Instability of AdS - one year later
Piotr Bizoń
Jagiellonian University
joint work with Andrzej Rostworowski, Maciej Maliborski, and Joanna Jalmużna
Oberwolfach, 3 August 2012
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
1 / 23
Anti-de Sitter (AdS) spacetime in d + 1 dimensions
AdS is the maximally symmetric solution of the vacuum Einstein equations
Rαβ − 12 gαβ R + Λgαβ = 0 with negative cosmological constant Λ
ds 2 = −(1 + r 2 /`2 )dt 2 +
dr 2
+ r 2 dΩ2S d−1
1 + r 2 /`2
where `2 = −d(d − 1)/2Λ, r ≥ 0, and −∞ < t < ∞.
Substituting r = ` tan x (0 ≤ x < π/2) we get
ds 2 =
`2
−dt 2 + dx 2 + sin2 x dΩ2S d−1
2
cos x
Conformal infinity x = π/2 is the timelike cylinder I = R × S d−1 with the
boundary metric dsI2 = −dt 2 + sin2 x dΩ2S d−1
AdS is the most popular spacetime on the arXiv (due to AdS/CFT)
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
2 / 23
Is AdS stable?
Surprisingly, this question seems to have been unexplored until the
past year (notable exception M. Anderson 2006)
In contrast, Minkowski spacetime (Λ = 0) is known to be stable
(actually asymptotically stable) – (Christodoulou&Klainerman 1993)
Key difference between Minkowski and AdS: the main mechanism of
stability of Minkowski - dissipation of energy by dispersion - is absent
in AdS because AdS is effectively bounded (for no flux boundary
conditions at I it acts as a perfect cavity)
Note that by positive energy theorems both Minkowski and AdS are
the unique ground states among asymptotically flat/AdS spacetimes
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
3 / 23
Model
The problem seems tractable only in 1 + 1 dimensions
⇒ spherical symmetry
Spherically symmetric vacuum solutions are static (Birkhoff’s
theorem) ⇒ we need matter to generate dynamics
Simple matter model: massless scalar field
1
Gαβ + Λgαβ = 8πG ∂α φ ∂β φ − gαβ (∂φ)2
2
g αβ ∇α ∇β φ = 0
In the asymptotically flat case (Λ = 0) this model has led to
important insights (proof of the weak cosmic censorship by
Christodoulou 1986-1999 and the discovery of critical phenomena at
the threshold for black hole formation by Choptuik 1993)
Henceforth we set d = 3 (similar behavior for d ≥ 3, but not for
d = 2). For even d ≥ 4 there is a way to bypass Birkhoff’s theorem.
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
4 / 23
Convenient parametrization of asymptotically AdS spacetimes
ds 2 =
`2 −2δ 2
−1
2
2
2
−Ae
dt
+
A
dx
+
sin
x
dΩ
cos2 x
where A and δ are functions of (t, x).
Auxiliary variables Φ = φ0 and Π = A−1 e δ φ̇ (0 = ∂x , ˙ = ∂t )
Field equations (using units where 4πG = 1)
0
Φ̇ = Ae −δ Π ,
0
1 2
−δ
tan
x
Ae
Φ
tan2 x
2
1 + 2 sin x
A0 =
(1 − A) − sin x cos x A Φ2 + Π2
sin x cos x
δ 0 = − sin x cos x Φ2 + Π2
Π̇ =
We want to solve the initial-boundary value problem for this system
for small perturbations of the AdS space φ = 0, A = 1, δ = 0
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
5 / 23
Initial and boundary conditions
We assume that initial data (φ, φ̇)|t=0 are smooth
Smoothness at the center implies that near x = 0
φ(t, x) = f0 (t) + O(x 2 ),
δ(t, x) = O(x 2 ),
A(t, x) = 1 + O(x 2 )
Smoothness at spatial infinity and finiteness of the total mass M
imply that near x = π/2 (using ρ = π/2 − x)
φ(t, x) = f∞ (t) ρ3 + O ρ5 , δ(t, x) = δ∞ (t) + O ρ6 ,
A(t, x) = 1 − 2Mρ3 + O ρ6
Remark: There is no freedom in prescribing boundary data. This is
directly related to the fact that the corresponding linearized operator
is essentially self-adjoint (Ishibashi&Wald 2004)
Local well-posedness (Holzegel&Smulevici 2011)
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
6 / 23
Weakly nonlinear perturbations
We seek an approximate solution starting from small initial data
(φ, φ̇)|t=0 = (εf (x), εg (x))
Perturbation series
φ = εφ1 + ε3 φ3 + ...
