Instability Stability Open Questions TTF & QP Fully Nonlinear Revisiting Scalar Collapse in AdS New Frontiers in Dynamical Gravity DAMTP, Cambridge Steve Liebling With: Venkat Balasubramanian (Western) Alex Buchel (Western/PI) Stephen Green (Guelph) Luis Lehner (Perimeter) Long Island University New York, USA March 24, 2014 Steven L. Liebling Revisiting Scalar Collapse in AdS 1 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Instability of Scalar Field in spher. symm. aAdS Evolution of a Scalar field in sph. symm., asympt. AdS (aAdS) [Bizo´ n -Rostworowski,2011] Fully nonlinear: Consider Gaussian-type initial data w/ amplitude ǫ and width σ Choptuik-type critical behavior However, sub-critical eventually collapses as well Perturbative about pure AdS: At linear order, uncoupled modes: oscillon Resonance at O ( ǫ 3 ): j r = j 1 + j 2 − j 3 Single mode stable, multiple modes unstable Conjecture: AdS generically unstable to collapse via weakly nonlinear turbulent cascade Steven L. Liebling Revisiting Scalar Collapse in AdS 2 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Instability in light of AdS/CFT Correspondence Holographic duality between ( d + 1)-dimensional global AdS (the bulk) and conformal field theory (CFT) on d − 1-dim. boundary ( S d − 1 × R ) Dictionary translates between bulk quantities of aAdS spacetime and quantum operators of CFT Interpretation of instability: initial data generically thermalizes by BH formation ...but are there non-thermalizing initial configurations in the CFT? Steven L. Liebling Revisiting Scalar Collapse in AdS 3 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Paths to Stability Perturbative analysis showing stable solutions [Dias,Horowitz,Marolf,Santos,1208.5772] Argue perturbatively for nonlinear stability Geons and boson stars, not necessarily spher. symm. Excite all modes [Buchel,Lehner,SLL,1210.0890] Perturbative argument for stability at O ( ǫ 3 ) A j 1 A j 2 A j 3 ω j r → ω j r + ǫ 2 � C j 1 j 2 j 3 j r , where the triple sum { j 1 , j 2 , j 3 } A jr is over all the resonance channels ω j 1 + ω j 2 = ω j r + ω j 3 Time-periodic solutions [Maliborski,Rostworowski,1303.3186] Construct time-periodic solutions Argue for nonlinear stability Frustrated resonance [Buchel,SLL,Lehner,1304.4166] Steven L. Liebling Revisiting Scalar Collapse in AdS 4 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Frustrated Resonance [Buchel,SLL,Lehner,1304.4166] Broadly distrib. energy perturbs AdS & introduces dispersion Dispersion competes with nonlinear sharpening BR data: increasing σ increases distribution of energy Issues 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 Steven L. Liebling Revisiting Scalar Collapse in AdS 5 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Perturbed Boson Star 4dmdr1.mpg Perturbed BS Steven L. Liebling Revisiting Scalar Collapse in AdS 6 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear 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 − µ 2 | φ | 2 No dispersion at linear order Mass changes location of transition σ c rit 0 . 8 0 . 7 0 . 6 σ crit 0 . 5 0 . 4 0 . 3 0 . 2 − 2 − 1 0 1 2 3 4 µ 2 Steven L. Liebling Revisiting Scalar Collapse in AdS 7 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Open Questions Among others, just two here: What’s stable and what’s unstable? ... in other words, can we identify whether initial data will collapse for any amplitude a priori? For ID that appears unstable, can we be sure whether it extends to ǫ → 0? ...using “unstable” as ID that collapses for ǫ → 0 but ǫ � = 0 Steven L. Liebling Revisiting Scalar Collapse in AdS 8 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Two-Time Formalism (TTF) Dynamics characterized by two time scales: fast time t –generally t < π where π is time for a bounce off boundary slow time τ –scale over which energy transfers among oscillons, τ ≡ ǫ 2 t Allow mode amplitudes A j ( t ) to be functions of both times A j ( t , τ ) Enforce at O ( ǫ 3 ) the absence of secular terms in the scalar field Advantages: Goes beyond initial transfer of energy (...to time t > 1 /ǫ 2 ) Conserves energy Both direct and inverse cascades Solve coupled, cubic, ODEs in complex mode amplitudes A j Resembles FPUT paradox Steven L. Liebling Revisiting