on the mathematical theory of black holes2018)_2.pdf · on the reality of black holes rigidity...

ON THE REALITY OF BLACK HOLES RIGIDITY STABILITY

ON THE MATHEMATICAL THEORY OFBLACK HOLES

Sergiu Klainerman

Princeton University

July 3, 2018


Outline

1 ON THE REALITY OF BLACK HOLES

2 RIGIDITY

3 STABILITY


KERR FAMILY K(a,m)

Boyer-Lindquist (t, r , θ, ϕ) coordinates.

−ρ2∆

Σ2(dt)2 +

Σ2(sin θ)2

ρ2

(dϕ− 2amr

Σ2dt)2

+ρ2

∆(dr)2 + ρ2(dθ)2,

∆ = r2 + a2 − 2mr ;

ρ2 = r2 + a2(cos θ)2;

Σ2 = (r2 + a2)2 − a2(sin θ)2∆.

Stationary. T = ∂tAxisymmetric. Z = ∂ϕ

Schwarzschild. a = 0,m > 0, static, spherically symmetric.

−∆

r2(dt)2 +

r2

∆(dr)2 + r2dσS2 ,

∆

r2= 1− 2m

r


SCHW. −(1− 2mr )dt

2 + (1− 2mr )−1dr 2 + r 2dσ2

S2

Event horizon r = 2m.Metric can be extended past it. Kruskal coordinates

Black and white holes r < 2m

Curvature singularity r = 0.

Exterior domains r > 2m.

Photon sphere r = 3m.

Null infinity (I+ ∪ I−) r =∞.


KERR SPACETIME K(a,m), |a| ≤ m

MAXIMAL EXT. ∆(r−) = ∆(r+) = 0, ∆ = r2 + a2 − 2mr

External region r > r+

Event horizon r = r+

Black Hole r < r+

Null Infinity r =∞


EXTERNAL KERR

Stationary, axisymmetric.

Nonempty ergoregion. Non- positive energy.

Region of trapped null geodesics


OTHER PRPERTIES OF KERR

PRINCIPAL NULL DIRECTIONS

e3 =r2 + a2

q√

∆∂t −

√∆

q∂r +

a

q√

∆∂ϕ

e4 =r2 + a2

q√

∆∂t +

√∆

q∂r +

a

q√

∆∂ϕ.

NULL FRAME S = {e3, e4}⊥, (ea)a=1,2 ∈ S orthonormal

αab := Ra4b4, αab := Ra3b3, βa :=1

2Ra434, β

a:=

1

2Ra334

ρ =1

4R3434,

?ρ =1

4∗R3434

FACT α = β = β = α = 0, ρ+ i ?ρ = − 2mr+iacosθ .


OTHER PROPERTIES OF KERR

STATIONARITY =⇒ Fαβ := DαTβ is a 2-form.

F = F + i ∗F , R = R + i ∗R,

DαFβγ = TλRλαβγ ⇒ d(iTF) = 0, 2iTF = dσ,

KERR σ = 1− 2mr+ia cos θ , F2 = − 4m2

(r+ia cos θ)4.

MARS-SIMON S = R+ 6(1− σ)−1Q,

Qαβµν = FαβFµν −1

3F2Iαβµν ,

Iαβµν = (gαµgβν − gανgβµ + i ∈αβµν)/4.

THEOREM(Mars 1999) S = 0 characterizes Kerr (locally)


I. RIGIDITY

RIGIDITY CONJECTURE.Kerr family K(a,m), 0 ≤ a ≤ m,exhaust all stationary, asymptot-ically flat, regular vacuum blackholes.

True in the axially symmetric case [Carter-Robinson]

True in general, under an analyticity assumption [Hawking]

True close to a Kerr space-time [Alexakis-Ionescu-Kl]


I. RIGIDITY MAIN NEW IDEAS

There exists a second Killingv-field along N ∪N .

Its natural extension leadsto an ill posed problem.

NEW APPROACH. Extend using a geometric continuationargument based on Carleman estimates.

Local extension based on Null Convexity.

Global extension based on T-Null Convexity,

MAIN OBSTRUCTION. Presence of T-trapped null geodesics.

No such objects in Kerr or close to Kerr!


