Asymptotic gluing of shear-free hyperboloidalinitial data.

Paul T. AllenLewis & Clark College

joint work withJames Isenberg, John M. Lee, Iva Stavrov Allen

JMM 2018

Outgoing radiation from a gravitational event

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Hyperboloidal foliations

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t.IE#I Geometry of slices is asymptotic to hyperbolic space

Hyperboloidal foliations in compactified spacetime

FEE:I Well-suited for future

evolution problem

I Compact domains aregood for numerics

CMC hyperboloidal data

I Metric g = ρ−2g is asymptotically hyperbolic

Riem[g] = − id + . . . ↔ |dρ|g = 1 + . . .

I Constant mean curvature

K = −g + Σ

I Constraint equations

R[g]− |Σ|2g + 6 = 0 divg Σ = 0

I Shear-free condition required for compactifiable spacetime

Σ = ρ−1(Hessg ρ− 1

3(∆gρ)g)

+ . . .

= ρ−1Hg(ρ) + . . .

Constructing hyperboloidal data

Andersson-Chrusciel-Friedrich, Andersson-Chrusciel,Gicquaud-Sakovich, Isenberg-Lee-Stavrov,. . .

I Start with “seed data”: metric λ, tensor µ

I Look for

g = φ4λ, Σ = φ−2(µ+DλW ), DλW = tracefree LWλ

I Constraints satisfied if φ and W satisfy the elliptic system

D∗λDλW = −divλ µ

∆λφ =1

8R[λ]φ− 1

8|µ+DλW |2gφ−7 +

3

4φ5

I Appropriate Holder theory available

Constructing shear-free hyperboloidal data

I To get boundary regularity use intermediate spaces

Ck,α(M) ⊂ C k,α;m(M) ⊂ Ck,α(M)

I Seed metric λ = ρ−2λ with

λ ∈ C k,α;2(M) ⊂ C1,1(M)

I Seed tensorµ = ρ−1Hλ(ρ) + . . .

I Solve for φ = 1 +O(ρ2) to get

g = φ4λ ∈ C k,α;2(M)

Constructing data for two-body problems I

*¥¥¥*¥ '

Isenberg-Lee-Stavrov gluing theorem

I For 0 < ε� 1 form connect sum Mε

I Construct seed data λε, Kε

I Apply conformal method for each ε

I Uniform estimates give convergence in exterior region

Friendly (retrospective) critique

I Shear-free condition missing

I Convergence in “physical topology”

I What’s happening in middle region?

A-Stavrov density theorem

I (g,K) “polyhomogeneous” data, not necessarily shear-free

I Construct shear-free (gε,Kε)→ (g,K) in physical topology

Applications

I Shear-free data is “sufficiently general”?

I Stronger topology needed in convergence results

Geometry of the gluing region

OA (( ¥ (I Data in gluing region approximates a slice of Minkowski

spacetime

Improved gluing theorem

it6 .

¥m¥I (Mε, gε,Kε) are shear-free

I Exterior regions converge strongly to original data

I Middle region converges strongly to Minkowski hyperboloid

A peak under the rug

Need estimates, uniform in ε, in each region:

I Construct approximate solution Nε(φapproxε ) ≈ 0

I Linearize about approximate solution φε = φapproxε + uε

0 = Nε(φapproxε + uε) = Nε(φapproxε ) + Lεuε +Qε(uε)

I Blowup analysis: uniform estimates for linearized operators

I Solve fixed-point problem in ε-ball

uε = L−1ε (Nε(φapproxε ) +Qε(uε))

I Function spaces with weights adapted to the gluing