Hidden symmetries and decay for the wave equation on the Kerr … · 2018-05-07 · symmetry operators. In particular, our method makes use of the second-order ... In this paper we

Hidden symmetries and decay for the wave

equation on the Kerr spacetime

L. Andersson and P. Blue

REPORT No. 44, 2008/2009, fall

ISSN 1103-467XISRN IML-R- -44-08/09- -SE+fall

HIDDEN SYMMETRIES AND DECAY FOR THE WAVE

EQUATION ON THE KERR SPACETIME

LARS ANDERSSON† AND PIETER BLUE‡

Abstract. Energy and decay estimates for the wave equation on the exterior

region of slowly rotating Kerr spacetimes are proved. The method used is ageneralization of the vector-field method, which allows the use of higher-order

symmetry operators. In particular, our method makes use of the second-order

Carter operator, which is a hidden symmetry in the sense that it does notcorrespond to a Killing symmetry of the spacetime.

The main result gives, in stationary regions, an almost inverse linear decay

rate and the corresponding decay rate at the event horizon and null infinity.Except for the small loss in the decay rate, this generalizes the known decay

results on the exterior region of the Schwarzschild spacetime.

Contents

1. Introduction 12. Notation and preliminaries 123. The bounded-energy argument 174. Decay estimates for the local energy 365. Pointwise decay estimates 54Appendix A. Relevance and global structure of Kerr 64References 67

1. Introduction

In this paper we prove boundedness and decay for solutions of the covariant waveequation

∇α∇αψ = 0

in the exterior region of the Kerr spacetime. In Boyer-Lindquist coordinates, theexterior region is given by (t, r, θ, φ) ∈ R× (r+,∞)× S2 with the Lorentz metric

gµνdxµdxν =−(

1− 2Mr

Σ

)dt2 − 4Mra sin2 θ

Σdtdφ

+Π sin2 θ

Σdφ2 +

Σ

∆dr2 + Σdθ2, (1.1)

where r+ = M +√M2 − a2 and

∆ = r2 − 2Mr + a2, Σ = r2 + a2 cos2 θ, Π = (r2 + a2)2 − a2 sin2 θ∆.

For 0 ≤ |a| ≤M , the Kerr metric describes a rotating black hole, with mass M andangular momentum Ma, and with horizon located at r = r+. The Schwarzschildspacetime is the subcase with a = 0. The exterior region is globally hyperbolic,with the surfaces of constant t, Σt, as Cauchy surfaces. Thus, the wave equation iswell posed in the exterior region, even though the Kerr spacetime can be extended.We consider initial data on the hypersurface Σ0.

Date: 17 August 2009 .

1

2 L. ANDERSSON AND P. BLUE

The Kerr black-hole spacetime is expected to be the unique, stationary, asymp-totically flat spacetime containing a nondegenerate Killing horizon [2]. Further,motivated by considerations including the weak cosmic censorship conjecture, theKerr black hole is expected to be the asymptotic limit of the evolution of asymptoti-cally flat, vacuum data in general relativity. An important step towards establishingthe validity of this scenario is to prove the stability of the Kerr solution, i.e. to showthat vacuum spacetimes evolving from data which represent a small perturbationof Kerr initial data asymptotically approach a Kerr solution. Decay for the scalarwave equation on the Kerr background is an important model problem for stability,and due to the importance of this application, the wave equation on black holebackgrounds has been actively studied in the last decade.

An essential tool in the analysis of both linear and nonlinear Lagrangian fieldequations is the use of Noetherian currents associated to Killing or conformal sym-metries of the background spacetime. In the relativistic setting, we interpret thesecurrents as momenta. A method, which may be referred to as the vector-fieldmethod, based on the systematic use of such currents, has been developed andhas played an essential role in the proof of the nonlinear stability of Minkowskispace [12], which built on earlier vector-field based estimates for the decay ratesof solutions to linear and nonlinear wave equations [28] and to Maxwell’s equa-tions and the spin-2 field equations [11]. Generalizations of these ideas have playeda central role in recent work concerning the wave and Maxwell equations on theSchwarzschild spacetime, and the wave equation on the Kerr spacetime. For thesenon-flat background spacetimes, the lack of symmetries presents an important newproblem.

The 10 dimensional group of isometries of the Minkowski space is broken to a 4dimensional group for the Schwarzschild spacetime, generated by ∂t and the spatialrotations. Further, the Schwarzschild spacetime contains orbiting null geodesics,located at the photon spere, the hypersurface with r = 3M . As high frequencywaves can track null geodesics for long times, an analysis of this feature is anessential step in a proof of decay for the wave equation.

