ON THE MATHEMATICAL THEORY OF BLACK HOLES2018)_3.pdf · ON THE REALITY OF BLACK HOLES RIGIDITY...

ON THE REALITY OF BLACK HOLES RIGIDITY STABILITY COLLAPSE

ON THE MATHEMATICAL THEORY OFBLACK HOLES

Sergiu Klainerman

Princeton University

July 5, 2018


Outline

1 ON THE REALITY OF BLACK HOLES

2 RIGIDITY

3 STABILITY

4 COLLAPSE


Outline


2 RIGIDITY

3 STABILITY

4 COLLAPSE


Outline


2 RIGIDITY

3 STABILITY

4 COLLAPSE


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).


STABILITY OF KERR

Ric[gm,a] = 0

Kerr family depends on two parameters a,m.

Ric′[∂gm,a∂m

]= Ric′

[∂gm,a∂a

]= 0

Einstein vacuum equations are diffeomorphism invariant

Ric′ [LXgm,a] = 0.

Dim(Ker Ric′

)=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, ωCurvature R = α, β, ρ, ∗ρ, β, α

5 Main Equations

6 S-foliations


BASIC QUANTITIES

NULL FRAME e3, e4, (ea)a=1,2, S = spane1, 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), . . .


BASIC EQUATIONS

NULL STRUCTURE EQTS. (Transport)

∇4Γ = R +∇/ Γ + Γ · Γ,∇3Γ = R +∇/ Γ + Γ · Γ

NULL STRUCTURE EQTS. (Codazzi)

∇/ Γ = R + Γ · Γ,

NULL BIANCHI

∇4R = ∇/R + Γ · R,∇3R = ∇/R + Γ · R


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.


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 MINKOWSKI

1 STABILITY OF VACUUM STATES FOR SCALAR WAVES

2 CLASSICAL VECTORFIELD METHOD

3 STABILITY OF MINKOWSKI- CHRISTODOULOU-KL.

4 STABILITY OF MINKOWSKI- NICOLO-KL.


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.


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 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 d R = 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

CONTROL OF CURVATURE.

1 All null components of R are O(ε2)-invariant

2 Bianchi identities d R = δR = 0

Bel-Robinson tensor Q = R · R + ∗R · ∗R.Symmetric, traceless, δQ = 0.Positive energy condition.

3 Energy type quantities.

ContractionsCommutations

4 Peeling estimates.


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(2001-2003)

THEOREM. The maximal development of any A.F. initial dataset converges to the flat space in the complement of the causalfuture of a sufficiently large compact set.

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

4 Improved peeling α = O(r−5), β = O(r−4).


STABILITY OF BLACK HOLES

1 MAIN DIFFICULTIES

2 MAIN ADVANCES

3 NONLINEAR STABILITY OF KERR UNDER POLARIZEDPERTURBATIONS


KERR STABILITY-MAIN DIFFICULTIES

UNLIKE STABILITY OF THE MINKOWSKI SPACE

1 Some null curvature components ρ, ρ∗ are nontrivial. Bianchisystem admits non-decaying states.

2 All other null components of the curvature tensor aresensitive to frame transformations.

3 Principal null directions are not integrable.

4 Have to track dynamically the parameters (af ,mf ) of thefinal Ker and the correct gauge condition.

5 Obstacles to prove decay for the simplest linear equationsKerrΦ = 0 on a fixed Kerr.


1’ST BREAKTHROUGH. TEUKOLSKI EQTS.

FACT[Teukolski 1973).] Extreme curvature components α, αverify, up to O(ε2)-errors, decoupled, albeit non-conservative,linear wave equations.

Whiting(1989). Mode Stability. Teukolski linearized gravityequations, have no exponentially growing modes. Spin-2

Y. Schlapentokh Rothman (2014). Quantitative modestability for Spin 0. a,mΦ = 0, |a| < m.

Dafermos-Rondianski-Rothman(2015) Make use of the NewVectorfield Method and Yacov’s result to deduce quantitativedecay estimates for a,mΦ = 0, |a| < m.


2’ND BREAKTHROUGH. VF. METHOD

CLASSICAL VF. METHOD.

Non perturbative method, based on the continuous symmetries ofMinkowski and adapted higher order energy estimates, to deriverobust uniform decay and peeling.

Null Condition. Structural, gauge dependent, condition on thequadratic part of a nonlinear system of wave equations whichinsures global regularity.

