local energy decay for maxwell fields on spherical black · pdf filefor scalar waves one can...

Spacetimes Equations Historical Remarks Pt 1: Spin Reduction Pt 2: Horizon Decomposition Pt 3: Bulk LE Pt 4: Elliptic part

Local Energy Decay for Maxwell Fields onSpherical Black Holes

Jacob Sterbenz,joint with Daniel Tataru

May 23, 2013


Spherical “Black Holes”

Metric Axioms

We consider space-times with coordinates (xa,X A) ∈ R2 × S2

such that the metric g = gabdxadxb + r2δABdxAdxB satisfies:

1 (Stationary asymptotic flatness)There exists a sphericallysymmetric time function t defined for t ⩾ 0, such that in the(t , r) coordinates one has hab = diag(−1,1) +O(r−1).Furthermore ∂thab ≡ 0, and ∂k

r hab = O(r−1−k) for k ⩾ 1.2 (Non-degenerate forward global hyperbolicity) There exists

a value r0 > 0 such that r = r0 is also space-like, and thesign of grr = ⟨dr ,dr⟩ changes only once in r ⩾ r0 while ∂r grr

never vanishes.3 (Strictly hyperbolic trapping) There is a one and only one

value rT in the region where ⟨dr ,dr⟩ > 0 such that thetime-like surface r = rT is trapped set for all null geodesicsinitially tangent to it, and the flow is normally hyperbolic.



Metric Axioms


such that the metric g = gabdxadxb + r2δABdxAdxB satisfies:1 (Stationary asymptotic flatness)There exists a spherically

symmetric time function t defined for t ⩾ 0, such that in the(t , r) coordinates one has hab = diag(−1,1) +O(r−1).Furthermore ∂thab ≡ 0, and ∂k

r hab = O(r−1−k) for k ⩾ 1.

2 (Non-degenerate forward global hyperbolicity) There existsa value r0 > 0 such that r = r0 is also space-like, and thesign of grr = ⟨dr ,dr⟩ changes only once in r ⩾ r0 while ∂r grr





Metric Axioms






never vanishes.

3 (Strictly hyperbolic trapping) There is a one and only onevalue rT in the region where ⟨dr ,dr⟩ > 0 such that thetime-like surface r = rT is trapped set for all null geodesicsinitially tangent to it, and the flow is normally hyperbolic.



Metric Axioms









Some Motivation

Relation to GRThese axioms are satisfied by spherical black holes withnon-degenerate horizons: Schwarzschild andnon-extremal Reissner-Nordström.

Stability of the AssumptionsIn general small stationary (metric) perturbations of suchspace-times lead to scalar wave equations with “good”dispersive properties:

The first and third assumptions are stable (easy; work ofHirsch-Pugh-Shub ’77).The second assumption is not stable in general (singularHJ problem). However the estimates near grr = 0 are allmultiplier based and don’t loose regularity so, they gothrough for small perturbations.For scalar waves one can combine horizon estimates withwork of Wunsch-Zworski (’11). (Horizons are modular).


Some Motivation



The first and third assumptions are stable (easy; work ofHirsch-Pugh-Shub ’77).

The second assumption is not stable in general (singularHJ problem). However the estimates near grr = 0 are allmultiplier based and don’t loose regularity so, they gothrough for small perturbations.For scalar waves one can combine horizon estimates withwork of Wunsch-Zworski (’11). (Horizons are modular).


Some Motivation



The first and third assumptions are stable (easy; work ofHirsch-Pugh-Shub ’77).The second assumption is not stable in general (singularHJ problem). However the estimates near grr = 0 are allmultiplier based and don’t loose regularity so, they gothrough for small perturbations.

For scalar waves one can combine horizon estimates withwork of Wunsch-Zworski (’11). (Horizons are modular).


Some Motivation



The first and third assumptions are stable (easy; work ofHirsch-Pugh-Shub ’77).The second assumption is not stable in general (singularHJ problem). However the estimates near grr = 0 are allmultiplier based and don’t loose regularity so, they gothrough for small perturbations.For scalar waves one can combine horizon estimates withwork of Wunsch-Zworski (’11). (Horizons are modular).


Maxwell Fields

Two Forms and Their Duals

Now orient (M,gαβ) so that (dt ,dr ,dx2,dx3) is a positivebasis of T ∗(M). There is a unique isomorphism⋆ ∶ Λp → Λ4−p such that ⟨ω,σ⟩gdVg = ω ∧ ⋆σ wheredVg =

√∣g∣dt ∧ dr ∧ dVS2 , and dVS2 is the standard volume

form on S2.

Let Fαβ be (any) antisymmetric two-tensor onM, anddefine F⋆ = ⋆F . We label its divergences as follows:

∇βF⋆αβ = Iα , ∇βFαβ = Jα .

We call Iα the magnetic source and Jα the electric sourceof Fαβ. On physical grounds one usually sets I ≡ 0, but formathematical purposes we will not do so here.


