On analysis and numerical treatment of Einstein s constraint equations

CRM/McGill Applied Mathematics Seminar

On analysis and numerical treatment of Einstein’s constraint equations

Gantumur Tsogtgerel

University of California, San Diego

Part 1: Joint with M. Holst and G. Nagy Part 2: Joint with M. Holst

March 13, 2009

Gravitational wave astronomy

Recently constructed gravitational wave detectors: LIGO, VIRGO, GEO600, TAMA300.

The two L-shaped LIGO observatories (in Washington and Louisiana), with legs at 4km, have phenomenal sensitivity, on the order of10⁻¹⁵m to10⁻¹⁸m. effective ranges (1.4Sol): 7-15MPc

Initial value formulation of the Einstein equations

The Lorentzian manifold(M, g)satisfies

G(g) := Ric(g) −¹₂R(g)g = 0.

SupposeM = R × Σ, eachΣt={t} × Σis spacelike. On eachΣt, one has R(g) −|K|²g+ (trgK)²=0,

divgK − d(trgK) =0. ^(C) Conversely, if (C) holds on some Riemannian manifold(Σ, g), then there are

• a Lorentzian manifold(M, g)

• and an embeddingθ : Σ→ M

such thatG(g) = 0and thatθ∗gandθ∗Kare the first and second fundamental forms ofθΣ⊂ M[Choquet-Bruhat ’52].

The conformal method

Let(Σ, ˆg)be a Riemannian manifold,σbe a symmetric tensor withdivgˆσ =0, trgˆσ =0, and letτ∈ C^∞(Σ). Withφa positive scalar, andwa vector field, put

g = φ⁴g,ˆ K = φ⁻²(σ + Lgˆw) + ¹₃τφ⁴g,ˆ whereLgˆw =£wg −ˆ ²₃gˆdivgˆw. Then (C) is equivalent to

−8∆gˆφ + R(ˆg)φ +²₃τφ⁵−

σ + Lgˆw 

2 ˆ

gφ⁻⁷=0,

−divgˆLgˆw +³₂φ⁶dτ =0.

Let us rewrite the above as

Aφ + Rφ +²₃τφ⁵− a(w)φ⁻⁷=: Aφ + f(w, φ) = 0, Bw + φ⁶dτ =0.

Note thattrgK = τand that ifτ =constthe system decouples.

Constant mean curvature solutions

[York, O’Murchadha, Isenberg, Marsden, Choquet-Bruhat, Moncrief, Maxwell, et al.]

Aφ + f(w, φ) = 0, Bw =0.

Sub- and super-solutions, or barriers:

Aφ−+ f(w, φ−)6 0, Aφ++ f(w, φ+)> 0.

For anys >0, the constraint equation is equivalent to

Aφ + sφ = sφ − f(w, φ) ⇔ φ = (A + sI)⁻¹(sφ − f(w, φ)).

Ifs >0is sufficiently large, the map

T : [φ−, φ+]→ [φ−, φ+] : φ7→ (A + sI)⁻¹(sφ − f(w, φ)) is monotone increasing. Also,T (φ−)> φ−andT (φ+)6 φ+. The iteration

φ_n+1= T (φ_n), φ₀= φ₋, converges to a fixed point ofT.

Super-solution

We want to findφ >0such that

Aφ + f(w, φ) = Aφ + Rφ +²₃τφ⁵− a(w)φ⁻⁷> 0.

Recalla(w) =

σ + Lgˆw 

2

gˆ, and assume thatwis fixed (w =0in CMC case).

Assume thatτ =const > 0,R =const, and letφ =const > 0. Rφ +²₃τφ⁵− a(w)φ⁻⁷ > ²₃τφ⁵+ Rφ − φ⁻⁷sup a(w)

= φ^{−7 2}₃τφ¹²+ Rφ⁸−sup a(w)
Choosingφ >0sufficiently large one can ensure that the above is nonnegative.

Near constant mean curvature solutions

[Isenberg, Moncrief, Choquet-Bruhat, York, Allen, Clausen, et al.]

Aφ + f(w, φ) = 0, Bw + φ⁶dτ =0.

WithS : φ7→ −B⁻¹(φ⁶dτ)this can be written as Aφ + f(S(φ), φ) = 0.

Sub- and super-solutions make sense, but in general T : φ7→ (A + sI)⁻¹(sφ − f(S(φ), φ))

is not monotone. Nevertheless, whendτis smallTis almost monotone, and the iterationφn+1= T (φn)converges.

Now one needs global sub- and super-solutions, e.g.,φ+>0such that Aφ++ f(w, φ+)> 0,

for allw∈ S([0, φ+]).

Global super-solution

We want to findφ >0such that

Aφ + f(w, φ) = Aφ + Rφ +²₃τφ⁵− a(w)φ⁻⁷> 0.

for allw∈ S([0, φ]). Recall thata(w) =

σ + Lgˆw 

2 ˆ

g. Elliptic estimates give a(w)6 p + qkφk¹²C⁰, with q∼ |dτ|²

Assume thatτ =const > 0,R =const, and letφ =const > 0, sokφk_C0= φ. Rφ +²₃τφ⁵− a(w)φ⁻⁷> ²₃τφ⁵+ Rφ − pφ⁻⁷− qφ⁻⁷φ¹²

= (²₃τ − q)φ⁵+ Rφ − pφ⁻⁷.

Ifq < ²₃τ, choosingφ >0sufficiently large one can ensure that the above is nonnegative.

