Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability

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1 Nonminimal derivative coupling scalar-tensor theories: odd-parity perturbations and black hole stability A. Cisterna 1 M. Cruz 2 T. Delsate 3 J. Saavedra 4 1 Universidad Austral de Chile 2 Facultad de Física, Universidad Veracruzana 3 University of Mons - UMONS, Bélgica 4 PUCV-Chile Escuela de Física Fundamental, 2016

2 Outline 1 Horndeski theories 2 The model 3 Application to BHs

3 Horndeski theory Inputs Modify gravity theory. Scalar-tensor theory. Non-minimal kinetic scalar field.

4 Modify gravity theory As we know GR enjoys the following main properties Einstein equations G µν = κt µν. Is invariant under diffeomorphisms µ G µν = 0 Possesses second order and symmetric EOM 2 g µν (x) x 2 The spacetime is four-dimensional Only one field enters in the purely gravitational description of the theory, the metric field, g µν (x)

5 In order to describe new gravitational phenomena through modifications of GR, we need to relax at least one of the previous features. Many examples of modified theories are known today: Higher dimensional theories as Lovelock Higher derivative theories Massive gravity theories, bigravity, f (R)-gravity... Brans-Dicke theory

6 Let us describe gravity with one extra degree of freedom ( S[g µν, φ, ψ] = φr ω(φ) ) φ µφ µ φ V (φ) d 4 x + S m [g, ψ], Here the gravity sector is described by g µν and φ(x). The matter field are coupled only with the metric tensor. Now we can ask ourself, which is the most general scalar-tensor theory which yields second order equations of motion, for both, the metric and the scalar field? This question was answered by Horndeski 40 years ago.

7 L H = k 1 (ρ, φ)δ µνρ αβγ µ α φr νσ βγ 4 3 k 1,ρ(φ, ρ)δ µνρ αβγ µ α φ ν β φ σ γ φ + k 3 (φ, ρ)δ µνρ αβγ αφ µ φr νσ βγ +... Here ρ = µ φ µ φ is the standard kinetic term of the scalar field.

8 Was proven that this Lagrangian is equivalent to the Lagrangian coming from covariantized Galileons L = K (φ, ρ) G 3 (φ, ρ) φ + G 4 (φ, ρ)r + G 4,ρ (φ, ρ)[( φ) 2 ( µ ν φ) 2 ] + G 5 (φ, ρ)g µν µ ν φ G 5,ρ 6 [( φ)3 3 φ( µ ν φ) 2 + ( µ ν φ) 3 ], Now commun sectors of the theory can be recognized easily, Brans-Dicke theory, K-essence, GR, etc. In particular our interest is focused on the theory described by the following term G 5 (φ, ρ)g µν µ ν φ which gives us non minimally kinetic coupled scalar fields G µν µ φ ν φ

9 Why second order theories? For the free particle we have: L fp = 1 2 mẋ 2 V }{{} Legendre transform H fp = p2 2m + V. Figure: Energy of particle

10 Ghost states... Figure: Energy of particle

11 Towards odd parity perturbations The model Equations of motion Odd-parity perturbations

12 The model The action we are working with take the form S[g µν, φ] = d 4 x [ 1 g 2κ (R 2Λ) 1 ] 2 (αgµν βg µν ) µ φ ν φ V (φ) We have: E µν = G µν + Λg µν κ [αt µν + βθ µν ] = 0, and µ (αg µν ν φ βg µν ν φ) dv (φ) dφ = 0,

13 For simplicity T µν = µφ νφ 1 2 gµν αφ α φ 1 gµνv (φ), α Θ µν = 1 2 µφ νφr 2 αφ (µφr α ν) α φ β φr µανβ Gµν αφ α φ ( µ α φ)( ν αφ) + ( µ νφ) φ + g µν [ 1 2 ( φ)2 + αφ β φr αβ + 1 ] 2 ( α β φ)( α β φ).

