decay of fields outside black holes: massless vlasov ...pblue/talks/warwickgpde2016abbrv.pdf ·...

Decay of fields outside black holes:Massless Vlasov outside a very slow Kerr

Pieter BlueUniversity of Edinburgh

Geometric PDEs Symposium2016 Dec 13, University of Warwick

Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr

THANK ORGANISERS


Overview

1. Inspired by question of black-hole stability

2. Very slowly rotating Kerr black hole, |a| M

3. Massless Vlasov

4. Energy bounds and Morawetz (integrated local energy) decayestimates

5. Illustrate method of previous work

6. Joint with L. Andersson and J. Joudioux


General relativity in one slide

I 4-dimensional space-time manifold: M = (t1, t2)× Σt .I Lorentz (-,+,+,+ signature) pseudometric: g .

I Timelike vector: g(v , v) < 0,I null vector: g(v , v) = 0,I spacelike vector: g(v , v) > 0.

I Einstein equation:

Ric[g ]ab −1

2R[g ]gab =Tab.

Tab from matter on (M, g)

I This is a wave-like equation, since Ric[g ]αβ is like gγδ∂γ∂δgαβplus other terms. (Assume summation convention.)


The Vlasov equation

I f on TM constant along geodesics. Equivalent PDE exists.

I Positive initial data, give positive solutions.

I In TM, the future-directed mass shell for m ≥ 0 is

Sm = V ∈ TM : g(V ,V ) = −m2, g(V ,T+) < 0.

Represents collection of particles of mass m.

I Massless Vlasov: Vlasov on S0.


Stability of Minkowski space

Theorem (Christodoulou-Klainerman)

Given a sufficiently small perturbation (in some moderately highweighted regularity space) of the t = 0 data in Minkowskispacetime, the unique solution of Ric[g ] = 0 has the propertiesthat

I it is geodesically complete, and

I the curvature goes to zero along any geodesic.

Proof via energy estimates from the vector-field method, based onmodels

I s = 0 wave equation

I s = 1 Maxwell equation

I “s = 2 Linearised Einstein equation”.

Can also consider geodesics, Vlasov


Some history: Einstein-Vlasov

Stability results for Minkowski as a solution of the Einstein-Vlasovequation:

I Spherical symmetry, m > 0 [Rendall-Rein]

I Spherical symmetry, m = 0 [Dafermos]

I Decay for Vlasov (m ≥ 0) in Minkowski via vector-fieldtechniques [Fajman-Joudioux-Smulevici]

I m = 0 [Taylor]


Some history: Energy bounds and Morawetz (integratedlocal energy decay) estimates outside black holes

Motivation:

I 2-parameter Kerr family of solutions to Einstein equation.Black holes for |a| ≤ M.

I Conjecture: the Kerr family of black holes is stable assolutions of the vacuum Einstein equation.

I Energy bounds and Morawetz (ILED) estimates have proveduseful for proving stronger estimates, particularly combinedwith the vector-field method.

History:Energy bounds and decay for wave on Kerr|a| M [Dafermos-Rodnianski, Tataru-Tohaneanu, Andersson-B]|a| < M [Dafermos–Rodnianski–Shlapentokh-Rothman]


Vector-field method for geodesics (particles)

(M, g) globally hyperbolic, foliated by Cauchy surfaces Σt .γ a future-directed, null geodesic.Define eX [γ](Σt) = −gαβ γ(t)αX β .

I T timelike, future-directed =⇒ eT [γ](t) ≥ 0.

I eX [γ](t2)− eX [γ](t1) =∫ t2

t1γαγβ∇(αXβ)dλ

I S(γ) constant along geodesics =⇒eX [γ]S(γ)2 has the same positivity and conservationproperties as eX [γ].


Vector-field method for PDEs

ϕ solving a PDE typically has Tαβ with “good properties”(Symmetric, dominant energy condition, divergence-free).

Define EX [ϕ](Σ) =∫

Σ TαβXαdνβ.

I T time-like, future-directed =⇒ ET [ϕ](t) ≥ 0.

I EX [ϕ](Σ2)− EX [ϕ](Σ1) =∫

Ω Tαβ∇(αX β)d4x

I Define a PDE symmetry to be a differential operator S suchthat for all ϕ satisfying the PDE, Sϕ satisfies the PDE. S is a

PDE symmetry =⇒ EX [Sϕ] has the same positivity andconservation properties as EX [ϕ].


The wave equation and energy and Morawetz (integratedlocal energy) estimates in R1+3

(− ∂2

∂t2+

∂2

∂x2+

∂2

∂y 2+

∂2

∂z2

)u = 0.

E∂t [u](t) =1

2

∫t×R3

|∂tu|2 +3∑

i=1

|∂iu|2d3x

0 =d

dtE [u](t) (Energy)

E∂t [u] &∫R

∫t×R3

|∂tu|2 +∑3

i=1 |∂iu|2

1 + |~x |2d3xdt. (Morawetz)


The wave equation L∞ estimates

∂1, ∂2, ∂3 are symmetries.Sobolev:

C‖u(t)‖2L∞x≤

2∑i=1

E [∂iu](t)

≤2∑

i=1

E [∂iu](0).


Black holes

A spacetime is asymptotically flat if there is an open set U∞diffeomorphic to U∞ = R× (R,∞)× S2 ⊂ R1+3 such that (in asuitable sense) the metric approaches the Minkowski metric onU∞.

The chronological past of a set U is the set of points p such thatfrom p there is a timelike curve to a point q ∈ U.