In the general Kerr spacetime, with a 6= 0, which we consider in this paper, thereare only two Killing fields, ∂t and ∂φ. In addition, in studying the wave equation onthe Kerr spacetime, one encounters several new phenomena which are not presentin the Schwarzschild case. There is an ergo-region outside the horizon, where thestationary Killing field ∂t fails to be timelike. Thus, the Kerr spacetime admits nopositive definite, conserved energy for the wave equation. Further, the orbiting nullgeodesics in Kerr fill an open region in spacetime. The lack of symmetries of theKerr spacetime is compensated for by the presence of a fundamentally new feature,which we make essential use of in this paper, a hidden symmetry.

By a hidden symmetry we mean an operator which commutes with the wave op-erator, not associated to a Killing vector field, but rather to a second-rank Killingtensor. For the Kerr spacetime, the Killing tensor and the related conserved quan-tity found by Carter [8] provides, via the associated second-order Carter operator,a hidden symmetry. The existence of the two Killing vectors and the Killing ten-sor imply the separability of many important equations on the Kerr spacetime,including the wave equation.

One of the fundamentally new ideas introduced in this work is a generalizationof the vector-field method which allows the use of not only Killing symmetries butalso hidden symmetries in the construction of suitable Noetherian currents for theanalysis of Lagrangian field equations. This allows us, in contrast to other, recentwork on the wave equation on Kerr, to carry out our proof of uniform boundednessand decay results exclusively in physical space, using only the coordinate functions

HIDDEN SYMMETRIES AND DECAY FOR THE WAVE EQUATION ON KERR 3

and differential operators. This technique almost eliminates the need for methodsinvolving separation of variables or Fourier analysis.1

To state our main results, we use the tortoise coordinate r∗, defined by

dr

dr∗= (r2 − 2Mr + a2)(r2 + a2), r∗(3M) = 0,

and the almost null coordinates u± given by

u+ = t+ r∗, u− = t− r∗.Our main results are:

Theorem 1.1 (Uniformly bounded, positive energy). There are positive constantsC1 and a1, and a nonnegative quadratic form on each hypersurface of constant t,ETχ [ψ](t), such that, if |a| < a1 and ψ : R× (r+,∞)× S2 → R is a solution of thewave equation, ∇α∇αψ = 0, then ∀t

ETχ [ψ](t) ≤ C1ETχ [ψ](0).

Theorem 1.2 (Decay estimates). There are positive constants C2, C ′2, and a2, andthere is a nonnegative quadratic form on each hypersurface of constant t, ‖ψ‖2(t),such that, if |a| < a2 and ψ : R×(r+,∞)×S2 → R is a solution of the wave equation,∇α∇αψ = 0, then there are the following decay estimates ∀t > 0, (θ, φ) ∈ S2:

(1) Decay in stationary regions: ∀r ∈ (3M, 4M):

|ψ(t, r, θ, φ)| ≤C2 max1, t−1+C′2|a|‖ψ‖(0).

(2) Near decay: ∀r < 3M :

|ψ(t, r, θ, φ)| ≤C2 max1, u+−1+C′2|a|‖ψ‖(0).

(3) Far decay: ∀r with r > 4M and r < t:

|ψ(t, r, θ, φ)| ≤C2r−1 max1, u−C

′2|a|(

u+ − u−u+ max1, u−

)1/2

‖ψ‖(0).

In particular, for t/2 < r < t:

|ψ(t, r, θ, φ)| ≤C2r−1 max1, u−−1/2+C′2|a|‖ψ‖(0).

(4) Decay near spatial infinity: ∀r with r > t:

|ψ(t, r, θ, φ)| ≤C2r−1 max1,−u−−1/2‖ψ‖(0).

Theorem 1.1 is the conclusion of section 3 and is given in theorem 3.13. Theorem1.2 follows from the conclusions of theorems 5.1, 5.2, and 5.4. Except for the lossin the exponent of C ′2|a|, the estimates stated in theorem 1.2 are the same as theresults proven using vector-field techniques in the Schwarzschild spacetime.2 Thedecay along null infinity is the same (modulo the loss in the exponent) as can beobtained in Minkowski space from initial data on t = 0 which decays like r−3/2.This is roughly the decay rate we require for the initial data in Kerr. Note that theKerr spacetime has a discrete, time reversal symmetry (t, φ) 7→ (−t,−φ), so thatthe results of theorems 1.1 and 1.2 can also be reversed to t < 0.

