Higher-Dimensional Kerr-Schild Spacetimes with (A)dS Background

Higher-Dimensional Kerr-Schild Spacetimes with (A)dS Background

Tomáš Málek

Institute of Theoretical Physics Faculty of Mathematics and Physics

Charles University in Prague

Spanish Relativity Meeting, ERE 2010 Granada, España

arXiv:1009.1727

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Kerr-Schild metric with (A)dS background

Kerr-Schild metricg_ab =ηab−2Hk_ak_b analytical tractable

KS class contains: Kerr, Myers-Perry, type N pp-waves . . .

Generalized Kerr-Schild metric

g_ab= ¯g_ab−2Hk_ak_b

KS vectork is null with respect to the both metric inverseg^ab = ¯g^ab+2Hk^ak^b

canonical form of the background metric g¯ = Ω(−dt²+dx₁²+. . .+dx_n−1² ) whereΩ_dS = ^{(n−2)(n−1)}_2Λt2 , Ω_AdS =−^{(n−2)(n−1)}_2Λx

12

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Newman-Penrose formalism in higher dimensions

real null framen≡m⁽⁰⁾,`≡m⁽¹⁾,m⁽ⁱ⁾

l^al_a=n^an_a=l^am_a⁽ⁱ⁾=n^am⁽ⁱ⁾_a =0 l^ana=1, m^(i)am_a^(j)=δ_ij

i,j, . . . range from 2 ton−1 a,b, . . . from 0 ton−1 metric in the null frame

g_ab =2n_(al_b)+δijm⁽ⁱ⁾_a m_b^(j) we identifyk with`

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Newman-Penrose formalism in higher dimensions

Ricci rotation coefficients

la;b=Lcdm^(c)_a m^(d)_b , na;b=Ncdm^(c)_a m^(d)_b , m⁽ⁱ⁾_a;b=Mⁱ cdm^(c)_a m^(d)_b ifk is geodetic (L_i0=L₁₀=0)

decomposition ofLij:

S_ij≡L_(ij)=σ_ij+θδ_ij, A_ij≡L_[ij]

θ,σ_ij andAijare theexpansion,shearandtwistwith corresponding scalars

σ²=σ_ijσ_ij, ω²=A_ijA_ij Directional derivatives along the frame vectors

D≡k^a∇_a, ∆≡n^a∇_a, δ_i≡m_(i)^a ∇_a.

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1

C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1

C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type G

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1

C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type I

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1

C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type I_i

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type II

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type II_i

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type III

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0

C_101i C_1ijk −1

C_1i1j −2

type III_i

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0 C_101i C_1ijk −1

C_1i1j −2

type N

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Generalized Kerr-Schild metric HD Newman-Penrose formalism HD Algebraic classification

Algebraic classification in higher dimensions

boost weightw:qˆ =λ^wq boost of the frame

`ˆ=λ`, nˆ=λ⁻¹n, mˆ⁽ⁱ⁾=m⁽ⁱ⁾

Weyl tensor frame components sorted by boost weight frame components boost weight

C_0i0j 2

C_010i C_0ijk 1 C₀₁₀₁ C_01ij C_0i1j C_ijkl 0 C_101i C_1ijk −1

C_1i1j −2

type D

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

KS vector k is geodetic iff T

vanishes

Ricci componentR₀₀=R_abk^ak^b R00=2Hk_c;ak^ak^c_;bk^b−1

2(n−2) Ω_,ab

Ω −3 2

Ω_,aΩ_,b Ω²

k^ak^b

Proposition

The vectork in the generalized Kerr-Schild metric is geodetic iff the component of the energy-momentum tensor

T₀₀=T_abk^ak^b=0 vanishes.

k geodetic for Einstein spaces

spacetimes with aligned matter fields (Maxwell field F_abk^a∝k_b, aligned pure radiationT_ab ∝k_ak_b)

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Geodetic and optical properties in both metrics

k is geodetic in the full metric

iff it is geodetic in the background metric k_a;bk^b=k_a,bk^b=k_a;bk^b, k^a_;bk^b=k^a_,bk^b+Ω_,b

Ω k^ak^b=k^a_;bk^b

null frame in the background

g¯_ab =2k_(a˜n_b)+δ_ijm_a⁽ⁱ⁾m^(j)_b , n˜a=na+Hk_a

optical matrices in the full and background metric are equal L_ij ≡k_a;bm^(i)am^(j)b=k_a;bm^(i)am^(j)b≡˜L_ij

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

Ricci tensor

let’s assume geodetick Ricci tensor

R_ab = (Hk_ak_b)_;cdg^cd−(Hk^sk_a)_;bs−(Hk^sk_b)_;as+_n−2^2Λ g¯_ab

−2H D²H+L_iiDH+2Hω² k_ak_b

k is eigenvector of Ricci R_abk^b =−h

D²H+ (n−2)θDH+2Hω²−_n−2^2Λ i k_a frame components

R₀₀ =0

R₀₁ =−D²H −(n−2)θDH −2Hω²+_n−2^2Λ R_ij =2HL_ikL_jk−2(DH+ (n−2)θH)S_ij+ _n−2^2Λ δ_ij

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

KS spacetime with geodetic k is at least of type II

positive boost weight frame components of Weyl vanish C_0i0j =0, C_010i =0, C_0ijk =0

Proposition

Generalized Kerr-Schild spacetime with geodetic vectork is algebraically special withk being the multiple WAND.

