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Correspondence Properties for Waves on Asymptotically Anti-de Sitter Spacetimes

Arick Shao

(joint work with Gustav Holzegel) Imperial College London

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Introduction

Section 1 Introduction

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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 (R4, g), with g = (1 + r2)−1dr2 − (1 + r2)dt2 + r2˚ γ. ˚ γ: round metric on S2. Generalises directly to higher dimensions.

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Introduction Anti-de Sitter Spacetime

AdS Infinity

r=0 I t r Compactified AdS, mod S2.

Consider “inverted radius” ρ := r−1 ⇒ g = ρ−2[(1 + ρ2)−1dρ2 − (1 + ρ2)dt2 + ˚ γ]. ρ is a “boundary defining function” ⇒ can think of “I := {ρ = 0}” as AdS infinity. I ≃ R × S2 has Lorentzian structure: ˚ g := −dt2 + ˚ γ. Spacetimes that “have same asymptotic infinity I” called asymptotically AdS (aAdS).

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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?

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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.

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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

• n fixed AdS (or aAdS) spacetime.

Consider analogous problem for scalar wave equation. Recently completed work.

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Introduction Problem Statements

The Main Problems

Question (1)

φ1, φ2: solutions on AdS of (g + σ)φ + aα∇αφ + V φ = 0, σ ∈ R, aα, V decay, with same Dirichlet and Neumann data on AdS infinity. Does φ1 = φ2 near infinity? 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 applied to solve Question (∞)?

I φ,dφ=0. φ=0? The uniqueness problem. Arick Shao (Imperial College London) Correspondence Properties 8 / 40

Results for Scalar Waves

Section 2 Results for Scalar Waves

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Results for Scalar Waves Initial Intuitions

Analytic Theory, Simplified

To get basic idea, assume: aα and V vanish (⇒ (g + σ)φ = 0). φ depends only on ρ. ⇒ 2nd-order ODE for φ: ρ2(1 + ρ2)∂2

ρφ − 2ρ∂ρφ + σφ = 0.

Frobenius method ⇒ two branches of solutions: φ± = ρβ±

∞

• k=0

a±

k ρk,

β± = 3 2 ±

• 9

4 − σ. Agrees with Breitenlohner-Freedman (5/4 < σ < 9/4).

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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 aα, V . 3 (Later) Consider other non-analytic metrics g.

Question: Similar results if φ, aα, V are only C ∞?

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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| ≤ ρ2+p(|∂tφ| + |∂ρφ| + |∇S2φ|) + ρp|φ|, where σ ∈ R and p > 0. Suppose that |ρ−β+φ| + |∇t,ρ,S2(ρ−β++1φ)| → 0, if σ ≤ 2 (β+ ≥ 2), |ρ−2φ| + |∇t,ρ,S2(ρ−1φ)| → 0, if σ ≥ 2 (β+ ≤ 2), as ρ ց 0, on a sufficiently large time interval 0 ≤ t ≤ t0, t0 > π. Then, φ vanishes in the interior of AdS, near I ∩ {0 < t < t0}. Furthermore, the results extend to (n + 1)-dimensional AdS spacetime for any n (with natural modifications to β±, ranges of σ, etc.).

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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 < t0}. AdS: can use global geometric properties to show “global” uniqueness (i.e., φ vanishes on {0 < t < t0}). Does not extend to general aAdS spacetimes. Remark Global uniqueness: t0 ≥ π necessary by finite speed of propagation. More surprisingly, this also seems necessary for local uniqueness.

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Results for Scalar Waves The Main Results

Remarks: Bounded Potentials

Question What if lower-order terms aα and V decay less? In particular, V only bounded? Proposition (Holzegel, S.; 2015)

Suppose φ is a C 2-solution of |gφ| ≤ ρ2(|∂tφ| + |∂ρφ| + |∇S2φ|) + |φ|. (p = 0) Suppose φ and ∇φ vanish to infinite order as ρ ց 0, on a sufficiently large time interval 0 ≤ t ≤ t0, t0 > π. Then, φ vanishes in the interior of AdS, near I ∩ {0 < t < t0}. Again, the results extend to (n + 1)-dimensional AdS spacetime for any n.

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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 H1-energy” E 1(t).

Roughly, like the H1-norm, but ∇ replaced by ρβ−∇ρ−β−.

Similarly defined “twisted H2-energy”, E 2(t), is also propagated. Assuming one of the following boundary conditions, ρ−β−φ → 0 (Dirichlet), ρ−2+2β−∂ρ(φρ−β−) → 0 (Neumann), then (g + σ)φ = 0 is well-posed in the twisted H1-norm.

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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| ≤ ρ2+p(|∂tφ| + |∂ρφ| + |∇S2φ|) + ρp|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).

• n a time interval 0 ≤ t ≤ t0 (t0 > π). Then, φ = 0 near I ∩ {0 < t < t0}.

Again, analogous results hold in other dimensions.