The norm ‖ψ‖(0) in theorem 1.2 is bounded if ψ is smooth on the hypersurfaceΣ0 and if ψ and its first nine derivatives (with respect to the Boyer-Lindquist

1We do not use separability or the Fourier transform (which are essentially equivalent to eachother) in the t coordinate. In the proof of lemma 3.12, we need to use separability in the φ

coordinate to control the axially symmetric component of the solution, but are otherwise able to

avoid using separability (or the Fourier transform) in φ. We never use separability in the r andθ coordinates, since this is only possible once separation in the t and φ coordinates has already

been performed.2Recently, better decay estimates have been proven in the Schwarzschild case, giving, for any

positive δ, decay rates of t−3/2+δ and u−3/2+δ+ in stationary and near regions respectively [32].


coordinates) decay like r−3/2+δ for some positive δ as r → ∞. By “smooth” wemean C∞ with respect to local coordinates. As r → r+, this is not the same as beingsmooth with respect to the Boyer-Lindquist coordinates, since they degenerate here.The quadratic form ETχ [ψ](0) in theorem 1.1 is bounded when ‖ψ‖2(0) in theorem1.2 is bounded.

We briefly comment on other related work. Estimates for the decay rate of so-lutions to the wave equation have been proven in the subcase of the Schwarzschildspacetime, where a = 0. Birkhoff’s theorem states that the Schwarzschild space-time is the unique spherically symmetric, vacuum spacetime solution of Einstein’sequation. For the coupled Einstein and scalar wave system, a decay rate and non-linear stability of the Schwarzschild solution have been proven in the sphericallysymmetric setting [13].

As mentioned earlier, for the wave equation without a symmetry assumptionbut on a fixed background spacetime, the case of the linear wave equation on theSchwarzschild spacetime is significantly simpler than the corresponding case in theKerr spacetime, since the ∂t Killing vector is timelike in the entire exterior regionand generates a conserved, positive energy, there is the full SO(3) group of rotationsymmetries available to generate higher energies, and the orbiting null geodesicsare restricted to r = 3M . The first two of these properties imply that solutionsremain bounded. Following the introduction of a Morawetz vector field and of theequivalent of an almost conformal vector field to the Schwarzschild spacetime [30],decay estimates for the wave equation were proven [7], proven with a weaker decayrate but less regularity loss [6], and proven separately with a stronger estimate nearthe event horizon [14]. These were extended to Strichartz estimates for the waveequation [33] and decay estimates for Maxwell’s equation [4]. The Morawetz vectorfield which made these estimates possible was centred about the orbiting geodesicsat r = 3M . This construction of a classical vector field fails when a 6= 0, since thereare orbiting geodesics filling an open set in spacetime.

Recently, new Morawetz vector fields with coefficients that depend on both space-time position and on Fourier operators have been introduced to construct a uni-formly bounded, positive energy and to show that solutions to the Kerr wave equa-tion remain uniformly bounded [17, 39]. These might reasonably be called Fourier-analytic, pseudodifferential, microlocal, or phase-space techniques, since the Fourieroperators represent coordinates in momentum space in contrast to spacetime co-ordinates in physical or configuration space. These results include a form of weakdecay, since the Morawetz estimate implies integrability of the local energy. In [16],it has been announced that these Fourier-analytic techniques can be extended todecay results very similar to ours in theorem 1.2. Like our work, these require that|a| is very small relative to M . We compare this work with our own in more detailin section 1.3. Fourier-analytic vector fields were used previously to prove Mourreestimates, which are similar to Morawetz estimates, in the proof of scattering forthe Klein-Gordon equation [25] and the Dirac equation [26].

Finally, we recall that decay for the wave equation has previously been obtained[21] from an explicit representation of solutions [20] using the complete separabilityof the Kerr wave equation. These decay results are of the form limt→∞ |ψ(t, r, θ, φ)| →0, where ψ(t, r, θ, φ) = ψLz (t, r, θ)e

iLzφ or where ψ is made up of a finite numberof azimuthal modes of this form. Decay rates have been obtained from this sepa-rability method for solutions to the Dirac equation [19] and spherically symmetricsolutions to the wave equation when a = 0 [29]. The decay without rate results forthe wave equation built on the earlier result that there are no exponentially growingmodes [43] and applies for all |a| ∈ [0,M ].

1.1. Hidden symmetries and the vector-field method. In the classical vector-field method, one seeks to control the solution of field equations by making use of


energy fluxes and deformation terms calculated for suitably chosen vector fields. Thevector fields used are often (approximate) conformal symmetries of the backgroundspacetime. Higher-order estimates are achieved by Lie differentiating the field alongfurther vector fields. An important advantage of the vector-field method is that oneworks entirely in terms of quantities in physical space.

A particularly clear example of the vector-field method was the use of the Lorentzgroup in the Minkowski spacetime, R1+3, to construct norms which could be usedin the Klainerman-Sobolev inequality and to use this to prove the well-posedness ofnonlinear wave equations [28]. The term “vector-field method” seems to have beenintroduced relatively recently, especially to describe generalisations of the work inR1+3 to situations where one lacks the full Lorentz group of symmetries. Applyingthis terminology retroactively, we would now describe the early uses of the radialMorawetz vector-field [35] and the conformal vector-field [24] as applications of thevector-field method. This may have previously been refered to as the method ofmultipliers or the Euler-Lagrange method, although these terms can also be appliedto more general techniques.

