Higher dimensional Kerr-Schild spacetimes 1

Transcription

1 Higher dimensional Kerr-Schild spacetimes 1 Marcello Ortaggio Institute of Mathematics Academy of Sciences of the Czech Republic Bremen August Joint work with V. Pravda and A. Pravdová, arxiv: [gr-qc]

2 Contents 1 Why Kerr-Schild spacetimes? Kerr-Schild in n = 4 dimensions Myers-Perry black holes in n > 4 dimensions 2 Formalism: null frames, classification of Weyl, geometric optics 3 Basic properties of Kerr-Schild in n 4 4 Vacuum Kerr-Schild solutions Non-expanding solutions Expanding solutions 5 Summary and outlook

3 Why Kerr-Schild spacetimes? Why Kerr-Schild spacetimes? Kerr-Schild ansatz g ab = η ab 2Hk a k b with k a k a = 0 Why Kerr-Schild spacetimes? original motivation: radiation [Trautman 62] contains various important solutions in n = 4 [Kerr-Schild 65, Debney-Kerr-Schild 69] rotating vacuum black holes in n > 4 [Myers-Perry 86] mathematical tractability: insight into general properties of exact solutions? find new solutions?

4 Why Kerr-Schild spacetimes? Kerr-Schild in n = 4 dimensions Schwarzschild ds 2 = dt 2 + dx 2 + dy 2 + dz 2 + 2m r k ak b dx a dx b Kerr ds 2 = dt 2 + dx 2 + dy 2 + dz 2 + Kerr-Newman 2mr3 r 4 + a 2 z 2 k ak b dx a dx b ds 2 = dt 2 + dx 2 + dy 2 + dz 2 + 2mr3 e 2 r 2 r 4 + a 2 z 2 k a k b dx a dx b k a dx a = dt + zdz r(xdx + ydy) a(xdy ydx) + r r 2 + a 2 r 2 + a 2, (x 2 + y 2 )(r 2 + a 2 ) 1 + z 2 r 2 = 1, A a dx a r = e 3 k r 4 +a 2 z 2 a dx a.

5 Why Kerr-Schild spacetimes? Myers-Perry black holes Myers-Perry black holes Black hole holes n 4: vacuum, rotating, asymptotically flat. For n even, i = 1,..., n 2 2 : ds 2 = dt 2 + dx i dx i + dy i dy i + dz 2 2Hk a k b dx a dx b k a dx a = dt + zdz + r 2H = µr 1 (n 2)/2 i=1 x 2 i + y2 i r 2 + a 2 i (n 2)/2 i=1 (n 2)/2 i=1 + z2 r 2 = 1. ( r(xi dx i + y i dy i ) r 2 + a 2 i a 2 i (x2 i + y2 i ) (r 2 + a 2 i )2 1 (n 2)/2 j=1 a ) i(x i dy i x i dy i ) r 2 + a 2, i 1 r 2 + a 2, j

6 Formalism: null frames, classification of Weyl, geometric optics Null frames Null frame (l, n, m (i) ) in an n-dimensional spacetime (n 4): l n = 1, m (i) m (j) = δ ij (i = 2,..., n 1) Metric g ab = 2l (a n b) + δ ij m (i) a m (j) b invariant under: Null rotations l = l, n = n + z i m (i) 1 2 zi z i l, m (i) = m (i) z i l Spatial rotations Boosts l = l, n = n, m (i) = X i jm (j) l = λl, n = λ 1 n, m (i) = m (i)

7 Formalism: null frames, classification of Weyl, geometric optics Classification of the Weyl tensor Frame decomposition of Weyl: C abcd = b=+2 z } { 4C 0i0j n {a m (i) b n cm (j) d } + + 4C 0101 n {a l b n c l d } + C 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) b=+1 z } { 8C 010i n {a l b n c m (i) d } + 4C 0ijk n {a m (i) {a m(j) b m(k) c m (l) d } ) b=0 b m(j) c + 8C 101i l {a n b l c m (i) 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, {z } {z } b= 1 b= 2 m (k) d } Algebraically special Weyl tensors If C 0i0j = 0, i.e. l is a WAND (Weyl Aligned Null Direction). Further alignment of l (and n) algebraic types (G,)I,II,(D,)III,N. [Coley-Milson-Pravda-Pravdová 04, Milson-Coley-Pravda-Pravdová 05]

