Revisiting Scalar Collapse in AdS - University of Cambridge · 2014. 3. 24. · InstabilityStabilityOpen QuestionsTTF & QPFully Nonlinear Two-Time Formalism (TTF) Dynamics characterized

Instability Stability Open Questions TTF & QP Fully Nonlinear

Revisiting Scalar Collapse in AdSNew Frontiers in Dynamical Gravity

DAMTP, Cambridge

Steve LieblingWith:

Venkat Balasubramanian (Western) Alex Buchel (Western/PI)

Stephen Green (Guelph) Luis Lehner (Perimeter)

Long Island UniversityNew York, USA

March 24, 2014

Steven L. Liebling Revisiting Scalar Collapse in AdS 1 / 23


Instability of Scalar Field in spher. symm. aAdS

Evolution of a Scalar field in sph. symm., asympt. AdS (aAdS)[Bizon -Rostworowski,2011]

Fully nonlinear:Consider Gaussian-type initial data w/ amplitude ε and width σChoptuik-type critical behaviorHowever, sub-critical eventually collapses as well

Perturbative about pure AdS:At linear order, uncoupled modes: oscillonResonance at O(ε3): jr = j1 + j2 − j3Single mode stable, multiple modes unstable

Conjecture:AdS generically unstable to collapse via weakly nonlinearturbulent cascade



Instability in light of AdS/CFT Correspondence

Holographic duality between(d + 1)-dimensional global AdS (the bulk)

andconformal field theory (CFT) on d − 1-dim. boundary (Sd−1 × R)

Dictionary translates between bulk quantities of aAdS spacetimeand quantum operators of CFT

Interpretation of instability:initial data generically thermalizes by BH formation

...but are there non-thermalizing initial configurations in the CFT?



Paths to Stability

Perturbative analysis showing stable solutions[Dias,Horowitz,Marolf,Santos,1208.5772]

Argue perturbatively for nonlinear stabilityGeons and boson stars, not necessarily spher. symm.

Excite all modes[Buchel,Lehner,SLL,1210.0890]

Perturbative argument for stability at O(ε3)

ωjr → ωjr + ε2∑{j1,j2,j3}

Aj1Aj2

Aj3

AjrCj1j2j3jr , where the triple sum

is over all the resonance channels ωj1 + ωj2 = ωjr + ωj3

Time-periodic solutions[Maliborski,Rostworowski,1303.3186]

Construct time-periodic solutionsArgue for nonlinear stability

Frustrated resonance[Buchel,SLL,Lehner,1304.4166]


http://arxiv.org/abs/arXiv:1208.5772

http://arxiv.org/abs/1210.0890

http://arxiv.org/abs/arXiv:1303.3186



Frustrated Resonance

[Buchel,SLL,Lehner,1304.4166]

Broadly distrib. energy perturbs AdS & introduces dispersionDispersion competes with nonlinear sharpeningBR data: increasing σ increases distribution of energyIssues with σ-parameterization:[Maliborski,Rostworowski,1307.2875]

Large-σ ceases to be broadly distributed (us and[Abajo-Arrastia,Silva,Lopez,Mas,Serantes,1403.2632])“window” in σ shrinks for higher dims but other ID stable






Perturbed Boson Star

4dmdr1.mpg Perturbed BS



Effect of Mass [Balasubramanian,Buchel,Green,Lehner,SLL,in prep]

Motivation: explore CFT operators of different weight...mass changes decay rate of SF at boundary

Introduce mass term −µ2|φ|2No dispersion at linear order

Mass changes location of transition σcrit

−2 −1 0 1 2 3 4

µ2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

σcr

it



Open Questions

Among others, just two here:

What’s stable and what’s unstable?...in other words, can we identify whether initial data willcollapse for any amplitude a priori?

For ID that appears unstable, can we be sure whether itextends to ε→ 0?

...using “unstable” as ID that collapses for ε→ 0 but ε 6= 0



Two-Time Formalism (TTF)

Dynamics characterized by two time scales:fast time t–generally t < π where π is time for a bounce offboundaryslow time τ–scale over which energy transfers among oscillons,τ ≡ ε2t

Allow mode amplitudes Aj(t) to be functions of both timesAj(t, τ)

Enforce at O(ε3) the absence of secular terms in the scalarfieldAdvantages:

Goes beyond initial transfer of energy (...to time t > 1/ε2)Conserves energyBoth direct and inverse cascades

Solve coupled, cubic, ODEs in complex mode amplitudes Aj

Resembles FPUT paradox



TTF and Fermi-Pasta-Ulam-Tsingou (FPUT, 1953)

Model 1D atoms in a crystal by masses linked by springs withnonlinear termAt linear order, Fourier modes decoupleNonlinear system may not not approach equipartition, aspredicted by classical stat. mech.