Nonlinear Stability of Minkowski. Uses generalized energyestimates, based on approximate symmetries, to get uniformdecay estimates for the curvature tensor.


NEW VECTORFIELD METHOD a,mΦ = 0

Compensates for the lack of enough symmetries of Kerr(a,m) byintroducing new geometric quantities to deal with:

Degeneracy of the horizon.

Trapped null geodesics.

Superradiance

Low decay at null infinity.


NEW VECTORFIELD METHOD a,mΦ = 0

The new method has emerged in the last 15 years in connection tothe study of boundedness and decay for the scalar wave equation,

a,mφ = 0

Partial Results. Soffer-Blue(2003), Blue-Sterbenz, Daf-Rod,MMTT, Blue-Anderson, Tataru-Tohaneanu, etc.

Full Range a < m.Dafermos-Rodnianski-Shlapentokh-Rothman (2014)


3’ND BREAKTHROUGH. CHANDRASKHAR TR.

FACT[Chandrasekhar(1975).] There exist a transformationα −→ P which takes solutions of the Teukolski equation on aSchwarzschild background to solutions of the Reggee-Wheelerequation,

SchwP + VP = 0.

Dafermos-Holzegel-Rodnianski(DHR 2016). Provequantitative decay for P and therefore also for α, α. They usethis as a first step to prove linear stability of Schwarzschild.

DHR, Ma(2017). Can extend the analysis to control theTeukolski equation for Kerr(a, m), |a| m.


LINEAR STABILITY OF SCHWARZSCHILD

Dafermos-Holzegel-Rodnianski(2016). Schwarzschild Kerr(0,m) islinearly quantitatively stable, once we mod out the unstablemodes related to:

Two parameter family of nearby stationary solutions.

Linearized gauge transformations

CHANDRASEKHAR TRANSF. Derive sharp decaybounds for α, α.

RECONSTRUCTION. Find appropriate gauge conditions,to derive bounds and decay for all other quantities of thelinearized Einstein equations on Schwarzschild.

Hung-Keller- Wang (2017). Alternative approach based theRegge-Weeler, Zerilli.Hung(2018), Johnson (2018). Wave coordinates


LINEAR STABILITY OF SCHWARZSCHILD

Dafermos-Holzegel-Rodnianski(2016). Schwarzschild Kerr(0,m) islinearly quantitatively stable, once we mod out the unstablemodes related to:

Two parameter family of nearby stationary solutions.

Linearized gauge transformations

CHANDRASEKHAR TRANSF. Derive sharp decaybounds for α, α.

RECONSTRUCTION. Find appropriate gauge conditions,to derive bounds and decay for all other quantities of thelinearized Einstein equations on Schwarzschild.

Hung-Keller- Wang (2017). Alternative approach based theRegge-Weeler, Zerilli.Hung(2018), Johnson (2018). Wave coordinates


AXIAL SYMMETRIC POLARIZED SPACETIMES

THEOREM[K-Szeftel] Small axial polarized perturbations of giveninitial conditions of an exterior Schwarzschild gm0 (m0 > 0) havemaximal future developments converging to another exterior Schw.solution gm∞ , m∞ > 0.

I+ H+ C C

C0 C0 T A (ext)M (int)M

1

I+

H+

C C

C 0

C 0

TA

(ext) M

(int

) M

1

I+

H+

C C

C0

C0 T

A(ext)M

(int)M

1

I +H +

C C

C 0C 0

TA

(ext)M

(int)M

1

I+H+

C C

C0C 0

T A(ex

t) M(in

t) M

1

I+ H+ C C


1

I+

H+

C C

C 0

C 0

TA

(ext

) M(int

) M

1

I+

H+

C

C

C0

C0

TA

(ext)M

(int)M

1


GEOMETRIC FEATURES

Spacetime M = (ext)M∪ (int)M

Optical function u in (ext)M initialized ”on I+”

Optical function u in (int)M initialized on T

An outgoing geodesic foliation in (ext)M - optical function uAn ingoing geodesic foliation in (int)M - optical function uNull frames in (int)M∪ (ext)M

