Correspondence and Rigidity Results on Asymptotically Anti-de Sitter - - PowerPoint PPT Presentation

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Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes

Arick Shao

Queen Mary University of London

International Conference on Nonlinear Waves and General Relativity 14 December, 2017 The Chinese University of Hong Kong

Joint work with G. Holzegel (Imperial College London)

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Introduction

Section 1 Introduction

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Introduction Physical Motivations

Correspondence Principles

Outstanding problem in theoretical physics:

Reconciling Einstein’s theory of gravity with quantum fjeld theories.

Infmuential research direction:

AdS/CFT correspondence (AdS: Anti-de Sitter) (CFT: Conformal fjeld theory)

AdS/CFT ⇒ holographic principle:

Gravitational theory on spacetime encoded in some theory on its boundary (of one less dimension). Original paper∗: 12154 12201 12381 12869 13156 13278 citations.†

∗ J. Maldacena, The large N limit of superconformal fjeld theories and supergravity (1999) † Data from http://inspirehep.net/record/451647/citations.

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Introduction Physical Motivations

What’s Missing?

In AdS context, little rigorous mathematics for:

Positive statements of this principle. Precise formulations of this principle.

In particular, in dynamical (non-static) settings. Main (long-term) questions:

1

Can rigorous statements toward holographic correspondences be formulated?

2

Can these statements be proved?

3

Can one understand the mechanisms behind such a correspondence?

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Introduction Anti-de Sitter Spacetime

General Relativity

Gravity described by Einstein’s theory of general relativity.

Spacetime: (n + 1)-dimensional Lorentzian manifold (M, g). g: Lorentzian metric, with signature (−, +, . . . , +). Gravity modelled by curvature of (M, g).

Gravity and matter coupled via the Einstein equations: Ricg −1 2 Scg g + Λg = T.

No matter (T ≡ 0) ⇒ Einstein-vacuum equations (EVE): Ricg = 2Λ n − 1g.

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Introduction Anti-de Sitter Spacetime

Anti-de Sitter Spacetime

Anti-de Sitter (AdS) spacetime:

Maximally symmetric solution of EVE... ... with negative cosmological constant Λ = −n(n−1)

2

. Lorentzian analogue of hyperbolic space.

Globally represented as (Rt × Rn

x, g), with

g := (1 + r2)−1dr2 − (1 + r2)dt2 + r2˚ γ.

r > 0, ω ∈ Sn−1: Polar coordinates on Rn. ˚ γ: Round metric for unit sphere Sn−1.

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Introduction Anti-de Sitter Spacetime

The Conformal Boundary

r=0 r=∞ ρ=0 I t r Compactifjed AdS, mod Sn−1.

Consider “inverted radius” ρ := r−1: g = ρ−2 [ (1 + ρ2)−1dρ2 − (1 + ρ2)dt2 + ˚ γ ] .

ρ2g smooth at ρ = 0 (r = ∞).

ρ is a boundary defjning function.

Conformal boundary “I := {ρ = 0}” of AdS: (I ≃ Rt × Sn−1,˚ g := −dt2 +˚ γ).

Asymptotically AdS (aAdS):

Spacetime with “similar conformal boundary”.

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Introduction The Main Question

The Correspondence Question

Question (0’) Is there some correspondence between:

aAdS solution of EVE (“gravitational dynamics”). Data prescribed at conformal boundary I.

(Ideally: boundary metric, stress-energy tensor.)

Attempt 1: Formulate this in terms of PDEs:

Given: “Cauchy” data on conformal boundary I. Question: Solve for unique solution of EVE in interior?

I Data Solve for g=? EVE, with data on I. Arick Shao (QMUL) Correspondence & Rigidity on AdS 8 / 39

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Introduction The Main Question

Ill-Posedness

Bad news: This problem is generally ill-posed.

AdS ≈ cylinder in Minkowski spacetime: C := {(t, x) ∈ R1+n | |x| < 1}. EVE ≈ wave equation. Wave equations ill-posed with Cauchy data on C.