8 Formalism: null frames, classification of Weyl, geometric optics Classification of the Weyl tensor Frame decomposition of Weyl: C abcd = b=+2 z } { 4C 0i0j n {a m (i) b n cm (j) d } + + 4C 0101 n {a l b n c l d } + C 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) b=+1 z } { 8C 010i n {a l b n c m (i) d } + 4C 0ijk n {a m (i) {a m(j) b m(k) c m (l) d } ) b=0 b m(j) c + 8C 101i l {a n b l c m (i) 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, {z } {z } b= 1 b= 2 m (k) d } Algebraically special Weyl tensors If C 0i0j = 0, i.e. l is a WAND (Weyl Aligned Null Direction). Further alignment of l (and n) algebraic types (G,)I,II,(D,)III,N. [Coley-Milson-Pravda-Pravdová 04, Milson-Coley-Pravda-Pravdová 05]

9 Formalism: null frames, classification of Weyl, geometric optics Classification of the Weyl tensor Frame decomposition of Weyl: C abcd = b=+2 z } { 4C 0i0j n {a m (i) b n cm (j) d } + + 4C 0101 n {a l b n c l d } + C 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) b=+1 z } { 8C 010i n {a l b n c m (i) d } + 4C 0ijk n {a m (i) {a m(j) b m(k) c m (l) d } ) b=0 b m(j) c + 8C 101i l {a n b l c m (i) 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, {z } {z } b= 1 b= 2 m (k) d } Algebraically special Weyl tensors If C 0i0j = 0, i.e. l is a WAND (Weyl Aligned Null Direction). Further alignment of l (and n) algebraic types (G,)I,II,(D,)III,N. [Coley-Milson-Pravda-Pravdová 04, Milson-Coley-Pravda-Pravdová 05]

10 Formalism: null frames, classification of Weyl, geometric optics Classification of the Weyl tensor Frame decomposition of Weyl: C abcd = b=+2 z } { 4C 0i0j n {a m (i) b n cm (j) d } + + 4C 0101 n {a l b n c l d } + C 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) b=+1 z } { 8C 010i n {a l b n c m (i) d } + 4C 0ijk n {a m (i) {a m(j) b m(k) c m (l) d } ) b=0 b m(j) c + 8C 101i l {a n b l c m (i) 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, {z } {z } b= 1 b= 2 m (k) d } Algebraically special Weyl tensors If C 0i0j = 0, i.e. l is a WAND (Weyl Aligned Null Direction). Further alignment of l (and n) algebraic types (G,)I,II,(D,)III,N. [Coley-Milson-Pravda-Pravdová 04, Milson-Coley-Pravda-Pravdová 05]

11 Formalism: null frames, classification of Weyl, geometric optics Geometric optics Frame components of the covariant derivative of l: 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 l b +L i0 m a (i) n b +L ij m (i) a m (j) b. Geodesic vector field l 0 = l a;b l b = L 10 l a + L i0 m (i) a L i0 = 0 = L 10 Matrix L ij l a;b m (i)a m (j)b characterizes the optics of a geodesic l: L ij = S ij + A ij, S ij L (ij) = σ ij + θδ ij, A ij L [ij]. Optical scalars shear σ 2 σ ij σ ij = l (a;b) l a;b 1 n 2 (la ;b )2 expansion θ = 1 n 2 S ii = 1 n 2 la ;a twist ω 2 A ij A ij = l [a;b] l a;b [Frolov-Stojković 03, Pravda-Pravdová-Coley-Milson 04, Lewandowski-Pawlowski 05]

12 Basic properties of Kerr-Schild in n 4 Kerr-Schild ansatz Take: define: n dim. flat metric η ab = diag( 1, 1,..., 1) null 1-form k a (η ab k a k b = 0) function H g ab = η ab 2Hk a k b then: k a = η ab k b = g ab k b g ab k a k b = 0: null in both geometries g ab = η ab + 2Hk a k b : simple (linear) inverse and Γ a bc = (Hka k b ),c (Hk a k c ),b + η ad (Hk b k c ),d + 2Hk a k d (Hk b k c ),d, R a b = (Hka k b ), s s (Hk a k s ),bs (Hk b k s ), a s, etc.

