Price’s law on nonstationary space–times

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Advances in Mathematics 230 (2012) 9951028www.elsevier.com/locate/aim

Prices law on nonstationary spacetimes

Jason Metcalfea, Daniel Tatarub, Mihai Tohaneanuc,a Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, United States

b Department of Mathematics, University of California, Berkeley, CA 94720, United Statesc Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States

Received 6 September 2011; accepted 2 March 2012Available online 21 April 2012

Communicated by C. Kenig

Abstract

In this article, we study the pointwise decay properties of solutions to the wave equation on a classof nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption thatuniform energy bounds and a weak form of local energy decay hold forward in time we establish a t3local uniform decay rate (Prices law, Price (1972) [54]) for linear waves. As a corollary, we also provePrices law for certain small perturbations of the Kerr metric.

This result was previously established by the second author in (Tataru [65]) on stationary backgrounds.The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutionsfor the vacuum Einstein equations, which seems to require a more robust approach to proving linear decayestimates.c 2012 Elsevier Inc. All rights reserved.

Keywords: Wave equation; Kerr; Nonstationary; Pointwise decay

1. Introduction

In this article, we consider the question of pointwise decay for solutions to the wave equationon certain asymptotically flat backgrounds. Our interest in this problem originates in generalrelativity, more precisely the wave equation on Schwarzschild and Kerr backgrounds. There theexpected local decay, heuristically derived by Price [54] in the Schwarzschild case, is t3. Thisconjecture was considered independently in two recent articles [27,65].

The work in [27] is devoted to the Schwarzschild spacetime, where separation of variablescan be used; in that context, local decay bounds are established for each of the spherical modes.

Corresponding author.E-mail addresses: [email protected] (J. Metcalfe), [email protected] (D. Tataru),

[email protected],[email protected], [email protected] (M. Tohaneanu).

0001-8708/$ - see front matter c 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2012.03.010

996 J. Metcalfe et al. / Advances in Mathematics 230 (2012) 9951028

Precisely, the sharp decay of t3 is shown for all solutions, and sharper t22k local decay isestablished for each k-th mode, k > 0.

The work in [65], on the other hand, applies to a large class of stationary asymptoticallyflat spacetimes, and asserts that if local energy decay holds then Prices law holds. Further,sharp pointwise decay rates are established in the full forward light cone; these have the formt1tr2. Local energy decay (described later in the paper) is known to hold in Schwarzschildand Kerr spacetimes.

Both of the above results involve taking the Fourier transform in time and hence rely heavilyon the stationarity assumption. The aim of this article is to prove the same result as in [65],namely that local energy decay implies Prices law, but without the stationarity assumption. Theproof below is more robust than the one in [65], and improves on the classical vector field method.As an application, in the last section of the paper we prove that local energy decay (and thusPrices law) holds for a class of nonstationary perturbations of the Kerr spacetime.

Just as in [65], this work is based on the idea that the local energy estimates contain all theimportant local information concerning the flow, and that only leaves the analysis near spatialinfinity to be understood. In the context of asymptotically flat metrics, this idea originates inearlier work [64,48] where it is proved that local energy decay implies Strichartz estimates in theasymptotically flat setting, first for the Schrodinger equation and then for the wave equation. Thesame principle was exploited in [45,67] to prove Strichartz estimates for the wave equation onthe Schwarzschild and then on the Kerr spacetime.

1.1. Notations and the regularity of the metric

We use (t = x0, x) for the coordinates in R1+3. We use Latin indices i, j = 1, 2, 3 forspatial summation and Greek indices , = 0, 1, 2, 3 for spacetime summation. In R3 we alsouse polar coordinates x = r with S2. By r we denote a smooth radial function whichagrees with r for large r and satisfies r 2. We consider a partition of R3 into the dyadic setsAR = {r R} for R 1, with the obvious change for R = 1.

To describe the regularity of the coefficients of the metric, we use the following sets of vectorfields:

T = {t , i }, = {xi j x ji }, S = tt + xx ,namely the generators of translations, rotations and scaling. We set Z = {T, , S}. Then wedefine the classes SZ (rk) of functions in R+ R3 by

a SZ (rk) |Z j a(t, x)| c j rk, j 0.By SZrad(r

k) we denote spherically symmetric functions in SZ (rk).The estimates in this article apply to solutions for an inhomogeneous problem of the form

(g + V )u = f, u(0) = u0, t u(0) = u1 (1.1)associated to dAlembertian g corresponding to a Lorentzian metric g, a potential V ,nonhomogeneous term f and compactly supported initial data u0, u1. For the metric g weconsider two cases:

Case A: g is a smooth Lorentzian metric in R+ R3, with the following properties.(i) The level sets t = const are space-like.(ii) g is asymptotically flat in the following sense:

g = m + gsr + glr ,

J. Metcalfe et al. / Advances in Mathematics 230 (2012) 9951028 997

where m stands for the Minkowski metric, glr is a stationary long range spherically symmetriccomponent, with SZrad(r

1) coefficients, of the form

glr = glr,t t (r)dt2 + glr,tr (r)dtdr + glr,rr (r)dr2 + glr,(r)r2d2and gsr is a short range component of the form

gsr = gsr,t t dt2 + 2gsr,ti dtdxi + gsr,i j dxi dx jwith SZ (r2) coefficients.

We remark that these assumptions guarantee that t is time-like near spatial infinity, but notnecessarily in a compact set. This leads us to the second case we consider.

Case B. g is a smooth Lorentzian metric in an exterior domain R R3 \ B(0, R0) whichsatisfies (i), and (ii) above in its domain, and in addition

(iii) the lateral boundary R B(0, R0) is outgoing space-like.This latter condition ensures that the corresponding wave equation is well-posed forward in

time. This assumption is satisfied in the case of the Schwarzschild and Kerr metrics (or smallperturbations thereof) in suitable advanced time coordinates. There the parameter R0 is chosenso that 0 < R0 < 2M in the case of the Schwarzschild metric, respectively r < R0 < r+ inthe case of Kerr (see [30] for the definition of r), so that the exterior of the R0 ball contains aneighborhood of the event horizon.

1.2. Normalized coordinates

Our decay results are expressed relative to the distance to the Minkowski null cone {t = |x |}.This can only be done provided that there is a null cone associated to the metric g whichis within O(1) of the Minkowski null cone. However, in general the long range componentof the metric produces a logarithmic correction to the cone. This issue can be remedied viaa change of coordinates which roughly corresponds to using ReggeWheeler coordinates inSchwarzschild/Kerr near spatial infinity. See [65]. After a further conformal transformation(which changes the potential V , see also [65]), the metric g is reduced to a normal form where

glr = g(r)r2d2, g SZrad(r1).In particular, we can replace g by an operator of the form

P = + Q (1.2)where denotes the dAlembertian in the Minkowski metric and the perturbation Q has the form

Q = g + gsr + V, gsr SZ (r2), g SZrad(r3), V SZ (r3). (1.3)We call these coordinates normal coordinates. All of the analysis in the paper is done in normalcoordinates and with g in normal form. The full perturbation Q above has only short rangeeffects.

1.3. Uniform energy bounds

The Cauchy data at time t for the evolution (1.1) is given by (u(t), t u(t)). To measure it weuse the Sobolev spaces H k , with the qualification that in Case A this means H k := H k(R3),while in Case B we use the obvious modification H k := H k(R3 \ B(0, R0)). We begin with thefollowing definition.