δ = ε2 δ2 + ε4 δ4 + ...
1 − A = ε2 A2 + ε4 A4 + ...
where (φ1 , φ̇1 )|t=0 = (f (x), g (x)) and (φj , φ̇j )|t=0 = (0, 0) for j > 1.
Inserting this expansion into the field equations and collecting terms
of the same order in ε, we obtain a hierarchy of linear equations
which can be solved order-by-order.
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
7 / 23
First order
Linearized equation (Ishibashi&Wald 2004)
φ̈1 + Lφ1 = 0,
L=−
1
∂x tan2 x ∂x
2
tan x
The operator L is essentially self-adjoint on L2 ([0, π/2], tan2 x dx).
Eigenvalues and eigenvectors of L are (j = 0, 1, . . . )
ωj2 = (3 + 2j)2 ,
( 1 , 32 )
ej (x) = dj cos3 x Pj 2
(cos 2x)
⇒ AdS is linearly stable
Linearized solution
φ1 (t, x) =
∞
X
aj cos(ωj t + βj ) ej (x)
j=0
where amplitudes aj and phases βj are determined by the initial data.
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
8 / 23
Second order (back-reaction on the metric)
A02 +
1 + 2 sin2 x
A2 = sin x cos x φ̇21 + φ01 2
sin x cos x
δ20 = − sin x cos x φ̇21 + φ01 2
so
cos3 x
A2 (t, x) =
sin x
Zx φ̇1 (t, y )2 + φ01 (t, y )2 tan2 y dy
0
δ2 (t, x) = −
Zx φ̇1 (t, y )2 + φ01 (t, y )2 sin y cos y dy
0
It follows that
ε2
M=
2
Zπ/2
φ̇1 (t, y )2 + φ01 (t, y )2 tan2 y dy + O(ε4 )
0
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
9 / 23
Third order
φ̈3 + Lφ3 = S(φ1 , A2 , δ2 )
(?)
where S := 2(A2 + δ2 )φ̈1 + (Ȧ2 + δ̇2 )φ̇1 + (A02 + δ20 )φ01 .
Projecting Eq.(?) on the basis {ej } we obtain an infinite set of
decoupled forced harmonic oscillations for the generalized Fourier
coefficients cj (t) := (φ3 , ej )
c̈j + ωj2 cj = Sj := (S, ej )
Let I = {j ∈ N0 : aj 6= 0} be aPset of indices of nonzero modes in the
linearized solution φ1 (t, x) = ∞
j=0 aj cos(ωj t + βj ) ej (x).
A lengthy but straightforward calculation yields that each triad
(j1 , j2 , j3 ) ∈ I 3 such that ωj = ωj1 + ωj2 − ωj3 gives rise to a resonant
term in Sj (i.e., a term proportional to cos ωj t or sin ωj t).
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
10 / 23
Example 1: single-mode data φ(0, x) = ε e0 (x)
First order φ1 (t, x) = cos(ω0 t)e0 (x), ω0 = 3
P
Third order φ3 (t, x) = ∞
j=0 cj (t)ej (x) where
c̈j + ωj2 cj = bj cos(ω0 t) + b̄j cos(3ω0 t)
3ω0 = ω3 but b̄3 = 0! Only j = 0 is resonant.
1
ε=0.25
log[Π2(t,0)]
0.5
0
The j = 0 resonance can be
easily removed by the
two-scale method (slow-time
phase modulation) which gives
2
φ ' ε cos(3t + 153
4π ε t) e0 (x).