Maxwell Fields

Two Forms and Their Duals

Now orient (M,gαβ) so that (dt ,dr ,dx2,dx3) is a positivebasis of T ∗(M). There is a unique isomorphism⋆ ∶ Λp → Λ4−p such that ⟨ω,σ⟩gdVg = ω ∧ ⋆σ wheredVg =

√∣g∣dt ∧ dr ∧ dVS2 , and dVS2 is the standard volume

form on S2.Let Fαβ be (any) antisymmetric two-tensor onM, anddefine F⋆ = ⋆F . We label its divergences as follows:

∇βF⋆αβ = Iα , ∇βFαβ = Jα .

We call Iα the magnetic source and Jα the electric sourceof Fαβ. On physical grounds one usually sets I ≡ 0, but formathematical purposes we will not do so here.


The Physical EnergyThe energy momentum tensor of a Maxwell field is givenby:Qαβ[F ] = FαγF γ

β − 14gαβFγδF γδ = 1

2(FαγF γβ +F⋆

αγF⋆ γβ ) .

Its divergence is: ∇αQαβ[F ] = 12(FαβJα + F⋆

αβIα) .For all future directed T which are “sufficiently timelike”with respect to −∇t and ∂r one has a uniform bound:

−Q(T ,∇t) ≈ ∑α<β ∣Fαβ ∣2 ,Thus, the energy of F on t = const is equivalent to∥F(t) ∥L2

x.

Since there are no time-like killing fields in a nbd of grr = 0we need some integrability of F in order to concludeenergy estimates. Note that there is no red shift at the levelof F or Q! (In fact component-wise some of F is blueshifted).




2(FαγF γβ +F⋆

αγF⋆ γβ ) .


αβIα) .

For all future directed T which are “sufficiently timelike”with respect to −∇t and ∂r one has a uniform bound:


x.





2(FαγF γβ +F⋆

αγF⋆ γβ ) .


αβIα) .For all future directed T which are “sufficiently timelike”with respect to −∇t and ∂r one has a uniform bound:


x.



Local Energy Decay?

A Fake Local Energy Decay EstimateBased on the previous slide one should expect a localenergy decay estimate roughly of the form:∥ r−

12−εF ∥L2(dVg)[0,T ] ⩽ Cε( ∥F ∣t=0 ∥L2(dVg∣t=0

) +

∥ r12+εI ∥L2(dVg)[0,T ] + ∥ r

12+εJ ∥L2(dVg)[0,T ]) .

Note that this is in perfect analogy with the correspondingbound for the scalar wave equation:∥ r−

12−ε∇φ ∥L2(dVg)[0,T ] ⩽

Cε( ∥∇φ∣t=0 ∥L2(dVg∣t=0) + ∥ r

12+ε2gφ ∥L2(dVg)[0,T ]) .

Two IssuesRegularity loss due to trapping. Not a problem just weakenthe norm a little bit in a compact set.The estimate is false in general (with any regularity loss)due to finite energy bound states.


Local Energy Decay?



) +

∥ r12+εI ∥L2(dVg)[0,T ] + ∥ r

12+εJ ∥L2(dVg)[0,T ]) .


12−ε∇φ ∥L2(dVg)[0,T ] ⩽

Cε( ∥∇φ∣t=0 ∥L2(dVg∣t=0) + ∥ r

12+ε2gφ ∥L2(dVg)[0,T ]) .

Two IssuesRegularity loss due to trapping. Not a problem just weakenthe norm a little bit in a compact set.

The estimate is false in general (with any regularity loss)due to finite energy bound states.


Local Energy Decay?



) +

∥ r12+εI ∥L2(dVg)[0,T ] + ∥ r

12+εJ ∥L2(dVg)[0,T ]) .


12−ε∇φ ∥L2(dVg)[0,T ] ⩽

Cε( ∥∇φ∣t=0 ∥L2(dVg∣t=0) + ∥ r

12+ε2gφ ∥L2(dVg)[0,T ]) .

Two IssuesRegularity loss due to trapping. Not a problem just weakenthe norm a little bit in a compact set.The estimate is false in general (with any regularity loss)due to finite energy bound states.


Charges

Local ChargesLet S ⊆M be a compact space-like two surface homotopicto some sphere t = const , r = const , and define thequantities (in terms of pull-backs of F ):

QS = ∫S F ∣S , Q⋆S = − ∫S F⋆∣S .

Let S ′ be some other space-like two surface such thatthere exists a tube Σ(S ′,S) with boundary∂Σ(S ′,S) = S ′ ⊔ S. Then by Stokes theorem one has:QS′ −QS = − ∫Σ(S′,S)(⋆I)∣Σ ,

Q⋆S′ −Q⋆

S = − ∫Σ(S′,S)(⋆J)∣Σ .

In particular if Fαβ is a free Maxwell field (I = J = 0) then QSand Q⋆

S are constants equal to:Q∞ = limr→∞ ∫St,r

FAB(t , r)dVS2 ,

Q⋆∞ = limr→∞ ∫St,r

Ftr(t , r)dVS2 .

Here eA,eB are an orthonormal basis of spheres St ,r withrespect to the restriction of g.