LOCAL EXTENSION

DEFINITION. O ⊂M is (T) strongly null-convex at p ∈ ∂O if,for any defining function f,

D2f (X ,X )(p) < 0, ∀X ∈ Tp(O), g(X ,X ) = 0, (g(X ,T ) = 0)

THEOREM (Ionescu-Kl ) (M, g) Ricci flat, pdo-riemannianmanifold. (O ⊂M, Z ) verify:

Z Killing v-field in O,

∂O is strongly null-convex at p ∈ ∂O

⇒ Z extends as a Killing vector-field to a neighborhood of p.


LOCAL EXTENSION

FACT. STRICT NULL CONVEXITY IS ESSENTIAL.

THEOREM (Ionescu-Kl ) ∃ smooth, stationary, extensions ofK(a,m), 0 < a < m, locally defined in a neighb. of a point on thehorizon p ∈ N ∪N \ N ∩N , which posesses no additionalKilling v-fields.


I. RIGIDITY CONCLUSIONS

There exist no other explicit stationary solutions.

There exist no other stationary solutions close to a nonextremal Kerr, (or Kerr-Newman).

The full problem is far from being solved. Surprises cannotbe ruled out for large perturbations.

Conjecture. [Alexakis-Ionescu-Kl]. Rigidity conjecture holds trueprovided that there are no T-trapped null geodesics.


RIGIDITY IN DEPTH

1 GENERAL LOCAL EXTENSION THEOREM

2 A COUNTEREXAMPLE

3 GLOBAL RIGIDITY

4 CARLEMAN ESTIMATES


GENERAL EXTENSION THEOREM

THEOREM I. (M, g) Ricci flat, pseudo-riemannian manifold;(O,Z ) verify:

Z Killing v-field in O,

∂O is strongly null-convex at p ∈ ∂O

⇒ Z extends as a Killing vector-field to a neighborhood of p.

DEFINITION O ⊂M is strongly null-convex at p ∈ ∂O if itadmits defining function f at p, s.t. for any X 6= 0 ∈ Tp(M),X (f )(p) = 0 and g(X ,X ) = 0, we have

D2f (X ,X )(p) < 0.


THEOREM 1 (MAIN IDEAS)

Consider L, DLL = 0, in a neighborhood of p ∈ ∂O.

Extend Z by the Jacobi equation

DLDLZ = R(L,Z )L (1)

Define π = LZg

(1) =⇒ Lβπαβ = 0

Define W = LZR− (B ∗ R)

B = 12(π + ω) appropriately defined s.t. W is Weyl.


THEOREM I (MAIN IDEAS)

Define

Pαβµ =1

2(Dαπβµ −Dβπαµ −Dµωαβ) , Pρ : gµνPµρν

where

DLωαβ = παρDβLρ − πβρDαL

ρ, ω|O = 0.

We have,

LµPαβµ = 0, Lβωαβ = 0 in M

We have,

DαWαβγδ =1

2

(BµνDνRµβγδ + PρR

ρβγδ + P ∗ R

)



We have

DLBαβ = LρPρβα −DαLρBρβ,

DLPαβµ = LνWαβµν + LνBµρRαβρν −DµL

ρPαβρ.

The last identity follows from

DνP′αβµ −DµP

′αβν = (LZR)αβµν

− (1/2)π ρα Rρβµν − (1/2)π ρ

β Rαρµν .

where

P ′αβµ := Pαβµ + (1/2)Dµωαβ.



Have derived a system of wave/transport equations

DLB =M(B, B,P,W ),

DLB =M(B, B,P,W ),

DLP =M(B, B,P,W ),

�gW =M(B,DB, B,DB,P,DP,W ,DW )

Use unique continuation argument to conclude that W , πvanish in a neighborhood of Z .

Role of strong null-convexity.


A COUNTEREXAMPLE

THEOREM II. ∃ smooth, sta-tionary, extensions of K(a,m),0 < a < m, locally defined in aneighb. of a point on the horizonp ∈ N ∪N \N ∩N , which fail tobe axi-symmetric

PROOF. Modify the Kerr metric g smoothly across the horizons.t. Ric(g) = 0, T = d/dt remains Killing, yet g admits noKilling extension for Z .

MORAL. No hope to construct an additional symmetry, as inHawking’s Rigidity Theorem, by relying only on local informations.


REDUCTION BY KILLING VECTORFIELDS

RECALL. DµDνTσ = RλµνσTλ, [LT,D] = 0,

DEFINITION. Given a Killing v-field T,

Fαβ : = Fαβ + i ∗Fαβ, Fαβ = DαTβ

σα := 2TλFλα

LEMMA. DαFβγ = TλRλαβγ and, in vacuum,Dµσν −Dνσµ = 0;

Dµσµ = −G2;

σµσµ = g(T,T)G2.