Definition 1.1. We say that the evolution (1.1) is forward bounded if the following estimateshold:

u(t1)H k ck(u(t0)H k + f L1([t0,t1];H k )), 0 t0 t1, k 0. (1.4)

It suffices to have this property when f = 0. Then the f term can be added in by theDuhamel formula. One case when the uniform forward bounds above are easy to establish iswhen t is a Killing vector which is everywhere time-like and V is nonnegative and stationary.Otherwise, there is no general result, but various cases have been studied on a case by casebasis.

We remark that in the case of the Schwarzschild and Kerr spacetimes t is not everywheretime-like, so the forward boundedness is not straightforward. However, it is known to hold forSchwarzschild (see [24,45]) as well as for Kerr with small angular momentum (see [29,21,66])and for a class of small stationary perturbations of Schwarzschild (see [21]).

The forward boundedness is not explicitly used in what follows, but it is defined here since itis usually seen as a prerequisite for everything that follows.

1.4. Local energy decay

A stronger property of the wave flow is local energy decay. We introduce the local energynorm LE

uLE = supR

r 12 uL2(RAR)

uLE[t0,t1] = supR

r 12 uL2([t0,t1]AR)

,(1.5)

its H1 counterpart

uLE1 = uLE + r1uLEuLE1[t0,t1] = uLE[t0,t1] + r1uLE[t0,t1],

(1.6)

as well as the dual norm

f LE =

R

r 12 f L2(RAR)

f LE[t0,t1] =

R

r 12 f L2([t0,t1]AR)

.(1.7)

These definitions are specific to (1 + 3)-dimensional problems. Some appropriate modificationsare needed in other dimensions; see for instance [48]. We also define similar norms for higherSobolev regularity

uLE1,k =||k

uLE1

uLE1,k [t0,t1] =||k

uLE1[t0,t1]

uLE0,k [t0,t1] =||k

uLE[t0,t1],


respectively

f LE,k =||k

f LE

f LE,k [t0,t1] =||k

f LE[t0,t1].

In Case A above this leads to the following.

Definition 1.2. We say that the evolution (1.1) has the local energy decay property if thefollowing estimate holds:

uLE1,k [t0,) ck(u(t0)H k + f LE,k [t0,)), t0 0, k 0 (1.8)in R R3.

The first local energy decay estimates for the wave equation were proved in the work ofMorawetz [5052]; estimates of this type are often called Morawetz estimates. By now there isan extensive literature devoted to this topic and its applications; without being exhaustive wemention [59,36,56,34,35,9,46,47,58,31].

The sharp form of the estimates as well as the notations above are from Metcalfe andTataru [48]; this paper also contains a proof of the local energy decay estimates for small (timedependent) long range perturbations of the Minkowski spacetime and further references. Seealso [49,1,46] for time dependent perturbations, as well as, e.g., [8,7,57] for time independent,nontrapping perturbations. There is a related family of local energy estimates for the Schrodingerequation. See, e.g., the original works [17,55,68] in this direction as well as [25,18] in variablegeometries. For notations and estimates most reminiscent to those used here, we refer the readerto [64,44].

In Case B an estimate such as (1.8) cannot hold due to the existence of trapped rays, i.e. nullgeodesics confined to a compact spatial region. However a weaker form of the local energydecay may still hold if the trapped null geodesic are hyperbolic. This is the case for both theSchwarzschild metric and for the Kerr metric with angular momentum |a| < M . To state suchbounds we introduce a weaker version of the local energy decay norm

uLE1w = (1 )uLE + r1uLEuLE1w[t0,t1] = (1 )uLE[t0,t1] + r1uLE[t0,t1]

for some spatial cutoff function which is smooth and compactly supported. Heuristically, is chosen so that it equals 1 in a neighborhood of the trapped set. We define as well a dual typenorm

f LEw = f L2 H1 + (1 ) f LE f LEw[t0,t1] = f L2[t0,t1]H1 + (1 ) f LE[t0,t1].

As before, we define the higher norms LE1,kw respectively LE,kw .

Definition 1.3. We say that the evolution (1.1) has the weak local energy decay property if thefollowing estimate holds:

uLE1,kw [t0,) ck(u(t0)H k + f LE,kw [t0,)), k 0, t0 0 (1.9)in either R R3 or in the exterior domain case.


Note that this implies in particular

uLE1,k [t0,) ck(u(t0)H k+1 + f LE,k+1[t0,)); (1.10)hence we can get rid of the loss at the trapped set by paying the price of one extra derivative.

Two examples where weak local energy decay is known to hold are the Schwarzschildspacetime and the Kerr spacetime with small angular momentum |a| M . In theSchwarzschild case the above form of the local energy bounds with k = 0 was obtained in [45],following earlier results in [41,36,24,20]. The number of derivatives lost in (1.10) can beimproved to any > 0 (see, for example, [5,45]), but that is not relevant for the problem athand.

In the case of the Kerr spacetime with small angular momentum |a| M the localenergy estimates were first proved in [66], in a form which is compatible with Definition 1.3.Stronger bounds near the trapped set as well as Strichartz estimates are contained in the paperof Tohaneanu [67]. For related work we also refer the reader to [67,23,2]. For a large angularmomentum |a| < M a similar estimate was proved in [22] for axisymmetric solutions.

The high frequency analysis of the dynamics near the hyperbolic trapped orbits is a veryinteresting related topic, but does not have much to do with the present article. For moreinformation we refer the reader to [16,14,53,70,10].

One disadvantage of the bound (1.10) is that it is not very stable with respect to perturbations.To compensate for that, for the present result we need to introduce an additional local energytype bound.

Definition 1.4. We say that the problem (1.1) satisfies stationary local energy decay bounds ifon any time interval [t0, t1] and k 0 we have

uLE1,k [t0,t1].k u(t0)H k + u(t1)H k + f LE,k [t0,t1] + t uLE0,k [t0,t1]. (1.11)

Unlike the weak local energy decay, here there is no loss of derivatives. Instead, the price wepay is the local energy of t u on the right hand side. Heuristically, (1.11) can be viewed as aconsequence of (1.9) whenever t is timelike near the trapped set. In the stationary case, this canbe thought of as a substitute of an elliptic estimate at zero frequency.

While one can view the stationary local energy decay as a consequence of the local energydecay, it is in effect far more robust and easier to prove than the weak local energy decay providedthat t is timelike near the trapped set. This allows one to completely sidestep all trapping relatedissues. This difference is quite apparent in the last section of the paper, where we separatelyestablish both stationary local energy decay and weak local energy decay for small perturbationsof Kerr. While the former requires merely smallness of the perturbation uniformly in time, thelatter needs a much stronger t1 decay to Kerr.

Because of the above considerations, for our first (and main) result in the theorem below weare only using as hypothesis the stationary local energy condition.

1.5. The main result

Given a multiindex we denote u = Zu. By um we denote the collection of all u with|| m. We are now ready to state the main result of the paper.Theorem 1.5. Let g be a metric which satisfies the conditions (i), (ii) in R R3,or (i), (ii), (iii) in R R3 \ B(0, R0), and V belonging to S(r3). Assume that the


evolution (1.1) satisfies the stationary local energy bounds from Definition 1.4. Suppose thatin normalized coordinates the function u solves u = f and is supported in the forward coneC = {t r R1} for some R1 > 0. Then the following estimate holds in normalized coordinatesfor large enough m:

|u(t, x)| . 1tt r2 , |u(t, x)| . 1

rt r3 (1.12)where

= umLE1 +t 52 fm

LE+r t 52 fm

LE.