Global Charges

Explicit FormulasThe charges are carried by two finite energy solutions tothe stationary homogeneous problem:

dF = dF⋆ = 0 , L∂t F = 0 .

They are explicitly given by:F

electric = 14πr2

√∣h∣dt ∧ dr ,

Fmagnetic = 1

4π

√∣δ∣dx2 ∧ dx3 .

Dynamic ProjectionFor a general two form F we project it dynamically onto thespan of F

electricand F

magneticas follows:

F = Q⋆(t , r)F electric +Q(t , r)F magnetic,

where Q⋆(t , r) and Q(t , r) denote the charge integralstaken over the spheres St ,r .


Global Charges

Explicit FormulasThe charges are carried by two finite energy solutions tothe stationary homogeneous problem:

dF = dF⋆ = 0 , L∂t F = 0 .They are explicitly given by:F

electric = 14πr2

√∣h∣dt ∧ dr ,

Fmagnetic = 1

4π

√∣δ∣dx2 ∧ dx3 .

Dynamic ProjectionFor a general two form F we project it dynamically onto thespan of F

electricand F

magneticas follows:

F = Q⋆(t , r)F electric +Q(t , r)F magnetic,

where Q⋆(t , r) and Q(t , r) denote the charge integralstaken over the spheres St ,r .


Main Theorem

NormsIt turns out to be very convenient to use asymptoticallyscale invariant norms (e.g. to handle “residual charges”).Thus, we define:

∥F ∥LE[0,T ] = supj ∥ r−12 F ∥L2(Rj)[r0,∞)×[0,T ] ,

∥J ∥LE⋆[0,T ] = ∑j ∥ r12 J ∥L2(Rj)[r0,∞)×[0,T ] ,

Here Rj = (t , r ,xA,xB) ∣ r ∼ 2j for j ∈ Z.

Main TheoremLet F be any two form on on a space-time (M,g) whichsatisfies the previous axioms. Define F as above. Then onehas the uniform bound:

∥ (wln)−1(F −F) ∥LE[0,T ] ≲ ∥F(0) ∥L2(dVg∣t=0)+∥wln(I,J) ∥LE∗[0,T ] ,

where w(r) = (1 + ∣ ln ∣r − rT ∣∣)/(1 + ∣ ln(r)∣).


An Application

Uniform decay estimatesA typical application of LE type estimates is to upgradethem to various types of stronger decay rates.

For example one may wish to use Klainerman-Sobolevtype inequalities, say applied to fields LI

X F in the energyspace. This amounts to controlling the commutatorsd[LX ,⋆] ≈ ⟨x⟩−2LX which leads to good LE∗ bounds.

Price Law type results (Metcalfe-Tataru-Tohaneanu in prep)Let F solve the Maxwell system with data (I,J) and F(0) on ageneral asymptotically flat space-time. Suppose F is truncatedso that all quantities are supported in the forward “cone”t > r −R (in normalized coords). Then if one assumes the LEestimate for all such fields F one has:∣F ∣ ≲ κ 1

⟨r⟩⟨t−r⟩3 , κ = Em(0) + ∥ t72 (I,J) ∥LE∗ + ∥ rt

72L∂t (I,J) ∥LE∗


An Application

Uniform decay estimatesA typical application of LE type estimates is to upgradethem to various types of stronger decay rates.For example one may wish to use Klainerman-Sobolevtype inequalities, say applied to fields LI

X F in the energyspace. This amounts to controlling the commutatorsd[LX ,⋆] ≈ ⟨x⟩−2LX which leads to good LE∗ bounds.

Price Law type results (Metcalfe-Tataru-Tohaneanu in prep)Let F solve the Maxwell system with data (I,J) and F(0) on ageneral asymptotically flat space-time. Suppose F is truncatedso that all quantities are supported in the forward “cone”t > r −R (in normalized coords). Then if one assumes the LEestimate for all such fields F one has:∣F ∣ ≲ κ 1

⟨r⟩⟨t−r⟩3 , κ = Em(0) + ∥ t72 (I,J) ∥LE∗ + ∥ rt

72L∂t (I,J) ∥LE∗


Local energy and related decay on various (non-BH) backgrounds

For the wave equation there is a very long history going back at least tothe work of Morawetz (’68).

Closely related to problems for Schrodinger equations, for exampleclassical resolvent estimates of Kato (’60s), Agmon (’75), ...

For the wave and Schrodinger equations on asym flat manifolds therehas been much recent work linking local energy decay (localsmoothing) and stronger estimates, for example: Smith-Sogge (2000),Smith-Sogge-Keel (’02), Tataru-Staffilani (’02), Metcalfe-Tataru (’12),Bony-Häfner (’10), ...

The high frequency problem for trapping geometries also well studiedIkawa (80s), Gérard-Sjöstrand (’88), Wunsch-Zworski (’11),Christianson (’07), Dyatlov (’13), ...

Related to a large body of work on resonances (and related resolventestimates): Burq, Sjöstrand, Vodev, Zworski, ...