For EVE to be well-posed, one requires:

Initial data at t = 0. Dirichlet or Neumann data on I.

r=1 r=1 The cylinder C. Arick Shao (QMUL) Correspondence & Rigidity on AdS 9 / 39

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Introduction The Main Question

A Unique Continuation Problem

Attempt 2: Formulate as unique continuation problem for PDEs.

If a solution exists, then must it be unique?

Question (0)

Suppose two aAdS solutions of the EVE have the same “boundary-Cauchy” data

• n their conformal boundaries. Then, must these solutions be isometric?

Is there a one-to-one correspondence between aAdS solutions of EVE and some space of “boundary-Cauchy” data?

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Introduction The Model Problem

The Wave Equation

Consider a model problem:

Wave equation on fjxed AdS/aAdS spacetime.

Question (1)

φ1, φ2: solutions on fjxed aAdS spacetime of (□g + σ)φ = G(φ, ∇φ), σ ∈ R. If φ1, φ2 have same Dirichlet and Neumann data on the boundary I, then is φ1 = φ2 locally near I? G linear: φ = 0 at I ⇒ φ = 0 near I?

I φ=0. φ=0? The uniqueness problem. Arick Shao (QMUL) Correspondence & Rigidity on AdS 11 / 39

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Introduction The Model Problem

Why the Wave Equation?

Question (1): essential step toward Question (0).

Wave equation: fjrst linearization of EVE. EVE ⇒ curvature satisfjes nonlinear wave equation.

Question (1) also has applications to rigidity results.

• Remark. Why □ + σ, not □?

σ determines asymptotics of φ near I.

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Results on AdS Spacetime

Section 2 Results on AdS Spacetime

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Results on AdS Spacetime The Main Result

Some Intuition

Consider: (□g + σ)φ = 0.

(Over)assume φ depends only on ρ ⇒ 2nd-order ODE for φ. Frobenius method ⇒ two branches of solutions: φ± = ρβ±

∞

∑

k=0

a±

k ρk,

β± = n 2 ± √ n2 4 − σ.

(Breitenlohner–Freedman)

Thus, for φ to vanish, we must eliminate both branches: ρ−β+φ → 0, ρ ↘ 0.

• Q. Is this condition suffjcient in general?

(A. Almost, for physically relevant σ.)

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Results on AdS Spacetime The Main Result

The Main Theorem

Theorem (Holzegel–S.; 2015)

Suppose φ is a C2-solution of |(□g + σ)φ| ≤ ρ2+p(|∂tφ| + |∂ρφ| + |∇S2φ|) + ρp|φ|, where σ ∈ R and p > 0. Assume the vanishing condition |ρ−β+φ| + |∇t,ρ,S2(ρ−β++1φ)| → 0, if σ ≤ n2 − 1 4 (β+ ≥ n + 1 2 ), |ρ− n+1

2 φ| + |∇t,ρ,S2(ρ− n−1 2 φ)| → 0,

if σ ≥ n2 − 1 4 (β+ ≤ n + 1 2 ), as ρ ↘ 0, on a suffjciently large time interval t ∈ [0, t0], t0 > π. Then, φ vanishes in the interior of AdS, near I ∩ {0 < t < t0}.

I φ=0 φ=0 Arick Shao (QMUL) Correspondence & Rigidity on AdS 15 / 39

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Results on AdS Spacetime The Main Result

Some Remarks

1

First such correspondence result in dynamical, non-analytic setting.

2

The suffjciently large time interval assumption is new. Clearly necessary for global uniqueness. Surprisingly, seems necessary even for local uniqueness.

3

Vanishing condition optimal when σ ≤ n2−1

4 .

σ = n2−1

4 : conformal mass.

• Q. (Open) Can we do better for σ > n2−1

4 ? 4

Result also holds for (appropriately defjned) tensor waves.

Useful for future applications to EVE.

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Results on AdS Spacetime The Main Result

Further Results

Theorem (Holzegel–S.; 2015)

On pure AdS, main theorem extends to global uniqueness result. Can show φ vanishes on all of {0 < t < t0}.

Theorem (Holzegel–S.; 2015)

When well-posedness theory∗ exists for □g + σ: For φ with fjnite (twisted) H2-energy... ... if φ has vanishing Dirichlet and Neumann data on I ∩ {0 < t < t0}... ... then φ = 0 near I ∩ {0 < t < t0}.

