Recent analytical and numerical studies of asymptotically AdS - - PowerPoint PPT Presentation

Recent analytical and numerical studies of asymptotically AdS spacetimes in spherical symmetry

Maciej Maliborski and Andrzej Rostworowski

Institute of Physics, Jagiellonian University, Krak´

• w

New frontiers in dynamical gravity, Cambridge, 24th March, 2014

Outline and motivation

◮ The phase space of solutions to the Einstein equations with Λ < 0

has a complicated structure. Close to the pure anti-de Sitter (AdS) space there exists a variety of coherent structures: geons

[Dias,Horowitz&Santos, 2011], [Dias,Horowitz,Marolf&Santos, 2012],

boson stars (standing waves) [Buchel,Liebling&Lehner, 2013] and time-periodic solutions [M&Rostworowski, 2013].

◮ The construction of non-generic configurations which stay close to

the AdS solution does not imply their stability.

◮ To understand the mechanism of (in)stability of asymptotically AdS

solutions we study a spherically symmetric complex self-gravitating massless scalar field (the simplest model possessing standing wave solutions).

◮ We conjecture that the dispersive spectrum of linear perturbations

• f standing waves makes them immune to the instability.

1/19

Complex (real) self-gravitating massless scalar field

Gαβ+Λ gαβ = 8πG

• ∇αφ ∇β ¯

φ − 1 2gαβ∇µφ∇µ ¯ φ

• , Λ = −d(d−1)/(2ℓ2) ,

gαβ∇α∇βφ = 0 .

◮ Spherically symmetric parametrization of asymptotically AdS

spacetimes ds2 = ℓ2 cos2 x

• −Ae−2δdt2 + A−1dx2 + sin2 x dΩ2

Sd−1

• ,

where (t, x) ∈ R × [0, π/2)

◮ Field equations with auxiliary variables: Φ = φ′ and Π = A−1eδ ˙

φ A′ = d − 2 + 2 sin2x sin x cos x (1 − A) + Aδ′ , δ′ = − sin x cos x

• |Φ|2 + |Π|2

, ˙ Φ =

• Ae−δΠ

′ , ˙ Π = 1 tand−1x

• tand−1x Ae−δΦ

′ .

◮ Units 8πG = d − 1 and notation ′ = ∂x, ˙= ∂t 2/19

Boundary conditions

◮ We require smooth evolution and finiteness of the total mass

m(t, x) = sind−2 x cosd x

• 1 − A(t, x)
• ,

M = lim

x→π/2 m(t, x) = π/2

• A
• |Φ|2 + |Π|2

tand−1 x dx . Then, there is no freedom in prescribing boundary data at x = π/2: reflecting boundary conditions

◮ Conserved charge for the complex field

Q = −ℑ π/2 φ¯ Π tand−1 x dx .

3/19

Linear perturbations of AdS

◮ Linear equation on an AdS background [Ishibashi&Wald, 2004]

¨ φ + Lφ = 0 , L = − 1 tand−1x ∂x

• tand−1x ∂x
• ,

◮ Eigenvalues and eigenvectors of L are (j = 0, 1, . . .)

ω2

j = (d+2j)2,

ej(x) = 2

• j!(j + d − 1)!

Γ(j + d/2) cosd x P (d/2−1,d/2)

j

(cos 2x) ,

◮ AdS is linearly stable, linear solution

φ(t, x) =

• j≥0

aj cos(ωjt + bj) ej(x) , with aj, bj determined by the initial data φ(0, x) and ˙ φ(0, x)

4/19

Real scalar field — time-periodic solutions

◮ We search for solutions of the form (|ε| ≪ 1)

φ(t, x) = ε cos(ωγt)eγ(x) + O(ε3) , solution bifurcating from single eigenmode

◮ We make an ansatz for the ε-expansion

φ= ε cos(τ)eγ(x) +

• dd λ≥3

ελ φλ(τ, x) , δ=

• even λ≥2

ελ δλ(τ, x) , 1 − A =

• even λ≥2

ελ Aλ(τ, x) . where we rescaled time variable τ = Ωt, Ω = ωγ +

• even λ≥2

ελ ωγ,λ ,

5/19

Time-periodic solution — perturbative construction

◮ We decompose functions φλ, δλ, Aλ in the eigenbasis

φλ =

• j≥0

fλ,j(τ)ej(x), δλ =

• j≥0

dλ,j(τ) (ej(x) − ej(0)) , Aλ =

• j≥0

aλ,j(τ)ej(x) , with expansion coefficients being periodic functions

◮ This reduces the constraint equations to algebraic system and the

wave equation to a set of forced harmonic oscillator equations

• ω2

γ∂ττ + ω2 k

• fλ,k =

π/2 Sλek(x) tand−1 x dx , with initial conditions fλ,k(0) = cλ,k, ˙ fλ,k(0) = ˜ cλ,k ,

