r 2, ψ pqr 1 r 2, mixed ψ wqr 1 r ; ( type II: Φ pq = Φ0 pq r + δ pq ) Φ 0 mr m+1 Φ0 ww m+1, Φ wv = Φ 0 wv

27 Conclusions Petrov classification and NP formalism can be generalized to arbitrary higher dimension. In 4D they reproduce the standard classification/formalism. Standard Golberg-Sachs does not hold in HD. Thus S ij is not proportional to δ ij as in 4D vacuum. Moreover in some (very special) cases multiple WANDs are not geodetic. Despite these complications, in some cases, one can obtain interesting results even without specifying dimension. See talk by A. Pravdová for a specific example.