∗ See (Warnick; 2013). Arick Shao (QMUL) Correspondence & Rigidity on AdS 17 / 39

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Results on AdS Spacetime Unique Continuation Theory

Unique Continuation

Pose as unique continuation (UC) problem: Problem (Unique Continuation) Suppose φ is a solution of □gφ − aα∇αφ − Vφ = 0.

If φ, dφ vanish on a hypersurface Σ... ... then must φ vanish on one side of Σ?

In our context: Σ = I.

Σ □gφ+...=0 φ,dφ=0 φ=0? UC problem. Arick Shao (QMUL) Correspondence & Rigidity on AdS 18 / 39

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Results on AdS Spacetime Unique Continuation Theory

Classical Theory

Analytic, linear wave equations:

Holmgren’s theorem ⇒ UC Assuming analyticity is too strong

Non-analytic: Crucial criterion is pseudoconvexity.

(Hörmander, Lerner–Robbiano) Σ := {f = 0} pseudoconvex ⇒ UC, Σ to {f > 0}. Purely local result (neighborhood of p ∈ Σ). (Alinhac, Alinhac–Baouendi) Σ not pseudoconvex ⇒ ... ... then ∃ aα, V for which UC from Σ to {f > 0} does not hold.

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Results on AdS Spacetime Unique Continuation Theory

Pseudoconvexity

Defjnition

Σ := {f = 0} is pseudoconvex (w.r.t. □g and sgn f) ⇔ ... ... −f is convex in tangent null directions to Σ.

Rough interpretation:

Any null geodesic hitting Σ tangentially... ... will lie in {f < 0} nearby.

Defjnition

Σ is zero pseudoconvex ⇔ Σ is ruled by null geodesics.

Σ f>0 f<0 null geodesic P Σ pseudoconvex at P. Σ f>0 f<0 null geodesic P Σ zero pseudoconvex at P. Arick Shao (QMUL) Correspondence & Rigidity on AdS 20 / 39

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Results on AdS Spacetime Unique Continuation Theory

Examples: Zero Pseudoconvexity

Zero pseudoconvex case is complicated:

Depends on geometry of spacetime near Σ. Result may be local, semi-global, or global.

1

(Global) Σ = timelike hyperplane in Rn+1: No local UC (Alinhac–Baouendi). Global UC from all of Σ (Kenig–Ruiz–Sogge).

2

(Semi-global) Σ = null infjnity of Rn+1: UC from > half of I± (Alexakis–Schlue–S.).

3

(Local) Σ = null infjnity of Schwarzschild: UC from I± near ι0 (Alexakis–Schlue–S.).

Σ ι0 (2) Σ ι0 (3) Arick Shao (QMUL) Correspondence & Rigidity on AdS 21 / 39

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Results on AdS Spacetime Pseudoconvexity in AdS

The Conformal Boundary

Bad news: The AdS conformal boundary I is zero pseudoconvex. OK news: Cylinders {ρ = ρ0}, ρ0 > 0, are (inward) pseudoconvex.

⇒ UC from I inward, provided φ decays as t → ±∞. (Since region {0 < ρ < ρ0} has boundary t = ±∞.)

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

1

They remain pseudoconvex.

2

They intersect I after a fjnite time interval.

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Results on AdS Spacetime Pseudoconvexity in AdS

A Pseudoconvex Foliation

Fix y > 0, and consider level sets of f := ρ sin(yt), 0 < t < y−1π. Lemma If y < 1, then {f = f0} is pseudoconvex for f0 ≪ 1.

⇒ Time interval [0, t0 := y−1π] has length > π. ⇒ “suffjciently long time interval assumption”.

I f=0 t=0 t=y−1π f=c>0 Level sets of f. Arick Shao (QMUL) Correspondence & Rigidity on AdS 23 / 39

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Results on AdS Spacetime Proving the Main Result

Carleman Estimates

Main analysis tool for UC: Carleman estimates.

Weighted spacetime integral estimate with free parameter. Pseudoconvexity + Carleman estimate + standard argument ⇒ UC.