If in addition the weak local energy bounds (1.9) hold then the same result is valid for all forwardsolutions to (1.1) with data (u0, u1) and f supported inside the cone C and

= u(0)Hm +t 52 fm

LE+r t 52 fm

LE.

We remark that we actually prove a slightly stronger result, with replaced by C4 inLemma 3.21.

As an application of this result, in the last part of the paper we prove Prices Law for certainsmall, time-dependent perturbations of Kerr spacetimes with small angular momentum (andV = 0).

The problem of obtaining pointwise decay rates for linear and nonlinear wave equations hashad a long history. Dispersive L1 L estimates providing t1 decay of 3 + 1 dimensionalwaves in the Minkowski setting have been known for a long time.

The need for weighted decay inside the cone arose in Johns proof [32] of the Straussconjecture in 3 + 1 dimensions. Decay bounds for + V with V = O(r3), similar to thosegiven by Prices heuristics, were obtained by Strauss and Tsutaya [60] and Szpak [62,61]. Seealso Szpak et al. [63].

A more robust way of proving pointwise estimates via L2 bounds and Sobolev inequalitieswas introduced in the work of Klainerman, who developed the so-called vector field method; seefor instance [37,38]. This idea turned out to have a myriad of applications and played a key rolein the Christodoulou and Klainerman [15] proof of the asymptotic stability of the Minkowskispacetime for the vacuum Einstein equations.

In the context of the Schwarzschild spacetime, Price was the first to heuristically computethe t3 decay rate for linear waves. More precise heuristic computations were carried out later byChing et al. [12,13]. Following the work of Wald [69], the first rigorous proof of the boundednessof the solutions to the wave equation was given in [33].

Uniform pointwise t1 decay estimates were obtained by Blue and Sterbenz [5] and alsoDafermos and Rodnianski [24]; the bounds in the latter paper are stronger in that they extend

uniformly up to the event horizon. A local t 32 decay result was obtained by Luk [43]. Theseresults are obtained using multiplier techniques, related to Klainermans vector field method; inparticular, the conformal multiplier plays a key role.

Another venue which was explored was to use the spherical symmetry in order to producean expansion into spherical modes and to study the corresponding ode. This was pursued byKronthaler [40,39], who in the latter article was able to establish the sharp Prices Law in thespherically symmetric case. A related analysis was carried out later by Donninger et al. [26] forall the spherical modes; they establish a t22k local decay for the k-th spherical mode. Later thesame authors obtain the sharp t32k in [27].


Switching to Kerr, the first decay results there were obtained by Finster et al. [28].Later Dafermos and Rodnianski [23] and Anderson and Blue [2] were able to extend theirSchwarzschild results to Kerr, obtaining almost a t1 decay. This was improved to t1 byDafermos and Rodnianski [19] and to t 32 by Luk [42].

Finally, the t3 decay result (Prices Law) was proved by the second author in [65]; the resultthere applies for a large class of stationary, asymptotically flat spacetimes. In addition, in [65]the optimal decay is obtained in the full forward light cone.

2. Vector fields and local energy decay

The primary goal of this section is to develop localized energy estimates when the vectorfields Z are applied to the solution u.

2.1. Vector fields: notations and definitions

For a triplet = (i, j, k) of multi-indices i, j and nonnegative integer k we denote || =|i | + 3| j | + 9k and

u = T i j Sku.On the family of such triplets we introduce the ordering induced by the ordering of integers.Namely, if 1 = (i1, j1, k1) and 2 = (i2, j2, k2) then we define

1 2 |i1| |i2|, | j1| | j2|, k1 k2.We use < for the case when equality does not hold. For an integer m, we denote

1 2 + m 1 2 + , || m.We also define

um = (u)||mu = (u)

and the analogues for < instead of .We now study the commutation properties of the vector fields with P . Denoting by Qsr the

class of all operators of the form

g + V, gsr SZ (r2), V SZ (r3),

we see that Q defined in (1.3) consists of R2 where R SZrad(r3) plus an element of Qsr .We now record the commutators of P with vector fields. The commutator with T yields

[P, T ] = Q, Q Qsr . (2.13)The same applies with ,

[P, ] = Q, Q Qsr . (2.14)However, in the case of S we get an extra contribution arising from the long range part of P ,

P S (S + 2)P = Q + R2, Q Qsr , R SZrad(r3). (2.15)


Further commutations preserve the Qsr class. Thus we can write the equation for j Sku in theform

P j Sku = j (S + 2)k Pu + Qu3 j+9k3 Q Qsr . (2.16)Suppose the function u solves the equation

Pu = f.Commuting with all vector fields, we obtain equations for the functions u . These can be writtenin the forms

Pu = Qu


2.3. The stationary local energy decay

Our first aim here is to include the vector fields S and in the stationary local energy decaybounds. A second aim is to derive a variation of the same bounds with different weights.

Lemma 2.7. Assume that the stationary local energy decay property (1.11) holds. Then for all0 t0 < t1 we also have

umLE1[t0,t1] .

i=0,1um(ti )L2 + fmLE + t umLE[t0,t1], (2.22)

respectively

umL2 .

i=0,1

r 12um(ti )L2+ r fmL2 + t umL2 . (2.23)

Proof. The proof of (2.22) is identical to the proof of Lemma 2.6 and is omitted. We now prove(2.23). We begin with the case m = 0, where we apply the classical method due to Morawetz.Assume first that we are in Case A. Multiplying the equation Pu = f by (xx + 1)u andintegrating by parts we obtain t1

t0

R3

Pu (xx + 1)udxdt = t1

t0

R3|u|2 + O(r1)|u|2 + O(r3)|u|2dxdt

+R3

O(r)|u|2 + O(r1)|u|2dxt=t1t=t0

.

Using the CauchySchwarz inequality on the left hand side and estimating the |u|2 terms by|u|2 terms via Hardy inequalities we are left with

LHS(2.23)(m = 0) . RHS(2.23)(m = 0)+ uL2(rR) + uL2(rR)for some fixed large R. Here the extraneous terms on the right hand side are only measured forsmall r , as the large r contribution can be absorbed on the left hand side. However, to bound themfor small r we have at our disposal the bound (2.22), whose right hand side is smaller than theright hand side of (2.23). The same outcome is reached in Case B by inserting a cutoff functionselecting the region {r 1} in the above computation.

To prove (2.23) we can use a simpler direct argument since there is no loss of derivatives onone hand and since we already have the bound (2.22) to use to estimate u


(i) Use the properties of the fundamental solution for the constant coefficient dAlembertian via Eq. (2.18). This yields improved bounds for r 1, but no improvement at allfor r 1.

(ii) Use the stationary local energy decay estimates for the operator P in the region r t . Thisallows us to obtain improved bounds for small r . The transition from L2 to pointwise boundsis done in a standard manner via Sobolev type estimates.

3.1. The cone decomposition and Sobolev embeddings

For the forward cone C = {r t} we consider a dyadic decomposition in time into setsCT = {T t 2T, r t}.