Results for BH: scalar fields

For scalar fields the first results in this direction are due to Laba-Soffer(’99) and Soffer-Blue (’03, ’07). for Schwarzschild. This largely boilsdown to ideas of Levine (’70).

Building on work of Soffer-Blue various improvements on Schwarzschildby Blue (’04) and Blue-Sterbenz (’06; conformal energies) andDafermos-Rodnianski (’09; stronger decay along horizon). These are allbased on local energy decay type estimates.

Energy boundedness was shown for small axi-symmetric perturbationsof Schwarzschild by Dafermos-Rodnianski (’11). Stronger resultsincluding for all ∣a∣ < M announced by Dafermos-Rodnianski (’10). Forbounded frequencies this is contained in the thesis ofShlapentokh-Rothman (’13).

Local energy decay (in the form we consider it here) was first proved byMarzoula-Metcalfe-Tataru-Tohaneanu (’10) for Schwarzschild andTataru-Tohaneanu (’11) for Kerr with ∣a∣≪ M. As far as I know for all∣a∣ < M LE decay (as on previous slide) is still an open problem.

Higher decay rates related to LE decay obtained by a number ofauthors: Tataru (’11), Metcalfe-Tataru-Tohaneau (’12), Luk (’12; firstapplication to a DNLW problem!), Blue-Andersson (’09). Higher decayrates also announced by Dafermos-Rodniaski (’09).


Results for BH: higher spin fields

For the case of higher spin field equations much less is known.

For a number of equations decay rates have been obtainedbased on a “pure fourier method” (integral equations) byFinster-Smoler-Yau (’06).For the Maxwell system on Schwarzschild uniform LEdecay obtained by Blue (’08).In the direction of Kerr with small ∣a∣ very recent work ofAndersson-Blue-Nicolas (’13) for complex potentials.Results for the full Maxwell system on Kerr with small ∣a∣also recently announced by Andersson-Blue-Nicolas (’13).




For a number of equations decay rates have been obtainedbased on a “pure fourier method” (integral equations) byFinster-Smoler-Yau (’06).

For the Maxwell system on Schwarzschild uniform LEdecay obtained by Blue (’08).In the direction of Kerr with small ∣a∣ very recent work ofAndersson-Blue-Nicolas (’13) for complex potentials.Results for the full Maxwell system on Kerr with small ∣a∣also recently announced by Andersson-Blue-Nicolas (’13).




For a number of equations decay rates have been obtainedbased on a “pure fourier method” (integral equations) byFinster-Smoler-Yau (’06).For the Maxwell system on Schwarzschild uniform LEdecay obtained by Blue (’08).

In the direction of Kerr with small ∣a∣ very recent work ofAndersson-Blue-Nicolas (’13) for complex potentials.Results for the full Maxwell system on Kerr with small ∣a∣also recently announced by Andersson-Blue-Nicolas (’13).




For a number of equations decay rates have been obtainedbased on a “pure fourier method” (integral equations) byFinster-Smoler-Yau (’06).For the Maxwell system on Schwarzschild uniform LEdecay obtained by Blue (’08).In the direction of Kerr with small ∣a∣ very recent work ofAndersson-Blue-Nicolas (’13) for complex potentials.

Results for the full Maxwell system on Kerr with small ∣a∣also recently announced by Andersson-Blue-Nicolas (’13).




For a number of equations decay rates have been obtainedbased on a “pure fourier method” (integral equations) byFinster-Smoler-Yau (’06).For the Maxwell system on Schwarzschild uniform LEdecay obtained by Blue (’08).In the direction of Kerr with small ∣a∣ very recent work ofAndersson-Blue-Nicolas (’13) for complex potentials.Results for the full Maxwell system on Kerr with small ∣a∣also recently announced by Andersson-Blue-Nicolas (’13).


First order equationsWe begin by introducing the field quantities:φ = 1

2εABFAB , φ⋆ = −1

2εABF⋆

AB ,/F aA = FaA , /F⋆

aA = F⋆aA .

Here ε is the volume form of δAB.

Then a portion of the Maxwell system can be written as:

dφ = ⋆δ/d /F − r2 ⋆h I , (1)

dφ⋆ = − ⋆δ /d /F⋆ − r2 ⋆h J , (2)

⋆hdφ⋆ = /d⋆ /F − r2J . (3)

From this one sees immediately that the local charge canbe subtracted off in terms of the averages:Q(t , r) = Q∞ + ∫ ∞r ∫S2(⋆hI)r(t ,s)s2dVS2 ,Q⋆(t , r) = Q⋆

∞ + ∫ ∞r ∫S2(⋆hJ)r(t ,s)s2dVS2 .

Thus w/o loss of generality we assume F ≡ 0.


First order equationsWe begin by introducing the field quantities:φ = 1

2εABFAB , φ⋆ = −1

2εABF⋆

AB ,/F aA = FaA , /F⋆

aA = F⋆aA .