A central result in mathematical relativity, and perhaps the most important ap-plication of the vector-field method, was the proof of the nonlinear stability of theMinkowski spacetime [12]. There had also been earlier work on the stability of theMinkowski spacetime, but this required hyperboloidal initial data [22]. The mon-umental proof of nonlinear stability built upon previous vector-field estimates forlinear and nonlinear wave equations [28], and for the Maxwell and spin-2 field equa-tions [11], which are better models for Einstein’s equations. This partly motivatesour work on the linear wave equation in the Kerr spacetime using generalisationsof the vector-field method. Since the original proof of nonlinear stability for theMinkowski spacetime, a simpler proof has been developed, but this also makes useof the vector-field method [31].

As mentioned above, in the Kerr spacetime, the lack of symmetries, as well asthe complicated nature of the orbiting null geodesics, makes it impossible to derivethe required estimates using only classical vector fields. In this section, we outlinea generalization of the vector-field method which allows us to take advantage of thepresence of hidden symmetries in the Kerr spacetime. In particular, we considerenergies based on operators of order greater than one, rather than just vector fields.

Let g = ∇α∇α. In the discussion here, we consider the scalar wave equationgψ = 0, but much of the discussion applies equally to general field equationsderived from a quadratic action. We define a symmetry operator to be a differentialoperator S such that if gψ = 0, then also gSψ = 0. The set of symmetryoperators is closed under scalar multiplication, addition, and composition, and eachsymmetry operator has a well-defined order as a differential operator. Thus, the setof symmetry operators forms a graded algebra. Given a set of generators of the setof symmetries, we can consider the subset consisting of generators of order n. Wedenote this subset of the generators of the symmetry operators by Sn and denotethe elements of Sn with an underlined index, e.g. Sa ∈ Sn.

If X is a conformal Killing field, then the operator LX generated by Lie differenti-ation with respect to X is clearly a symmetry operator. We take a hidden symmetryto be a symmetry operator which is not in the algebra generated by the Killing vec-tor fields. Since the Minkowski spacetime saturates the Delong-Takeuchi-Thompsoninequality, there are no hidden symmetries [10]. In the Schwarzschild spacetime,there are no hidden symmetries [9]. In the Kerr spacetime, it is well-known thatthere is Carter’s Killing 2 tensor and that this generates a hidden symmetry [8, 42].

The energy-momentum tensor for the wave equation is

T [ψ]αβ = ∇αψ∇βψ −1

2gαβ (∇γψ∇γψ) . (1.2)


The momentum associated with a vector field X and the energy associated with avector field X and evaluated on a hypersurface Σ are

PX[ψ]α =T [ψ]αβXβ ,

EX[ψ](Σ) =

∫

Σ

PX[ψ]αdηα,

where dη is integration with respect to the surface volume induced by g on Σ. Inthe following, unless there is room for confusion, we will drop reference to ψ in thenotation for momentum and energy. When the spacetime is foliated by surfaces ofconstant time, we will denote these surfaces by Σt and typically denote the energyon such a surface by EX(t) = EX(Σt).

The energy momentum tensor (1.2) satisfies the dominant energy condition, andhence for X timelike, the energy induced on a hypersurface with a timelike normal(i.e. a spacelike hypersurface) is positive definite. The energy conservation lawtakes the form

EX(Σ2)− EX(Σ1) =

∫

Ω

(∇αPαX)√−|g|d4x,

where Ω is the region enclosed between Σ1 and Σ2. This is often referred to as thedeformation formula. Energy estimates are often performed by controlling the bulk(also called deformation) terms ∇αPαX. However, for the Morawetz estimate (e.g.inequality (3.6)), one makes use of the sign of the bulk term itself.

By estimating higher-order energies one may, via Sobolev estimates, get pointwisecontrol of the fields. Higher-order energies may be defined by using symmetries. Iffor 0 ≤ i ≤ n, there is a collection of order-n differential operators, Si, then we candefine the higher-order energy (of order n) for a vector field X to be

EX,n[ψ](Σ) =n∑

i=0

∑

S∈SiEX[Sψ](Σ).

Since the energy momentum tensor is quadratic in ψ, we can define a bilinearform of the energy momentum by

T [ψ1, ψ1]αβ =1

4(T [ψ1 + ψ2]αβ − T [ψ1 − ψ2]αβ) .

It is convenient to define an index version of the bilinear energy momentum, withrespect to a set of symmetry operators Sa by

T [ψ]abαβ =T [Saψ, Sbψ]αβ .

Given a double-indexed collection of vector fields, Xab, we define the associatedgeneralized momentum and energy to be

PXab [ψ]α =T [ψ]abαβXabβ ,

EXab [ψ](Σ) =

∫

Σ

PXab [ψ]αdηα.