For each CT we need a further double dyadic decomposition of it with respect to either the sizeof t r or the size of r , depending on whether we are close or far from the cone,

CT =

1RT/4C RT

1U 1 we set

C RT = CT {R < r < 2R}, CUT = CT {U < t r < 2U }while for R = 1 and U = 1 we have

C R=1T = CT {0 < r < 2}, CU=1T = CT {0 < t r < 2}with the obvious change for C1T in Case B. By C

RT and C

UT we denote enlargements of these sets

in both space and time on their respective scales. We also define

C


Expressed in terms of the local energy norm, the estimate (3.24) yields the following.

Corollary 3.9. We have

wL(C


to obtain bounds for u for large r , while for small r , we shall rely on the Sobolev-type estimatesof Lemma 3.8.

3.3. An initial decay bound

Here we combine the above one dimensional reduction with the local energy bounds in orderto obtain an initial pointwise decay estimate for the functions u . This has the following form.

Lemma 3.10. The following estimate holds:

|u| . logt rrt r 12(u+nLE1 + r f+nLE). (3.30)

Here and later in this section n represents a large constant which does not depend on but mayincrease from one subsection to the next. For the above lemma we can take n = 25 for instance,but later on it becomes tedious and not particularly illuminating to keep track of the exact valueof n. We shall do so similarly for the enlargements of the cones C RT and C

UT . We shall not track,

though we note that only a finite number will ever be required, each subsequent enlargementwhich is needed and shall allow the enlargement to change from line to line while maintainingthe same notation.

Proof. We assume that r 1. For r . 1 we instead use directly the Sobolev type embedding(3.26).

For large r , we bound u by v and the function H , using (2.20), by

r2 HLE + r2SHLE . u+25LE1 + r f+25LE .Hence it remains to show that the solution v to (3.28) satisfies

|v| . logt rrt r 12(r2 HLE + r2SHLE). (3.31)

We now prove (3.31). The index plays no role in it so we drop it. One can then estimate |rv|as

|rv(t, r)| .

DtrH(s, )dsd. (3.32)

We assume that r t , as there is no further gain for smaller r in estimating the integral on theright hand side.

We partition the set Dtr into a double dyadic manner as

Dtr =Rt

DRtr , DRtr = Dtr {R < r < 2R}

and estimate the corresponding parts of the above integral. We consider two cases.(i) R < (t r)/8. Then we need to use the information about Su. For any (s, ) C , let

s,( ) be the integral curve corresponding to the vector field S, parametrized by time, satisfyings,(0) = (s, ). The fundamental theorem of calculus combined with the CauchySchwarzinequality gives

|H(s,(0))|2 1s s

0|H(s,( ))|2 d + 1s

s0|(SH)(s,( ))|2 d.


We apply this for (s, ) DRtr and integrate. In the region DRtr , we have R and|s (t r)| . R; therefore we obtain

DRtr

|H |2dsd . Rt r

BR|H |2 + |SH |2dsd

where

BR = {(s, ) : R/8 < < 8R}.Hence by the CauchySchwarz inequality we conclude that

DRtr

Hdsd . R52

t r 12(HL2(BR) + SHL2(BR))

. 1t r 12

(r2 HLE + r2SHLE).

The logarithmic factor in (3.31) arises in the dyadic R summation. We note that in the L2(BR)norm above H is viewed as a two dimensional function, whereas the LE norm applies to H as aradial function in 3+ 1 dimensions.

(ii) (t r)/8 < R < t . Then we neglect SH and simply use the CauchySchwarz inequality,DRtr

Hdds . R 32 (t r) 12 HL2(BR) . R1(t r)12 r2 HLE.

The dyadic R summation is again straightforward.

3.4. Improved L2 gradient bounds

The bounds obtained in the previous step for u apply as well to u . However, u and udo not play symmetrical roles in the expressions for F and G . In particular, the weights thatcome with u are worse than the ones that come with u . Hence, when we seek to reiterate andimprove the initial pointwise bound (3.30) it pays to have better bounds for u . This is the aimof this step in the proof. Our dyadic L2 gradient bound is contained in the next lemma.

Lemma 3.11. For 1 U, R T/4 we havewL2(C RT ) . R

1wL2(C RT ) + T1SwL2(C RT ) + RPwL2(C RT ) (3.33)

respectively

wL2(CUT ) . U1(wL2(CUT ) + SwL2(CUT ))+ T PwL2(CUT ). (3.34)

Applied to u this gives the following corollary.

Corollary 3.12. For 1 U, R T/4 we haveuL2(C RT ) . R

1u+nL2(C RT ) + R fL2(C RT ) (3.35)respectively

uL2(CUT ) . U1u+nL2(CUT ) + T fL2(CUT ). (3.36)


Applied to u we also obtain the following corollary.Corollary 3.13. For 1 U, R T/4 we have

2uL2(C RT ) . R1u+nL2(C RT ) + R fL2(C RT ) (3.37)

respectively

2uL2(CUT ) . U1u+nL2(CUT ) + T fL2(CUT ). (3.38)

Proof of Lemma 3.11. To keep the ideas clear we first prove the lemma with P replaced by .We consider a cutoff function supported in C RT which equals 1 on C

RT . Integration by parts

gives (|xw|2 |tw|2)dxdt =

w wdxdt 1

2

()w2dxdt. (3.39)

To estimate w we use the pointwise inequality|w|2 M 1

(t r)2 |Sw|2 + t

t r (|xw|2 |tw|2)

which is valid inside the cone C for a fixed large M . Hence|w|2dxdt .

1

(t r)2 |Sw|2 + t

t r ||w2 + t

t r |w||w|dxdtwhere all weights have a fixed size in the support of . The function can be further chosen sothat || . r2. Then the conclusion of the lemma follows by applying the CauchySchwarzinequality to the last term. The argument for CUT is similar, with the only difference that now wehave || . t1(t r)1.

Now consider the above proof but with replaced by P . Then, given the form of P in (1.2),and (1.3), the relation (3.39) is modified as follows:

(|xw|2 |tw|2)dxdt =

Pw wdxdt 12

((P + V ))w2dxdt

+

O(r1)|w|2dxdt.The bound for (P + V ) is similar to the bound for, and one can easily see that the last errorterm is harmless. The proof of the lemma is concluded.

3.5. Improved L2 bounds for small r

A very unsatisfactory feature of our first pointwise bound (3.30) is the r1 factor which isquite bad for small r . Here we devise a mechanism which allows us to replace this factor by t1.Our main bound is an L2 local energy bound, derived using the stationary local energy decayassumption. We have the following proposition.

Proposition 3.14. Assume that the problem (1.1) satisfies stationary local energy decaybounds (1.11). Then the following estimates hold:

umLE1(C


Proof. We first observe that we can harmlessly truncate u to C 0 wehave

|u(T, x)|2 . 1s

s0|(u)(T,x ( ))|2 d + s

T 2

s0|(Su)(T,x ( ))|2 d. (3.41)

This holds uniformly with respect to s, so we have the freedom to choose 0 < s T favourablydepending on x . Suppose we take s = T . Integrating the above estimate over |x | T for thischoice of s and applying a change of coordinates yields

CT{t=T }|um(T )|2 dx . T1

CT|um |2 + |Sum |2 dx dt.

