the wave equation in spherically symmetric spacetimesmasarikm/waveeqn_sphsymm.pdf · 2011-09-12 ·...

athematics

Background and Geometry PreliminariesThe Black Hole Case

The Particle-like Case

The Wave Equation in Spherically SymmetricSpacetimes

Matthew P. Masarik

Department of MathematicsUniversity of Michigan

Matthew P. Masarik The Wave Equation in Spherically Symmetric Spacetimes

athematics



Outline

1 Background and Geometry Preliminaries

2 The Black Hole CaseBasicsThe Wave EquationSpectral AnalysisDecayApplication to Einstein/Yang-Mills

3 The Particle-like CaseBasicsThe Wave EquationSpectral Analysis


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Introduction

There has been much recent effort devoted to studying thestability of several spacetime metrics coming from GR. Inparticular, the Minkowski spacetime, the Schwarzschildspacetime, and the Kerr spacetime. The ultimate goal in thisstudy is to prove the nonlinear stability of these metrics.

For Minkowski, this was done by Christodoulou andKlainerman in 1993.

It is a great open problem in GR to prove the nonlinearstability of the Kerr metric; nonlinear stability of theSchwarzschild metric remains open as well (this would ofcourse follow from a result on Kerr).


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An Apology

This talk will focus on the linear wave equation in a generalspherically symmetric spacetime. Why do we care about thelinear wave equation?

In short, the nonlinear stability of these metrics is too hard.Instead, we first investigate the linear stability of thesemetrics. We see three obvious reasons for this:

Decay of the linear wave equation implies linear stability ofthe metric under scalar wave perturbations; results on thelinear wave equation are important to obtain results innonlinear regime; and it is worthwhile understanding the linearwave equation.


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Review of Results

There is a plethora of results concerning the decay of thelinear wave equation in the Schwarzschild and Kerr metrics,and the methods break down into roughly two classes.

The first class is basically a harmonic analysis approach wherethe main tool is Strichartz estimates. This is the idea behindDafermos & Rodnianski (2008), Donninger, Schlag, Soffer(2009), Luk (2010), etc. These results are very technicalworks – even reading the introduction and discerning theresults can be a challenge.

The second class uses spectral methods to obtainrepresentation formulae, and then analyzes the formulae. Thisapproach is very classical, and we obtain robust, generalresults.


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Review of Results, continued

Kronthaler showed decay of the linear wave equation in theSchwarzschild metric.

Kronthaler, Blue-Sterbenz, Dafermos-Rodnianski get rates:

t−3−2l for modes, t−32 for full solution

Dafermos & Rodnianski obtain pointwise boundedness in Kerr

Finster, Kamran, Smoller, Yau showed decay of the linearwave equation in Kerr

Dafermos & Rodnianski obtain t−1 rate for Kerr


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What is Left

It may seem that most of the decay questions have beenanswered, but we now wish to study more general spacetimemetrics. For example, perturbations on Minkowski orperturbatoins on Schwarzschild. Solutions of theEinstein-Yang/MIlls equations fall into this category.

Right now we restrict our attention to spherically symmetriccases – the positive angular momentum case may be futurework.

Much of the work by Kronthaler (2006) can be extended toapply in a more general setting, but only after a few key ideas.


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BasicsThe Wave EquationSpectral AnalysisDecayApplication to Einstein/Yang-Mills

Outline





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Schwarzschild: Basic facts

Let us recall some facts about the Schwarzschild metric:

The Schwarzschild metric is given by

ds2 = −(

1− 2m

r

)dt2 +

(1− 2m

r

)−1

dr 2 + r 2dΩ2,

where r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, and

dΩ2 = dθ2 + sin2 θdφ2.

The wave equation in Schwarzschild is given by[∂2

t −(

1− 2m

r

)1

r 2

(∂r (r 2 − 2mr)∂r + ∆S2

)]ζ = 0.

Kronthaler analyzed this PDE in the exterior region (r > 2m)with data compactly supported away from the horizon(r = 2m).