13 Basic properties of Kerr-Schild in n 4 Geometric optics Γ a bc =... L i0 k a;b m (i)a k b = L i0, L ij k a;b m (i)a m (j)b = L ij Optics of k inherited from flat space k geodetic in g ab k geodetic in η ab same shear, expansion and twist in both geometries

14 Basic properties of Kerr-Schild in n 4 Geometric optics Γ a bc =... L i0 k a;b m (i)a k b = L i0, L ij k a;b m (i)a m (j)b = L ij Optics of k inherited from flat space k geodetic in g ab k geodetic in η ab same shear, expansion and twist in both geometries and R ab k a k b = 2HL i0 L i0 Geodesic condition k geodetic T ab k a k b = 0 In particular: vacuum Λ-term aligned Maxwell (F a b kb k a ) aligned pure radiation (T ab k a k b ), etc....

15 Basic properties of Kerr-Schild in n 4 Geometric optics Γ a bc =... L i0 k a;b m (i)a k b = L i0, L ij k a;b m (i)a m (j)b = L ij Optics of k inherited from flat space k geodetic in g ab k geodetic in η ab same shear, expansion and twist in both geometries and R ab k a k b = 2HL i0 L i0 Geodesic condition k geodetic T ab k a k b = 0 In particular: vacuum Λ-term aligned Maxwell (F a b kb k a ) aligned pure radiation (T ab k a k b ), etc.... From now on: k geodesic + affinely parametrized (L i0 = 0 = L 10 ).

16 Basic properties of Kerr-Schild in n 4 Algebraic type of Weyl With k geodesic R 0i0j = R 010i = R 0ijk = 0, R 00 = R 0i = 0 C 0i0j = 0 ( Ψ 0 = 0 ) C 010i = 0, C 0ijk = 0 ( Ψ 1 = 0 ). Algebraic type for KS KS, k geodesic type II (or more special), k WAND

17 Basic properties of Kerr-Schild in n 4 Algebraic type of Weyl With k geodesic R 0i0j = R 010i = R 0ijk = 0, R 00 = R 0i = 0 C 0i0j = 0 ( Ψ 0 = 0 ) C 010i = 0, C 0ijk = 0 ( Ψ 1 = 0 ). Algebraic type for KS KS, k geodesic type II (or more special), k WAND Consequences for black hole spacetimes no black rings in KS (since of type I i [Pravda-Pravdová 05]) Myers-Perry BHs and uniform black strings are type D ( BHs can be only type G, I i, D, O [Pravda-Pravdová-M.O. 07b]) cf. [Hamamoto-Houri-Oota-Yasui 07]

18 Vacuum Kerr-Schild solutions Vacuum equations Ricci tensor R 00 = 0 = R 0i identically. R 01 and R ij are simple and depend only on the optics of k (and on H), i.e. (D k a a ) Basic equations Tracing (2): D 2 H + (n 2)θDH + 2Hω 2 = 0 (1) (D ln H)S ij = L ik L jk (n 2)θS ij (2) (n 2)θ(D ln H) = σ 2 + ω 2 (n 2)(n 3)θ 2. (3)

19 Vacuum Kerr-Schild solutions Vacuum equations Ricci tensor R 00 = 0 = R 0i identically. R 01 and R ij are simple and depend only on the optics of k (and on H), i.e. (D k a a ) Basic equations Tracing (2): Two families: D 2 H + (n 2)θDH + 2Hω 2 = 0 (1) (D ln H)S ij = L ik L jk (n 2)θS ij (2) (n 2)θ(D ln H) = σ 2 + ω 2 (n 2)(n 3)θ 2. (3) 1 θ = 0 non-expanding (3) σ = 0 = ω ( Kundt) (1) D 2 H = 0 2 θ 0 expanding (3) D ln H =... (2)...

20 Vacuum Kerr-Schild solutions Non-expanding solutions Non-expanding solutions θ = σ = ω = 0, D 2 H = 0 KS θ=0 Kundt-N all vacuum Kundt-N in [Coley-Fuster-Hervik-Pelavas 06] easy to see that vacuum Kundt-N KS θ=0 (e.g.: ds 2 = 2dudr + δ ij dx i dx j +Hdu 2 = ds 2 0 +Hdu2 ) Non-expanding vacuum KS E.g., type N pp -waves. KS + non-expanding Kundt + type N