Carleman estimate roughly of the form ∥wλ,f(□g + σ)φ∥2

L2(f<f0) ≳ λ∥wλ,f∇φ∥2 L2(f<f0) + λ3∥wλ,fφ∥2 L2(f<f0).

λ ≫ 1: free chosen constant. wλ,f: weight depending on f, λ.

Some technical diffjculties:

1

Zero pseudoconvexity ⇒ dealing with degenerating weights.

2

Infjnite domains ⇒ infjnite volume ⇒ integrability issues.

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Results on AdS Spacetime Proving the Main Result

Short Time Intervals

Conjecture UC does not generally hold if t0 < π. Special property of AdS geometry:

∃ family of future null geodesics from I ∩ {t = 0}... ... which refocus at I ∩ {t = π}.

Idea: Counterexamples via geometric optics.

Similar to (Alinhac–Baouendi). Solutions concentrated near such a geodesic... ... arbitrarily close to zero data for φ.

I t=0 t=π φ,dφ=0 Null geodesics near I. Arick Shao (QMUL) Correspondence & Rigidity on AdS 25 / 39

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Results on aAdS Spacetimes

Section 3 Results on aAdS Spacetimes

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Results on aAdS Spacetimes Constructing aAdS Spacetimes

Toward the EVE

Main goal: UC problem for the EVE.

For EVE, the geometry itself is the unknown. Results on AdS spacetime must be robust. Methods must apply to general aAdS spacetimes.

(Holzegel–S.; 2015) aAdS spacetimes with static conformal boundary.

In context of solving the EVE (as initial-boundary problem)... ... one also encounters non-static conformal boundaries.

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Results on aAdS Spacetimes Constructing aAdS Spacetimes

Defjnition of aAdS

Step 1: Construction of conformal boundary and spacetime.

Conformal boundary: I := Rt × S.

S: (n − 1)-dimensional manifold—cross-section of I.

Spacetime (near boundary): M := (0, ρ0)ρ × I.

Step 2: Construction of aAdS metric.

Adopt Fefgerman–Graham (FG) gauge. Expand remaining (t, S)-components of g about I. g = ρ−2{dρ2 + [˚ gab + ¯ gabρ2 + O(ρ3)]dxadxb}. Conformal boundary: (I,˚ g).

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Results on aAdS Spacetimes Constructing aAdS Spacetimes

Features of Construction

1

Allow general boundary topology and geometry: (I,˚ g).

Example: Can consider planar AdS.

2

No loss of generality in choosing FG gauge.

Can change coordinates from more general gauge to FG...

3

For Einstein-vacuum spacetimes in FG gauge:

EVE connects ¯ g to geometry of conformal boundary. −¯ g is precisely the Schouten tensor ˚ P for (I,˚ g).

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Results on aAdS Spacetimes Extending Unique Continuation

The Pseudoconvexity Criterion

Question Do the previous results on AdS extend to aAdS spacetimes?

In particular, can we still fjnd pseudoconvexity near I?

Lemma (Pseudoconvexity Criterion) Suppose the following conditions hold:

1

−¯ g satisfjes a (pseudo-)positivity condition: −¯ g − ζ˚ g ≥ c > 0 for some function ζ.

2

|L∂t˚ g| is suffjciently small (depending on ¯ g).

Then, there is a pseudoconvex foliation near I...

... spanning a suffjciently long time interval on I.

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Results on aAdS Spacetimes Extending Unique Continuation

An Additional Diffjculty

Previous choice of f := ρ/ sin(yt) generally fails. sin(yt) shows up in AdS computations as solution of ψ′′ + y2ψ = 0.

aAdS setting: Nonzero L∂t˚ g ⇒ extra ψ′-term. Idea: Replace sin(yt) by function ψ resembling damped oscillator.

Lemma Assuming the pseudoconvexity criterion:

For an appropriate ψ, the level sets of f := ρ/ψ are pseudoconvex for f ≪ 1.