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The Regge-Wheeler Coordinate

The starting point for studying the wave equation inSchwarzschild is change coordinates tou(r) = r + 2m log

(r

2m − 1)

and letζ(t, r(u), θ, φ) = r(u)ψ(t, u, θ, φ).

Then the wave equation becomes[∂2

t − ∂2u +

(1− 2m

r

)(2m

r 3− ∆S2

r 2

)]ψ = 0

on R× R× S2.

One then couples this with data(ψ, iψt)(u, 0, θ, φ) = Ψ0(u, θ, φ) ∈ C∞0 (R× S2)2.


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The Regge-Wheeler Coordinate, continued

Why is the Regge-Wheeler coordinate ubiquitous in Schwarzschild?

It yields a beautiful energy, the density given by

ψ2t +ψ2

u+

(1− 2m

r

)[2m

r 3ψ2 +

1

r 2

(1

sin2 θ(∂φψ)2 + sin2 θ(∂cos θψ)2

)].

Unfortunately, when working in a more general sphericallysymmetric black hole metric, the generalization of theRegge-Wheeler coordinate fails to yield a positive definiteenergy density.

The positive energy density is imperative to usingenergy/spectral methods.


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Essential Properties of the Blackhole Geometry: Near theHorizon

Consider a metric ds2 = −T−2(r)dt2 + K 2(r)dr 2 + r 2dΩ2. Forthis to be a generalized Schwarzschild blackhole, we requireT ,K ∈ C∞(r0,∞) with T ,K > 0 for r > r0 > 0. In addition, werequire for r near r0,

T (r) ∼ c1(r − r0)−12 + O(1), c1 > 0

K (r) ∼ c2(r − r0)−12 + O(1), c2 > 0

T ′(r) ∼ c3(r − r0)−32 + O(r − r0)−

12

K ′(r) ∼ c4(r − r0)−32 + O(r − r0)−

12


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Essential Properties of the Blackhole Geometry: TheFar-Field

We must also impose conditions on the far-field behavior of themetric. We shall insist that for large r ,

T (r) ∼ 1 + O(

1r

)K (r) ∼ 1 + O

(1r

)T ′(r)T (r) + K ′(r)

K(r) ∼ O(

1r2

)Remark

The above conditions are satisfied by the Schwarzschild metric, thenon-extreme Reissner-Nordstrom metric, and black hole solutionsof the Einstein-Yang/Mills (EYM) equations.


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The wave equation in a geometry ds2 = gijdx idx j is given by

0 = ζ := g ij∇i∇jζ =1√−g

∂

∂x i

(√−gg ij ∂

∂x j

)ζ,

where g ij is the inverse of the metric gij and g = det(gij).

So in the geometry ds2 = −T−2(r)dt2 + K 2(r)dr 2 + r 2dΩ2,the d’Alembert operator takes the form

−T 2∂2t +

1

r 2

(r 2∂r

K 2

)+

T

K 3∂r

(K

T

)∂r +

1

r 2∆S2 ,

where we have dropped the arguments of T ,K and ∆S2 is thestandard Laplacian on the sphere S2.


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The Change of Coordinate

Upon making the change of coordinate

u(r) := −∫ ∞

r

K (α)T (α)

α2dα,

the wave equation ζ = 0 becomes(−r 4∂2

t + ∂2u +

r 2

T 2∆S2

)ψ = 0 on R× (−∞, 0)× S2.

Note: If we set T = K =(1− 2m

r

)− 12 (Schwarzschild coefficients),

then we find

u(r) =1

2mlog

(1− 2m

r

).


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Existence/Uniqueness

Theorem

The Cauchy problem(−r 4∂2

t + ∂2u + r2

T 2 ∆S2

)ψ = 0 on R× (−∞, 0)× S2

(ψ, iψt)(0, u, θ, φ) ∈ C∞0 ((−∞, 0)× S2)2

has a global, smooth, unique solution ψ that is compactlysupported in (−∞, 0)× S2 for each time t.

To prove it, one applies the theory of symmetric hyperbolicsystems to the equation for ξ = rψ in the coordinate

s(u) :=

∫ u

ur 2(α)dα.