In practice it is convenient to consider momenta with lower-order terms, designedto improve certain deformation terms in ∇αPαX. For a scalar function, q ([33], butpreviously appearing in [15]), or a double-indexed collection of functions, qab, theassociated momenta are defined to be

Pq[ψ]α =q(∇αψ)ψ − 1

2(∂αq)ψ

2,

Pqab [ψ]α =qab(∇αSaψ)Sbψ −1

2(∂αq

ab)(Saψ)(Sbψ).

For a pair consisting of a vector field and a scalar function, (X, q), the associatedmomentum is defined to be the sum of the momenta associated with the vector fieldand the scalar. For a pair of collections, (Xab, qab), again the momentum is defined


to be the sum of the momenta. In all cases, the energy on a hypersurface is given bythe flux, defined with respect to the momentum vector, through the hypersurface.

It is important to point out, as we show in lemma 2.1, that the deformation termsfor the generalized momenta are computationally not much more difficult to handlethan the classical ones. As for the classical momenta and energies, in defining thegeneralized vector fields, momenta, and energies as outlined above, one is interestedin getting positive definiteness of the energies or bulk terms. Here, an additionalsubtlety arises. Namely, in the Morawetz estimate presented in equation (3.6),one achieves positive definiteness only modulo boundary terms. We generate theseboundary terms when we integrate by parts to use the formal self-adjointness of thesecond-order symmetry operators. These boundary terms can then be controlledby the energy. The presence of these boundary terms is a completely new featurecompared to the classical energies and momenta.

1.2. Symmetries and null geodesics of Kerr. For any geodesic, the quantitygαβ γ

αγβ is a constant of the motion. Given a Killing field ξ, the quantity pξ =gαβ γ

αξβ is an additional conserved quantity. In Kerr, we have the Killing fields ∂tand ∂φ with the associated constants of motion pt and pφ. For a timelike or nullgeodesic, these correspond to the energy and the angular momentum of a particleor photon with world line γ and are denoted E and Lz.

More generally, if the spacetime admits a Killing k-tensor, i.e. a symmetrictensor Kα1···αk which solves the Killing equation ∇(α1

Kα2···αk+1) = 0, then K =Kα1···αk γ

α1 · · · γαk is a conserved quantity. In the particular case of Killing 2-tensors, which is the only case we are interested in here, there is associated to theKilling tensor a symmetry operator K = ∇αKαβ∇β , such that [K,g] = 0 [8, 42].Since the commutator is zero, this operator is clearly a symmetry in the slightlyweaker sense defined in the previous section.

In Kerr, Carter’s Killing 2-tensor, provides a fourth constant of the motion Q =Qαβ γ

αγβ . For a null geodesic, we have

Q = p2θ +

cos2 θ

sin2 θp2φ + a2 sin2 θp2

t .

A similar expression exists for timelike or spacelike geodesics. Any linear combi-nation of E2, ELz, and L2

z can be added to Q to give an alternate choice for thefourth constant of the motion. The form we have chosen is uncommon, but usefulfor our purposes because it is nonnegative.

As was demonstrated by Carter, the presence of the extra conserved quantityallows one to separate the equations of geodesic motion. Of most interest to us isthe equation for the r coordinate of null geodesics,

Σ2

(dr

dλ

)2

=−R(r;M,a;E,Lz, Q), (1.3)

where

R(r;M,a;E,Lz, Q) =− (r2 + a2)2E2 − 4aMrELz + (∆− a2)L2z + ∆Q. (1.4)

One finds that orbiting null geodesics, i.e. ones which do not fall into the black holeor escape to infinity, must have orbits with constant r. The r values allowing orbitingnull geodesics are solutions to the equations R = 0, ∂R/∂r = 0. The solutions tothis system in the exterior region turn out to be unstable, which corresponds withour conventions to ∂2R/∂r2 < 0.

In the Schwarzschild case, i.e. for a = 0, there are only orbits on the sphereat r = 3M , which is called the photon sphere. For nonzero a, the orbiting nullgeodesics fill up an open region in spacetime which we shall also refer to as thephoton sphere in the Kerr case. As a → 0, the photon sphere tends to r = 3M .


There are many descriptions of the Kerr spacetime and its geodesics, including[3, 23, 40].

In Boyer-Lindquist coordinates, the d’Alembertian g = ∇α∇α takes the form

g =1

Σ

(∂r∆∂r +

1

∆R(r;M,a; ∂t, ∂φ, Q)

), (1.5)

where R is given by (1.4) with the conserved quantities E,Lz, Q replaced by theircorresponding operators ∂t, ∂φ, and the second-order Carter operator3 Q,

Q =1

sinθ∂θ sin θ∂θ +

cos2 θ

sin2 θ∂2φ + a2 sin2 θ∂2

t ,

R(r;M,a; ∂t, ∂φ, Q) =− (r2 + a2)2∂2t − 4aMr∂t∂φ + (∆− a2)∂2

φ + ∆Q. (1.6)

We have used some unusual sign conventions in defining R to avoid factors of iwhen replacing the constants of motion by differential operators.