A similar bound holds for um(2T ). Combining the last three estimates we obtainumLE1(CT ) . fm+nLE(CT ) + T

12 um+nL2(CT ) (3.42)

which brings us half of the way to the proof of (3.40).We now consider the second expression on the right hand side above, and we estimate it via

the same argument as before, but using (2.23) instead of (2.22). Indeed, (2.23) yields

umL2(CT ) . r fmL2(CT ) +r 12um(T )

L2

+r 12um(2T )

L2+ t umL2(CT ).

The last term is controlled as above by

t umL2(CT ) . T1(SumL2(CT ) + rumL2(CT )). T 12 (rSumLE1(CT ) + rumLE1(CT )).

The initial and final terms are estimated by a more careful use of (3.41). Namely, we integrate

(3.41) with s(x) = r 12 T 12 . This yieldsr|um(T )|2 dx . T 12

CTr 12 |um |2 + T 32

CTr 32 |Sum |2 dx dt

which implies thatr 12 um(T )L2. T 14

r 14umL2(CT )

+ T 12 rSumLE1(CT ).


Combining the last three bounds leads to

umL2(CT ) . r fm+nL2(CT ) + T12 rum+nLE1(CT )

+ T 14r 14um

L2(CT ).

We can discard the last term on the right hand side by absorbing it into the left hand side forr T and into the second right hand side term for r T . Thus we obtain

umL2(CT ) . r fm+nL2(CT ) + T12 rum+nLE1(CT ). (3.43)

Finally, the bound (3.40) is obtained by combining (3.42) and (3.43).

3.6. Improved pointwise bounds for u for small r

Here we use Sobolev type inequalities to make the transition from L2 local energy bounds topointwise bounds for small r . Our main estimate has the following form.

Proposition 3.15. We have

umL(C


3.7. Improved pointwise estimates for u near the coneHere we convert the improvement in the L2 bounds for um given by Lemma 3.11 near the

cone into a similar L bound. This shows that the pointwise bounds for um are better thanthose for um by a factor of t r1.Proposition 3.16. We have

UumL(CUT ) . um+nL(CUT ) + T 12 U

12 fm+nL2(CUT )

+ T 12 U 32 fm+nL2(CUT ). (3.47)

Proof. The proof is similar to the proof of Proposition 3.15. In view of the Sobolev inequality(3.25), the bound (3.47) would follow from

UumL2(CUT ) +U22umL2(CUT ) . um+nL2(CUT ) +U T fm+nL2(CUT )

+U 2T fm+nL2(CUT ).To prove the bound on um we use (3.36), which shows that

UumL2(CUT ) . um+nL2(CUT ) +U T fm+nL2(CUT ).To prove the bound on 2um we use (3.38) to obtain

U 22umL2(CUT ) . Uum+nL2(CUT ) +U2T fm+nL2(CUT ).

3.8. Gradient bounds associated to the first decay bound

Here we start with the bound (3.30) for um and improve it for small r , as well as complementit with gradient bounds.

Lemma 3.17. The following pointwise estimates hold:

|um | . C1 logt rtt r 12

, |um | . C1 logt rrt r 32(3.48)

where

C1 = um+nLE1 + supR,U

T12 R

12 U

12 fm+nL2(C R,UT ) + T

12 R32 U

32 fm+nL2(C R,UT ).

Here for brevity C R,UT stands for either CRT or C

UT , with the natural convention that R T

in CUT and U T in C RT . The proof is a direct application of Propositions 3.15 and 3.16, using(3.30) as a starting point.

3.9. The second decay bound

Lemma 3.18. The following decay estimate holds:

|um | . C2 logt rtt r , |um | . C2logt rrt r2 (3.49)


where

C2 = um+nLE1 + supR,U

T R12 U

12 fm+nL2(C R,UT ) + R

32 U

32 fm+nL2(C R,UT ).

Proof. By the Sobolev embeddings of Lemma 3.8 we have

| fm | . 1tr2t rC2.

Also by (3.48) we have

|um+6| . logt rtt r 12

C2, |um+6| . logt rrt r 32C2.

Hence for the functions Gm we obtain

|Gm | . 1r3t rC2.Computing via the one dimensional reduction, this leads to a bound for um of the form

|um | . C2 logt rrt rwhich is comparable to (3.49) near the cone, but it is weaker for r t . Then the small r boundfor um and the bound for um are obtained from Propositions 3.15 and 3.16 (we need toincrease n appropriately at this stage).

3.10. The third decay bound


|um | . C3 log3t r

tt r2 , |um | . C3log3t rrt r3 (3.50)

where

C3 = um+nLE1 + sup T 2 R12 U

12 fm+nL2(C R,UT ) + T R

32 U

32 fm+nL2(C R,UT ).

Proof. As before, the main step in the proof is to obtain a pointwise bound for um whichcoincides with (3.50) near the cone,

|um | . log3t r

rt r2 C3. (3.51)Once this is done, the full estimate (3.50) follows easily by a direct application ofPropositions 3.15 and 3.16. However, at this stage there is a new twist, namely that the onedimensional reduction no longer suffices for the proof of (3.51).

The pointwise bound for fm has the form

| fm | . 1t2r2t rC3.

Also by (3.49) we have

|um+6| . logt rtt r C3, |um+6| .logt rrt r2 C3.


We use the wave equation for um given by (2.18), and rewrite Gm in the form

Gm = fm + aum+6 + t (bu


where the main contribution comes from the cone. The above left hand side dominates trtv;therefore the proof of the lemma is complete.

3.11. The fourth (and final) decay bound

Here we reiterate one last time to remove the logarithms in (3.50) and establish the finalbound, which concludes the proof of the theorem.


|um | . C4 1tt r2 , |um | . C4

1

rt r3 (3.56)

where

C4 = um+nLE1 + supT

R,U

T 2 R12 U

12 fm+nL2(C R,UT ) + T R

32 U

32 fm+nL2(C R,UT ).

Proof. The argument is similar to the previous step, but with some extra care in order to avoid thelogarithmic losses. The main goal is again to obtain a pointwise bound for um which coincideswith (3.50) near the cone,

|um | . 1rt r2 C4 (3.57)

after which the full estimate (3.50) follows from Propositions 3.15 and 3.16.The pointwise bound for fm still has the form

| fm | . 1t2r2t rC4,

but now in addition we have dyadic summability with respect to R and U . Also by (3.50) wehave

|um+6| . log3t r

tt r2 C4, |um+6| .log3t rrt r3 C4.

We split Gm = G1m + G2m exactly as before.For G1m we use the one dimensional reduction, based on the bound

|G1m | . | fm | +log3t rr3t r3 C4,

which gives the pointwise bound (3.57) for u1m . The dyadic summability for f causes theabsence of logarithms in the bound for the contribution of f . The contribution of the secondterm is of the order of

log3t rrt r4

where we have a full extra power of t r available to control the logarithms.Finally, the contribution of G2m is controlled by Lemma 3.20.


4. Perturbations of Kerr spacetimes

We now present an application of Theorem 1.5 to general relativity. We are able to recoverPrices Law not only for Schwarzschild and Kerr spacetimes, but also for certain small, time-dependent perturbations thereof. We begin by presenting the results obtained in [45,66] for theSchwarzschild and Kerr metrics. We continue in 4.2 with a proof of stationary local energy decayestimates for perturbations of Schwarzschild; while the perturbations are required to be small,no decay to Schwarzschild is assumed. This result applies as well to small perturbations of Kerrwith small angular momentum. Finally, in 4.3 we establish weak local energy decay estimatesfor small perturbations of Kerr; here, we essentially require a t1 decay rate of the perturbedmetric to Kerr.

