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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
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THANK ORGANISERS
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.)
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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]
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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]
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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].
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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].
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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)
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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).
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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)
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr
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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.
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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)
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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.
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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.
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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.
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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.
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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.]
Pieter Blue Decay of fields outside black holes: Massless Vlasov outside a very slow Kerr