In an asymptotically flat spacetime, a black hole is thecomplement of the past of U∞. The event horizon is theboundary of the black hole.


Geometry of the Kerr spacetime

I Mass M, rotational parameter a.

I Black hole for |a| < M.

I Schwarzschild is a = 0.

I Spherical co-ordinates, (t, r , θ, φ):

I Exterior: r > r+ = M +√

M2 − a2.Globally hyperbolic. Σt′ = t = t ′ are Cauchy.

I ∂t is not timelike for r − r+ . a.


Hidden symmetries and the geodesic equation

I Symmetries: ∂t , ∂φ.Killing tensor/ Hidden symmetry: ∇(γKαβ) = 0.

I Conserved quantities for null geodesics

e = vt , lz = vφ, q = v 2θ +

1

sin2 θv 2φ + a2 sin2 θv 2

t .

Let S2 = e2, elz , l2z , q = Saa.

I Geodesic equation

∆

Σ

(dr

dλ

)2

= −R(r ; M, a; e, lz , q),

R(r ; M, a; e, lz , q) = −(r 2 + a2)2e2 + 4aMrelz + ∆q + (∆− a2)l2z

= R(r ; M, a)aSa.

I Orbiting geodesics (“trapping”) at R = 0 = ∂rR. (Problem)


Stress-energy tensor and symmetries for Vlasov

I Stress-energy for field f at p ∈M.

Tab[f ]p =

∫(S0)p

vavbf

√−detg

v0dv 1dv 2dv 3.

I Tab is symmetric, divergence-free, dominant energy condition.

I Multiplication by conserved quantities for null geodesics is asymmetry for the Vlasov equation.


2-symmetry-strengthening

I Let S = Saa∈A be a collection of symmetries.

I A 2-symmetry-strengthened vector-field is a collection ofvector fields X aab.

I A 2-symmetry-strengthened vector-field is timelike if for allreals σa, the vector X aabσaσb is timelike. It isfuture-directed if for all reals σa, the vector X aabσaσb isfuture-directed.

I A 2-symmetry-strengthened stress-energy-tensor is acollection of stress-energy tensors Tabab.

I Tabab is symmetric if Tabab = Tbaab. It is divergence-freeif, for all f solving the PDE, ∇aTabab = 0. It satisfies thedominant energy condition if for all future-directed, timelikeX aab and Y b, TababY bX aab ≥ 0.


Energy bounds and Morawetz estimates for the Vlasovequation: the strategy

Construct a timelike 2-symmetry-strengthened vector-field T andanother 2-symmetry-strengthened vector-field A such that

ET ≥ 0, (1a)

Tabab∇aAbab ≥ 0, (1b)

Tabab∇aTbab .|a|M

Tabab∇aAbab, (1c)

ET & |EA|. (1d)


The timelike 2-symmetry-strenthened vector-field

I If T a is a timelike and future-directed vector-field, thenTaab = T aδab is a timelike and future-directed2-symmetry-strengthened vector-field.

I In Kerr with |a| M, for ΩH = a/2Mr+,

T = ∂t + χr≤10MΩH∂φ

is timelike everywhere and only fails to be Killing nearr ∼ 10M.

I Take T = (∂t + χr≤10MΩH∂φ)aδab.


The Morawetz 2-symmetry-strengthened vector-field

Given a vector field A = h∂r , shematically Tab∇aAb has the form

−(∂rh)vrvr + h(∂rRa)Saba vavb. (2)

I h = ∂rR, first factor is (∂rR)2.Second factor is positive because the orbiting geodesics areunstable ∂2

rR < 0.

I If A is a 2-symmetry-strengthened vector-field, we can takeh = L(a∂rRb) with LaSa = e2 + l2

z + q.


The result

There is a C > 0 such that for f a nonnegative solution of themassless Vlasov equation∫

(S0)M

∆2

(r 2 + a2)2v 2r |f |2 + r 5R′R′Lf dµ(S0) + ET[f ](t)

≤ CET[f ](0).

where

ET[f ](0)

∼∫

Σt

∫(S0)p

((r 2 + a2)2

∆|vt |2 + ∆|vr |2 + v 2

θ +1

sin2 θv 2φ

)|f |2d3vd3x ,

|f |2 =

∣∣∣∣M2v 2t + v 2

θ +1

sin2 θv 2φ

∣∣∣∣2 f ,

d3x = sin θdrdθdφ, d3v =1

|vt |r 2 sin θdv rdvθdvφ.


Killing things & Algebraic nonsense

Graded algebra: vector space with product, decomposes intogrades indexed by integer, and Va × Vb → Vc .

I T: algebra of all formal sums of tensors satisfying conformalKilling tensor condition Kα1...αk

= K(α1...αk ) and∇(βKα1...αk ) = g(βα1

pα2...αk ), graded by valence.

I V: algebra generated by conformal Killing vectors.

I G: Conserved quantities for null geodesics.

I W: PDE symmetries (taking solutions to solutions) for thewave equation.

I M: PDE symmetries for the Maxwell equation.

Obvious results:

I G = T by Kα1...αk7→ Kα1...αk

γα1 . . . γαk .

I V →W, V →M by Lie differentiation.


Algebraic nonsense: nonobvious results

I In flat Ri+j , T = V = G. [Eastwood]

I In Kerr, T 6= V, V 6= W, V 6= M.

I An additional necessary and sufficient condition is required forT = W; there are examples where this fails. [Michel, Radoux,Silhan]

I In 1 + 3 dimensions, an additional necessary and sufficientcondition is required for T = M. [Andersson, Backdahl, B.]


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