Here ε is the volume form of δAB.Then a portion of the Maxwell system can be written as:

dφ = ⋆δ/d /F − r2 ⋆h I , (1)

dφ⋆ = − ⋆δ /d /F⋆ − r2 ⋆h J , (2)

⋆hdφ⋆ = /d⋆ /F − r2J . (3)

From this one sees immediately that the local charge canbe subtracted off in terms of the averages:Q(t , r) = Q∞ + ∫ ∞r ∫S2(⋆hI)r(t ,s)s2dVS2 ,Q⋆(t , r) = Q⋆

∞ + ∫ ∞r ∫S2(⋆hJ)r(t ,s)s2dVS2 .

Thus w/o loss of generality we assume F ≡ 0.


Reduction to Spin ZeroUsing the first order system and basic Hodge theory onehas:∥ (wln)−1 /F ∥LS[0,T ] ≲∥ (wln)−1r−1(− /∆)− 1

2 (dφ,dφ⋆) ∥LS[0,T ] + ∥ (I,J) ∥LS∗[0,T ] ,

under the assumption:∫S2 φdVS2 =∫S2 φ⋆dVS2 =∫S2 IadVS2 =∫S2 JadVS2 ≡0 .

Here − /∆ = /d⋆d is the scalar Laplace-Beltrami operator onS2. Here the components of /F are taken in a normalizedbasis (∂a, r−1∂A).Thus all the dynamical information is contained in φ and φ⋆

and the game switches to estimating the RHS normsabove involving these quantities.



2 (dφ,dφ⋆) ∥LS[0,T ] + ∥ (I,J) ∥LS∗[0,T ] ,under the assumption:∫S2 φdVS2 =∫S2 φ⋆dVS2 =∫S2 IadVS2 =∫S2 JadVS2 ≡0 .

Here − /∆ = /d⋆d is the scalar Laplace-Beltrami operator onS2. Here the components of /F are taken in a normalizedbasis (∂a, r−1∂A).

Thus all the dynamical information is contained in φ and φ⋆




2 (dφ,dφ⋆) ∥LS[0,T ] + ∥ (I,J) ∥LS∗[0,T ] ,under the assumption:∫S2 φdVS2 =∫S2 φ⋆dVS2 =∫S2 IadVS2 =∫S2 JadVS2 ≡0 .

Here − /∆ = /d⋆d is the scalar Laplace-Beltrami operator onS2. Here the components of /F are taken in a normalizedbasis (∂a, r−1∂A).Thus all the dynamical information is contained in φ and φ⋆



Second order equationsTo prove estimates we’ll use second order equations.These are easiest to compute with respect to a conformalmetric.

If Ω is a non-zero a weight function set g = Ω2g. Then theMaxwell system can be written as:

dF = −Ω−2⋆I , dF⋆ = Ω−2⋆J ,

d ⋆F = Ω−2J , d ⋆F⋆ = Ω−2I .Combining we have:2hodgeF = d ⋆(Ω−2⋆I) − d(Ω−2J) ,2hodgeF⋆ = −d ⋆(Ω−2⋆J) − d(Ω−2I) ,where 2hodge = −(dd ⋆ + d ⋆d) is the Hodge Laplacian of g.If we choose Ω = r−1 then a little bit of calculation gives:20φ = −∇a(r2 ⋆h I)a − ⋆δ/dJ ,20φ⋆ = −∇a(r2 ⋆h J)a + ⋆δ/dI .Here 20 = ∇a∇a + r−2 /∆ is the “spin zero” wave operator.


Second order equationsTo prove estimates we’ll use second order equations.These are easiest to compute with respect to a conformalmetric.If Ω is a non-zero a weight function set g = Ω2g. Then theMaxwell system can be written as:

dF = −Ω−2⋆I , dF⋆ = Ω−2⋆J ,

d ⋆F = Ω−2J , d ⋆F⋆ = Ω−2I .

Combining we have:2hodgeF = d ⋆(Ω−2⋆I) − d(Ω−2J) ,2hodgeF⋆ = −d ⋆(Ω−2⋆J) − d(Ω−2I) ,where 2hodge = −(dd ⋆ + d ⋆d) is the Hodge Laplacian of g.If we choose Ω = r−1 then a little bit of calculation gives:20φ = −∇a(r2 ⋆h I)a − ⋆δ/dJ ,20φ⋆ = −∇a(r2 ⋆h J)a + ⋆δ/dI .Here 20 = ∇a∇a + r−2 /∆ is the “spin zero” wave operator.



dF = −Ω−2⋆I , dF⋆ = Ω−2⋆J ,

d ⋆F = Ω−2J , d ⋆F⋆ = Ω−2I .Combining we have:2hodgeF = d ⋆(Ω−2⋆I) − d(Ω−2J) ,2hodgeF⋆ = −d ⋆(Ω−2⋆J) − d(Ω−2I) ,where 2hodge = −(dd ⋆ + d ⋆d) is the Hodge Laplacian of g.

