Recent analytical and numerical studies

I To understand the mechanism of instability of asymptotically AdSsolutions we study a spherically symmetric complex self-gravitatingmassless scalar field the simplest model possessing s

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

Maciej Maliborski and Andrzej Rostworowski

Institute of Physics, Jagiellonian University, Krak´ ow

New frontiers in dynamical gravity,

Cambridge, 24th March, 2014

Outline and motivation

I The phase space of solutions to the Einstein equations withΛ < 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]

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

the AdS solution does not imply their stability

I To understand the mechanism of (in)stability of asymptotically AdSsolutions we study a spherically symmetric complex self-gravitatingmassless scalar field (the simplest model possessing standing wave

solutions)

I We conjecture that the dispersive spectrum of linear perturbations

of standing waves makes them immune to the instability

Complex (real) self-gravitating massless scalar field

Boundary conditions

I We require smooth evolution and finiteness of the total mass

m(t, x) = sin

d −2xcosdx 1− A(t, x)
,

Then, there is no freedom in prescribing boundary data at

x = π/2: reflecting boundary conditions

I Conserved charge for the complex field

Q =−=

Z π/2

0

φ ¯Π tand −1x dx

Linear perturbations of AdS

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

Real scalar field — time-periodic solutions

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

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

I We make an ansatz for theε-expansion

Time-periodic solution — perturbative construction

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

I This reduces the constraint equations to algebraic system and the

wave equation to a set of forced harmonic oscillator equations

fλ,k(0) = cλ,k, f˙λ,k(0) = ˜cλ,k,

I We use the integration constants {cλ,k, ˜cλ,k} and frequency

expansion coefficientsωγ,λ to remove all of the resonant terms

cos(ωk/ωγ)τ or sin(ωk/ωγ)τ

Time-periodic solution — numerical construction

I One extra equation for the

dominant mode condition

Time-periodic solution — perturbative and numerical

results

Ford = 4, γ = 0

Ε‡1 ´ 10-10

0 10 20 30 40

i

5 10 15

j

-150 -100 -50 0

Time-periodic solution — non-linear stability

Sections of the phasespace spanned by the set

of Fourier coefficients{fj(t), pk(t)} ,

Thed = 4, γ = 0, ε = 0.01 case

Complex scalar field — standing waves

I The standing wave ansatz

φ(t, x) = eiΩtf (x), Ω > 0,δ(t, x) = d (x), A(t, x) = A(x),withf (x) a real function The field equations reduce to

−Ω2e

d

A f =

1tand −1x tan

I Boundary conditions

f (π/2) = 0, A(π/2) = 1, d0(π/2) = 0,

f0(0) = 0, A(0) = 1, d (0) = 0

Standing waves — perturbative construction

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

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

I Decomposition into the eigenbasis ej(x)

Standing waves — numerical construction

I Solution given by the set of 3N + 1 numbers

I Three equations on each ofN collocation points

Standing waves — perturbative and numerical results

7 10 13

The Fourier coefficients ˆfλ,j of a

ground state standing wave ind = 4

0 2 4 6 8 10

¶

FrequencyΩ of a fundamentalstanding wave versusf (0) = ε for

d = 2, 3, 4, 5, 6

Standing waves — linear perturbation

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

I 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)ei X t+ ψ2(x)e−iX t,α(t, x) = α(x) cosX t ,

β(t, x) = β(x) cosX t

I Set of linear algebraic-differential equations forα(x), β(x) and

ψ1(x), ψ2(x)

Standing waves — linear perturbation

I Numerical approach or perturbativeε-expansion

I Linear problem – the condition for the χ0

Standing waves — spectrum of ground state

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

Resonant vs asymptotically resonant spectrum

t0

1 2

Ricci scalarR(t, 0) =−`−2 3|Π(t, 0)|2+ 20
of perturbed ground statestanding wave withf (0) = 0.16 in d = 4 [Animation]

Resonant vs asymptotically resonant spectrum

t0.01

Ricci scalarR(t, 0) =−`−2 3|Π(t, 0)|2+ 20
of perturbed AdS in d = 4

Resonant vs asymptotically resonant spectrum

Ricci scalarR(t, 0) =−`−2 3|Π(t, 0)|2+ 20
of perturbed AdS in d = 4.Onset of instability at timet =O(ε−2)

I Using nonlinear perturbation expansion we derived explicitly the

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

I This spectrum is only asymptotically resonant, in contrast to thefully resonant spectrum around the pure AdS space (for reflecting

boundary conditions)

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

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

I 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