4.1. The Schwarzschild and Kerr metrics

The Kerr metric in BoyerLindquist coordinates is given by

ds2 = gt t dt2 + gtdtd + grr dr2 + gd2 + gd2where t R, r > 0, (, ) are the spherical coordinates on S2 and

gt t = a2 sin2

2, gt = 2a 2Mr sin

2

2, grr =

2

g = (r2 + a2)2 a21 sin2

2sin2 , g = 2

with

= r2 2Mr + a2, 2 = r2 + a2 cos2 .Here M represents the mass of the black hole, and aM its angular momentum.

A straightforward computation gives us the inverse of the metric:

gt t = (r2 + a2)2 a2 sin2

2, gt = a 2Mr

2, grr =

2,

g = a2 sin2

2 sin2 , g = 1

2.

The case a = 0 corresponds to the Schwarzschild spacetime. One can view M as a scalingparameter, and a scales in the same way as M . Thus M/a is a dimensionless parameter. We shallsubsequently assume that a is small a/M 1, so that the Kerr metric is a small perturbationof the Schwarzschild metric. One could also set M = 1 by scaling, but we prefer to keep M inour formulae. We let gS and gK denote the Schwarzschild, respectively Kerr metric, and S,Kdenote the associated dAlembertians.

In the above coordinates the Kerr metric has singularities at r = 0 on the equator = /2 andat the roots of , namely r = M

M2 a2. The singularity at r = r+ is just a coordinate

singularity, and corresponds to the event horizon. The singularity at r = r is also a coordinatesingularity; for a further discussion of its nature, which is not relevant for our results, we referthe reader to [11,30]. To remove the singularities at r = r we introduce functions r, v+ and+ so that (see [30])

dr = (r2 + a2)1dr, dv+ = dt + dr, d+ = d + a1dr.


We call v+ the advanced time coordinate. The metric then takes the EddingtonFinkelstein form

ds2 =

1 2Mr2

dv2+ + 2drdv+ 4a2 Mr sin2 dv+d+ 2a sin2 drd+

+ 2d2 + 2[(r2 + a2)2 1a2 sin2 ] sin2 d2+which is smooth and nondegenerate across the event horizon up to but not including r = 0.

In order to talk about perturbations of Kerr we need to settle on a suitable coordinate frame.The BoyerLindquist coordinates are convenient at spatial infinity but not near the event horizonwhile the EddingtonFinkelstein coordinates are convenient near the event horizon but not atspatial infinity. To combine the two we replace the (t, ) coordinates with (t, ) as follows.

As in [45,66], we define

t = v+ (r)where is a smooth function of r . In the (t, r, +, ) coordinates the metric has the form

ds2 =

1 2Mr2

dt2 + 2

1

1 2Mr

2

(r)

dtdr 4a2 Mr sin2 dtd+

+

2(r)

1 2Mr2

((r))2

dr2 2a(1+ 22 Mr(r)) sin2 drd+

+ 2d2 + 2[(r2 + a2)2 1a2 sin2 ] sin2 d2+.On the function we impose the following two conditions.

(i) (r) r for r > 2M , with equality for r > 5M/2.(ii) The surfaces t = const are space-like, i.e.

(r) > 0, 2

1 2Mr2

(r) > 0.

For convenience we also introduce

= (r)+ + (1 (r))where is a cutoff function supported near the event horizon and work in the (t, r, , )coordinates which are identical to (t, r, , ) outside of a small neighborhood of the eventhorizon.

Given r < re < r+ we consider the Kerr metric and the corresponding wave equation

Ku = f (4.1)in the cylindrical region

M = {t 0, r re} (4.2)with initial data on the space-like surface

=M {t = 0}. (4.3)The lateral boundary ofM,

+ =M {r = re}, +[t0,t1] = + {t0 t t1} (4.4)


is also space-like and can be thought of as the exit surface for all waves which cross the eventhorizon. This places us in the context of Case B in Section 1.1. The choice of re is not important;for convenience one may simply use re = M for all Kerr metrics with a/M 1 and smallperturbations thereof.

A main difficulty in proving local energy decay in Schwarzschild/Kerr spacetimes is due tothe presence of trapped rays (null geodesics). In the Schwarzschild case, this occurs on the photonsphere {r = 3M}. Consequently, the local energy bounds have a loss at r = 3M . To localize therewe use a smooth cutoff function ps(r) which is supported in a small neighborhood of 3M andwhich equals 1 near 3M . By ps(r) we denote a smooth cutoff that equals 1 on the support ofps . Then we define suitable modifications of the LE1, respectively LE norms by

uL E1S = r uLE +1 3Mr

u

LE+ r1uLE

f L ES = (1 ps) f L E + ps f H1+1 3Mr L2 .With these notations, the local energy decay estimates established in [45] have the followingform.

Theorem 4.1. Let u solve Su = f inM. ThenuL E1S + sup

v0u(t)L2 . u(0)L2 + f L ES . (4.5)

As written there is a loss of one derivative at r = 3M . This can be improved to an loss, oreven to a logarithmic loss, see [45], but that is not so relevant for our purpose here.

In the case of Kerr, the trapped rays are no longer localized on a sphere. However, if a/M 1then they are close to the sphere {r = 3M}.

We now recall the setup and results from [66] for the Kerr spacetime. Let , , and bethe Fourier variables corresponding to t, r, and , and

pK(r, , , ,,) = gt t 2 + 2gt + g2 + grr2 + g2= gt t ( 1(r, , ,,))( 2(r, , ,,))

be the principal symbol of K. Here 1 and 2 are real distinct smooth 1-homogeneous symbols.It is known that all trapped null geodesics in r > r+ satisfy

Ra(r, ,) = 0 (4.6)where

Ra(r, ,) = (r2 + a2)(r3 3Mr2 + a2r + a2 M) 2 2aM(r2 a2) a2(r M)2.

Let ra(,) be the root of (4.6) near r = 3M , which can be shown to exist and be unique forsmall a. Then define the symbols

ci (r, , ,,) = r ra(i ,), i = 1, 2and the associated spacetime norms:

u2L2ci = ci (D, x)u2L2 + u2H1


g2ci L2 = infci (x,D)g1+g2=g(g12L2 + g22H1).

The replacements of the LE and LE norms are

uL E1K = ps(Dt 2(D, x))c1(D, x)psu2L2

+ps(Dt 1(D, x))c2(D, x)psu2L2+r uLE + (1 2)uLE + r1uLE

f L EK = (1 ps) f L E + ps f c1 L2c2 L2 .Then the main result in [66] asserts the following theorem.

Theorem 4.2. Let u solve Ku = f inM. ThenuL E1K + sup

v0u(t)L2 . u(0)L2 + f L EK . (4.7)

This is the direct counterpart of (4.5), which corresponds to 1 = 2 and c1 = c2 = r 3M .Again, the loss of one derivative can be improved to an loss, or even to a logarithmic loss;see [67].

4.2. Stationary local energy decay for small perturbations of Schwarzschild

Here we consider a Lorentzian metric g inM which is a small perturbation of Schwarzschildexpressed in the (t, r, , ) coordinates. Our main result is as follows.

