The trapped region - aei.mpg.delaan/talks/Salamanca.pdf · The trapped region Lars Andersson Albert Einstein Institute and University of Miami ERE, Sept. 18, 2008 (joint work with

The trapped region

Lars Andersson

Albert Einstein Instituteand

University of Miami

ERE, Sept. 18, 2008

(joint work with Marc Mars, Walter Simon, Jan Metzger)

Lars Andersson (AEI and UM) The trapped region ERE 1 / 41

Outline

1 Marginally outer trapped surfaces

2 Curvature bounds

3 Existence of MOTS

4 Area bound

5 The trapped region


Overview

I will discuss some recent results on marginally outer trappedsurfaces (MOTS), dynamical horizons and the trapped region.

MOTS are black hole boundaries in GRThey are analogues of minimal surfaces in Riemannian geometry

Curvature boundsBarriers give existence (Proof uses Jang’s equation)

Area bounds, “maximum principle” hold for outermost MOTSApplication: coalescence of black holes

The trapped region can be characterized

Exterior Cauchy problem


MOTS

Notation

(M, g, K ) ⊂ L Cauchy hypersurface in a 3+1 dimensional Lorentzspacetime. Σ spacelike surface in M.

`− `+n

ν

Σ

Declare `+ to be outer

A = 〈∇·ν, ·〉,

χ± = K Σ ± A,

K Σ = K |TΣ×TΣ,

H = ΣtrA, P = ΣtrK Σ = tr K − K (ν, ν), θ± = Σtrχ± = P ± H.

Definition

Σ is a marginally outer trapped surface (MOTS) if θ+ = 0


MOTS

MOTS and singularities

θ+ is the logarithmic variation of area: θ+ = δ`+µΣ/µΣ

Σ MOTS ↔ outgoing null rays marginally collapsing

We call Σ (weakly) outer trapped if (θ+ ≤ 0) θ+ < 0.

Recall NEC: G(v , v) ≥ 0 for any null vector v .

Theorem

Suppose NEC holds. If a marginally outer trapped surface Σ separatesand has noncompact exterior, then L is null geodesically complete.

⇒ MOTS can be viewed as black hole boundaries

The usual definition of trapped surface is θ+ < 0, θ− < 0.


MOTS

Ideas

θ+ = P + H is elliptic

H = δνµΣ/µΣ

A graph in R3 is minimal iff

H[f ] :=∑

i

Di

(Di f√

1 + |Df |2

)= 0

but by Raychaudhuri

δf`+θ+ = −Wf = −(|χ+|2 + G(`+, `+)f

so |Σ| is not an elliptic functional w.r.t. null variations.

However, Lf = δfνθ+ is elliptic.


MOTS

Barriers

If there are barriers, there is a minimal (H=0) surface between

H > 0

H = 0

H < 0

Proof by minimization


MOTS

Persistence of MOTS

Suppose we have analogue for MOTS of existence in thepresence of barriers.By Rauchaudhuri, MOTS should persist if NEC holds.

MASigma0

`+

θ+ < 0 barrier

Mt

Σ MOTS

θ < 0 θ > 0

MOTT

Therefore expect MOTS are generically in a marginally outertrapped tube (MOTT) which is spacelike if NEC holds: Dynamicalhorizon


MOTS

Trapped region

The trapped region T is the union of all weakly outer trapped domains

T = ∪Ω ⊂ M : ∂Ω is weakly outer trapped

Theorem (Andersson & Metzger,2007)

∂T is a MOTS, the uniqueoutermost MOTS

∂T MOTS

θ+ ≤ 0

T

Ω

Theorem (Galloway, 2008)

Suppose NEC holds. Then the outermost MOTS is an S2


MOTS

Stability operator

Let Lf = δfνθ+. Then

Lf = −∆f + 2S(∇f ) + f [div S − |S|2 −12|χ+|2 +

12

Σ Sc +(µ − J(ν)]

where S(X ) = K (X , ν).L is the analogue of the minimal surface stability operator.Facts:

L is 2:nd order elliptic, non-self adjoint in general,∃! principal eigenvalue λ ∈ R, with positive eigenfunction φΣ locally outermost ⇒ λ ≥ 0λ ≥ 0 ⇔ ∃f ≥ 0 : Lf ≥ 0 ⇒ if λ ≥ 0 max. principle holds.

Definition

Σ is stable if λ ≥ 0

Σ locally outermost ⇒ Σ is stable.


MOTS

Local existence of horizons

Theorem (Andersson, Mars, & Simon, 2005)

Suppose Σ is strictly stable (λ > 0). Then ∃ MOTT H containing Σ. His weakly spacelike if NEC holds.

Proof is an application of the implicit function theorem.

θ+ ≤ 0 θ+ ≥ 0MOTT H outermost MOTS jumps

`+

Mt

This is a local result. The outermost MOTS can jump.Lars Andersson (AEI and UM) The trapped region ERE 12 / 41

MOTS

Bifurcation

As a MOTS is created, the MOTT bifurcates in general.

θ+ ≤ 0 θ+ ≥ 0

Mt

TheoremThe above picture holds assuming W 6= 0.