ε=0.1768
-0.5
ε=0.125
-1
-1.5
-2
0
200
400
600
800
1000
1200
t
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
11 / 23
Example 2: two-mode data φ(0, x) = ε (e0 (x) + e1 (x))
First order
φ1 (t, x) = cos(ω0 t)e0 (x) + cos(ω1 t)e1 (x), ω0 = 3, ω1 = 5
P
Third order φ3 (t, x) = ∞
j=0 cj (t)ej (x) where
X
(j)
(j)
(j)
c̈j +ωj2 cj =
b~ cos(Ω~ t), Ω~ = ωj1 ±ωj2 ±ωj3 , jn ∈ {0, 1}
k
k
k
~k=(j1 ,j2 ,j3 )
(j)
(2)
Ω~ ∈ {1, 3, 5, 7, 9, 11, 13, 15}. The resonance Ω(1,1,0) = 7 = 5 + 5 − 3
k
cannot be removed, hence we get the secular term
c2 (t) ∼ t sin(7t)
For n-mode initial data the number of triads satisfying the
diophantine resonance condition grows rapidly with n and hence, so
does the number of unremovable secular terms. We believe that
these secular terms are progenitors of the higher-order resonant mode
mixing which shifts the energy spectrum to higher frequencies.
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
12 / 23
√
√
Let Φj := ( A Φ, ej0 ) and Πj := ( A Π, ej ). Then
Zπ/2
1
M=
2
∞
X
AΦ2 + AΠ2 tan2 x dx =
Ej (t) ,
j=0
0
where Ej := Π2j + ωj−2 Φ2j can be interpreted as the j-mode energy.
1
0.95
φ(0, x) = ε (e0 (x) + e1 (x))
Σk
0.9
0.85
k
1 X
Σk :=
Ej
M
k=1
k=2
k=4
k=8
k=16
0.8
0.75
j=0
0.7
0
20
40
60
80
100
120
140
160
180
t
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
13 / 23
Gaussian initial data
2x
Φ(0, x) = 0 , Π(0, x) = ε exp − tan
2
σ
108
ε=6*21/2
ε=6
ε=3*21/2
ε=3
(a)
7
10
Π2(t,0)
106
105
104
103
102
101
0
10
0
200
400
600
800
1000
1200
1400
1600
t
All solutions eventually collapse to the Schwarzschild-AdS black hole
φ = 0, δ = 0, A = 1 − M cos3 x/ sin x (generic endstate of evolution)
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
14 / 23
Key evidence for instability
106
(b)
105
ε=6*21/2
ε=6
ε=3*21/2
ε=3
103
102
-2
2 2
ε Π (ε t,0)
4
10
101
100
-1
10
0
2000
4000
6000
8000
10000
12000
2
ε t
Onset of instability at time t = O(ε−2 )
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
15 / 23
Conjectures
Our numerical and formal perturbative computations led us to:
Conjecture 1
Anti-de Sitter space is unstable against the formation of a black hole
under arbitrarily small generic perturbations
Note that we did not claim that all perturbed solutions end up as black
holes. On the contrary:
Conjecture 2
Einstein-scalar-AdS equations admit time-quasiperiodic solutions
Analogous conjecture (existence of geons) for vacuum Einstein’s equations
was made later by Dias, Horowitz and Santos (2011)
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
16 / 23
Past year
Answering emails from string community
Computation of energy spectra
Building up evidence for the instability of AdS
Studies of turbulence in toy models
Construction of quasiperiodic solutions
Optimizing numerical codes
Multi-scale analysis of the onset of turbulence
Trying to incorporate ”deformed” boundary conditions
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
17 / 23
Energy spectrum
1
Ej/M
0.1
t=0
t=1100
t=1495
0.01
0.001
0.0001
10
100
1000
j
Ej ∼ j −α
Piotr Bizoń (Jagiellonian University)
Fit gives α ≈ 1.2 (6/5??)