Fixed point approach

[Holst, Nagy, GT ’07, ’08]

Let0 < φ−6 φ+<∞be global barriers, i.e.,

Aφ−+ f(w, φ−)6 0, Aφ++ f(w, φ+)> 0, for allw∈ S([φ−, φ+]). Then fors >0large, and anyw∈ S([φ−, φ+])

Tw: φ7→ (A + sI)⁻¹(sφ − f(w, φ)) is monotone increasing onU = [φ−, φ+], and forφ∈ U

T (φ)≡ TS(φ)(φ)6 TS(φ)(φ+)6 φ+, T (φ)> φ−, soT : U→ U. SinceTis compact, there is a fixed point inU.

Global super-solution

[Holst, Nagy, GT ’07, ’08]

We want to findφ >0such that

Aφ + f(w, φ) = Aφ + Rφ +²₃τφ⁵− a(w)φ⁻⁷> 0.

for allw∈ S([0, φ]). Recall that

a(w)6 p + qkφk¹²C⁰

Assume thatR =const > 0,τ =const, and letφ =const > 0. Rφ +²₃τφ⁵− a(w)φ⁻⁷> ²₃τφ⁵+ Rφ − pφ⁻⁷− qφ⁻⁷φ¹²

> φ⁻⁷ Rφ⁸− (q −²₃τ)φ¹²− p

Ifpis small enough (depending on how largeqis), choosingφ >0sufficiently small one can ensure that the above is nonnegative.

Extensions

• The framework is extended to allow for rough data, e.g., metrics inH^swith s > ⁵₂

• The global super-solution construction is extended to all metrics in the positive Yamabe class (closed manifolds)

Ongoing work / wish list

• Asymptotically flat manifolds

• Manifolds with boundary, black hole initial data

• Zero and negative Yamabe classes, large data

• Full parameterization of the solution space

Finite element methods

Model problem:−∆u = f, or

a(u, v) := (∇u, ∇v) = (f, v) for all v ∈ H LetS⊂ Hbe a linear subspace. Consideru˜∈ Ssuch that

a(˜u, v) = (f, v) for all v ∈ S This gives the Galerkin orthogonality

a(u − ˜u, v) = 0 for all v ∈ S

oru − ˜u⊥aS.u˜is called the Galerkin approximation ofufromS.

Typical finite element mesh

Sis the space of continuous functions which are linear on each triangle.

-5 0 5

X -6

-4 -2 0 2 4 6

Z

Linear vs. nonlinear approximation

LetS0⊂ S1⊂ . . . ⊂ Hwith corresponding meshesT0, T1, . . ., and Galerkin approximationsu0, u1, . . ..

ku − uika=dist(u, Si)6 Ch^s−1i kukH^s

wherehiis the maximum diameter of the triangles inTi. IfTj+1is the uniform refinement ofTj, thenhi∼ 2⁻ⁱand the number of vertices ofTiisNi∼ 2ⁱⁿin n-dimension.

ku − uika=dist(u, Si)6 C2^−i(s−1)kukH^s6 CN^−(s−1)/ni kukH^s

IsTioptimal among meshes withNivertices?

Given a mesh, letS(T )be the corresponding FE space. Let ΣN=∪{S(T) : T is a refinement of T0and #T 6 N}

Then with_p¹ = ¹₂+^s−1_n

dist(u, ΣN)6 CN^−(s−1)/nkukW^s,p

Adaptive finite element methods

In a typical AFEM, the sequenceuiis generated as follows. Start with some initial meshT0. Seti =0, and repeat

• Solve foru_i

• Estimate the distribution ofui− uover the triangles ofTi

• Refine the triangles ofTiwith largest error, to getTi+1

• i + +

We say the method is optimal if

kui− uka6 CN^−(s−1)/nkukW^s,p

Linear convergence

From the Galerkin orthogonality

a(u − u_i+1, v) = 0 for all v ∈ S_i+1, takingv = ui+1− ui, we have

ku − uik²_a=ku − ui+1k²_a+kui+1− uik²_a. So if

kui+1− uika> cku − uika, with constantc∈ (0, 1), we have

ku − ui+1k²_a=ku − uik²_a−kui+1− uik²_a6 (1 − c²)ku − uik²_a.

Quasi-orthogonality for semilinear problems

Let us consider

a(u, v) + (f(u), v) = 0, ∀v ∈ H We have

ku − uik²_a=ku − ui+1k²_a+kui+1− uik²_a+2a(u − ui+1, ui+1− ui)

a(u − ui+1, ui+1− ui) = (f(u) − f(ui+1), ui+1− ui) 6 Ckf(u) − f(ui+1)kkui+1− uik 6 Cku − ui+1kkui+1− uik

6 Ch_i+1ku − ui+1kH¹kui+1− u_ikH¹

Ongoing work / Open problems

• Geometry

• Boundary conditions

• Coupled system

• Fast solution of the discretized system

• Higher order elements, flexible mesh

• Problems with genuinely critical exponent

Manuscripts, Collaborators, Acknowledgments

HNT2 M. HOLST, G. NAGY,ANDGT, Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics. Phys. Rev. Let., 100:161101, 2008. Also available as arXiv:0802.1031[gr-qc]

HNT1 M. HOLST, , G. NAGY,ANDGT, Rough solutions of the Einstein constraint equations on closed manifolds without near-CMC conditions. arXiv:0712.0798[gr-qc]. To appear in Comm. Math. Phys..

HT1 M. HOLST,ANDGT, Convergent adaptive finite element approximation of the Einstein constraints. In preparation.

Acknowledgments:

NSF: DMS 0411723, DMS 0715146 (Numerical geometric PDE) DOE: DE-FG02-05ER25707 (Multiscale methods)