◮ We use the integration constants {cλ,k, ˜

cλ,k} and frequency expansion coefficients ωγ,λ to remove all of the resonant terms cos(ωk/ωγ)τ or sin(ωk/ωγ)τ

6/19

Time-periodic solution — numerical construction

We make an ansatz (τ = Ωt) φ=

• 0≤i<N
• 0≤j<K

fi,j cos((2i + 1)τ)ej(x) , Π=

• 0≤i<N
• 0≤j<K

pi,j sin((2i + 1)τ)ej(x) .

◮ Solution given by the set of

2 × K × N + 1 numbers

◮ Two equations on each of

K × N collocation points

◮ One extra equation for the

dominant mode condition

• 0≤i<N

fi,γ = ε .

Π2 Π2

x Τ

xk, Τn

7/19

Time-periodic solution — perturbative and numerical results

For d = 4, γ = 0

Ε 1.1010

10 20 30 40

i

5 10 15

j

150 100 50

log10fi,jΕ

φ=

• 0≤i<N
• 0≤j<K

fi,j cos((2i + 1)τ)ej(x) ,

8/19

Time-periodic solution — non-linear stability

4.9 1011 4.9 1011 4.7 1013 4.7 1013

f6 p10

• 2. 1014
• 2. 1014

4.7 1013 4.7 1013

f10 p10

1.7 106 1.7 106 4.9 1011 4.9 1011

f2 f6

1.6 108 1.6 108 6.7 1012 6.7 1012

f4 f7

Closed curves on the slices of phase space – strong evidence for the non-linear stability. Sections of the phase space spanned by the set

• f Fourier coefficients

{fj(t), pk(t)} , φ =

• 0≤j<K

fj(t)ej(x) , Π =

• 0≤j<K

pj(t)ej(x) . [Animation] The d = 4, γ = 0, ε = 0.01 case.

9/19

Complex scalar field — standing waves

◮ The standing wave ansatz

φ(t, x) = eiΩtf (x), Ω > 0, δ(t, x) = d (x), A(t, x) = A(x), with f (x) a real function. The field equations reduce to −Ω2 ed A f = 1 tand−1 x

• tand−1 xAe−d f ′′ ,

A′ = d − 2 + 2 sin2 x sin x cos x (1 − A) + Ad ′, d ′ = − sin x cos x

• f ′2 +

Ωed A f 2 ,

◮ Boundary conditions

f (π/2) = 0, A(π/2) = 1, d ′(π/2) = 0, f ′(0) = 0, A(0) = 1, d (0) = 0.

10/19

Standing waves — perturbative construction

◮ We look for solutions of the system of the form (|ε| ≪ 1)

f (x)=

• dd λ≥1

ελ fλ(x), f1(x) = eγ(x) eγ(0) , d (x)=

• even λ≥2

ελ dλ(x), 1 − A(x) =

• even λ≥2

ελ Aλ(x), Ω = ωγ +

• even λ≥2

ελ ωγ,λ , where eγ(x) is a dominant mode in the solution in the limit ε → 0

◮ Decomposition into the eigenbasis ej(x)

fλ(x)=

• j

ˆ fλ,jej(x), dλ(x)=

• j

ˆ dλ,j(ej(x) − ej(0)), Aλ(x) =

• j

ˆ Aλ,jej(x) .

◮ System of differential equations → set of algebraic equations for the

Fourier coefficients

11/19

Standing waves — numerical construction

◮ Ansatz

f (x)=

N−1

• j=0

ˆ fjej(x), d (x)=

N−1

• j=0

ˆ dj (ej(x) − ej(0)) , A(x) = 1 −

N−1

• j=0

ˆ Ajej(x) .

◮ Solution given by the set of 3N + 1 numbers ◮ Three equations on each of N collocation points

{xj ∈ (0, π/2) : eN(xj) = 0 , j = 0, . . . , N − 1} .

◮ One extra equation for the value of central density

f (0) = ε .

◮ Non-linear system solved with the Newton-Raphson algorithm 12/19

Standing waves — perturbative and numerical results

• 13

10

• 7

10 7 10 13 10

6 4 2 2 4 6 10 20 30 36 1 5 10 15 19

j Λ

The Fourier coefficients ˆ fλ,j of a ground state standing wave in d = 4.