Curvature bounds

Curvature bounds

Theorem (Andersson & Metzger, 2005)

Let Σ be a stable MOTS. Then

|A| ≤ C(|Riem |C0 , |K |C1 , inj(M))

Proof ingredients:

Many ideas from (Schoen, Simon, & Yau, 1975).

symmetrized stability (Galloway & Schoen, 2006)

Simons identity

Kato inequality

Hoffmann-Spruck Sobolev inequality

Stampacchia iteration

local area bounds (Pogorelov)


Existence of MOTS

Jang’s equation

Consider R × M,with metric ds2 + g

Define K by pullbackLet M be the graph of f .

s M = s = f

ν

M

On M we have induced mean curvature H and P = trM K .

Jang’s equation is J [f ] := H−P = 0

Analogue of the equation θ+ = 0


Existence of MOTS

Jang’s equation

Translation invariance: J[f ] = J[f + t] ⇒ stability operator hasLφ = 0, with φ = 〈ν, e4〉

L is the analogue of the minimal surface stability operator for M(Schoen & Yau, 1981):

local curvature bounds (⇒ compactness)existence proof using capillarity deformation & Leray-SchauderH− σP = τ fσ,τ

deform σ from 0 to 1 → fτlet τ 0→ get converence of subsequence of fτ to solution f .


Existence of MOTS

Jang’s equation

Solution has blowups in general. Blowups project to MOTS.

νν

Mθ+ = 0

νν

Mθ− = 0

Therefore Jang’s equation can be used to prove existence of MOTS.(Andersson & Metzger, 2007): Blowup surfaces are stable MOTS


Existence of MOTS

Existence of MOTS

Theorem (Schoen, 2004; Andersson & Metzger, 2007)

Suppose M is compact with barrier boundaries ∂±M, such that

θ+[∂−M] < 0, θ+[∂+M] > 0

Then M contains a MOTS Σ

This is the analogue of the barrier argument for existence of minimalsurfaces.Proof makes use of a Dirichlet problem for Jang’s equation.(Eichmair, 2007) has studied the Plateau problem for MOTS usingPicard’s method.


Existence of MOTS

Existence of MOTS

∂+M

∂−M

θ+ > 0

θ+ < 0

θ+ = 0Σ


Existence of MOTS

Existence of MOTS

Proof is by solving a sequence of Dirichlet problems for Jangs equationwhich forces a blowup solution.

J [f ] = 0, f

∣∣∣∣∂±M

= ∓Z

Let Z → ∞

Converges tosolution with blowups

Z

−Z

∂−M ∂+M

J = 0


Existence of MOTS

Existence of MOTS

Proof uses bending to get H > 0 at ∂M (needed to have barriersfor gradient control at the boundary)

M

∂+Mθ+ > 0

(M, g, K )

H > 0

Solution must blow up somewhere ⇒ ∃ MOTS

We have foliation by barriers near ∂M

⇒ can show the MOTS constructed are in the undeformed regionof M

Can allow θ+[∂−M] ≤ 0: deform data inside ∂−M


Existence of MOTS

Application: Persistence of MOTS

MASigma0

`+

θ+ < 0 barrier

Mt

Σ MOTS

θ < 0 θ > 0MOTT

Theorem (Andersson, Mars, Metzger, & Simon, 2008)

Let L be a spacetime which satisfies NEC. Let Mt, be a Cauchyfoliation of L, and assume we have outer barriers.If M0 contains a MOTS, then each Mt , t ≥ 0 contains a MOTS.

It seems natural to view the collection of MOTS as the black holeboundary in L.Further regularity and continuation results for MOTT, cf. (Andersson etal., 2008).


Area bound

Area bound


Suppose M has outer barrier. There is a constantC = C(|Riem |C0 , |K |C1 , inj(M), Vol(M)) such that for a boundingMOTS Σ in M, either

|Σ| ≤ C

or there is a MOTS Σ′ outside Σ.

If |Σ| is very large, due to curvature bounds and the bounded Vol(M), itmust nearly meet itself from the outside ⇒ outer injectivity radius i+(Σ)must be small.


Area bound

Area bound

Surgery and heat flow:

Σ

ν

ν

θ+ < 0θ+ < 0

ν

ν

i+(Σ)

Glue in a neck with θ+ < 0. θ+ heat flow: x = −θ+ν gives Σs,s ≥ 0.

Maximum principle ⇒ Σs outside Σ, with θ+[Σs] < 0 for s > 0.


Area bound

Area bound

Σs is an inner barrier ⇒ apply existence result

⇒ ∃ MOTS Σnew outside Σ

Each step eats up at least δVol of the outside volume:Voloutside(Σnew) ≤ Voloutside(Σ) − δVol

⇒ only finitely many steps

⇒ eventually get Σ0 outside Σ with outer injectivity radiusi+(Σ0) > δ∗.

Σ0 has the claimed area bound. To estimate the area, use theestimates on curvature and i+ to estimate the volume of a tubearound Σ from below (divergence theorem) in terms of |Σ|.

This tube must have volume bounded by Vol(M).