Theorem 4.3. Let g be a Lorentzian metric onM and u a smooth function inM.(a) Let ps be a smooth cutoff function which selects a small neighborhood of the photon sphere

{r = 3M}. If|[g (gK)]| . r||, 0 || 1 (4.8)

for a small enough , then the stationary local energy bound holds for u:

uL E1[t0,t1] . u(t0)L2 + u(t1)L2 + guL E[t0,t1] + pst uL E[t0,t1].(4.9)

(b) If in addition

|g | . r1, 1 || k + 1 (4.10)then (1.11) also holds.

Proof. (a) The argument is based on the computation in [45], which only requires the use ofvector fields. We will first prove the estimate in the Schwarzschild case, but do it in such a wayso that the transition to g is perturbative.

Let X be a differential operator

X = b(r)r + c(r)t + q(r) (4.11)for some smooth functions b, c, q : [re,) R with c constant outside a compact region andb, q satisfying

|r b| cr|||r q| cr1||.

(4.12)


Let M[t0,t1] = {t0 < t < t1, r > re}, and let dVS = r2drdtd denote the Schwarzschildinduced measure. It was shown in [45] that one can find X as above so that t1

t0QS(t)dt =

M[t0,t1]

Su Xu dVS BDRS[u]t=t1t=t0

BDRS[u]r=re

(4.13)

with t1t0

QS(t)dt & u2L E1S,w[t0,t1]

where

uL E1S,w = (r 3M)u2L2 + r2r u2L2 + r1u2L2 + (1 2)u2L2

and the boundary terms satisfy

BDRS[u]t=ti

u(ti )2L2 , i = 1, 2

BDRS[u]r=re

u2H1+[t0,t1]

.Comparing the LE1S,w norm we obtain from this computation with the LE

1 norm which we need,one sees that two improvements are necessary, one near r = 3M and another for large r .

The improvement for large r is a consequence of the fact that for large r one can viewthe Schwarzschild metric as a small perturbation of the Minkowski metric. Precisely, from theestimate [49, (2.3)] (see also [48]) we have for large R

>RuL E1[t0,t1] . uL E1S,w[t0,t1] + >RSuL E[t0,t1] + u(t0)L2 . (4.14)The similar bound for the metric g is also valid.

To gain the improvement for r close to 3M we add a Lagrangian correction term. Precisely, adirect integration by parts yields

M[t0,t1]Su 2psu dVS =

M[t0,t1]

2ps g

S uu + (S2ps)u2 dVS

+Mt

2ps g00S ut udx

t=t1

t=t0.

Since t is timelike in the support of ps , we can write the pointwise bound

|psu|2 . 2ps gS muuu + C |pst u|2

for some large constant C . Then the previous identity yields

psu2L2(M[t0,t1]) .M[t0,t1]

Su 2psu dVS

+Cpst u2L2(M[t0,t1]) +

i=1,2u(ti )L2 . (4.15)


Combining (4.13)(4.15) we obtain

u2L E1(M[t0,t1]) + u2H1+[t0,t1]

+ u(t1)2L2.

M[t0,t1]

Su X1u dVS + u(t0)2L2

+Cpst u2L2(M[t0,t1]) + >RSu2L E(M[t0,t1])

(4.16)

where

X1 = X + 2ps(r)with a fixed small constant . At this point, the desired conclusion (4.9) for the Schwarzschildmetric follows if we estimate the integral term by SuL EuL E1 .

It remains to show that a similar estimate holds with S replaced by g . This substitution iseasily made in the last term on the right hand side by performing a similar substitution in (4.14).Consider now the difference in the integral term,

D =M[t0,t1]

(S g)u X1u dVS.

To estimate this we use the bound (4.8) to write

S g = (gS gi j ) + O(r1).Then we integrate by parts in a standard manner. Using also Hardy type inequalities we obtain

|D| .

i=1,2u(ti )2L2 + u2H1+[t0,t1] + u

2L E1(M[t0,t1])

.

Hence (4.16) for g follows, and the proof of the stationary local energy bound (4.9) isconcluded.

(b) The proof follows closely that of Theorem 4.5 in [66], but for the sake of completenesswe include it here. The result will follow by induction on k. The case k = 0 is part (a) of thetheorem. We will prove the case k = 1, and the rest follows in a similar manner.

We need to estimate 2uL E[t0,t1]. We already control t uL E[t0,t1], therefore it remainsto estimate the second order spatial derivatives. We write

gu = L1t u + L2uwhere L1 is a first order operator and L2 is a purely spatial second order operator. Then we have

L2uL E[t0,t1] . guL E[t0,t1] + L1t uL E[t0,t1]which is favorable where L2 is elliptic. But L2 is elliptic wherever t is time-like. Since g isa small perturbation of gS, this happens everywhere outside a small neighborhood of r = 2M .Thus we have the elliptic estimate

out2uL E[t0,t1] . guL E[t0,t1] + t uL E[t0,t1] + uL E[t0,t1] (4.17)where out selects the region {r > 2M + }.


It remains to estimate 2ehuL2 where eh is a smooth cutoff function which selects theregion {r < 2M + 2} near the event horizon. The function v = ehu solves the equation

gv = h := eh f + [g, eh]where the second term on the right hand side is controlled in H1 via (4.17). Hence the conclusionof part (b) of the proposition would follow from the following.

Lemma 4.4. Let Meh = {re < r < 2M + 3, t 0} with a fixed small . Let g be an O()perturbation of gS in Cm+1(Meh) with sufficiently small. Then for all functions v with supportin {r < 2M + 3} we have

vHm (Meh) . v(0)Hm + gvHm (Meh). (4.18)The similar estimate holds in any interval [t0, t1].Proof. This is an estimate which is localized near the event horizon, and we will prove ittaking advantage of the red shift effect. In microlocal terms, the red shift effect is equivalentto exponential energy decay along the light rays which stay on the event horizon, and smallperturbations thereof. But for this estimate, these are all light rays of interest. All others exit thedomainMeh in a finite time.

We begin with a simplification. If is small enough then for r < 2M the r spheres areuniformly time-like; therefore we can use standard local energy estimates for the wave equationto reduce the problem to the case when re = 2M 2.

For m = 0 the above bound follows from part (a) of the proposition. For m = 1 we commuteg with the vector fields t , and r . We have

[t ,g] = O()Q2, [ ,g] = O()Q2for some second order partial differential operator Q2 with bounded coefficients. Hence applying(4.18) with m = 0 to tv and v we obtain

vH1(Meh) + tvH1(Meh) . hH1(Meh) + vH2(Meh). (4.19)We still need to bound rv. For that we compute the commutator

[g, r ] = (r grrS )2r + O()Q2 + N2 (4.20)where N2 stands for a second order operator with no 2r terms. The key observation here is that = r grrS > 0 near r = 2M . We can now write

(g 1 X)rv = r h + (O()Q2 + N2)v, 1 > 0with N2 as above and most importantly, a positive coefficient 1. We recall here that X looks liker near the event horizon. Because of this the operator

B = g 1 Xsatisfies the same estimate (4.9) asg for functions supported near the event horizon. To see thisit suffices to examine (4.16) with S replaced by g . Since X = X1 near the event horizon, itfollows that the contribution of 1 X has the right sign and can be discarded. Hence we obtain

rvH1(Meh) . hH1(Meh) + vH2(Meh) + vH1(Meh)+tvH1(Meh) + vH1(Meh) (4.21)


where the last three terms account for the effect of N2. Combining the bounds (4.19) and (4.21)we obtain

vH2(Meh) . hH1(Meh) + vH2(Meh) + vH1(Meh).If is sufficiently small then the conclusion (4.18) follows for m = 1. The argument for m > 1is similar.