Note: s is a generalization of the Regge-Wheeler coordinate.Matthew P. Masarik The Wave Equation in Spherically Symmetric Spacetimes

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Energy and the Hamiltonian

The solution ψ admits a conserved energy:

E (ψ) =

∫ 2π

0

∫ 1

−1

∫ 0

−∞r 4 (ψt)2 + (ψu)2

+r 2

T 2

(1

sin2 θ(∂φψ)2 + sin2 θ(∂cos θψ)2

)dud(cos θ)dφ.

Recast the PDE as i∂tΨ = HΨ for Ψ = (ψ, iψt)T and

H =

(0 1A 0

)where A = − 1

r 4∂2

u −∆S2

r 2T 2.


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Energy and the Hamiltonian, continued

The solution of the original Cauchy problem yields a solutionΨ of the Hamiltonian system with data inC∞0 ((−∞, 0)× S2)2.

The energy functional induces an inner product 〈·, ·〉 on thespace C∞0 ((−∞, 0)× S2)2, with respect to which theHamiltonian is symmetric:

0 =d

dt〈Ψ,Ψ〉

= 〈∂tΨ,Ψ〉+ 〈Ψ, ∂tΨ〉= i〈HΨ,Ψ〉 − i〈Ψ,HΨ〉.


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Energy and the Hamiltonian, continued

The above implies that 〈HΨ,Ψ〉 = 〈Ψ,HΨ〉 for a solution Ψ.In particular, 〈HΨ0,Ψ0〉 = 〈Ψ0,HΨ0〉.But the data is arbitrary in C∞0 ((−∞, 0)× S2)2 – a simplepolarization argument then yields the symmetry of H onC∞0 ((−∞, 0)× S2)2.


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Projecting Onto Spherical Harmonics

Projecting onto spherical harmonics yields, for each angularmomentum number l = 0, 1, 2, . . . , the Cauchy problem

i∂tΨlm = HlΨlm on R× (−∞, 0)

Ψlm(0, u) = Ψlm0 ∈ C∞0 (−∞, 0)2

where

Hl =

(0 1

− 1r4∂

2u + l(l+1)

r2T 2 0

).

We also note that Hl is symmetric on C∞0 (−∞, 0)2 with respect tothe inner product

〈Ψ, Γ〉l =

∫ 0

−∞r 4ψ2γ2 + (∂uψ1)(∂uγ1) +

r 2

T 2l(l + 1)ψ1γ1du.


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Essential Self-Adjointness of Hl

We consider the Hilbert space H := H1Vl ,0⊗Hr2,0, where

Vl(u) = l(l+1)r2T 2 and

H1Vl ,0

is the completion of C∞0 (−∞, 0) within the Hilbertspace(ψ : ψu ∈ L2(−∞, 0) and r 2Vlψ ∈ L2(−∞, 0)

, 〈·, ·〉l1

)and

Hr2,0 is the completion of C∞0 (−∞, 0) within the Hilbertspace

(ψ : r 2ψ ∈ L2(−∞, 0)

, 〈·, ·〉l2

).

Proposition

The operator Hl with domain D(Hl) = C∞0 (−∞, 0)2 is essentiallyself-adjoint in the Hilbert space H.


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Stone’s Theorem

Theorem (Stone’s Theorem)

Let U(t) be a strongly continuous one-parameter unitary group ona Hilbert space H. Then there is a self-adjoint operator A on H sothat U(t) = e itA.Furthermore, let D be a dense domain which is invariant underU(t) and on which U(t) is strongly differentiable. Then i−1 timesthe strong derivative of U(t) is essentially self-adjoint on D and itsclosure is A.

For proof, we refer to Reed and Simon.


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Essential Self-Adjointness of Hl , continued

Let U(t) be the solution operators: U(t)Ψlm0 = Ψlm(t). Then

the U(t) extend to a one-parameter unitary group on H thatis strongly continuous on H and strongly differentiable onC∞0 (−∞, 0)2 (due to the smoothness of the solution Ψlm andthe energy conservation).

Moreover, U(t) leave the dense subspace C∞0 (−∞, 0)2

invariant for all times t and the strong-derivative of U(t) isjust −Hl .