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Results on aAdS Spacetimes Extending Unique Continuation

The Main Theorem

Theorem (Holzegel–S.; 2016)

Let (M, g) be an aAdS spacetime satisfying the pseudoconvexity

• criterion. Suppose φ is a C2-solution of

|(□g + σ)φ| ≤ ρ2+p(|∂tφ| + |∂ρφ| + |∇S2φ|) + ρp|φ|, where σ ∈ R and p > 0, and suppose that |ρ−β+φ| + |∇t,ρ,S2(ρ−β++1φ)| → 0, if σ ≤ n2 − 1 4 , |ρ− n+1

2 φ| + |∇t,ρ,S2(ρ− n−1 2 φ)| → 0,

if σ ≥ n2 − 1 4 , as ρ ↘ 0, on a suffjciently large time interval 0 ≤ t ≤ t0. Then, φ vanishes in the interior of AdS, near I ∩ {0 < t < t0}.

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Results on aAdS Spacetimes Extending Unique Continuation

Some Remarks

Pseudoconvexity criterion directly connects:

Pseudoconvexity (and hence UC) Asymptotics on conformal boundary (L∂t˚ g and ¯ g).

For vacuum spacetimes:

˚ Ric(v, v) ≥ c > 0 for null v. Static boundary: positive Ricci curvature on S.

Many, but not all, well-known aAdS spacetimes satisfy this criterion.

AdS, Schwarzschild-AdS, and Kerr-AdS. Planar AdS (R × Sn−1 → R × Rn−1) fails the criterion.

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Results on aAdS Spacetimes Extending Unique Continuation

Some More Remarks

How to interpret the pseudoconvexity criterion? −¯ g − ζ˚ g ≥ c > 0.

Consider the truncated metric ˜ g = ρ−2[dρ2 + (˚ gab + ¯ gabρ2)dxadxb]. If pseudoconvexity criterion holds, then...

˜ g-null geodesics from I remaining close to I... ... will return to I within some time t0.

The pseudoconvexity criterion is gauge-dependent.

Not invariant under change of ρ. (Work in progress) Is there a gauge-independent statement? (Work in progress) Or, is there an “optimal” gauge?

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Results on aAdS Spacetimes Some Applications

Applications to Vacuum Spacetimes

1

Linearised EVE: (Holzegel–S.; 2015) applies to linearised EVE about AdS.

2

Rigidity of AdS: If the Weyl curvature W vanishes at I, ...

... then the spacetime must be pure AdS.

3

Rigidity of Kerr-AdS: Is there an analogous result?

4

Extension of symmetry: Given (appropriately defjned) symmetry of (I,˚ g): Can it be extended to the spacetime?

5

Correspondence for EVE: (Original question)

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Results on aAdS Spacetimes Some Applications

The Vacuum Setting

Assume FG expansion for (vacuum) g: ρ2g = dρ2 + [g(0)

ab + g(2) ab ρ2 + · · · + g(n) ab ρn + O(ρn+1)]dxadxb.

g(0) = ˚ g, g(2) = ¯ g. All coeffjcients below ρn determined by g(0). g(n): connected to stress-energy tensor.

For full nonlinear problem (5):

Goal: Prove g(0) and g(n) determine g (near I). (On a large enough time interval.) (Work in progress) Previous linear results are a key step.

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Results on aAdS Spacetimes Some Applications

Symmetry Extension

Symmetry extension (4): more immediately tractable problem

Assume: aAdS, symmetry at conformal boundary. Killing vector fjeld Z with LZg(0) = 0 and LZg(n) = 0. Vanishes on large enough time interval. Goal: Symmetry extends to aAdS bulk (near I).

Extend Z to spacetime Killing fjeld.

(Work in progress):

n = 3: Pieces in place. n > 3: Some issues.

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Results on aAdS Spacetimes Some Applications

Key Steps

• 1. Guess an extension of Z.
• 2. Derive closed system of:

Wave equations (for LZW). Transport equations (for LZg). Adapt (Ionescu–Klainerman, Alexakis–Ionescu–Klainerman), (Alexakis–Schlue)

Apply UC results (from I) to this system.

• 3. Prove LZg(0) = 0 and LZg(n) = 0 ⇒ ...

... Solution of system vanishes at I ... ... at suffjcient rate for UC.

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The End The End

The End

Thank you for your attention.

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