ERNST POTENTIAL. σ = −(X + iY ), X = g(T,T).


THEOREM II (MAIN IDEAS)

STATIONARITY. Ernst potentials X = g(T,T), Y = . . .

SYMMETRY REDUCTION. g = (Φ, h), Φ = (X + iY ) = −σ

hαβ : = Xgαβ − TαTβ

Ric(h)αβ = < DαΦ,DβΦ >

�hΦ = X−1DµΦDµΦ

WAVE MAP. Φ : (N, h) −→ H2.

RECONSTRUCTION

M = R×N1+2.

g = (Xdϕ+ ωαdxα)2 + X−1hαβdx

αdxβ, α, β = 0, 1, 2

curl ω = X−2dY , div ω = 0.


THEOREM II (MAIN IDEAS)

Reduce the construction to solving a local characteristic Cauchyproblem for a wave map equation Φ = (X + iY ) : (N 2+1, h)→ Hcoupled to a 2 + 1 dimensional Einstein equations.

Rαβ(h) =1

X 2

(∇αX∇bX +∇aY∇bY )

�h(X + iY ) =1

Xhab∇a(X + iY )∇b(X + iY )

where X = g(T,T), Y Ernst potential and,

hαβ := Xgαβ − TαTβ

CHARACTERISTIC PROBLEM. Prescribe X ,Y . Solve locallythe equations in a neighborhood of a point p ∈ N ∪N away fromthe bifurcate sphere.

IMPORTANT. Vectorfield T must be strictly spacelike at p!


RETURN TO MAIN RESULTS

RIGIDITY CONJECTURE. Kerr family K(a,m), 0 ≤ a ≤ m,exhaust all stationary, asymptotically flat, regular vacuum blackholes.

True if coincides with Kerr on N ∩N [Ionescu-Kl]

True if close to a Kerr space-time [Alexakis-Ionescu-Kl]


ALEXAKIS IONESCU Kl. RIGIDITY

MARS-SIMON TENSOR. There exists a complex valued4-tensor S, defined in terms of the curvature R and DT.

S ≡ 0⇐⇒ (M, g) locally isometric to Kerr

THEOREM. (M, g) regular, stationary, non-extremal with Marstensor S sufficiently small ⇒ M isometric to external Kerr.

MAIN IDEAS:

1 Hawking argument provides a second Killing v-field K∣∣∣N∪N

.

2 Strict null convexity on N ∩N allows one to extend K to asmall neighborhood

3 Smallness of S allows us to extend K everywhere based onT-null convexity

4 A linear combination of T,K generates a rotational Killingv-field Z.


I. RIGIDITY CONCLUSIONS

There exist no other explicit stationary solutions.

There exist no other stationary solutions close to a nonextremal Kerr, (or Kerr-Newman).

The full problem is far from being solved. Surprises cannotbe ruled out for large perturbations.

Conjecture. [Alexakis-Ionescu-Kl]. Rigidity conjecture holds trueprovided that there are no T-trapped null geodesics.


STABILITY OF KERR

CONJECTURE[Stability of (external) Kerr].

Small perturbations of a given exterior Kerr (K(a,m), |a| < m)initial conditions have max. future developments converging toanother Kerr solution K(af ,mf ).


GENERAL STABILITY PROBLEM N [φ] = 0.

NONLINEAR EQUATIONS. N [φ0 + ψ] = 0, N [Φ0] = 0.

1 ORBITAL STABILITY(OS). ψ small for all time.

2 ASYMPT STABILITY(AS). ψ −→ 0 as t →∞.

LINEARIZED EQUATIONS. N ′[φ0]ψ = 0.

1 MODE STABILITY (MS). No growing modes.

2 BOUNDEDNESS.

3 QUANTITATIVE DECAY.


GENERAL STABILITY PROBLEM N [φ = 0

STATIONARY CASE. Possible instabilities for N ′[φ0]ψ = 0:

Family of stationary solutions φλ, λ ∈ (−ε, ε),

N [φλ] = 0 =⇒ N ′[φ0](d

dλΦλ)|λ=0 = 0.

Mappings Ψλ : R1+n → R1+n, Ψ0 = I taking solutions tosolutions.

N [φ0 ◦Ψλ] = 0 =⇒ N ′[φ0]d

dλ(φ0 ◦Ψλ)|λ=0 = 0.

Intrinsic instability of φ0. Negative eigenvalues of N ′(φ0).