Thus, by Stone’s theorem, Hl is essentially self-adjoint andU(t) = e−itH l .


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A Representation Formula

By Stone’s theorem, we can write Ψlm(t, u) = e−itH l Ψlm0 (u).

To obtain an explicit representation of Ψlm, we shall expressΨlm

0 in terms of the spectral projections of H l .

For, if we had Ψlm0 (u) =

∫R Ψlm

0 (v)dµ(v , u), where dµ are the

spectral measures of H l , then the spectral theorem wouldimply

Ψlm(t, u) =

∫R

e−iωtΨlm0 (v)dµ(v , u).


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Stone’s Formula

In general, the spectral measures are difficult to compute. But fora self-adjoint operator A, Stone’s formula relates the spectralprojections to the resolvent:

(P(a,b)+P[a,b]

)= lim

ε0

1

πi

∫ b

a(A− ω − iε)−1 − (A− ω + iε)−1dω,

where the limit is taken in the strong operator topology.So, in order to derive a useful representation formula, we shallinvestigate the resolvent of H l .


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The Resolvent

We will construct the resolvent out of special “solutions” ofthe eigenvalue equation H lΨ = ωΨ.

The eigenvalue equation is equivalent to the ODE−ξ′′(u)− ω2r 4ξ + r2

T 2 l(l + 1)ξ = 0 on (−∞, 0).

Change to the s coordinate and consider η(s) = rξ(u(s)).The equation for η is

−η′′(s)− ω2η(s) +

(l(l + 1)

r 2T 2− 1

rT 2K 2

(T ′

T+

K ′

K

))η = 0

on (−∞,∞).

Note: s(r) =

∫ r

rK (r ′)T (r ′)dr ′


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Jost Solutions

Let us consider Im ω < 0.

We seek linearly independent solutions η1,ω, η2,ω satisfying theboundary conditions

lims→−∞

e−iωsη1,ω(s) = 1, and lims→∞

e iωsη2,ω(s) = 1.

These solutions are referred to as the Jost solutions.

We have existence, uniqueness, smoothness, and analyticity inω for Im ω < 0, and we can also extend η1,ω, η2,ω

continuously up to the real axis.

For Im ω > 0, we obtain solutions via the definitionηi ,ω := ηi ,ω.


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Jost Solutions, continued

One finds the Jost solutions by converting the ODE for η1,ω

into the integral equation

η1,ω(s) = e iωs +1

ω

∫ s

−∞sin[ω(s − s)]V (s)η1,ω(s)ds,

where V is the potential term in the η ODE.

One solves this integral equation by a perturbation series:

η1,ω(s) =∑∞

n=0 η(n)1,ω(s), where η

(0)1,ω(s) = e iωs and

η(n+1)1,ω (s) =

1

ω

∫ s

−∞sin[ω(s − s)]V (s)η

(n)1,ω(s)ds.


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Integral Representation of the Resolvent

Corresponding to ηi ,ω, there are solutions ζi ,ω of the ODE

−ζ ′′(u)− ω2r 4ζ +r 2

T 2l(l + 1)ζ = 0.

Let hω(u, v) := − 1w(ζ1,ω ,ζ2,ω)

ζ1,ω(u)ζ2,ω(v), u ≤ v

ζ1,ω(v)ζ2,ω(u), u > v .

We then have(H l − ω

)−1Γ(u) =

∫ 0−∞ kω(u, v)Γ(v)dv , where

kω(u, v) = δ(u − v)

(0 01 0

)+ r 4(v)hω(u, v)

(ω 1ω2 ω

).


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Spectral Projections

From Stone’s formula, we then get(P[a,b]+P(a,b)

)Ψ(u) = lim

ε0

1

πi

∫ b

a

∫ 0

−∞(kω+iε(u, v)−kω−iε(u, v))Ψ(v)dvdω.

We have kω(u, v) = kω(u, v), k can be continuously extendedto Im ω = 0, and we consider Ψ ∈ C∞0 (−∞, 0)2. Thus,

1

2

(P[a,b] + P(a,b)

)Ψ(u) = − 1

π

∫ b

a

∫supp Ψ

Im(kω(u, v))Ψ(v)dvdω.