0.0 0.1 0.2 0.3 0.4 0.5 2 4 6 8 10

• Frequency Ω of a fundamental

standing wave versus f (0) = ε for d = 2, 3, 4, 5, 6.

13/19

Standing waves — linear perturbation

[Gleiser&Watkins, 1989], [Choptuik&Hawley, 2000], [Buchel,Liebling&Lehner, 2013]

◮ Perturbative ansatz (|µ| ≪ 1)

φ(t, x) = eiΩt f (x) + µ ψ(t, x) + · · ·

• ,

A(t, x) = A(x) (1 + µ α(t, x) + · · · ) , δ(t, x) = d (x) + µ (α(t, x) − β(t, x)) + · · · , with a harmonic time dependence ψ(t, x) = ψ1(x)eiXt + ψ2(x)e−iXt, α(t, x) = α(x) cos Xt , β(t, x) = β(x) cos Xt .

◮ Set of linear algebraic-differential equations for α(x), β(x) and

ψ1(x), ψ2(x)

14/19

Standing waves — linear perturbation

◮ Numerical approach or perturbative ε-expansion ◮ Linear problem – the condition for the χ0

• Lψ1,0 − (χ0 − ωγ)2ψ1,0 = 0 ,

Lψ2,0 − (χ0 + ωγ)2ψ2,0 = 0 , ⇒

• ω2

j − (χ0 − ωγ)2 = 0 ,

ω2

k − (χ0 + ωγ)2 = 0 . ◮ Thus

◮ ψ1,0 = ej(x), ψ2,0 ≡ 0, and χ0 = ωγ ± ωj , ◮ ψ1,0 ≡ 0, ψ2,0 = ek(x), and χ0 = −ωγ ± ωk .

◮ From a form of perturbative ansatz we take

ψ1,0 = eζ(x), ψ2,0 = 0, χ±

0 = ωγ ± ωζ . 15/19

Standing waves — spectrum of ground state

For d = 4, γ = 0 we get (an asymptotic form for ζ → ∞)

◮ For χ+ 0 = ωγ + ωζ

X + ≈

• 8 + 1207ε2

112 + 908257501ε4 86929920

• +
• 2 + 81ε2

32 + 706663ε4 322560

• ζ

− 105ε2 64 + 29319ε4 28672

• ζ−1 +

165ε2 16 + 472547ε4 28672

• ζ−2 + O
• ζ−3

,

◮ For χ− 0 = ωγ − ωζ

X − ≈ 73ε2 112 + 48824929ε4 28976640

• +
• −2 − 81ε2

32 − 706663ε4 322560

• ζ

+ 105ε2 64 + 29319ε4 28672

• ζ−1 +

15ε2 4 + 50753ε4 4096

• ζ−2 + O
• ζ−3

,

16/19

Resonant vs. asymptotically resonant spectrum

500 1000 t 10 20 (|Π(t, 0)|2 − (Ωf(0))2)/ε 8 4 2 1

1 2

Ricci scalar R(t, 0) = −ℓ−2 3|Π(t, 0)|2 + 20

• f perturbed ground state

standing wave with f (0) = 0.16 in d = 4. [Animation]

17/19

Resonant vs. asymptotically resonant spectrum

10 100 1000 10000 t 0.01 1 100 Π(t, 0)2 8 4 2 1

1 2 1 4

Ricci scalar R(t, 0) = −ℓ−2 3|Π(t, 0)|2 + 20

• f perturbed AdS in d = 4.

18/19

Resonant vs. asymptotically resonant spectrum

500 1000 1500 ε2t 0.35 0.4 0.45 0.5 ε−2Π(ε2t, 0)2 8 4 2 1

1 2 1 4

Ricci scalar R(t, 0) = −ℓ−2 3|Π(t, 0)|2 + 20

• f perturbed AdS in d = 4.

Onset of instability at time t = O(ε−2)

18/19

Summary

◮ Using nonlinear perturbation expansion we derived explicitly the

spectrum of linear perturbations of standing waves in even d – first step towards the understanding stability of time-periodic solutions.

◮ This spectrum is only asymptotically resonant, in contrast to the

fully resonant spectrum around the pure AdS space (for reflecting boundary conditions).

◮ In this situation the energy transfer to higher frequencies is less

effective and the dispersion relation causes the wave packet to spread out in space, which prevents the gravitational collapse.

◮ For pure AdS there is instability for arbitrary small perturbations.

For asymptotically AdS solutions with non-resonant spectrum there is a threshold for triggering the instability.

19/19