GENERAL STABILITY PROBLEM N [φ0] = 0

STATIONARY CASE. Expected linear instabilities due tonon-decaying states in the kernel of N ′[φ0]:

1 Presence of continuous family of stationary solutions φλimplies that the final state φf may differ from initial state φ0

2 Presence of a continuous family of invariant diffeomorphismrequires us to track dynamically the gauge condition to insuredecay of solutions towards the final state.

QUANTITATIVE LINEAR STABILITY. After accounting for(1) and (2), all solutions of N ′[φ0]ψ = 0 decay sufficiently fast.

MODULATION. Method of constructing solutions to thenonlinear problem by tracking (1) and (2).


GENERAL STABILITY PROBLEM

TO PROVE NONLINEAR STABILITY.

NEED: Dynamically defined gauge condition and mechanismto track the final state

NEED: Robust mechanism for deriving “sufficient” decay forthe main linearized quantities–with respect to the gauge

NEED: A version of the null condition–with respect to the gauge

NEED: A strategy to disentangle the nonlinearinterdependence of the above.


STABILITY OF KERR

Ric[gm,a] = 0

Kerr family depends on two parameters a,m.

δRic

[∂gm,a

∂m

]= δRic

[∂gm,a

∂a

]= 0

Einstein vacuum equations are diffeomorphism invariant

δRic [LXgm,a] = 0.

Dim(Ker δRic

)=4×∞+ 2


GEOMETRIC FRAMEWORK

1 Null Pair e3, e4.

2 Horizontal structures {e3, e4}⊥

3 Null Frames e3, e4, (ea)a=1,2.

4 Null decompositions

Connection Γ = {χ, ξ, η, ζ, η, ω, ξ, ω}

Curvature R = {α, β, ρ, ∗ρ, β, α}

5 Main Equations

6 S-foliations


KERR FAMILY K(a,m)

BOYER-LINDQUIST (t, r , θ, ϕ).

−ρ2∆

Σ2(dt)2 +

Σ2(sin θ)2

ρ2

(dϕ− 2amr

Σ2dt)2

+ρ2

∆(dr)2 + ρ2(dθ)2,

∆ = r2 + a2 − 2mr ;

q2 = r2 + a2(cos θ)2;

Σ2 = (r2 + a2)2 − a2(sin θ)2∆.

STATIONARY, AXISYMMETRIC. T = ∂t , Z = ∂ϕ

PRINCIPAL NULL DIRECTIONS.

e3 =r2 + a2

q√

∆∂t −

√∆

q∂r +

a

q√

∆∂ϕ

e4 =r2 + a2

q√

∆∂t +

√∆

q∂r +

a

q√

∆∂ϕ.


BASIC QUANTITIES

NULL FRAME e3, e4, (ea)a=1,2, S = span{e1, e2}

CONNECTION COEFFICIENTS. χ, ξ, η, ζ, η, ω, ξ, ω

χab = g(Dae4, eb), ξa =1

2g(D4e4, ea), ηa =

1

2g(ea,D3e4),

ζa =1

2g(Dae4, e3), ω =

1

4g(D4e4, e3) . . .

CURVATURE COMPONENTS. α, β, ρ, ?ρ, β, α

αab = R(ea, e4, eb, e4), βa =1

2R(ea, e4, e3, e4),

ρ =1

4R(e4, e3, e4, e3), . . .


CRUCIAL FACT.

1 In Kerr relative to a principal null frame we have

α, β, β, α = 0, ρ+ i ?ρ = − 2m

(r + ia cos θ)3

ξ, ξ, χ, χ = 0.

2 In Schwarzschild we have, in addition,

{e3, e4}⊥ is integrable?ρ = 0, η, η, ζ = 0

The only nonvanishing components of Γ are

tr χ, tr χ, ω, ω

3 In Minkowski we also have ω, ω, ρ = 0.


O(ε) - PERTURBATIONS

ASSUME. There exists a null frame e3, e4, e1, e2 such that

ξ, ξ, χ, χ, α, α, β, β = O(ε)

FRAME TRANSFORMATIONS, (f , f )a=1,2, log λ = O(ε)

e ′4 = λ(e4 + faea + O(ε2)

)e ′3 = λ−1

(e3 + f aea + O(ε2)

)e ′a = ea +

1

2f ae4 +

1

2fae3 + O(ε2)

Curvature components α, α are O(ε2) invariant.

For perturbations of Minkowski all curvature components areO(ε2)-invariant.