21 Vacuum Kerr-Schild solutions Expanding solutions Expanding solutions R ij = 0 splits into DH = f(θ, σ, ω) and Optical constraint (OC) L ik L jk = L lkl lk (n 2)θ S ij LL T = Tr(LLT ) 2(n 2)θ (L + LT ) independent of H purely geometric restriction on the choice of k in flat space purely algebraic non-trivial (counterexamples)

22 Vacuum Kerr-Schild solutions Expanding solutions Algebraic type With OC, by reductio ad absurdum types III and N not possible type II, D only (e.g., Myers-Perry)

23 Vacuum Kerr-Schild solutions Expanding solutions Algebraic type With OC, by reductio ad absurdum types III and N not possible type II, D only (e.g., Myers-Perry) How to proceed then? 1 solve the OC form of L ij 2 solve Sachs equations r-dependence of L ij (and θ, σ, ω) (extending [Sachs 61, Newman-Penrose 62]) 3 solve DH = f(θ, σ, ω) r-dependence of H

24 Vacuum Kerr-Schild solutions Expanding solutions Algebraic type With OC, by reductio ad absurdum types III and N not possible type II, D only (e.g., Myers-Perry) How to proceed then? 1 solve the OC form of L ij 2 solve Sachs equations r-dependence of L ij (and θ, σ, ω) (extending [Sachs 61, Newman-Penrose 62]) 3 solve DH = f(θ, σ, ω) r-dependence of H This enables one to discuss: weak Goldberg-Sachs theorem? curvature singularities special subfamilies (not in this talk)

25 Vacuum Kerr-Schild solutions Expanding solutions Solving the optical constraint and the Sachs equations Optical constraint: LL T = 1 2 F(L + LT ) Sachs equations [Pravda-Pravdová-M.O. 07a]: DL = L 2

26 Vacuum Kerr-Schild solutions Expanding solutions Solving the optical constraint and the Sachs equations Optical constraint: LL T = 1 2 F(L + LT ) Sachs equations [Pravda-Pravdová-M.O. 07a]: DL = L 2 then in an appropriate parallely transported frame L (1) ( )... 1 r b 0 L (µ) = µ L ij = L (p) r 2 + (b 0 µ) 2 b 0, µ r, L L = 1 r diag(1, }. {{.., 1, 0,..., 0), }}{{} (m 2p) (n 2 m) with µ = 1,..., p, 0 2p m n 2, m rank(l).

27 Vacuum Kerr-Schild solutions Expanding solutions Solving the optical constraint and the Sachs equations Optical constraint: LL T = 1 2 F(L + LT ) Sachs equations [Pravda-Pravdová-M.O. 07a]: DL = L 2 then in an appropriate parallely transported frame L (1) ( )... 1 r b 0 L (µ) = µ L ij = L (p) r 2 + (b 0 µ) 2 b 0, µ r, L L = 1 r diag(1, }. {{.., 1, 0,..., 0), }}{{} (m 2p) (n 2 m) with µ = 1,..., p, 0 2p m n 2, m rank(l). Generically σ 0, but no shear in each 2-block: generalized Goldberg-Sachs?

28 Vacuum Kerr-Schild solutions Expanding solutions Solving DH = f(θ, σ, ω): r-dependence of H KS function H = H 0 r m 2p 1 p µ=1 1 r 2 + (b 0 µ) 2

29 Vacuum Kerr-Schild solutions Expanding solutions Solving DH = f(θ, σ, ω): r-dependence of H KS function H = H 0 r m 2p 1 p 1 r 2 + (b 0 µ) 2 H 0 for r rm 1 µ=1

30 Vacuum Kerr-Schild solutions Expanding solutions Solving DH = f(θ, σ, ω): r-dependence of H KS function Examples: H = H 0 r m 2p 1 p 1 r 2 + (b 0 µ) 2 H 0 for r rm 1 µ=1 n = 4 Kerr (m = 2, p = 1): H µr r 2 + a 2 cos 2 θ n = 5 Myers-Perry (m = 3, p = 1): H µ r 2 + a 2 cos 2 θ + b 2 sin 2 θ Cf. also coordinates of [Chen-Lü-Pope 06].

31 Vacuum Kerr-Schild solutions Expanding solutions Curvature singularities Singularities at r = 0 (caustic) of the metric function H = H p 0 1 r m 2p 1 r 2 + (b 0 µ) 2. µ=1 Confirmed by Kretschmann R abcd R abcd.