If we choose Ω = r−1 then a little bit of calculation gives:20φ = −∇a(r2 ⋆h I)a − ⋆δ/dJ ,20φ⋆ = −∇a(r2 ⋆h J)a + ⋆δ/dI .Here 20 = ∇a∇a + r−2 /∆ is the “spin zero” wave operator.



dF = −Ω−2⋆I , dF⋆ = Ω−2⋆J ,

d ⋆F = Ω−2J , d ⋆F⋆ = Ω−2I .Combining we have:2hodgeF = d ⋆(Ω−2⋆I) − d(Ω−2J) ,2hodgeF⋆ = −d ⋆(Ω−2⋆J) − d(Ω−2I) ,where 2hodge = −(dd ⋆ + d ⋆d) is the Hodge Laplacian of g.If we choose Ω = r−1 then a little bit of calculation gives:20φ = −∇a(r2 ⋆h I)a − ⋆δ/dJ ,20φ⋆ = −∇a(r2 ⋆h J)a + ⋆δ/dI .Here 20 = ∇a∇a + r−2 /∆ is the “spin zero” wave operator.


The Main Estimate

Let φ and H be a scalar functions on [0,T ] × [r0,∞) × S2. LetGa be a one form in the xa variables, depending also onxA ∈ S2, with ⋆hdG = K .

Suppose that all of these objects obeythe moment condition:∫S2 φdVS2 =∫S2 GbdVS2 =∫S2 HdVS2 ≡ 0 .

If φ,G,H are all supported in r ≤ CT and solve:

20φ = ∇aGa +H ,

Then one has the local energy decay type estimate estimate:∥ (wln)−1r−1(d(− /∆)−

12φ, r−1φ) ∥LS[0,T ]

≲ ∥ (− /∆)−12 r−1(dφ(0) −G(0)) ∥L2(dVg∣t=0

) + ∥ r−2φ(0) ∥L2(dVg∣t=0)

+ ∥wlnr−1(r−1G, (− /∆)−12 K , (− /∆)−

12 H) ∥LS∗[0,T ] .

Here w(r) = (1 + ∣ ln ∣r − rT ∣∣)/(1 + ∣ ln(r)∣) as usual.


The Main Estimate

Let φ and H be a scalar functions on [0,T ] × [r0,∞) × S2. LetGa be a one form in the xa variables, depending also onxA ∈ S2, with ⋆hdG = K . Suppose that all of these objects obeythe moment condition:∫S2 φdVS2 =∫S2 GbdVS2 =∫S2 HdVS2 ≡ 0 .

If φ,G,H are all supported in r ≤ CT and solve:

20φ = ∇aGa +H ,

Then one has the local energy decay type estimate estimate:∥ (wln)−1r−1(d(− /∆)−

12φ, r−1φ) ∥LS[0,T ]

≲ ∥ (− /∆)−12 r−1(dφ(0) −G(0)) ∥L2(dVg∣t=0

) + ∥ r−2φ(0) ∥L2(dVg∣t=0)

+ ∥wlnr−1(r−1G, (− /∆)−12 K , (− /∆)−

12 H) ∥LS∗[0,T ] .

Here w(r) = (1 + ∣ ln ∣r − rT ∣∣)/(1 + ∣ ln(r)∣) as usual.


Further reduction of the main theoremTo simplify matters introduce rescaled spaces:∥φ ∥LS0 = supj⩾0 2−

j2 ∥χjφ ∥L2(dV) ,

∥φ ∥LS∗0= ∑j⩾0 2

j2 ∥χjg ∥L2(dV) ,

and:E(φ[t]) = ∫[r0,∞]×S2(φ2

t + φ2r + r−2∣/dφ(t)∣2)drdVS2 ,

where ∣/dφ∣2 = gAB /∇Aφ /∇Bφ and dV = dVhdVS2 .

Then the core part of our main estimate becomes: Let20φ = G, then one has:

∥∂rφ ∥LS0[0,T ] + ∥ (wln)−1(∂tφ, r−1/dφ) ∥LS0[0,T ] ≲

E12 (φ[0]) + ∥wlnG ∥LS∗0 [0,T ] ,

where we assume ⟨∂r , ∂t⟩h = 0 at r = rT .


Further reduction of the main theoremTo simplify matters introduce rescaled spaces:∥φ ∥LS0 = supj⩾0 2−

j2 ∥χjφ ∥L2(dV) ,

∥φ ∥LS∗0= ∑j⩾0 2

j2 ∥χjg ∥L2(dV) ,

and:E(φ[t]) = ∫[r0,∞]×S2(φ2

t + φ2r + r−2∣/dφ(t)∣2)drdVS2 ,

where ∣/dφ∣2 = gAB /∇Aφ /∇Bφ and dV = dVhdVS2 .Then the core part of our main estimate becomes: Let20φ = G, then one has:

∥∂rφ ∥LS0[0,T ] + ∥ (wln)−1(∂tφ, r−1/dφ) ∥LS0[0,T ] ≲

E12 (φ[0]) + ∥wlnG ∥LS∗0 [0,T ] ,

where we assume ⟨∂r , ∂t⟩h = 0 at r = rT .