Scalar Collapse in AdS 9 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear TTF and Fermi-Pasta-Ulam-Tsingou (FPUT, 1953) Model 1D atoms in a crystal by masses linked by springs with nonlinear term At linear order, Fourier modes decouple Nonlinear system may not not approach equipartition , as predicted by classical stat. mech. ...apparently still debated more than 50 years later! For N → ∞ (nonlinear string) and small energies, system is similarly resonant [D. Campbell’s APS 2010 talk] Steven L. Liebling Revisiting Scalar Collapse in AdS 10 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear TTF and Fully Nonlinear... two-mode ID Similar “evolutions”... both direct and inverse cascades TTF convergent with increasing j max In ǫ → 0 and j max → ∞ limits, TTF & NL should converge Steven L. Liebling Revisiting Scalar Collapse in AdS 11 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear TTF and Quasi-Periodic (QP) Solutions Specify A qp j ( τ ) = α j e − i β j τ Solutions branching from dominant mode j r ...two branches for j r > 0 One-parameter generalizations of MR time-periodic solutions Such solutions balance direct and inverse transfers...no energy transfer among modes ˙ E j = 0 Approximated by exponential spectrum E j = e − µ j Steven L. Liebling Revisiting Scalar Collapse in AdS 12 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear (Approximate) QP Solution Evolved Fully Nonlinearly 1 . 75 1 . 70 Log 10 | Π 2 ( x, 0) /ǫ 2 | 1 . 65 1 . 60 1 . 55 1 . 50 1 . 45 0 5 10 15 20 25 30 35 40 ǫ 2 t lnasqj qp.mpg QP NL Steven L. Liebling Revisiting Scalar Collapse in AdS 13 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Engineered Initial Data (ID) Form of ID for fully nonlinear evolutions Specify amplitudes c j of oscillons present in ID: Π( x , 0) = 0 φ ( x , 0) = Σ j ( c j e j ( x )) Examples: Equal energy 2-mode ID: c j = δ i j / (3 + 2 i ) + δ k j / (3 + 2 k ) Exponential amplitude ID: c j = e − α j Exponential energy ID: c j = e − α j / (3 + 2 j ) Steven L. Liebling Revisiting Scalar Collapse in AdS 14 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Two-mode Stable Solutions Equal-Energy Two-Mode Initial Data: Modes 0 and 1: c j = e 0 / 3 + e 1 / 5 10 9 8 Log 10 | Π 2 ( x, 0) /ǫ 2 | 7 6 5 4 3 2 1 0 2 4 6 8 10 12 ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 15 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Two-mode Stable Solutions Equal-Energy Two-Mode Initial Data: Modes 0 and 1: c j = e 0 / 3 + e 1 / 5 10 BH Formation 9 8 Frustrated Resonance Log 10 | Π 2 ( x, 0) /ǫ 2 | 7 6 5 4 3 2 1 0 2 4 6 8 10 12 ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 16 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Two-mode Stable Solutions Equal-Energy Two-Mode Initial Data: Modes 0 and 1: c j = e 0 / 3 + e 1 / 5 Left from: Benettin,Carati,Galgani,Giorgilli, 2008] Steven L. Liebling Revisiting Scalar Collapse in AdS 17 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Other stable solutions Three-Mode Initial Data: Modes 1, 3, and 8: c j = e 1 / 10 + e 3 + e 8 / 10 5 . 4 5 . 2 Log 10 | Π 2 ( x, 0) /ǫ 2 | 5 . 0 4 . 8 4 . 6 4 . 4 4 . 2 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 18 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Other stable solutions Three-Mode Initial Data: Modes 1, 3, and 8: c j = e 1 / 10 + e 3 + e 8 / 10 5 . 4 5 . 2 Log 10 | Π 2 ( x, 0) /ǫ 2 | 5 . 0 4 . 8 4 . 6 4 . 4 4 . 2 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 19 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Other (possibly) stable solutions Three-Mode Initial Data: Modes 1, 3, and 8: c j = e 1 + e 3 + e 8 not one-mode dominant 12 12 11 11 10 10 Log 10 | Π 2 ( x, 0) /ǫ 2 | Log 10 | Π 2 ( x, 0) /ǫ 2 | 9 9 8 8 7 7 6 6 5 5 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 00 0 . 01 0 . 02 0 . 03 0 . 04 0 . 05 ǫ 2 t ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 20 / 23
Instability Stability Open Questions TTF & QP Fully Nonlinear Other (possibly) stable solutions Exponential Amplitude: c j = e − α j for α = 0 . 575 10 9 8 Log 10 | Π 2 ( x, 0) /ǫ 2 | 7 6 5 4 3 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 ǫ 2 t Steven L. Liebling Revisiting Scalar Collapse in AdS 21 / 23
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