Area bound

Application: Coalescence of black holes

Σ1 Σ2


Area bound

Application: Coalescence of black holes

By the gluing result: if Σ1,Σ2 are sufficiently close, there is a MOTS Σsurrounding them.

Σ

This gives a “maximum principle for MOTS”.The usual maximum principle does not apply for MOTS whichmeet on the outside.


The trapped region

The trapped region

Definition

Let (M, g, K ) be an AF data set. The trapped region is

T = ∪Ω⊂M∂Ω is weakly outer trapped


If ∃ Ω ⊂ M, with ∂Ω weakly outer trapped, then T has smoothboundary ∂T, with θ+[∂T] = 0. In particular, ∂T is the uniqueoutermost MOTS in M.


The trapped region

The trapped region

Replace T by

T = ∪Ωθ+[∂Ω] ≤ 0, and i+(∂Ω) ≥ δ∗

For the collection of subsets defining T we have compactness ⇒∂T = Σ is a MOTS.

LemmaT ⊂ T .


The trapped region

The trapped region

To see this, suppose ∃ WOT Ω * T . We can argue that this meansΩ ∩ T 6= 0.

θ+ < 0 by smooting

ΩT

Smoothing gives a barrier and hence there is a MOTS outside. Thiscan be taken to be in T . So T = T .


The trapped region

MOTS and MITS

We may have weakly inner trapped S in the region outside theoutermost MOTS Σout.

Ωout

Σout

θ− ≤ 0

|x | = R


The trapped region

MOTS and MITS

Then ∃! outermost MITS Sout in Ωout

By (Galloway, 2008),Σout, Sout ∼ S2.

Ωout

Sout

Σout

θ− ≤ 0

|x | = R

There is a solution to Jang’s equation which blows up precisely atΣout and Sout and → 0 at |x | = ∞.

This result can be used to complete the proof of the positive masstheorem (Schoen & Yau, 1981) without deforming the Cauchydata.


The trapped region

Exterior Cauchy problem

Suppose NEC holds + suitable matter equation.

Exterior region

Mt

Σ = ∂T

Expect: local well-posedness for the exterior IVP, due to curvaturebounds for ∂T

Potentially interesting for black hole evolutions and stability of Kerr.


The trapped region

Concluding remarks

Bray proposed Generalized Apparent Horizons: H = |P| and ageneralized Jang’s equation, as part of an approach to the generalPenrose Inequality. (Eichmair, 2008) proved existence ofoutermost GAH. These are outer minimizing. However, not clearhow they are related to black holes.Large families of GAH conditions can be treated using thetechniques discussed hereGlobal properties of MOTT may be relevant for understanding thestrong field Cauchy problem for the Einstein equations.The known conditions for existence of MOTS in a Cauchy data set(Schoen & Yau, 1983; Yau, 2001; Galloway & O’Murchadha, 2008)involve nonvacuum data.A better understanding of conditions for the existence of MOTS invacuum, due to concentration of curvature terms of, say, curvatureradii, conformal spectral gap, etc. is needed.


The trapped region

References I

Andersson, L., Mars, M., Metzger, J., & Simon, W. (2008). Timeevolution of marginally trapped surfaces.

Andersson, L., Mars, M., & Simon, W. (2005). Local existence ofdynamical and trapping horizons. Phys. Rev. Lett., 95, 111102.

Andersson, L., & Metzger, J. (2005). Curvature estimates for stablemarginally trapped surfaces.

Andersson, L., & Metzger, J. (2007). The area of horizons and thetrapped region.

Eichmair, M. (2007). The plateau problem for apparent horizons.(arXiv.org:0711.4139)

Eichmair, M. (2008). Existence, regularity, and properties ofgeneralized apparent horizons. (arXiv.org:0805.4454)

Galloway, G. J. (2008). Rigidity of marginally trapped surfaces and thetopology of black holes. Comm. Anal. Geom., 16(1), 217–229.


The trapped region

References II

Galloway, G. J., & O’Murchadha, N. (2008). Some remarks on the sizeof bodies and black holes. Class. Quant. Grav., 25, 105009.

Galloway, G. J., & Schoen, R. (2006). A generalization of Hawking’sblack hole topology theorem to higher dimensions. Comm. Math.Phys., 266(2), 571–576.

Schoen, R. (2004). talk given at the Miami Waves conference, Jan.2004.

Schoen, R., Simon, L., & Yau, S. T. (1975). Curvature estimates forminimal hypersurfaces. Acta Math., 134(3-4), 275–288.

Schoen, R., & Yau, S. T. (1981). Proof of the positive mass theorem. II.Comm. Math. Phys., 79(2), 231–260.

Schoen, R., & Yau, S. T. (1983). The existence of a black hole due tocondensation of matter. Comm. Math. Phys., 90(4), 575–579.


The trapped region

References III

Yau, S. T. (2001). Geometry of three manifolds and existence of blackhole due to boundary effect. Adv. Theor. Math. Phys., 5(4),755–767.


The trapped region - aei.mpg.delaan/talks/Salamanca.pdf · The trapped region Lars Andersson Albert Einstein Institute and University of Miami ERE, Sept. 18, 2008 (joint work with

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