This formula yields Pa = 0 for any a ∈ R, and thus

P(a,b) = − 1

π

∫ b

a

∫supp Ψ

Im(kω(u, v))Ψ(v)dvdω.


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A Representation Formula

By the spectral theorem, we then have

Ψlm(t, u) = e−itH l Ψlm0 (u)

= − 1

π

∫R

e−iωt

∫supp Ψ

(Im(kω(u, v))Ψ(v)dvdω.

We next wish to analyze the integrand, so let us note that thepair ζ1,ω, ζ1,ω form a fundamental set for the ζ ODE. Thus,we have

ζ2,ω(u) = λ(ω)ζ1,ω(u) + µ(ω)ζ1,ω(u)

where µ(ω) is never zero.


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A Representation Formula, continued

We make the definitions

γ1,ω = Re ζ1,ω, γ2,ω = Im ζ1,ω, and Γaω = (γa,ω, ωγa,ω)T ,

as well as

α11(ω) = 1 + Re

(λ(ω)

µ(ω)

), α22(ω) = 1 + Re

(λ(ω)

µ(ω)

),

and α12(ω) = α21(ω) = −Im

(λ(ω)

µ(ω)

).

Some calculation then yields

Ψlm(t, u) =1

2π

∫R

e−iωt 1

ω2

2∑a,b=1

αab(ω)Γaω(u)〈Γb

ω,Ψ0〉ldω.


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Decay for the Modal Solutions

Theorem

For fixed u ∈ (−∞, 0), the integrand in the representation formulafor Ψlm is in L1(R,C2).

This follows from the following facts for large |ω|:λ(ω) = O(1) and µ(ω) = 1 + O

(1ω

)|ζ1,ω(u)| ≤ C + O

(1ω

)〈Γbω,Ψ

lm0 〉l exhibits arbitrary polynomial decay in ω.

The Riemann-Lebesgue lemma guarantees then that for fixedu ∈ (−∞, 0), Ψlm(t, u)→ 0 as t →∞.


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Decay of the Full Solution

Let ΨL :=∑∞

l=L

∑|m|≤l ΨlmYlm. For any arbitrary compact subset

K of (−∞, 0)× S2, with smooth boundary and any ε > 0, we canfind an L ∈ N so that ‖ΨL(t)‖H2(K) < ε for all t.Idea:

There exists L0 so that‖ΨL0(t)‖2 =

∑∞l=L0

∑|m|≤l ‖Ψlm

0 ‖2 < ε for all t.

Next, the problem with data HΨ0 =∑∞

l=0

∑|m|≤l(H lΨ

lm0 )Ylm

has the solution HΨ. So there exists an L1 so that‖HΨL1(t)‖ < ε for all t.

Proceeding inductively, for each N ∈ N there is an LN so that‖HnΨLN (t)‖ < ε for each t.


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Decay of the Full Solution, continued

Looking at the energy, there is a constant C0(K ) > 0 so thatfor ΨLN = (ψLN

1 , ψLN2 )T we have

‖ψLN1 ‖H1(K) + ‖ψLN

2 ‖L2(K) < C0(K )‖ΨLN‖.

Similarly, applying this to HΨLN = (ψLN2 ,AψLN

1 )T , we findC1(K ) > 0 so that

‖AψLN1 ‖L2(K) + ‖ψLN

2 ‖H1(K) < C1(K )‖HΨLN‖.

The ellipticity of A guarantees that for smooth h andK ⊂⊂ K ⊂⊂ (−∞, 0)× S2

‖h‖Hk+2(K) < C‖Ah‖HK (K) + C‖h‖Hk+1(K).


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Thus, there exist new constants C0(K ),C1(K ) > 0, so that

‖ψLN1 ‖H2(K) + ‖ψLN

2 ‖H1(K) < C0(K )‖ΨLN‖+ C1(K )‖HΨLN‖.