4.3. Local energy decay for small perturbations of Kerr

Here we consider small perturbations of a Kerr spacetime with small angular momentum,|a| M . Our main result is as follows.

Theorem 4.5. Let g be a Lorentzian metric onM and u a function inM solving gu = f .(a) Assume that g satisfies (4.8) and decays to gK near the photon sphere,

ps |[g (gK)]| c(t), 0 || 1 (4.22)where c L1t (in particular, we can take c = t1). Then the weak local energy estimateholds:

u2H1+R

+ supt0

u(t)2L2 + u2L E1K . u(0)2L2 + f 2L EK . (4.23)

(b) Assume in addition that (4.10) holds. Then

u2H k+R

+ supt0

u(t)2H k + u2L E1,kK . u(0)2H k + f 2L E,kK , (4.24)

and thus (1.9) holds.

Proof. On any fixed compact time interval we have uniform energy estimates. Eliminating acompact time interval, we can assume without any restriction in generality that the integrabilitycondition on c(t) is strengthened to

0c(t)dt . , |c(t)| . (4.25)

where (4.8) was also used.(a) The proof of (4.23) is similar to the proof of Theorem 4.3, but it requires the use of

pseudodifferential operators. Let us start by recalling the idea behind the proof of Theorem 4.2.It is shown in [66] that there exists a pseudodifferential operator S1 of order 1 that satisfies thefollowing.

(a) S1 is a differential operator in t of order 1.(b) S1 = X + psswps , where X is defined as in (4.11) and s S1.(c) LetM[0,t0] = {0 < t < t0, r > re} and dVK = 2drdtd denote the Kerr induced measure.

Then one has t00

QK(t)dt = M[0,t0]

(Ku)(S1u)dVK BDRK[u]t=t0t=0 BDR

K[u]r=re

(4.26)


with t00

QK(t)dt & u2L E1K,w[0,t0]

u2L E1K,w

= ps(Dt 2(D, x))c1(D, x)psu2L2+ps(Dt 1(D, x))c2(D, x)psu2L2+r2r u2L2 + r1u2L2 + (1 2ps)u2L2

and the boundary terms satisfying

BDRK[u]t=ti

u(ti )L2BDRK[u]

r=re

u2H1+[0,t0]

. (4.27)Note that conditions (a) and (b) guarantee that the boundary terms are well-defined afterintegrating by parts.

The same reasoning as in Theorem 4.3 leads to the counterpart of (4.16), namely

u2L E1K(M[0,t0])

+ u2L2+[0,t0]

+ u(t1)2L2.

M[0,t0]

Ku S1u dVK + u(t0)2L2 + >RKu2L E(M[0,t0]). (4.28)

Here we seek to replace the Kerr metric by g. As discussed in Theorem 4.3, the bound (4.14)holds as well for the metric g; therefore the last term is not an issue. Hence it remains to considerthe difference

D =M[0,t0]

(K g)u S1u dVK.

We split S1 = X1 + Sps whereX1 = X ps Xps, Sps = ps Xps + psswps .

Thus X1 is a first order differential operator which is supported away from the photon sphere.Correspondingly, we split D = D1 + Dps . For D1, integration by parts using (4.8) leads to

|D1| . u(0)2L2 + u(t0)2L2 + u2H1+[0,t0] + u

2L E1K(M[0,t0])

.

Here it is essential that the outcome of the integration by parts is supported away from the photonsphere {r = 3M}, where the LE1K and LE1 norms are equivalent.

To estimate Dps we need to use the stronger bound (4.22). Then near the photon sphere wecan write

K g = 1|gK||gK|(gK g) + O(c(t))

where we also have gK g = O(c(t)) and (gK g) = O(c(t)). Thus integrating byparts we obtain

|Dps | .M[0,t0]

psc(t)(|u|2 + |u|2)dVK + c(0)u(0)2L2 + c(t0)u(t0)2L2 .


By (4.25) it follows that

|Dps | .

supt[0,t0]

u(t)2L2 + u2L E1K(M[0,t0]).

Applying the bounds for D1 and Dps , we complete the replacement of K by g in (4.28),obtaining

u2L E1K(M[0,t0])

+ u2L2+[0,t0]

+ u(t0)2L2.

M[0,t0]

gu S1u dVK + u(0)2L2

+>Rgu2L E(M[0,t0]) + supt[0,t0]u(t)2L2 . (4.29)

To conclude the proof of (4.23) we estimate the integral term in the last inequality byuL E1K(M[0,t0])guL EK and use the CauchySchwarz inequality. The last term on the righthand side is eliminated by taking the supremum in the resulting estimate over t0 [0, t1] forarbitrary t1 > 0.

(b) We prove the estimate (4.24) for k = 1; the argument for larger k is identical. We beginby applying the estimate (4.23) to t u. Commuting g with t we have

gt u = tgu + O(r1)[(1 ps)Q2u + ps Q1u] + O(c(t))ps Q2uwhere Q1 and Q2 stand for second order operators with bounded coefficients. Here we have used(4.8) for first derivatives of g away from the photon sphere, (4.22) for first derivatives of g nearthe photon sphere, and (4.10) for second order derivatives of g. We estimate the second term ingt u in LEK and the third in L1t L

2x . This gives

t uL E1K . uL E1,1K + uL E1K + guL E,1K .Away from the event horizon the vector field t is timelike; therefore, arguing as in the proof ofTheorem 4.3(b), we can use an elliptic estimate to conclude that

outuL E1,1K . uL E1,1K + uL E1K + guL E,1K .On the other hand, near the event horizon we use Lemma 4.4 to obtain

ehuH2 . outuH2 + ehguH1 + u(0)H1 .Combining the last three estimates we obtain (4.24) for k = 1.

4.4. Conclusion

We can now prove Prices law for certain perturbations of the Kerr spacetimes.

Theorem 4.6. Let g be a Lorentzian metric close to gK in the sense that it satisfies (4.8), (4.10)and (4.22). Let u solve (1.1) with smooth, compactly supported initial data and V = 0. Then(1.12) holds.

Proof. This is an obvious consequence of Theorems 1.5, 4.3 and 4.5.


Acknowledgments

The first author was partially supported by NSF grant DMS0800678. The second author waspartially supported by NSF grant DMS0354539 and by the Miller Foundation.

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Price's law on nonstationary space--timesIntroductionNotations and the regularity of the metricNormalized coordinatesUniform energy boundsLocal energy decayThe main result

Vector fields and local energy decayVector fields: notations and definitionsThe weak local energy decayThe stationary local energy decay

The pointwise decayThe cone decomposition and Sobolev embeddingsThe one dimensional reductionAn initial decay boundImproved L2 gradient boundsImproved L2 bounds for small r Improved pointwise bounds for u for small r Improved pointwise estimates for nablau near the coneGradient bounds associated to the first decay boundThe second decay boundThe third decay boundThe fourth (and final) decay bound

Perturbations of Kerr space--timesThe Schwarzschild and Kerr metricsStationary local energy decay for small perturbations of SchwarzschildLocal energy decay for small perturbations of KerrConclusion

AcknowledgmentsReferences

Documents

Price’s law on nonstationary space–times