Waves close to the horizonThere are three types of waves one needs to contend withclose to grr = 0. Call value r = rM .

1 Waves which propagate parallel to the horizon. These arethe easiest to control due to the red shift (microlocally itsjust the wave equation with positive friction).

2 Pure incoming waves from r = r2 with r2 > rM . These are thenext easiest to control, and are estimated well in terms ofthe ∂t ≈ ∇r energy flux across the horizon r = rM .

3 “Turning waves” on which r is concave down. These are byfar the most complicated, and are related to trapping. Tohandle them one needs essentially microlocal ideas.

One very useful features of (regular) horizons is that whenproperly understood these three mechanism allow one to“truncate” the horizon from the problem.


Pure multiplier estimatesLet r1, r2, r3 be three parameters such thatr0 ⩽ r1 < rM < r2 < r3 < rT . Then there exists a fixed value of r3such that for every k > 1 one has the following estimates, wherethe implicit constants are uniform in r1, r2 (but may depend onr3,k ):

For parallel waves we use the estimate ofDafermos-Rodnianski:

∥χ[r1,r3](Lφ, /dφ) ∥LS0[0,T ] ≲ ∥χ[r3,rT ](r − rT )/dφ ∥LS0[0,T ]

+E12⩾r1

(φ[0]) + ∥χ[r1,rT ]20φ ∥LS∗0 [0,T ] ,

For incoming waves we use:∥χ[r1,r2]

wkLφ ∥LS0[0,T ] ≲ ε∥χ[r1,∞)(wln)−1∂tφ ∥LS0[0,T ]

+ ∥χ[r1,r2]wk(r − rM)Lφ ∥LS0[0,T ]

+ (rM − r1)12 ∥χ[r3,rT ](r − rT )/dφ ∥LS0[0,T ]

+E12⩾r1

(φ[0]) + ε−1∥χ[r1,∞)wln20φ ∥LS∗0 [0,T ] .


The “Microlocal” EstimateTo estimate the remaining (turning) waves we use:

∥χ[r2,r3]wk(Lφ, (r − rM)Lφ) ∥LS0[0,T ] + ∥χ[r2,r3]

/dφ ∥LS0[0,T ]

+ ∥χ[r3,∞)∂rφ ∥LS0[0,T ] + ∥χ[r3,∞)(wln)−1(∂tφ, r−1/dφ) ∥LS0[0,T ]

≲ ∣ ln(r2 − rM)∣k∥χ[rM ,r2]wk(Lφ, (r − rM)Lφ) ∥LS0[0,T ]

+Cr2(∥χ[rM ,∞)wln20φ ∥LS∗0 [0,T ] +E

12⩾rM

(φ[0])) .


Normalized CoordinatesIn order to prove the last estimate we introduce somenormalized coordinates in r > rM . Note that to prove thelast estimate above we really only need them in r > rM + ε,so we are free to let them degenerate at r = rM .

Then there exists two functions s = t + b(r) and r∗ = r∗(r)defined in the region r > rM , such that r∗(rT ) = 0, and suchthat (s, r∗) are smooth coordinates in r > rM with:r∗ → −∞ as r → rM , r∗ → ∞ as r → ∞ .

These have the following asymptotics as r → rM :∂r s = −∣det(gab)∣−

12 (grr)−1 + s(r) ,

∂r r∗ = ∣det(gab)∣−12 (grr)−1 ,

where s is uniformly bounded with all of its derivatives onr > rM . They are also “asymptotically flat” as r →∞.With respect to (s, r∗) the (1 + 1) dimensional Lorentzianmetric h can be written as:h = Ω2(−ds2 + dr2

∗) , Ω2 = −g00 = ∣det(gab)∣grr .


Normalized CoordinatesIn order to prove the last estimate we introduce somenormalized coordinates in r > rM . Note that to prove thelast estimate above we really only need them in r > rM + ε,so we are free to let them degenerate at r = rM .Then there exists two functions s = t + b(r) and r∗ = r∗(r)defined in the region r > rM , such that r∗(rT ) = 0, and suchthat (s, r∗) are smooth coordinates in r > rM with:r∗ → −∞ as r → rM , r∗ → ∞ as r → ∞ .


12 (grr)−1 + s(r) ,

∂r r∗ = ∣det(gab)∣−12 (grr)−1 ,


∗) , Ω2 = −g00 = ∣det(gab)∣grr .




12 (grr)−1 + s(r) ,

∂r r∗ = ∣det(gab)∣−12 (grr)−1 ,

where s is uniformly bounded with all of its derivatives onr > rM . They are also “asymptotically flat” as r →∞.

With respect to (s, r∗) the (1 + 1) dimensional Lorentzianmetric h can be written as:h = Ω2(−ds2 + dr2

∗) , Ω2 = −g00 = ∣det(gab)∣grr .




12 (grr)−1 + s(r) ,

∂r r∗ = ∣det(gab)∣−12 (grr)−1 ,


∗) , Ω2 = −g00 = ∣det(gab)∣grr .


