Correspondence Properties for Waves on Asymptotically Anti-de Sitter Spacetimes Arick Shao (joint work with Gustav Holzegel) Imperial College London Arick Shao (Imperial College London) Correspondence Properties 1 / 40
Introduction Section 1 Introduction Arick Shao (Imperial College London) Correspondence Properties 2 / 40
Introduction Anti-de Sitter Spacetime Anti-de Sitter Spacetime Anti-de Sitter ( AdS ) spacetime: Maximally symmetric solution of Einstein vacuum equations ( EVE ). With negative cosmological constant Λ . For convenience, fix Λ := − 3. Globally represented as manifold ( R 4 , g ) , with g = ( 1 + r 2 ) − 1 dr 2 − ( 1 + r 2 ) dt 2 + r 2 ˚ γ . γ : round metric on S 2 . ˚ Generalises directly to higher dimensions. Arick Shao (Imperial College London) Correspondence Properties 3 / 40
Introduction Anti-de Sitter Spacetime AdS Infinity Consider “inverted radius” ρ := r − 1 ⇒ t g = ρ − 2 [( 1 + ρ 2 ) − 1 d ρ 2 − ( 1 + ρ 2 ) dt 2 + ˚ γ ] . ρ is a “boundary defining function” ⇒ can think of “ I := { ρ = 0 } ” as AdS infinity. r = 0 I I ≃ R × S 2 has Lorentzian structure: g := − dt 2 + ˚ r γ . ˚ Spacetimes that “have same asymptotic infinity I ” called asymptotically AdS ( aAdS ). Compactified AdS, mod S 2 . Arick Shao (Imperial College London) Correspondence Properties 4 / 40
Introduction Problem Statements Motivations Question ( ∞ ) Does “geometric boundary data” prescribed at AdS infinity determine interior dynamics of EVE? If two aAdS vacuum spacetimes have identical “Dirichlet and Neumann data” at infinity, then must they be isometric? (If not globally, then at least locally near infinity?) In other words: Is there some correspondence between boundary data at infinity and interior gravitational dynamics? Arick Shao (Imperial College London) Correspondence Properties 5 / 40
Introduction Problem Statements Difficulties Bad news: Initial value problems for hyperbolic equations generally ill-posed on timelike hypersurfaces (such as I ). Thus, may not expect to solve EVE. However, can still ask whether existing solutions are unique. Also, the EVE are highly nonlinear. This is work in progress: Expect to prove various positive results. Arick Shao (Imperial College London) Correspondence Properties 6 / 40
Introduction Problem Statements A Linear Model Problem EVE is hard ⇒ consider first a model problem. “(Very) poor man’s linearisation” of EVE: scalar wave equation � φ + σφ = 0, σ ∈ R on fixed AdS (or aAdS) spacetime. Consider analogous problem for scalar wave equation. Recently completed work. Arick Shao (Imperial College London) Correspondence Properties 7 / 40
Introduction Problem Statements The Main Problems Question (1) I φ 1 , φ 2 : solutions on AdS of ( � g + σ ) φ + a α ∇ α φ + V φ = 0 , a α , V decay, σ ∈ R , with same Dirichlet and Neumann data on AdS infinity. Does φ 1 = φ 2 near infinity? φ = 0? φ, d φ = 0. Equivalently: If φ solves the above, then does φ vanishing at I ⇒ φ vanishes near I ? Question (2) Can we generalise solution to Question (1) so it can be The uniqueness problem. applied to solve Question ( ∞ )? Arick Shao (Imperial College London) Correspondence Properties 8 / 40
Results for Scalar Waves Section 2 Results for Scalar Waves Arick Shao (Imperial College London) Correspondence Properties 9 / 40
Results for Scalar Waves Initial Intuitions Analytic Theory, Simplified To get basic idea, assume: a α and V vanish ( ⇒ ( � g + σ ) φ = 0). φ depends only on ρ . ⇒ 2nd-order ODE for φ : ρ 2 ( 1 + ρ 2 ) ∂ 2 ρ φ − 2 ρ∂ ρ φ + σφ = 0. Frobenius method ⇒ two branches of solutions: � � ∞ β ± = 3 9 φ ± = ρ β ± a ± k ρ k , 2 ± 4 − σ . k = 0 Agrees with Breitenlohner-Freedman (5 / 4 < σ < 9 / 4). Arick Shao (Imperial College London) Correspondence Properties 10 / 40
Results for Scalar Waves Initial Intuitions Removal of Analyticity Analytic theory ⇒ for φ to vanish, must eliminate both branches: ρ − β + φ → 0, ρ ց 0. Goal: Remove analyticity assumptions. 1 Consider non-analytic φ (depending on all variables). 2 Consider non-analytic a α , V . 3 (Later) Consider other non-analytic metrics g . Question: Similar results if φ , a α , V are only C ∞ ? Arick Shao (Imperial College London) Correspondence Properties 11 / 40