We can iterate this argument to obtain constantsC0(k), . . . ,Ck(K ) > 0 so that

‖ψLN1 ‖Hk+1(K) + ‖ψLN

2 ‖Hk (K) <

k∑n=0

Cn(K )‖HnΨLN‖.

In particular, given any ε > 0 there is an L so that‖ΨL(t)‖H2(K) < ε for all t.


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The Sobolev embedding theorem then yields, for a possiblylarger L, that ‖ΨL(t)‖L∞(K) < ε for all t.

Coupling this with the decay of the fixed modes guaranteesthat for any ε > 0, we may find an L ∈ N and a t0 so that

|Ψ(t, u, θ, φ)| ≤L−1∑l=0

∑|m|≤l

|Ψlm(t, u)Ylm(θ, φ)|+|ΨL(t, u, θ, φ)| < ε

for t > t0.


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EYM Equations

Coupling the Einstein equations with the Yang-Mills equationswith gauge group G yields

Rij −1

2Rgij = σTij , d ∗ Fij = 0,

where Tij is the stress-energy tensor associated to theg -valued curvature 2-form Fij and g is the Lie algebra of G .

Letting G = SU(2), restricting to static, spherically symmetricsolutions:

ds2 = −T−2(r)dt2 + A−1(r)dr 2 + r 2dΩ2 and

F = w ′τ1dr∧dθ+w ′τ2dr∧(sin θdφ)−(1− w 2

)τ3dθ∧(sin θdφ).

Thus, the static, spherically symmetric EYM equations overSU(2) reduce to three unknowns: T ,A,w .


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EYM Equations, continued

We obtain three coupled ODE’s for the unknowns T ,A,w :

rA′ +(

1 + 2(w ′)2)

A = 1−(1− w 2

)2

r 2,

r 2Aw ′′ +

[r(1− A)−

(1− w 2

)2

r

]w ′ + w

(1− w 2

)= 0,

2rAT ′

T=

(1− w 2

)2

r 2+(

1− 2(w ′)2)

A− 1.

Smoller et al. proved the existence of black hole solutions: i.e.for any r0 > 0, there exist smooth solutions of these ODE’s on(r0,∞), A > 0 on (r0,∞), A(r0) = 0, and A,T → 1 asr →∞.


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The EYM Metric as a Generalized Schwarzschild Blackhole

We make the identification K 2 = A−1. Then, the metricds2 = −T−2(r)dt2 + K 2(r)dr 2 + r 2dΩ2 is a generalizedSchwarzschild blackhole.

This follows easily from the facts (shown by Smoller, et al.):

limr→∞

w 2(r) = 1,

limr→∞

rw ′(r) = 0,

limrr0

w 2(r) < 1,

limrr0

|w ′(r)| <∞,(r0 −

(1− w 2(r0)

)2

r0

)6= 0.


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BasicsThe Wave EquationSpectral Analysis

Outline





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Essential Properties: At the Origin

Here again we consider a metric of the form

ds2 = −T (r)2dt2 + K (r)2dr 2 + r 2dΩ2,

where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π, but this time we letr ∈ [0,∞).

We assume that T ,K ∈ C∞[0,∞) and that the metric isnonsingular: so we assume that T ,K > 0 for r ∈ [0,∞).Since we will assume that T ,K → 1 as r →∞, this impliesthat T ,K are uniformly bounded and uniformly bounded awayfrom zero.

We must also enforce the smoothness of the metric at theorigin: K (0) = 1, T ′(0) = 0 = K ′(0).


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Essential Properties: The Far-field

Interestingly, the far-field behavior we require is the same as inthe black hole case. To remind ourselves, we insist for large rthat

T (r) ∼ 1 + O(

1r

)K (r) ∼ 1 + O

(1r

)T ′(r)T (r) + K ′(r)

K(r) ∼ O(

1r2

).


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The Boundary at r = 0

In the black hole case, due to the horizon being pushed to−∞ and the compact support of the solution, we didn’tconsider a boundary condition at r = r0.

In the particle-like case, the boundary r = 0 must beconsidered. To figure out what boundary condition to impose,we change to Cartesian coordinates (t, x , y , z) where theboundary disappears. We use the theory of symmetrichyperbolic systems in these coordinates to find a smoothsolution of ζ = 0 that is compactly supported for each timet.