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Ideas Behind the Proof

Section 3 Ideas Behind the Proof

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Ideas Behind the Proof Classical Theory

Unique Continuation

View question as unique continuation (UC) problem—a classical problem in PDEs: Suppose (g + aα∇α + V )φ = 0 on a domain. Suppose φ, dφ = 0 on hypersurface Σ. Must φ vanish (locally) on one side of Σ? Cauchy-Kovalevskaya: g, aα, V analytic ⇒ can solve for unique power series solutions (if Σ noncharacteristic). Holmgren’s theorem: Solution unique in class of distributions.

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Ideas Behind the Proof Classical Theory

Non-Analytic Theory

However, this is not a satisfactory answer: Cannot deal with non-analytic g, aα, V . Cannot deal with nonlinear wave equations. In the non-analytic setting, UC depends on geometry near Σ. (H¨

• rmander, Lerner-Robbiano) Main criterion is pseudoconvexity:

Σ := {f = 0} pseudoconvex ⇒ UC from Σ to {f > 0}.

(Alinhac) Σ not pseudoconvex ⇒ ∃ aα, V for which UC from Σ to {f > 0} does not hold.

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Ideas Behind the Proof Classical Theory

Pseudoconvexity

Definition (Lerner-Robbiano)

Σ := {f = 0} is pseudoconvex (w.r.t. g and sgn f ) iff ∇2f (X, X) < 0 on Σ, if g(X, X) = Xf = 0. (−f convex on Σ in tangent null directions.)

Visually: any null geodesic hitting Σ tangentially will lie in {f < 0} nearby. Definition

If Σ ruled by null geodesics, it is called zero pseudoconvex.

Σ f >0 f <0 null geodesic P

Σ pseudoconvex at P.

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Ideas Behind the Proof Classical Theory

Applications in Relativity

UC results have been applied in general relativity:

1 Rigidity of Kerr black holes: (Alexakis-Ionescu-Klainerman;

Carter-Robinson, Hawking)

2 UC for waves from infinity in asymptotically flat spacetimes:

(Alexakis-Schlue-S.)

3 Non-existence of time-periodic spacetimes: (Alexakis-Schlue;

Papapetrou, Biˇ c´ ak-Scholtz-Tod)

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Ideas Behind the Proof Some Examples

Example: Pseudoconvexity

Example Consider the finite timelike cylinder Σ := {t0 < t < t1, |r| = r0} ⊆ Rn+1. Σ is pseudoconvex (w.rt. , inward). ⇒ (local) UC from Σ into the interior.

UC

Cylinder.

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Ideas Behind the Proof Some Examples

Examples: Zero Pseudoconvexity

Example Zero pseudoconvex—depends on geometry near Σ:

1 Timelike hyperplane in Rn+1: no local UC

(Alinhac-Baouendi).

2 Null infinity of Rn+1: UC from “at least half of

I±” (Alexakis-Schlue-S.).

3 Null infinity of Schwarzschild: UC from I± near

ι0 (Alexakis-Schlue-S.). (2) demonstrates non-locality in Σ.

Σ ι0

(2)

Σ ι0

(3)

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Ideas Behind the Proof The Search for Pseudoconvexity

AdS Infinity

AdS infinity, I, is zero pseudoconvex. Must examine geometry near I more carefully. However, cylinders {ρ = ρ0} are pseudoconvex. ⇒ UC from I to the interior, provided φ decays as t → ±∞. Extra decay needed, since region {0 < ρ < ρ0} has boundary t = ±∞. Of course, extra decay condition is undesirable.

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Ideas Behind the Proof The Search for Pseudoconvexity

Improving Pseudoconvexity

Question Can we “bend” the hypersurfaces {ρ = ρ0} back toward I, so that:

1 They remain pseudoconvex (inward). 2 They intersect I after a finite time interval.

More precisely, fix y > 0, and consider level sets of f := ρ sin(yt), 0 < t < y−1π.

I f =0 t=0 t=y−1π f =c>0

Level sets of f .

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Ideas Behind the Proof The Search for Pseudoconvexity

Improved Pseudoconvexity

Lemma If y < 1, then {f = f0} is pseudoconvex (inward) for f0 ≪ 1. i.e., time interval [0, t0 := y−1π] must have length > π. Remark Lemma ⇒ “sufficiently long time interval” assumption in main theorems. ⇒ Optimism that some UC from I ∩ {0 < t < y−1π} holds.

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Ideas Behind the Proof The Carleman Estimate

Carleman Estimates

As usual, prove UC via a Carleman estimate. Carleman estimate + standard argument ⇒ UC. Carleman estimate is roughly of the form e−Fλ(f )(g +σ)φ2

L2(f <f0) λe−Fλ(f )Dφ2 L2(f <f0) +λ3e−Fλ(f )φ2 L2(f <f0).

λ ≫ 1: constant. Fλ(f ): reparametrisation of f . Some technical difficulties:

1 Infinite domains ⇒ infinite volume ⇒ integrability issues. 2 Zero pseudoconvexity ⇒ have to balance decaying weights. Arick Shao (Imperial College London) Correspondence Properties 28 / 40

Ideas Behind the Proof The Carleman Estimate

Proof of Carleman Estimate

Carleman estimate can be thought of as an energy estimate for g, but:

1 We want boundary terms to vanish. 2 We want bulk terms to be positive.

Objective (1) from vanishing assumptions for φ as ρ ց 0. Objective (2) achieved using a positive commutator: Consider wave equation not for φ, but for ψ = e−Fλ(f )φ. Multiplier method: integrate by parts

• f <f0

gψ(Sψ + hψ), Sα := ∇αf .