Results for Scalar Waves The Main Results Main Theorem, I Theorem (Holzegel, S.; 2015) Suppose φ is a C 2 -solution of | ( � g + σ ) φ | ≤ ρ 2 + p ( | ∂ t φ | + | ∂ ρ φ | + | ∇ S 2 φ | ) + ρ p | φ | , where σ ∈ R and p > 0 . Suppose that | ρ − β + φ | + | ∇ t ,ρ, S 2 ( ρ − β + + 1 φ ) | → 0 , if σ ≤ 2 ( β + ≥ 2 ), | ρ − 2 φ | + | ∇ t ,ρ, S 2 ( ρ − 1 φ ) | → 0 , if σ ≥ 2 ( β + ≤ 2 ), as ρ ց 0 , on a sufficiently large time interval 0 ≤ t ≤ t 0 , t 0 > π . Then, φ vanishes in the interior of AdS, near I ∩ { 0 < t < t 0 } . Furthermore, the results extend to ( n + 1 ) -dimensional AdS spacetime for any n (with natural modifications to β ± , ranges of σ , etc.). Arick Shao (Imperial College London) Correspondence Properties 12 / 40
Results for Scalar Waves The Main Results Remarks: Comparisons Comparisons with the analytic theory: 1 σ ≤ 2: vanishing condition is optimal. 2 σ > 2: require more vanishing than expected. Question: Can this be improved? 3 We also require vanishing conditions for ∇ φ . Analytic case: redundant information, not needed. 4 New: “Sufficiently large time interval” assumption. Arick Shao (Imperial College London) Correspondence Properties 13 / 40
Results for Scalar Waves The Main Results Remarks: Local and Global Uniqueness Result is “local”: only show φ vanishes near I ∩ { 0 < t < t 0 } . AdS: can use global geometric properties to show “global” uniqueness (i.e., φ vanishes on { 0 < t < t 0 } ). Does not extend to general aAdS spacetimes. Remark Global uniqueness: t 0 ≥ π necessary by finite speed of propagation. More surprisingly, this also seems necessary for local uniqueness. Arick Shao (Imperial College London) Correspondence Properties 14 / 40
Results for Scalar Waves The Main Results Remarks: Bounded Potentials Question What if lower-order terms a α and V decay less? In particular, V only bounded? Proposition (Holzegel, S.; 2015) Suppose φ is a C 2 -solution of | � g φ | ≤ ρ 2 ( | ∂ t φ | + | ∂ ρ φ | + | ∇ S 2 φ | ) + | φ | . (p = 0 ) Suppose φ and ∇ φ vanish to infinite order as ρ ց 0 , on a sufficiently large time interval 0 ≤ t ≤ t 0 , t 0 > π . Then, φ vanishes in the interior of AdS, near I ∩ { 0 < t < t 0 } . Again, the results extend to ( n + 1 ) -dimensional AdS spacetime for any n. Arick Shao (Imperial College London) Correspondence Properties 15 / 40
Results for Scalar Waves Connections to Well-Posedness Well-Posedness Goal: Connect results to (non-analytic) local well-posedness theory. Connect vanishing conditions to zero Dirichlet and Neumann data. Theorem (Warnick) Let 5 / 4 < σ < 9 / 4 . Then: � g + σ propagates a “twisted H 1 -energy” E 1 ( t ) . Roughly, like the H 1 -norm, but ∇ replaced by ρ β − ∇ ρ − β − . Similarly defined “twisted H 2 -energy”, E 2 ( t ) , is also propagated. Assuming one of the following boundary conditions, ρ − β − φ → 0 (Dirichlet), ρ − 2 + 2 β − ∂ ρ ( φρ − β − ) → 0 (Neumann), then ( � g + σ ) φ = 0 is well-posed in the twisted H 1 -norm. Arick Shao (Imperial College London) Correspondence Properties 16 / 40
Results for Scalar Waves Connections to Well-Posedness Main Theorem, II Main idea : Given extra regularity, Dirichlet and Neumann conditions ⇒ vanishing assumptions in first theorem. Theorem (Holzegel, S.; 2015) Suppose φ is a C 2 -solution of | ( � g + σ ) φ | ≤ ρ 2 + p ( | ∂ t φ | + | ∂ ρ φ | + | ∇ S 2 φ | ) + ρ p | φ | , where 5 / 4 < σ < 9 / 4 and p > 0 . Suppose φ satisfies both vanishing Dirichlet and Neumann conditions. φ has finite twisted energy E 2 ( t ) . on a time interval 0 ≤ t ≤ t 0 (t 0 > π ). Then, φ = 0 near I ∩ { 0 < t < t 0 } . Again, analogous results hold in other dimensions. Arick Shao (Imperial College London) Correspondence Properties 17 / 40
Ideas Behind the Proof Section 3 Ideas Behind the Proof Arick Shao (Imperial College London) Correspondence Properties 18 / 40
Ideas Behind the Proof Classical Theory Unique Continuation View question as unique continuation (UC) problem—a classical problem in PDEs: Suppose ( � g + a α ∇ α + V ) φ = 0 on a domain. Suppose φ, d φ = 0 on hypersurface Σ . Must φ vanish (locally) on one side of Σ ? Cauchy-Kovalevskaya : g , a α , V analytic ⇒ can solve for unique power series solutions (if Σ noncharacteristic). Holmgren’s theorem : Solution unique in class of distributions. Arick Shao (Imperial College London) Correspondence Properties 19 / 40
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