Instability of AdS
Oberwolfach, 3 August 2012
18 / 23
Minimally coupled scalar field in d + 1 dimensions
Qualitatively the same phenomenology for any d ≥ 3
(Jalmużna,Rostworowski,B)
d = 2 (Jalmużna 2012)
ds 2 = −(1 − M + r 2 )dt 2 + (1 − M + r 2 )−1 dr 2 + r 2 dφ2
There is a mass gap for the formation of black holes:
I
I
I
M=0
AdS
0 < M < 1 naked singularities
M>1
BTZ black holes
Results:
I
I
I
turbulent instability for small perturbations but no collapse
no indication for singularity formation
evidence for equipartition of energy (almost flat spectrum of energy)
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
19 / 23
Vacuum in 4 + 1 dimensions
In 4 + 1 dimensions there is a way to bypass Birkhoff’s theorem:
cohomogeneity-two (biaxial) Bianchi IX ansatz (B&Schmidt 2005)
sin2 x 2B 2
`2
−2δ 2
−1
2
2
2
−4B 2
−Ae dt + A dx +
ds =
e (σ1 + σ2 ) + e
σ3
cos2 x
4
where A, δ, B are functions of (t, x). Inserting this ansatz into 5D
Einstein’s vacuum field equations with Λ = −6/`2 one gets
0
4e −δ −2B
1 3
−δ 0
−8B
tan
x
Ae
B
−
e
−
e
tan3 x
3 sin2 x
2(4e −2B − e −8B − 3A)
A0 = 4 tan x (1 − A) − 2 sin x cos x A B 02 + P 2 +
3 tan x
δ 0 = −2 sin x cos x B 02 + P 2
Ḃ = Ae −δ P,
Ṗ =
This system exhibits very similar turbulent behavior to the self-gravitating
scalar field (blowup of the Kretschmann scalar in finite time).
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
20 / 23
Instability of Minkowski spacetime enclosed in a cavity
Spherically symmetric metric ds 2 = −A N −2 dt 2 + A−1 dr 2 + r 2 dΩ2
Einstein-scalar field equations (Φ := φ0 and Π := A−1 e δ φ̇)
0
Φ̇ = Ae −δ Π ,
1 2 −δ 0
r Ae Φ
r2
1
A0 = (1 − A) − r A Φ2 + Π2
r
0
δ = −rA2 e −δ Φ2 + Π2
Π̇ =
Recently Maliborski solved this system numerically inside a spherical
cavity r ≤ R with no-flux boundary conditions (Dirichlet Π(t, R) = 0
or Neumann Φ(t, R) = 0) and found that arbitrarily small data evolve
into a black hole (phenomenology very similar to the AdS case)
This indicates that Λ < 0 plays no dynamical role in instability of AdS
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
21 / 23
Nonlinear waves on bounded domains
On unbounded domains the ground state is typically asymptotically
stable due to the dissipation of energy by dispersion. On bounded
domains asymptotic stability is excluded and it is natural to expect
that generic perturbations will exhibit weakly turbulent behavior, that
is the energy will move from low to high frequencies.
Simple example: scalar wave u(t, x) with defocusing nonlinearity on
the interval 0 ≤ x ≤ π with fixed endpoints
utt − uxx + m2 u + u 3 = 0,
u(t, 0) = 0 = u(t, π)
Existence of periodic-in-time solutions depends
p crucially on the
spectrum of the linearized problem (ωj = j 2 + m2 )
I
I
m > 0 (nonresonant case) ⇒ periodic solutions from linear modes by
KAM theory (or Poincare-Lindstedt)
m = 0 (resonant case) ⇒ periodic solutions are hard to construct
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
22 / 23
Conclusions
Dynamics of asymptotically AdS spacetimes is an exceptional meeting
meeting point of fundamental problems in general relativity, PDE
theory, theory of turbulence, and high energy physics. Understanding
of these connections is at its infancy
Insight into this problem can be gained by studying propagation of
nonlinear waves on bounded domains
For defocusing wave equations on fixed backgrounds the turbulent
behavior may last forever but for Einstein’s equations formation of a
black hole provides a cutoff for the process of energy concentration on
exceedingly small spatial scales (in analogy to viscosity for the
Navier-Stokes equation)
From numerical explorations of Einstein’s equations there can grow
understanding, conjectures, and roads to proofs and phenomena that
would not have been imaginable in the pre-computer era. In my
opinion, the role of computation within general relativity seems
destined to expand in future.
Piotr Bizoń (Jagiellonian University)
Instability of AdS
Oberwolfach, 3 August 2012
23 / 23