ON THE MATHEMATICAL THEORY OF BLACK HOLES2018)_1.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 2, 2018


Outline

1 ON THE REALITY OF BLACK HOLES

2 RIGIDITY

3 STABILITY


KERR FAMILY K(a,m)

COORDINATES (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 ;

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, sph. 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 EXTENSION∆(r−) = ∆(r+) = 0, ∆ = r2 + a2 − 2mr

EXTERNAL REGION r > r+EVENT HORIZON r = r+BLACK HOLE r < r+NULL INFINITY r =∞


EXTERNAL KERR

ERGOREGION g(T ,T ) < 0

TRAPPED NULL GEODESICS


TESTS OF REALITY Ric(g) = 0

1 RIGIDITY. Does the Kerr family K(a,m), 0 ≤ a ≤ m,exhaust all possible vacuum black holes ?

2 STABILITY. Is the Kerr family stable under arbitrarysmall perturbations ?

3 COLLAPSE. Can black holes form starting fromreasonable initial data configurations ? Formation oftrapped surfaces.

Ric(g)=0


STATIONARY BLACK HOLES

DEFINITION [External Black Hole]

Asymptoticaly flat, globally hyperbolic, Lorentzian manifoldwith boundary (M, g), diffeomorphic to the complement of acylinder ⊂ R1+3.

Metric g has an asymptotically timelike, Killing vectorfield T ,

LTg = 0.

Completeness (of Null Infinity)


RIGIDITY

RIGIDITY CONJECTURE. Kerr family K(a,m), 0 ≤ a ≤ m,exhaust all stationary regular vacuum black holes.

True in the axially symmetric case [Carter-Robinson]

True in general, under an analyticity assumption [Hawking]

IS ANALYTICITY REASONABLE ?


RIGIDITY - NEW 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]

CONJECTURE. [Alexakis-Ionescu-Kl]. Rigidity conjecture holdstrue provided that there are no T-trapped null geodesics.


COLLAPSE

GOAL. Investigate the mechanism of formation of black holesstarting with reasonable initial data configurations.

TRAPPED SURFACE. Concept introduced by Penrose inconnection to his incompleteness theorem.

FACT. Incompleteness theorem + weak cosmic censorship

⇒TRAPPED SURFACES DETECT BLACK HOLES.


PENROSE SINGULARITY THEOREM

THEOREM. Space-time (M, g) cannot be future null geodesicalycomplete, if

Ric(g)(L, L) ≥ 0, ∀L null

M contains a non-compact Cauchy hypersurface

M contains a closed trapped surface S

Null expansions tr χ, tr χ

Raychadhouri dds tr χ+ 1

2 tr χ2 ≤ 0.


QUESTIONS

Can trapped surfaces form in evolution? In vacuum?

Can trapped surfaces form starting with non-isotropic, initialconfigurations?


MAIN RESULTS

THEOREM[Christ(2008)]. (∃) open set of regular, vacuum, datawhose MGFHD contains a trapped surface.

1 Specify short pulse characteristic data, for which one canprove a general semi-global result, with detailed control.

2 If, in addition, the data is sufficiently large, uniformly alongall its null geodesic generators, a trapped surface must form.

3 Similar result for data given at past null infinity.


FORMATION OF TRAPPED SURFACES

THEOREM[ Kl-Luk-Rodnianski(2013)] Result holds true fornon-isotropic data concentrated near one null geodesic generator.

1 Combines all ingredients in Christodoulou’s theorem with adeformation argument along incoming null hypersurfaces.

2 Reduces to a simple differential inequality on S0,0 = H0 ∩ H0.


II. STABILITY

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 ).


STABILITY

Despite 50 years of theoretical progress and a wealth of numericaland indirect astrophysical observations the conjecture is still open.

Schwarzschild. Regge-Wheeler(1957), Vishvevshara(1970),Zerilli(1970)

Kerr. Teukolski, Press-Teukolski(1973)

Whiting(1989) The Teukolski linearized gravity system(LGS), have no exponentially growing modes.

If lack of exponential growing modes for the linearized equations,at a fixed state, was enough to deduce stability,

formation of shock waves or emergence of instabilities in fluids,would be ruled out!


STABILITY- FAR FROM BEING SETTLED!

Lack of exponentially growing modes is far from sufficient toestablish even the boundedness of solutions to the linearizedequations.

Weak linear instabilities are to be expected in view of

1 Final Kerr differs from the one we perturb.2 Diffeomorphism covariance of EVE

NONLINEAR STABILITY REQUIRES QUANTITATIVE DECAYTO THE FINAL STATE IN THE CORRECT “CENTER OFMASS” FRAME.


STABILITY- FAR FROM BEING SETTLED!

To make sure that the nonlinear terms remain negligiblethrough the entire evolution we need

1 Mechanism for deriving sufficiently strong quantitative decay.

2 Structure of nonlinear terms. Null condition

3 Modulation procedure to produce the final state and center ofmass frame.

Quantitative decay for even the simplest linear wave equationson fixed Kerr backgrounds was only recently understood.


MAIN RESULTS ON STABILITY

Mode stability of the Kerr family.[Whiting(1989)]

Global Stability of Minkowski space.[Christodoulou-Kl(1990)]

Stability for scalar linear waves K(a,m), 0 ≤ a < m.[Dafermos-Rodnianski-Schlapentokh Rothman (2014)]

Linear stability near Schwarzschild.[Dafermos-Holzegel-Rodnianski(2016)]

Quantitative decay for Teukolski eqts. K(a,m), 0 ≤ a� m.[ Ma, DHR (2017)]


AXIAL SYMMETRIC POLARIZED SPACETIMES

THEOREM[K-Szeftel] Small, axial polarized, perturbations ofgiven initial conditions of an exterior Schwarzschild gm0 (m0 > 0)have maximal future developments converging to another exteriorSchw. 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⇤

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

TA

(ext)M

(int)M

1


CONCLUSIONS

1 RIGIDITY. Understood only perturbatively, close to the Kerrfamily! General case is wide open.

2 STABILITY.Stability of Minkowski space is understood.

“Poor man’s linear stability” is fully understood.

Linear Stability of Schwarzschild is well understood.

Nonlinear stability of Schwarzschild under restrictiveperturbations

Nonlinear stability of axially symmetric perturbations could besettled in the near future.

3 COLLAPSE. Major results have been obtained but the fullscope of the problem remains to be explored.


CONCLUSIONS.

Development of new mathematical methods and strategies to dealwith weak and strong gravitational fields.

1 VECTORFIELD METHOD. Powerful mechanism to derivedecay estimates for geometric wave equations.

2 NULL STRUCTURE. Einstein equations have a uniquegeometric structure which allows, in conjunction with thevectorfield method, to control the nonlinear equations.

3 LONG TIME CONTROL. Understanding of a generalmechanism for long time control of the Einstein equations.Stability of Minkowski space.

4 BLACK HOLE FORMATION. Understanding of a powerfulnew mechanism (in vacuum, non-isotropic) for the formationof black holes.

5 UNIQUE CONTINUATION. Powerful method to deal withill posed problem in General Relativity, most importantly inthe rigidity problem.


Outline

1 ON THE REALITY OF BLACK HOLES

2 RIGIDITY

3 STABILITY

ON THE MATHEMATICAL THEORY OF BLACK HOLES2018)_1.pdf · on the reality of black holes rigidity...

Documents

Transcript of ON THE MATHEMATICAL THEORY OF BLACK HOLES2018)_1.pdf · on the reality of black holes rigidity...