Chulalongkorn
University
จุ ฬ าลงกรณ์ มหาวิ ท ยาลั ย
Non-linear dynamics of AdS perturbations
Oleg Evnin
Ben Craps, OE, Joris Vanhoof, JHEP 1410 (2014) 48
Ben Craps, OE, Joris Vanhoof, JHEP 1501 (2015) 108
OE, Chethan Krishnan, Phys. Rev. D91 (2015) 12, 126010
Ben Craps, OE, Joris Vanhoof, arXiv:1508.04943
Ben Craps, OE, Puttarak Jai-akson, Joris Vanhoof, arXiv:1508.05474
+ groups at Krakow, Perimeter Institute, etc...
Selected Topics in Theoretical High Energy Physics, 21-27 September, 2015, Tbilisi
Minkowski space is stable due to
dispersion of energy to infinity
Consider, for instance, an infalling spherical shell of massless
scalar field minimally coupled to gravity.
For sufficiently large amplitude: black hole formation.
For sufficiently small amplitude: shell scatters back to infinity. In
general, Minkowski space has been shown to be stable under
small perturbations.
[Christodoulou, Klainerman 1993]
AdS acts like a box: Is it stable?
If a shell has too small amplitude to
form a black hole right away, it
scatters away to the boundary. But
the boundary will reflect the shell. It
will continue scattering, slightly
changing its shape every time,
possibly forming a small black hole
in the center after multiple
reflections.
Holography relates black hole
formation to thermalization
AdS
CFT
• Anti-de Sitter spacetime
• Conformal Field Theory
• Black hole
• Thermal state
• Black hole formation
• Thermalization
Conjecture: arbitrarily small perturbations
may lead to black hole formation
For Gaussian shells with amplitude (and specific width), Bizon
and Rostworowski numerically found BH formation at times
scaling like
.
Naive perturbation theory in shows energy transfer to high
frequencies, but breaks down at times of order
.
Non-linear Instability – Weak Turbulence - Collapse
Non-collapsing configurations exist!
Complex interplay of stable and
unstable behavior!
Study scalar in AdSd+1 perturbatively
Spherically symmetric perturbations
and
Metric determined by constraints  Solve e.o.m. for
Perturbative expansion
Expansion in normal modes
with

specific complicated integrals involving AdS mode functions
Resonances (→secular terms)
Resonant if
Integer normal mode spectrum
 many resonances!
Resonances lead to secular terms
They invalidate naive perturbation theory on time scales
[Bizoń, Rostworowski 2011]
.
Time-averaging and flow equations
approximate as cubic
Hamiltonian form:
Introduce new (complex) variables
(“interaction picture”)

Average
over explicit time-dependence:
(“integrate out fast oscillations”)
“flow equations”
Flow equations are reliable on time intervals of order
[Craps, OE, Vanhoof 2015]
Many flow channels are closed
Normal mode spectrum
 resonances if
[Craps, OE, Vanhoof 2014]
Flow equations conserve three quantities
“Free” energy

“Particle number”

(closed flow channels crucial!)
“Interaction” energy

(quartic “interaction energy”)
[Craps, OE, Vanhoof 2015]; cf. [Basu, Krishnan, Saurabh 2014] for probe scalar field
Exact conservation laws in approximate equations valid
on time-scales of order
Simultaneous conservation of E and J implies dual cascades
Transferring all energy to higher-n modes (which have more
energy per particle) would decrease J, hence
some of the energy must flow to lower-n modes
[Buchel, Green, Lehner, Liebling 2015]
Flow equations have quasiperiodic solutions
Flow equations in terms of real amplitudes
and phases
:
Families of quasiperiodic (QP) solutions, which have constant
amplitudes,
, have been found numerically.
[Balasubramanian, Buchel, Green, Lehner, Liebling 2014]
Some recent analytic considerations available...
QP solutions seen as “anchors” of stability islands.
Flow equations see onset of collapse
Extrapolating numerical GR to arbitrarily small amplitude is tricky.
In [Bizoń, Rostworowski 2011],
scaling of the collapse time was
observed, but does the scaling persist to arbitrarily small ?
cf. [Dimitrakopoulos, Freivogel, Lippert, Yang 2014]
The effective flow equations have
scaling built in. If collapse
is captured by these equations, extrapolation becomes possible.
Numerical study of flow equations (truncated to 172 normal modes) for
collapsing initial data in AdS5 suggests (for the infinite system) a
finite-time oscillatory singularity.
Fit to ansatz
that tends to zero in finite time.
[Bizoń, Maliborski, Rostworowski 2015]
shows
Analytic methods complement numerics
Using numerically derived UV asymptotics of the interaction
coefficients in the flow equations, [Bizoń, Maliborski, Rostworowski 2015]
found that their amplitude spectrum seemed consistent with the
flow equations.
The UV asymptotics of the interaction coefficients have now been
derived analytically for arbitrary dimension. [Craps, OE, Vanhoof 2015]
The bottleneck in numerical studies of the flow equations is the
evaluation of interaction coefficients up to high mode number. A
new recursive method is hoped to speed this up.
[Craps, OE, Vanhoof 2015]
General qualitive features of quasiperiodic solutions have been
[Craps, OE, Jai-akson, Vanhoof 2015]
explained analytically.
Other developments and open questions
• Massive scalar fields coupled to gravity.
[Kim 2014]
[Okawa, Lopes, Cardoso 2015]
[Deppe, Frey 2015]
• Scalar fields in Gauss-Bonnet gravity.
[Deppe, Kolly, Frey, Kunstatter 2015]
• Pure gravity in 4+1d.
[Bizoń, Rostworowski] (talk at Strings 2014)
• Spherical cavity in Minkowski space.
[Maliborski 2012]
[Okawa, Cardoso, Pani 2014]
scalar field. [Basu, Krishnan, Saurabh 2014]
[Maliborski, Rostworowski 2014]
• Self-interacting probe
[Yang 2015] [BC, Evnin, Jai-akson, Vanhoof 2015]
• Stability of spacetimes with asymptotically resonant spectra.
[Dias, Horowitz, Marolf, Santos 2012]
[Menos, Suneeta 2015]
Other developments and open questions
• Are all stability islands anchored on quasiperiodic solutions?
[Deppe, Frey 2015]
• Fate of quasiperiodic solutions/stability islands for
?
• Can one prove collapse for arbitrarily small initial data?
[Bizoń, Maliborski, Rostworowski 2015]
• Deeper reason for closed flow channels?
• Beyond spherical symmetry?
[Evnin, Krishnan 2015]
[Dias, Horowitz, Santos 2012]
• Analytic handle on the turbulent regime?
LOCAL ORGANIZERS: Auttakit CHATRABHUTI and Oleg EVNIN.
INTERNATIONAL ADVISORY COMMITTEE: Adi ARMONI (Swansea U),
Eric BERGSHOEFF (Groningen U), Martin CEDERWALL (Chalmers U),
Jarah EVSLIN (IMP-CAS, Lanzhou), Amihay HANANY (IC London),
Sergey KETOV (Tokyo Metro U), Nobuyoshi OHTA (Kin-dai U, Osaka).
http://www.thaihep.phys.sc.chula.ac.th/BKK2016/