...apparently still debated more than 50 years later!

For N →∞ (nonlinear string) and small energies, system issimilarly resonant

[D. Campbell’s APS 2010 talk]



TTF and Fully Nonlinear...two-mode ID

Similar “evolutions”...both direct and inverse cascades

TTF convergent with increasing jmax

In ε→ 0 and jmax →∞ limits, TTF & NL should converge



TTF and Quasi-Periodic (QP) Solutions

Specify Aqpj (τ) = αje

−iβjτ

Solutions branching from dominant mode jr ...two branches forjr > 0One-parameter generalizations of MR time-periodic solutionsSuch solutions balance direct and inverse transfers...no energytransfer among modes Ej = 0Approximated by exponential spectrum Ej = e−µj



(Approximate) QP Solution Evolved Fully Nonlinearly

0 5 10 15 20 25 30 35 40ε2t

1.45

1.50

1.55

1.60

1.65

1.70

1.75

Log

10|Π

2(x,0

)/ε2|

lnasqj qp.mpg QP NL



Engineered Initial Data (ID)

Form of ID for fully nonlinear evolutions

Specify amplitudes cj of oscillons present in ID:

Π(x , 0) = 0

φ(x , 0) = Σj (cjej(x))

Examples:

Equal energy 2-mode ID: cj = δij/(3 + 2i) + δkj /(3 + 2k)

Exponential amplitude ID: cj = e−αj

Exponential energy ID: cj = e−αj/(3 + 2j)



Two-mode Stable SolutionsEqual-Energy Two-Mode Initial Data: Modes 0 and 1: cj = e0/3 + e1/5

0 2 4 6 8 10 12ε2t

1

2

3

4

5

6

7

8

9

10L

og10|Π

2(x,0

)/ε2|




0 2 4 6 8 10 12ε2t

1

2

3

4

5

6

7

8

9

10L

og10|Π

2(x,0

)/ε2|

BH FormationF

rustratedR

esonance




Left from: Benettin,Carati,Galgani,Giorgilli, 2008]



Other stable solutionsThree-Mode Initial Data: Modes 1, 3, and 8: cj = e1/10 + e3 + e8/10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5ε2t

4.2

4.4

4.6

4.8

5.0

5.2

5.4

Log

10|Π

2(x,0

)/ε2|



Other stable solutionsThree-Mode Initial Data: Modes 1, 3, and 8: cj = e1/10 + e3 + e8/10

0.0 0.1 0.2 0.3 0.4 0.5ε2t

4.2

4.4

4.6

4.8

5.0

5.2

5.4

Log

10|Π

2(x,0

)/ε2|



Other (possibly) stable solutionsThree-Mode Initial Data: Modes 1, 3, and 8: cj = e1 + e3 + e8

not one-mode dominant

0.00 0.05 0.10 0.15 0.20ε2t

5

6

7

8

9

10

11

12

Log

10|Π

2(x,0

)/ε2|

0.00 0.01 0.02 0.03 0.04 0.05ε2t

5

6

7

8

9

10

11

12

Log

10|Π

2(x,0

)/ε2|



Other (possibly) stable solutionsExponential Amplitude: cj = e−αj for α = 0.575

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4ε2t

3

4

5

6

7

8

9

10

Log

10|Π

2(x,0

)/ε2|



Take-home Points

TTF formalism most useful, perturbative approach

System demonstrates both direct- and indirect cascades

Similarity to FPUT system...identical structure of equationsas TTF; resonant and nonresonant regimes

Stability regionsExistence of stable, quasi-periodic solutionsID w/ broadly distributed energy immune frustratedresonance...(regardless of deficiencies in large-σparameterization)Frustrated resonance continues to higher dimensionsFrustrated resonance extends to massive case (qualitativelysimilar)Other, not just one-mode dominant solutions...equal energy,two-mode IDPoses questions about thermalization/equilibration in CFT



Uncanny resemblance!FPUT and 2-Mode 0-1 Equal Energy FNL

Plot of energy in each mode Ej(t)


Documents

Revisiting Scalar Collapse in AdS - University of Cambridge · 2014. 3. 24. · InstabilityStabilityOpen QuestionsTTF & QPFully Nonlinear Two-Time Formalism (TTF) Dynamics characterized