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The Boundary at r = 0, continued

A straight-forward application of the divergence theorem thenshows that ζr (t, 0, θ, φ) = 0. So this is the boundary conditionwe impose.

Changing coordinates, we have a solution of ζ = 0 in thecoordinates (t, r , θ, φ). This solution is not compactlysupported (since it need not vanish at the origin), but for eacht it vanishes for large r .


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The Change of Coordinates

As in the black hole case, we change variables to thecoordinate u(r) = −

∫∞r

T (α)K(α)α2 dα.

This maps the interval (0,∞) to (−∞, 0).

Some asymptotics: r →∞/u → 0 yields u = −1r + O

(1r2

)⇔

1u = −r + O(1)

r → 0/u → −∞ yields u = −1r + O(1) ⇔ 1

u = −r + O(r 2)

Letting ψ(t, u, θ, φ) = ζ(t, r(u), θ, φ) we find(−r 4∂2

t + ∂2u +

r 2

T 2∆S2

)ψ = 0

on R× (−∞, 0)× S2


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The Cauchy Problem

One can then show that ψ is the global, smooth solution ofthe Cauchy problem

(−r 4∂2

t + ∂2u + r2

T 2 ∆S2

)ψ = 0 on R× (−∞, 0)× S2

ψu = O(

1u3

)as u → −∞

(ψ, iψt)(0, rθ, φ) = Ψ0(r , θ, φ) ∈ B2

B is the set of smooth functions on (−∞, 0) that aresupported away from zero, ψu = O

(1u3

)as u → −∞, and ψ

and all its derivatives have finite limits at −∞


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Energy & Hamiltonian Reformulation

As in the black hole case, the solution to this has a conservedenergy.

Also as in the black hole case, this problem can be recast as aHamiltonian system (the form of the Hamiltonian H is thesame), and there exists a Hilbert space H on which H isessentially self-adjoint due to energy conservation. We alsoproject to spherical harmonics, as before.

The analysis to find a representation formula and show decayis similar, but there is a marked difference.


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The Resolvent

We wish to construct the resolvent; to that end we considerthe eigenvalue equation HlΓ = ωΓ. Changing to the variables(u) =

∫ u−∞ r 2(u′)du′ and letting

η(s) = r(u(s))gamma(u(s)), this is equivalent to the ODE

−η′′(s)− ω2η(s) +

(l(l + 1)

r 2T 2− 1

rT 2K 2

(T ′

T+

K ′

K

))η = 0

on (0,∞).

The form of the ODE is the same as in the black hole case,but the domain is different. We can construct a solution tothis with boundary conditions at s =∞ as before, but we needto construct a solution with boundary conditions at s = 0.Fortunately, this can be done and the rest of the analysis canalso be done (with a couple of nontrivial modifications).


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References

[1] [Smoller et al., 1993]Existence of Black Hole Solutions for the Einstein-Yang/MillsEquations. Communications in Mathematical Physics, 154:377-401.

[2] [Kronthaler, 2006]The Cauchy problem for the wave equation in theSchwarzschild geometry. Journal of Mathematical Physics,47(4):042501-+, April 2006.

[3] [Finster et al., 2006]Decay of Solutions of the Wave Equation in the KerrGeometry. Communications in Mathematical Physics,264:465-503, June 2006.


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Summary & Future Work

Summary: We have demonstrated that in a general class ofspherically symmetric spacetimes (including both black hole andparticle like cases) that scalar waves decay as t →∞.Future Work:

Obtain sharp rates of decay

Extend to less regular coefficients and data

Extend to axially symmetric case (generalize Kerr)


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Acknowledgments

Thanks to Joel Smoller for introducing this problem and hishelpful discussions.

Special acknowledgment to Johann Kronthaler, whose (2006)work served as a roadmap in solving these problems.


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Thanks for your attention!


the wave equation in spherically symmetric spacetimesmasarikm/waveeqn_sphsymm.pdf · 2011-09-12 ·...

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