STABILITY OF VACUUM STATES

�φ = F (φ, ∂φ, ∂2φ) in R1+n.

FACT. Vacuum state Φ = 0,

Unstable for most equations for n = 3.

Stable if n = 3 and F verifies the null condition.

Dimension n = 3 is critical

FACT. Geometric nonlinear wave equations verify some, gaugedependent, version of the null condition.


VECTORFIELD METHOD

Use well adapted vectorfields, related to

1 Symmetries,

2 Approximate, symmetries,

3 Other geometric features

to derive generalized energy bounds and robust L∞-quantitativedecay.

It applies to tensorfield equations such as Maxwell and Bianchitype equations,

dF = 0, δF = 0.

and nonlinear versions such as Einstein field equations.


STABILITY OF MINKOWSKI Rµν = 0

1 INITIAL DATA SETS. (Σ3, g , k)

Constraint Eqts.

Maximal trgk = 0.

Flat. (R3, e, 0)

A. F. gij − (1 + 2mr )δij = Ok+1(r−2), kij = Ok(r−2)

Smallness assumption

2 MGHD DEVELOPMENT.


STABILITY OF MINKOWSKI SPACE

THEOREM[Christodoulou-K 1993] Any A.F. initial data set, closeto the flat one, has a complete, maximal development convergingto Minkowski.

I. GAUGE CONDITION .

optical function u -properly initialized.time function t -maximal

II. ROBUST DECAY.

Bianchi identities δR = δ ∗R = 0Vectorfield method

Construct approximate Killing and conformal Killing fieldsadapted to the foliation

III. NULL CONDITION.


CHRISTODOULOU-K THEOREM (1990)

1 Maximal foliation Σt .2 Null foliation Cu.3 Adapted frame

L = e4, L = e3, (ea)a=1,2 ∈ T(S(t, u))

St,u = Σt ∩ Cu 4πr2 = Area(St,u).

4 Weak peeling decay, as r →∞ along Cu.

αab = R(L, ea, L, eb) = O(r−7/2)

2βa = R(L, L, L, ea) = O(r−7/2)

4ρ = R(L, L, L, L) = O(r−3)

4 ∗ρ = ∗R(L, L, L, L) = O(r−7/2)

2βa

= R(L, L, L, ea) = O(r−2)

αab = R( L, ea, L, eb) = O(r−1)


GLOBAL STABILITY OF MINKOWSKI


STABILITY OF MINKOWSKI SPACE

ROLE OF CURVATURE.

All null components of R, relative to adapted null frames, staywithin O(ε) of their initial values.

All null components of R are O(ε2)-invariant with respect toO(ε) frame transformations. Effectively gauge independent.

BIANCHI IDENTITIES δR = δ ∗R = 0

Bel-Robinson tensor Q = R · R + ∗R · ∗R.

Q symmetric, traceless, δQ = 0.

Energy type quantities1 Contractions2 Commutations

Effective, invariant, way to treat the wave character of EVE.


GLOBAL STABILITY OF MINKOWSKI

Proof is based on a huge bootstrap with three major steps,

Assume the boundedness of the curvature norms and deriveprecise decay estimates for the connection coefficients of thet, u foliations.

Use the connection coefficients estimates to derive estimatesfor the deformation tensors of the approximate Killing andconformal Killing vectorfields

Use the latter to derive energy-like estimates for the curvatureand thus close the bootstrap.


THEOREM K-NICOLO(2001)

ASSUME: (Σ, g , k) A.F. maximal,

CONSTRUCT:

1 Double null foliation C (u), C (u) with complete outgoingleaves C (u).

gαβ∂αu∂βu = gαβ∂αu∂βu = 0.

2 Null frame L, L; (ea)a=1,2 adapted to foliation

Su,u = Cu ∩ C (u), 4πr2 = Area(St,u).

3 Weak peeling r →∞ along C (u)

α, β, ρ− ρ, ?ρ = O(r−7/2), β = O(r−2), α = O(r−1)


THEOREM K-NICOLO(2003)

ASSUME

g − gS = Oq+1(r−(32+γ)), k = Oq(r−(

52+γ)), γ >

3

2, q ≥ 5.

THEN

α = O(r−5), β = O(r−4), r →∞ along C (u).

INTERPOLATED RESULTS


Outline

1 ON THE REALITY OF BLACK HOLES

2 RIGIDITY

3 STABILITY

on the mathematical theory of black holes2018)_2.pdf · on the reality of black holes rigidity...

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