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Ideas Behind the Proof The Carleman Estimate

Carleman Estimates, Continued

To ensure bulk terms are positive:

1 Bulk terms containing derivative of φ tangent to level sets of f :

Positive only when level sets of f are pseudoconvex.

2 Bulk terms containing φ and normal derivatives:

Use freedom to choose reparametrization Fλ(f ): Fλ(f ) = κ log f + λp−1f p, κ ∈ R, p > 0.

Precise vanishing assumption needed for φ depends on κ: ⇒ Must carefully optimise κ in Carleman estimate.

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Ideas Behind the Proof The Case Against Short Intervals

The Short Interval Conjecture

Conjecture UC (or at least the Carleman estimate) does not generally hold if t0 < π. Observation: Family of future null geodesics from I ∩ {t = 0} which bend back to I. Any such geodesic hits I at time π. Geometric optics solutions near these geodesics: Support of φ arbitrarily close to I. But, φ vanishes on I ∩ {ε < t < π − ε}.

I t=0 t=π φ,dφ=0

Null geodesics near I.

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Ideas Behind the Proof The Case Against Short Intervals

Comparison with Timelike Cylinders

Can contrast with cylinder C = {r = r0} in Rn+1 (inwardly pseudoconvex). Timespan of analogous null geodesics depends

• n angle made with C.

Observation also important for studying linear waves in AdS (Holzegel-Luk-Smulevici-Warnick). Explains loss of derivatives in energy decay.

C t=0

Null geodesics near C.

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Toward the Einstein Equations

Section 4 Toward the Einstein Equations

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Toward the Einstein Equations Generalizations: Wave Equations

Tensor Waves

Problem: EVE is tensorial, not scalar. Scalarising tensorial quantities ⇒ angular frames degenerate. Solution: Generalise results to spherical tensorial waves. Application: treat linearised EVE about AdS (L-EVE). Corollary (Holzegel, S.; 2015) Suppose a Weyl field W = Wαβγδ on AdS spacetime satisfies L-EVE. If W vanishes to sufficient order at I, then W vanishes in the interior.

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Toward the Einstein Equations Generalizations: Wave Equations

Asymptotically AdS Spacetimes

To deal with EVE, we must handle other aAdS spacetimes. Theorem (Holzegel, S.; 2015)

Results extend to large subclass of aAdS spacetimes: General boundary topology allowed (replace S2 by another surface). Metrics (in Fefferman-Graham gauge) of form g = ρ−2{dρ2 + [˚ gab + ¯ gabρ2 + O(ρ3)]dxadxb}. ˚ g is static, and ¯ g satisfies a positivity condition. Moreover, for vacuum spacetimes: Condition for ¯ g ⇔ positive curvature on sections of I.

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Toward the Einstein Equations Generalizations: Wave Equations

Nonstatic Spacetimes

Static assumption on boundary metric ˚ g too restrictive. Q: Can we remove this assumption? A: Yes (work in preparation with G. Holzegel). Main idea: alter level sets of f = ρ/ sin(yt). sin(yt) arises in computations as harmonic oscillator (ψ′′ + y2ψ = 0). Roughly: replace sin(yt) by function resembling damped oscillator.

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Toward the Einstein Equations Generalisations: Einstein Equations

Rigidity of AdS

Question Can we apply previous results to treat EVE itself? Wave nature of EVE: curvature satisfies nonlinear wave equation. Caveat: background spacetime depends on wave. Simplest case: rigidity result for AdS. (Work in preparation with G. Holzegel). Assume aAdS spacetime, with Weyl curvature W vanishing on I. Then, spacetime must be AdS (at least near I).

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Toward the Einstein Equations Generalisations: Einstein Equations

Extension of Symmetries

Question For vacuum aAdS spacetimes, are symmetries on I inherited inside? Metric level: no (e.g., Kerr-AdS). What about at curvature level? Conjecture (Work in progress with G. Holzegel) If L∂tW vanishes on I, then spacetime is stationary near I. Similar results expected to hold for other symmetries (spherical, axial).

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Toward the Einstein Equations Generalisations: Einstein Equations

Rigidity of Kerr-AdS

Extension of spherical symmetry ⇒ rigidity result for Schwarzschild-AdS. Can we push this further? Question Rigidity result for Kerr-AdS? (Work in progress with G. Holzegel) Does not yet follow from extension of axial symmetry.

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Toward the Einstein Equations Generalisations: Einstein Equations

Future Work

Conjecture (Question (∞)) If two aAdS vacuum spacetimes have same Dirichlet and Neumann data

• n I, then they must be isometric near I.

In other words: there exists a correspondence between vacuum aAdS spacetimes and boundary data. Question What about existence of solutions to EVE from boundary data? For which Dirichlet + Neumann data on I is there a solution to EVE? In other words: for what space of boundary data do we have the above correspondence?

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