The conformal equation and estimatesWe can now prove estimates in terms of a rescaled versionof 20:20

RW = −∂2s + ∂2

r∗ +V(r∗) /∆ = −⟨∂t , ∂t⟩g20 ,

Where V = −r−2⟨∂t , ∂t⟩g .

The normal hyperbolicity assumption is simply that V havea single non-degenerate critical point. Thus, one can prove(rough) estimates in the standard way.For applications we need sharper bounds. This iscontained in work of Marzoula-Metcalfe-Tataru-Tohaneanu:Let φ be a space-time function which is supported in theregion ∣r∗∣ ⩽ C for some fixed C > 0. Then one has thebound:∥∂r∗φ ∥L2[0,T ] + ∥ (wln)−1(∂sφ, /dφ) ∥L2[0,T ] ≲E

12 (φ[0]) + ∥wln2

0RWφ ∥L2[0,T ] .



RW = −∂2s + ∂2

r∗ +V(r∗) /∆ = −⟨∂t , ∂t⟩g20 ,

Where V = −r−2⟨∂t , ∂t⟩g .The normal hyperbolicity assumption is simply that V havea single non-degenerate critical point. Thus, one can prove(rough) estimates in the standard way.

For applications we need sharper bounds. This iscontained in work of Marzoula-Metcalfe-Tataru-Tohaneanu:Let φ be a space-time function which is supported in theregion ∣r∗∣ ⩽ C for some fixed C > 0. Then one has thebound:∥∂r∗φ ∥L2[0,T ] + ∥ (wln)−1(∂sφ, /dφ) ∥L2[0,T ] ≲E

12 (φ[0]) + ∥wln2

0RWφ ∥L2[0,T ] .



RW = −∂2s + ∂2

r∗ +V(r∗) /∆ = −⟨∂t , ∂t⟩g20 ,

Where V = −r−2⟨∂t , ∂t⟩g .The normal hyperbolicity assumption is simply that V havea single non-degenerate critical point. Thus, one can prove(rough) estimates in the standard way.For applications we need sharper bounds. This iscontained in work of Marzoula-Metcalfe-Tataru-Tohaneanu:Let φ be a space-time function which is supported in theregion ∣r∗∣ ⩽ C for some fixed C > 0. Then one has thebound:∥∂r∗φ ∥L2[0,T ] + ∥ (wln)−1(∂sφ, /dφ) ∥L2[0,T ] ≲E

12 (φ[0]) + ∥wln2

0RWφ ∥L2[0,T ] .


“Elliptic Renormalization” of residual chargesWe still need to go back and deal with the equation:

20φ = ∇aGa +H .

The main issue here is that division by (− /∆)− 12 is not

compatible with the (t , r) derivatives on the RHS.

However, there is a simple “gauge structure” to thisequation which is:φ→ ψ ∶= φ − χ Ô⇒ 20ψ = ∇a(G − dχ)a +H − r−2 /∆χ .We can now try to chose χ to as to minimize G − dχ. Thisturns out to be an elliptic problem reminiscent of theCoulomb gauge.The estimates are provided via rescaling and standardelliptic bounds. The main thing here is that we need w(r)to be an A2 weight. For this we would just needw ≈ ∣r − rT ∣−

12+ε so there is plenty of room.



20φ = ∇aGa +H .


compatible with the (t , r) derivatives on the RHS.However, there is a simple “gauge structure” to thisequation which is:φ→ ψ ∶= φ − χ Ô⇒ 20ψ = ∇a(G − dχ)a +H − r−2 /∆χ .

We can now try to chose χ to as to minimize G − dχ. Thisturns out to be an elliptic problem reminiscent of theCoulomb gauge.The estimates are provided via rescaling and standardelliptic bounds. The main thing here is that we need w(r)to be an A2 weight. For this we would just needw ≈ ∣r − rT ∣−




20φ = ∇aGa +H .


compatible with the (t , r) derivatives on the RHS.However, there is a simple “gauge structure” to thisequation which is:φ→ ψ ∶= φ − χ Ô⇒ 20ψ = ∇a(G − dχ)a +H − r−2 /∆χ .We can now try to chose χ to as to minimize G − dχ. Thisturns out to be an elliptic problem reminiscent of theCoulomb gauge.

The estimates are provided via rescaling and standardelliptic bounds. The main thing here is that we need w(r)to be an A2 weight. For this we would just needw ≈ ∣r − rT ∣−




20φ = ∇aGa +H .


compatible with the (t , r) derivatives on the RHS.However, there is a simple “gauge structure” to thisequation which is:φ→ ψ ∶= φ − χ Ô⇒ 20ψ = ∇a(G − dχ)a +H − r−2 /∆χ .We can now try to chose χ to as to minimize G − dχ. Thisturns out to be an elliptic problem reminiscent of theCoulomb gauge.The estimates are provided via rescaling and standardelliptic bounds. The main thing here is that we need w(r)to be an A2 weight. For this we would just needw ≈ ∣r − rT ∣−


local energy decay for maxwell fields on spherical black · pdf filefor scalar waves one can...

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