32 Vacuum Kerr-Schild solutions Expanding solutions Curvature singularities Singularities at r = 0 (caustic) of the metric function H = H p 0 1 r m 2p 1 r 2 + (b 0 µ) 2. µ=1 Confirmed by Kretschmann R abcd R abcd. Not valid for for m = 2p, m = 2p + 1 still singularities at possible special points where b 0 µ = 0: n = 4 Kerr (m = 2, p = 1): µr H r 2 + a 2 cos 2 θ n = 5 Myers-Perry with b = 0: µ H r 2 + a 2 cos 2 θ For BHs, more details in [Myers-Perry 86]. r = 0 and θ = π/2 r = 0 and θ = π/2

33 Summary and outlook Summary and outlook studied basic properties of KS ansatz g ab = η ab 2Hk a k b in n > 4 dimensions, e.g.: - k geodetic T ab k a k b = 0 - k geodetic type II or more special

34 Summary and outlook Summary and outlook studied basic properties of KS ansatz g ab = η ab 2Hk a k b in n > 4 dimensions, e.g.: - k geodetic T ab k a k b = 0 - k geodetic type II or more special focused on vacuum solutions: two invariant subfamilies

35 Summary and outlook Summary and outlook studied basic properties of KS ansatz g ab = η ab 2Hk a k b in n > 4 dimensions, e.g.: - k geodetic T ab k a k b = 0 - k geodetic type II or more special focused on vacuum solutions: two invariant subfamilies θ = 0: complete solution, equivalent to type N Kundt

36 Summary and outlook Summary and outlook studied basic properties of KS ansatz g ab = η ab 2Hk a k b in n > 4 dimensions, e.g.: - k geodetic T ab k a k b = 0 - k geodetic type II or more special focused on vacuum solutions: two invariant subfamilies θ = 0: complete solution, equivalent to type N Kundt θ 0: richer and more complex structure - type II and D only - found r-dependence of L ij and H (R ij = 0, R 01 = 0) - weak (and partial) Goldberg-Sachs theorem - curvature singularities - subfamilies ω = 0, σ = 0 ( with Robinson-Trautman) next: - (in principle) find explicit k and solve R 11 = 0 and R 1i = 0 - anything like Kerr s theorem in HD? - find new solutions? (symmetries, simple optics )

37 Riemann and Ricci tensors Riemann tensor k geodesic, affinely parametrized: R 0i0j = 0, R 010i = 0, R 0ijk = 0, R 0101 = D 2 H, R 01ij = 2A ji DH + 4HS k[j A i]k, R 0i1j = L ij DH 2HA ki L kj, R ijkl = 4H(A ij A kl + A k[j A i]l + S l[i S j]k ), R 011i = δ i (DH) + 2L [i1] DH + L ji δ j H + 2H(2L ji L [1j] + L j1 A ji ), R 1ijk = 2L [j i δ k] H + 2A jk δ i H 2H ( δ i A kj + L 1j L ki L 1k L ji L j1 A ki + L k1 A ji + 2L [1i] A kj + A lj M ki A lk M ji), R 1i1j = δ (i (δ j) H) + M k (ij) δ k H + (2L 1j L j1 )δ i H + (2L 1i L i1 )δ j H + N (ij) DH S ij H + 2H ( δ (i L 1 j) S ij 2L 1(i L j)1 + 2L 1i L 1j L k(i N k j) 2HL k(i A j)k k k k ) + L 1kM (ij) 2HA ik A jk L k(im j)1 L (i km j)1. l l

38 Riemann and Ricci tensors Ricci tensor k geodesic, affinely parametrized: R 00 = 0, R 0i = 0, R 01 = [D 2 H + (n 2)θDH + 2Hω 2 ], R ij = 2HL ik L jk 2[DH + (n 2)θH]S ij, R 11 = δ i (δ i H) + (N ii 2HL ii )DH + (4L 1j 2L j1 i M ji)δ j H L ii H + 2H ( 2δ i L [1i] + 4L 1i L [1i] + L i1 L i1 L 11 L ii + 2L [1j] j M ii 2A ij N ij 2Hω 2), R 1i = δ i (DH) + 2L [i1] DH + 2L ij δ j H L jj δ i H + 2H ( j i ) δ j A ij + A ij M kk A kj M kj L jj L 1i + 3L ij L [1j] + L ji L (1j).