I+ H+ C C


1

I+

H+

C C

C 0

C 0

TA

(ext) M

(int

) M

1

I+

H+

C C

C0

C0 T

A(ext)M

(int)M

1

I +H +

C C

C 0C 0

TA

(ext)M

(int)M

1

I+H+

C C

C0C 0

T A(ex

t) M(in

t) M

1

I+ H+ C C


1

I+

H+

C C

C 0

C 0

TA

(ext

) M(int

) M

1

I+

H+

C

C

C0

C0

TA

(ext)M

(int)M

1


BOOTSTRAP SPACETIME

I+ H+ C C


1

I+

H+

C C

C 0

C 0T

A(ext) M

(int

) M

1

I+

H+

C C

C 0

C 0

TA

(ext) M

(int

) M

1

I+

H+

C C

C0

C0 T

A(ext)M

(int)M

1

I +H +

C C

C 0C 0

TA

(ext)M

(int)M

1

I+H+

C C

C0C 0

T A(ex

t) M(in

t) M

1

I+ H+ C C


1

I+

H+

C

C

C0

C0

TA

(ext)M(int)M

1

I+

H+

C C

C 0

C 0

TA

(ext

) M(int

) M

1

I+

H+

C

C

C0

C0

TA

(ext)M

(int)M

1

Sigma*

Outgoing optical function u initialized on the GCM spacelikehypersurface Σ∗Incoming optical function u initialized on the timelikehypersurface T .


KEY FEATURES OF THE CONSTRUCTION

1. Identify a quantity which converges to the final mass.

HAWKING MASS 2mH(u,r)r = 1 + 1

16π

∫S tr χtr χ

2. Final mass

m∞ = limu→∞

limr→∞

mH(u, r).

3. Spacelike boundary Σ∗ foliated by GCMS 2-spheres.

Used to initialize the spacetime foliation. The constructionmakes use of the full number of degrees of freedom of thediffeomorphism group .



GCMS. Given S0 ⊂M construct a deformation S0 → S and aunique compatible frame eS3 , e

S4 , (eSA)A=1,2 such that certain key

quantities vanish identically on S.

3a. D/S,?1 tr χS = D/S,?

2 D/S,?1 tr χS = D/S,?

2 D/S,?1 µS = 0

3b. Projection of βS on ker D/S,?2 and similar other quantities also

vanish.

4. Construction of GCMS based on solving an Elliptic Hodgesystem coupled with transport equations.



4. Together with the knowledge of α, α the GCMS determineall connection and curvature components on Σ∗ by solvingvarious Hodge systems.

5. All connection and curvature components are determined bytransport equations and Hodge systems from their initialvalues on Σ∗.



6. Spacetime M, spacelike like hypersurface Σ∗ and the twogeodesic foliations are constructed by a continuity argumentstarting with an initial data layer L0 ∪ L0.

7. Initial layer L0 ∪ L0 constructed by K-Nicolo(2001-2003).

8. Derive sufficient decay for Γ,R to be able to estimate P fromthe master equation

gP + VP = N(Γ,R).


COMPLETE STATEMENT OF THEOREM

INITIAL LAYER ASSUMPTION. Iklarge+5 ≤ ε0

CONCLUSIONS. There exists a future globally hyperbolicdevelopment with complete future null infinity I+ and futurehorizon H+ which verifies

N (En)klarge

+N (Dec)ksmall

≤ Cε0, ksmall =

⌊1

2klarge

⌋+ 1.

In particular,

On (ext)M, we have,

|α|, |β| . ε0 min 1

r3(u + 2r)12

+δdec,

1

r2(u + 2r)1+δdec

,

|β| . ε0

r2u1+δdec, |α| . ε0

ru1+δdec,

|χ|, |ζ| . ε0 min 1

r2u12

+δdec,

1

ru1+δdec

, |χ| . ε0

ru1+δdec.


On (int)M,

|α|, |β|, |β|, |α|, |χ|, |ζ|, |χ| . ε0

u1+δdec.

m∞ = limu→∞ limr→∞mH(u, r), |m∞ −m0| . ε0.

On the future Horizon H+,

r = 2m∞ + O

( √ε0

u1+δdec

2

).

On (ext)M,∣∣∣∣ρ+2m∞r3

∣∣∣∣ . ε0 min 1

r2u1+δdec,

1

r3u1/2+δdec

∣∣∣∣tr χ− 2

r

∣∣∣∣ . ε0

r2u1+δdec,

∣∣∣∣∣tr χ+2(1− 2m∞

r

)r

∣∣∣∣∣ . ε0

r2u1+δdec.


On (int)M, we have∣∣∣∣ρ+2m∞r3

∣∣∣∣ , ∣∣∣∣κ+2

r

∣∣∣∣ ,∣∣∣∣∣κ− 2

(1− 2m∞

r

)r

∣∣∣∣∣ . ε0

u1+δdec.