It is clear from the above that ∂t, ∂φ, and Q are symmetry operators for thewave equation on Kerr. We denote the set of order-n generators of the symmetryalgebra generated by these operators by

Sn = ∂ntt ∂nφφ QnQ |nt + nφ + 2nQ = n;nt, nφ, nQ ∈ N. (1.7)

In particular,

S0 =Id, S1 =∂t, ∂φ.Of particular importance in our analysis will be the set of second-order symmetryoperators,

S2 = ∂2t , ∂t∂φ, ∂

2φ, Q = Sa,

and underlined indices always refer to the index in this set.The function R is polynomial in its last three variables, so R(r;M,a; ∂t, ∂φ, Q) is

well defined. Furthermore, it can be written as a linear combination of the second-order symmetries with coefficients which are rational in r, M , and a,

R(r;M,a; ∂t, ∂φ, Q) =RaSa.1.3. Strategy of proof and further results. Recall from earlier in the introduc-tion that there are three major problems in the Kerr spacetime:

(1) No positive, conserved energy: There is no timelike, Killing vector. Inparticular, the vector field ∂t, which is Killing, is only timelike outside theergosphere, r > M +

√M2 − a2 cos2 θ.

(2) Lack of sufficient classical symmetries: The higher energies generated theLie derivatives in the ∂t and ∂φ directions do not control enough directionsto control Sobolev norms or the function.

(3) Complicated trapping: There are null geodesics which orbit the black hole,in the sense that they neither escape to null infinity nor enter the blackhole. Since solutions to the wave equation can follow null geodesics foran arbitrarily long time, this presents an obstacle to decay. Furthermore,there are orbiting geodesics occuring over a range of r in the Kerr spacetime(with |a| > 0), which makes the situation more complicated than in theSchwarzschild spacetime (a = 0), where there are only orbiting geodesics atr = 3M .

To overcome the first problem, we first observe that the vector ∂t is timelike forsufficiently large r; that, if

ωH =a

r2 + a2

3Since the Carter operator Q and the Carter constant are closely related, we use the samenotation for both.


denotes the angular velocity of the horizon, then the vector ∂t + ωH∂φ is null onthe horizon and timelike for sufficiently small r > r+; that the regions where ∂t and∂t + ωH∂φ are timelike overlap when |a| is sufficiently small; and that both ∂t and∂t + ωH∂φ are Killing. Thus, if we let

Tχ = ∂t + χωH∂φ, (1.8)

where χ is identically 1 for r < rχ for some constant rχ, identically 0 for r > rχ+M ,and smoothly decreases on [rχ, rχ + M ], then, for sufficiently small a, this vector-field will be timelike everywhere and will be Killing outside the fixed region r ∈[rχ, rχ+M ]. Thus, to prove the boundedness of this positive, it will be sufficient tocontrol the behaviour of solutions in this fixed region through a Morawetz estimate.

To overcome problem (2), we note that the second-order operator Q is a symme-try and is a weakly elliptic operator. Using Q, ∂2

φ, and ∂2t as symmetries to generate

higher energies, we can control energies of the the spherical Laplacian of ψ. Thesecontrol Sobolev norms which are sufficiently strong to control |ψ|2.

To handle the complicated trapping, we will use our extension of the vector-fieldmethod to include hidden symmetries. To construct a Morawetz multiplier, wewould like to construct a vector field with a weight that changes sign at the orbitinggeodesic, but this is not possible using a classical vector-field. If we introduceL = LaSa = ∂2

t +∂2φ+Q to give us an elliptic operator and an extra, free, underlined

index, we can take as our collection of Morawetz vector fields

Aab =− zwR′(aLb)∂r,

qabA =− 1

2z(∂r

(wR′(a

))Lb),

R′a =∂r

( z∆Ra),

with z and w smooth, positive functions to be chosen. Applying the analogy of thedeformation formula, the difference between the energies on one hypersurface andanother is

E(Aab,q

abA )

(Σ2)− E(Aab,q

abA )

(Σ1) =

∫ (∇αPα(Aab,q

abA )

)√−|g|d4x.

Ignoring several distracting details, the deformation is of the form

1

2zR′aR′bLαβ(∂αSaψ)(∂βSbψ)

+ z1/2∆3/2

(−∂r

(wz1/2

∆1/2R′a

))Lb(∂rSaψ)(∂rSbψ)

+1

4(∂r∆∂rz(∂rwR′a))Lb(Saψ)(Sbψ).