14 The perturbed metric ds 2 = A(r)dt 2 + B(r)dr 2 [ ] dz 2 + C(r) 1 kz 2 + (1 kz2 )(dϕ + k 1 dt + k 2 dr + k 3 dz) 2, φ = φ 0 (r) + ɛφ(t, r, z)

15 Considering the Einstein field equations only at first order in ɛ, we find that E t r = ɛκ dφ 0 dr E r z = ɛκ dφ 0 dr + O(ɛ 2 ) = 0, [ ( α t Φ A β 1 ABC 2 [ ( α zφ B β B 2 C A r C r A + 1 Cr 2 4 C 1 2 kb Cr r A r C r A + 1 Cr 2 4 C 1 2 ) t Φ ] + O(ɛ 2 ) = 0, ( ) ) ] ACr + CA r r zφ A Φ = 0

16 BUT! [ { Eϕ r = A 1 + βκ ( ) } ] 2 dφ0 (1 kz 2 ) 2 ( zk 2 r k 3 ) z C 2B dr [{ βκ ( ) } ] 2 dφ0 (1 kz 2 ) ( r k 1 t k 2 ) = 0, t 2B dr [ Eϕ t = { B C 1 βκ ( ) } ] 2 dφ0 (1 kz 2 ) 2 ( zk 1 t k 3 ) z A 2B dr [ { + C βκ ( ) } ] 2 dφ0 (1 kz 2 )( r k 1 t k 2 ) = 0, r AB 2B dr [ Eϕ z = { A C 1 + βκ ( ) } ] 2 dφ0 ( zk 2 r k 3 ) r B 2B dr [ + { B C 1 βκ ( ) } ] 2 dφ0 ( t k 3 zk 1 ) = 0. t A 2B dr

17 The important thing is... [ C 2 P (+) AB r 1 C A Q = C A B P(r) (±) = 1 ± βκ 2B ] 1 Q + (1 kz 2 ) 2 [ P ( ) r z 1 (1 kz 2 ) B P (+)(1 kz 2 ) 2 ( z k 2 r k 3 ), ) 2. ( dφ0 dr The mathematical methods courses are useful! Q = Q(r, t)d(z) ] Q = C z A 2 t Q. (1)

18 The master equation Ψ(r, t) = [ CP ( ) ] 1/2 Q(r, t) Fourier decomposition: Ψ = Ψ ω e iωt dt ( HΨ ω := 2 Ψ ω r 2 + A γ + 3 CP (+) 4C 2 1 2P ( ) d 2 P ( ) dr 2 + = 2 Ψ ω r 2 1 dc dp ( ) 2CP ( ) dr dr + V Ψω = ω2 eff Ψω, ( ) dc 2 1 dr 2C ) d 2 C dr P( ) 2 Ψ ω = ω 2 eff Ψω, ( ) 2 dp( ) dr ω 2 eff = P ( ) P (+) ω 2.

19 The spectrum dr (Ψ ω ) HΨ ω = dr [ DΨ ω 2 +V S Ψ ω 2] (Ψ ω DΨ ω ) Boundary, where D = r + S and If we choose S = 1 dc 2C dr + 1 V S = V + ds dr S2. (2) dp ( ) 2P ( ) dr, we find A V S = γ. (3) CP (+)

20 The solution For spherically symmetric spacetimes ( ) αβk A(r) = r 2 L 2 + k ( ) α + βλ 2 arctan r βk αβk µ α α βλ r r + 3α + βλ α βλ k, and B(r) = α2 ((α βλ) r 2 + 2βk) 2 (α βλ) 2 (αr 2 + βk) 2 A(r), C(r) = r, α + βλ < 0.

21 Slowly-rotating objects k 1 = ω(r). Then the frame dragging function satisfies [ { C βκ ( ) } ] 2 dφ0 (1 kz 2 ) r r AB 2B dr ω(r) = 0. Solving we have The GR case ω(r) = c 1 + c 2 r.

22 Thank you for your attention!