On (ext)M, in u, r , θ, ϕ coordinates

g = gm∞, (ext)M + O( ε0

u1+δdec

)gm∞, (ext)M = −2dudr −

(1− 2m∞

r

)(du)2 + r2dσ2.

On (int)M, in u, r , θ, ϕ coordinates

g = gm∞, (int)M + O

(ε0

u1+δdec

)gm∞, (int)M = 2dudr −

(1− 2m∞

r

)(du)2 + r2dσ2


OTHER CONCLUSIONS

BONDI MASS. MB(u) = limr→+∞m(u, r) for all 0 ≤ u < +∞

BONDI MASS LAW FORMULA.

∂uMB(u) = − 1

16

∫S∞(u)

Θ2(u, ·) for all 0 ≤ u < +∞.

with

Θ(u, ·) = limr→+∞

r χ(r , u, ·) for all 0 ≤ u < +∞.

FINAL BONDI MASS.

MB(+∞) = limu→+∞

MB(u) = m∞


MAIN INTERMEDIATE STEPS

THM 1. Given a GCM admissible spacetime verifying

N (En)klarge

(0) . ε0 and BA, we deduce, for q + V q = err,

N (dec)ksmall+20[q] . ε0.

THM 2-3. Under the same assumptions we have in(int)M∪ (ext)M,

N (dec)ksmall+15[α, α] . ε0.

THM 4, 5. Under the same assumptions, we have in(int)M∪ (ext)M,

N (dec)ksmall+5[R, Γ] . ε0.

THM 6. Under the same assumptions as above we have in M,

N (En)klarge

+N (Dec)ksmall+5 . ε0.


MAIN INTERMEDIATE STEPS

DEFINITION. Let U ⊂ R+ the set all values of u∗ such anadmissible spacetime exists for u ∈ [0, u∗] verifying BA.

THM 7. There exists δ0 > 0 small enough such that forsufficiently small ε0 > 0, ε > 0, [0, δ0] ⊂ U .

THM 8. Given a GCM admissible spacetime with 0 < u∗ < +∞such that

N (En)klarge

+N (Dec)ksmall

. ε0,

we can find u′∗ > u∗ such that u′∗ ∈ U .


CONSTRUCTION OF GCMS

METRIC.

g = −2duds + Ωdu2 + γ

(dθ − 1

2bdu

)2

+ e2Φdϕ2.

FRAME TRANSFORMATIONS. f , f , a = O(ε),

e ′3 = ea(e3 + f eθ +1

4f 2e4)

e ′θ = (1 +1

2f f )eθ +

1

2(fe3 + f e4) + l.o.t.

e ′4 = e−a(

(1 +1

2f f )e4 + feθ +

1

4f 2e3

)+ l.o.t.

DEFORMATIONS. Ψ :S −→ S

u =u + U(θ), s =

s + S(θ), θ ∈ [0, π].



PROPOSITION. GivenS with

r = 2m0(1 + δH) there exists a

nearby deformed sphere S of area radius rS =r + O(ε), and a

compatible frame

(e ′3 = eS3 , e′4 = eS4 , e

′θ = eSθ )

which verifies the GCM conditions

κS =2

rS, κS = µS = 0.



ADAPTED FRAME TRANSFORMATIONS

Ψ∗(eθ) = eSθ

COMPATIBILITY. U,S : [0, π] −→ [0, π], U(0) = S(0) = 0,uniquely determined in terms of a, f , f by transport typeequations.

GCMS- CONDITION. Leads to a nonlinear elliptic Hodge system

on S for a, f , f which has trivial kernel ifS is sufficiently close to

r = 2m0.



ITERATION. Define iteratively quintetsQ(n) = (U(n),S (n), a(n), f (n), f (n)) starting with Q(0).

Q(0) represents the trivial deformation.

(U(n),S (n)) defines the map Ψ(n) :S −→ S(n). Define the

triplet (a(n+1), f (n+1), f (n+1)) by solving the nonlinear ellipticsystem on S(n),

D(n)(f (n+1), f (n+1), a(n+1)) = 0

Construct the pair U(n+1),S (n+1)by solving a transportequation defined by the triplet (a(n+1), f (n+1), f (n+1)).

Contraction Argument. Need to compare the pull backs toS.


Outline


2 RIGIDITY

3 STABILITY

4 COLLAPSE

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