In the first line, one factor of R′ arises from the wave equation, and the other fromour choice of the Morawetz vector field Aab, which allows us to construct a perfectsquare to obtain positivity. In the second line, the term involves two derivatives of−R. Near the photon orbits, the convexity properties of R, which ensured that theorbits are unstable, ensure that this term is positive. We are free to choose z andw to get positivity away from the photon orbits. The fourth term is lower-order,since it involves fewer derivatives.

For small a, with v denoting terms of the form Saψ, and with our choices of zand w, the sum of the second and third terms is of the form

M

(∆2

r2(r2 + a2)(∂rv)2 +

9r2 − 46Mr + 54M2

r4v2

)(1.9)

with small perturbations on the coefficients. The coefficient on v2 is positive outsidea compact interval in (r+,∞). As shown in [5], it is sufficient to prove a Hardy


estimate which bounds the quadratic form in (1.9) from below by a sum of positiveweights times (∂rv)2 and v2.

The positive terms arising from the deformation of Aab dominate the deformationterms (with extra derivatives) terms arising in the failure of Tχ to be Killing.(In fact, the terms in the Morawetz estimate only control the second and thirdderivatives of ψ, where as the deformation terms from the third-order Tχ energyalso involve the first derivatives and undifferentiated factors of ψ. It is at thispoint, in the proof of lemma 3.12, where we are forced to make a decompositionin harmonics of ∂φ, i.e. to separate variables in φ, to obtain additional control onthe rotationally symmetric components.) On the other hand, the energy associatedwith Aab is dominated by the (third-order) energy associated with Tχ. Since thereis a factor of a on the Tχ deformation terms, we have a small parameter, whichallows us to close the boot-strap argument in which the Tχ energy is controlledby the integral of its deformation, which is controlled by the integral of the Aab

deformation, which is controlled by the Aab energy, which is finally controlled bythe Tχ energy. This allows us to establish theorem 1.1.

To prove decay, we introduce the vector field

K = (t2 + r2∗ + 1)∂t + 2tr∗

((r2 + a2)2

Π

)∂r∗ .

In the Minkowski spacetime, the corresponding vector field is a conformal Killingvector field and generates a positive, conserved energy, sometimes called the con-formal energy or conformal charge. In the Schwarzschild spacetime, it is nowwell known that the corresponding vector field has a deformation tensor whichis favourable outside a compact region in (r+,∞), and that growth in the K energycan only occur as a result of the wave remaining inside the compact interval for longperiods of time. Since the Morawetz estimate rules this out, after another boot-strap argument, it is possible to bound the K energy. After changing variables,we are able to mimic most of the argument from the Schwarzschild spacetime. Weneed to introduce some lower-order terms, qK, to remove the worst terms in thedeformation expression, but are still left with terms, in the K deformation, of theform

at2(∂rψ)(∂φψ).

since these involve a quadratic expression in the derivatives times a factor of t2,these are of the same order as the K energy itself. Even after applying the Morawetzestimate, one is left with an estimate of the form

EK(t2)− EK(t1) ≤ C|a|∫ t2

t1

EK(t)

tdt+ (more easily controlled terms),

where EK(t) denotes the energy evaluated on the hypersurface t × (r+,∞)× S2.Although one cannot obtain a uniform bound on the K energy from this, if thesurfaces are of constant t, then the growth cannot be faster than tC|a|. Since, inregions of fixed r, the K energy is like the Tχ energy times t2, this means that the

energy density in regions of fixed r (away from the horizon) decays like t−2+C|a|.Since the energy density is quadratic, if we use the third-order K energy, we cancontrol the solution ψ in regions of fixed r by t−1+C|a|, which proves the first partof theorem 1.2.

We also prove decay near to and far from the black hole. In both cases, we dropa null geodesic (or almost-null curves) from the point q at (t, r∗, θ, φ) to a pointp at r = 3M , where we already have decay by theorem 1.2, and then estimate|ψ(q)− ψ(p)|, where ψ = (r2 + a2)1/2ψ. The point p at r = 3M has a t coordinategiven by the u+ or u− coordinate of q in the near or far cases respectively. Inthe near case, following [4, 38], we consider the region in the (u+, u−) plane and


enclosed by p, q, curves of constant u+ and of constant u−, and t = 0. We apply

Stokes’ theorem in this region to the one-form (∂−ψ)du− and estimate the bulk

term using the Morawetz and K estimates. This allows us to control |ψ(q)− ψ(p)|as one piece of the boundary term. In the far case, we drop an almost-null curvefrom q to the surface r = t/2, and we call this intersection point p. Although p is

not in a stationary region, we are still able to show sufficiently strong decay for ψ(p)directly from the K energy bound. We evolve the solution from the hypersurfacet = 0 to the hypersurface of constant t through p. Since the deformation of the Kenergy is easily controlled when r > t/2, we can deform the hypersurface so thatit becomes almost null and passes through q. We then integrate the derivative ofψ along an almost null curve in this hypersurface to estimate |ψ(q) − ψ(p)|. Byapplying the Cauchy-Schwarz inequality, we are able to control the integral of thisderivative by the product of the K energy on the hypersurface and by the desireddecaying factor. The contribution from the endpoint p at r = 3M decays muchfaster. These estimates give the remaining parts of theorem 1.2.