[Pravda Pravdová Ortaggio 2007]: static (or stationary with

“reflection symmetry”) expanding spacetimes are of types G, I_i, D or O

Corollary

Static (or stationary with “ref. sym.”) generalized Kerr-Schild spacetimes with geodetic vectork are of type D or O.

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

Vacuum equations

from now on we restrict us to Einstein spaces n-dimensional EFEs:

R_ab= 2 n−2Λg_ab R_ij = _n−2² Λδ_ij component can be written as

(n−2)θ(DlogH) =σ²+ω²−(n−2)(n−3)θ²

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

Nonexpanding Einstein KS spaces belong to Kundt N

vanishing expansionθ=0 implies vanishing shearσ=0 and twistω =0, thusL_ij =0

substituting EFEs to Weyl: boost weight 0 and−1 components vanish

Proposition

Einstein Kerr-Schild spacetimes with non-expanding KS congruencek are of type N withk being the multiple WAND.

Twist and shear of the KS congruencek necessarily vanish and these solutions thus belong to the class of Einstein type N Kundt spacetimes.

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

Optical constraint

equationR_ij= _n−2² Λδ_ij now nontrivial

Optical constraint

L_ikL_jk = L_ikL_ik (n−2)θS_ij L_ijnormal matrix:[L,L^T] =0

in appropriate frameL_ijhas block-diagonal form consisting of 2x2 blocks:

„ S A

−A S

«

types III and N not compatible with expanding Einstein KS

Proposition

Einstein Kerr-Schild spacetimes with expanding KS congruence k are of Weyl types II or D or conformally flat

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

r -dependece of optical matrix L

block-diagonal form + Sachs equation(DLik=−L_ikLkj):

Lij= 0 B B B B B B B B B B B

@ L₍₁₎

. .. L_(p)

L˜ 1 C C C C C C C C C C C A

L(µ)=

„ s_(2µ) A_2µ,2µ+1

−A2µ,2µ+1 s_(2µ)

«

(µ=1, . . . ,p),

s_(2µ)= r

r²+ (a⁰_(2µ))², A_2µ,2µ+1= a_(2µ)⁰ r²+ (a⁰_(2µ))²,

L˜=1

rdiag(1, . . . ,1

| {z }

(m−2p)

,0, . . . ,0

| {z }

(n−2−m)

)

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

General properties

Nonexpanding Einstein KS spacetimes Expanding Einstein KS spacetimes

Singularities

integratingR_ij = _n−2² Λδ_ij: H= _rm−2p−1^H0

Qp µ=1

1 r²+(a⁰_(2µ))²

divergences atr =0

1 2p6=m, 2p6=m−1

2 2p=m(meven), 2p=m−1 (modd)

if some ofa⁰_(2µ)vanish somewhere in spacetime

Kretschmann scalar

R_abcdR^abcd=4Φ²+8Φ^S_ijΦ^S_ij −24Φ^A_ijΦ^A_ij +C_ijklC_ijkl+ 8n

(n−1)(n−2)²Λ²

whereΦij≡C0i1j,thenΦ =C0101,Φ^S_ij =−¹

2Cikjk,Φ^A_ij =¹₂C01ij

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Kerr-de Sitter

Kerr-(A)dS in KS form [Gibbons et al. (2004)]

g_ab = ¯g_ab+2M ρ² k_ak_b in case of 5D

g¯=− (1−λr²)∆

(1+λa²)(1+λb²)dt²+ r²ρ²

(1−λr²)(r²+a²)(r²+b²)dr²

+ρ²

∆dθ²+(r²+a²)sin²θ

1+λa² dφ²+(r²+b²)cos²θ 1+λb² dψ²

k= ∆

(1+λa²)(1+λb²)dt+ r²ρ²

(1−λr²)(r²+a²)(r²+b²)dr

−asin²θ

1+λa²dφ−bcos²θ 1+λb²dψ

whereρ²=r²+a²cos²θ+b²sin²θ , ∆ =1+λa²cos²θ+λb²sin²θ

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Comparison of 5D Kerr-de Sitter with general results

5D Kerr-de Sitter: general KS (n=5,m=3,p=1):

L= 0 B B B B B

@

r ρ²

√

ρ²−r²

ρ² 0

−

√

ρ²−r² ρ²

r

ρ² 0

0 0 ¹_r

1 C C C C C A

L= 0 B B B B

@

s(2) A2,3 0

−A_2,3 s₍₂₎ 0

0 0 ¹_r

1 C C C C A

H=−M ¹

r²+a²cos²θ+b²sin²θ H=H₀ ¹

r²+(a⁰₍₂₎)²

s(2)= ^r

r²+a²cos²θ+b²sin²θ s(2)= ^r

r²+(a⁰₍₂₎)²

A_(2,1)=

√

a²cos²θ+b²sin²θ

r²+a²cos²θ+b²sin²θ A_(2,1)= ^a

0 (2) r²+(a⁰₍₂₎)²

H0=−M, a⁰₍₂₎=a²cos²θ+b²sin²θ a6=0,b6=0: no singularity

a6=0,b=0: singularity ifθ= ^π₂

a=b=0 (de Sitter-Schwarzschild-Tangherlini) p=0: singularity

Introduction General KS vector field Geodetic KS vector field Example of expanding solution

Conclusions

GKS are of type II or more special KS vectork is geodetic iffT₀₀=0

non-expanding Einstein GKS belong to type N Einstein Kundt class (equivalence at least in Ricci-flat case) expanding Einstein GKS

are of type II, D or O

r-dependence ofLijand Weyl bw 0 components explicitly determined

condition for presence of a singularity atr =0