The small a condition which we impose is significantly stronger than the conditionthat |a| ≤M which implies the existence of a black hole and which might be ideallyimposed. There are several fundamental and technical reasons for this small acondition. Perhaps most importantly, the construction of Tχ relies on there beinga region where both ∂t+χωH∂φ and ∂t are timelike in which to perform the blending.When a is sufficiently large, but still smaller than M , there is no such overlappingregion, so this particular construction fails. In addition, we use the assumption onthe smallness of a to close the bounded Tχ energy argument and the K energygrowth argument. If a is not small relative to the absolute constants appearing inthose estimates, it would not be possible to close the boot strap. A clear technicalobstacle is that, in the proof of the Morawetz estimate, we perturb the Hardyestimate in (1.9). If a were too large, the perturbation argument would fail, andour numerical investigation suggests that when a is larger than about .9M , thereare no longer positive solutions of the associated ODE, which we use to prove theestimate. These obstacles are the most fundamental obstacles to extending therange of a, but there are also numerous other, technical estimates in which we havemade use of the smallness of a.

Having summarized our method, we will now compare it with methods used inrecent, related work. Recently, others have constructed a bounded energy [17, 39].To make a comparison, we point to several features which they share but which aredifferent from those in our approach.

Since our energy is based on Tχ, which becomes null on the event horizon, theenergy we control has a weight which vanishes linearly at r = r+. The other worksmake use of the horizon-penetrating vector field, first introduced in [14]. This isdenoted Y [17] or X2 [39]. By combining the horizon-penetrating vector field withthe equivalent of Tχ, they are able to construct a timelike vector field and an energywhich do not degenerate near the event horizon. This is clearly advantageous. Sincethe properties of the Y or X2 vecter field are quite stable [16], it can be used in aseparate step, which we omit, following our result.

Neither [17] nor [39] use Q to generate higher energies. Away from the eventhorizon, they use the symmetries ∂2

t , ∂t∂φ, and ∂2φ and the fact that ψ satisfies the

wave equation. Near the event horizon, they generate higher energies using ∂t anda horizon-penetrating, radial vector field (e.g. Y in [17]). This is possible becauseof a favourable sign in the error terms arising from the failure of the radial vectorfield to be a symmetry.

Both of [17, 39] perform a Fourier transform in the t and φ variables to con-struct a pseudodifferential Morawetz multiplier, which we have avoided in favour ofdifferential operators.


Less importantly, both avoid surfaces of constant t in favour of surfaces andcoordinates which go through the event horizon. Since vector-field arguments canbe deformed from one surface to another, this is a minor difference, although, thepresence of the functions, q, slightly complicates this. Although all known Morawetzarguments have, in some sense, a troublesome lower-order term, [17, 39] use adifferent construction so that they can use positivity arising from Y or X2, insteadof the Hardy estimate we use to control the negativity in (1.9).

In [16], a decay rate of t−1+δ(a) is stated with δ(a) → 0 as a → 0 and with thecorresponding decay rates along the event horizon and null infinity. The detailedoutline of a proof focuses on using a vector field K, which is very similar to the oneused previously in Schwarzschild and, hence, to the one we use in this paper.

The structure of this paper is as follows: In section 2, we provide some furthernotation which we use in this paper. In section 3, the main argument of this paper,we expand the energy associated with Tχ and prove the Morawetz estimate usingthe symmetry-indexed vector fields. This is followed by the K argument to provelocal energy decay in section 4, and then by the decay estimate for ψ itself in section5.

2. Notation and preliminaries

In this section, we present some more notation and basic estimates which we willuse through out the paper.

To begin, we note that, in estimates, C is used to denote an absolute constantor a constant which depends only on M . The notation x . y means x ≤ Cy, andthe notation x h y means x . y and y . x. Further, it is convenient to introducethe following notation

d2µ =µdθdφ, µ = sin θ,

d3µ =d2µdr, d3µ∗ =d2µdr∗,

d4µ =d2µdrdt, d4µ∗ =d2µdr∗dt.

2.1. Canonical analysis. The volume element for the Kerr metric in Boyer-Lindquistcoordinates is

√−|g| =Σ sin θ.

It is convenient to consider instead of the covariant d’Alembertian g, the trans-formed d’Alembertian = Σg. Recalling (1.5), we can write in the form

= ∂r∆∂r +1

∆R(r;M,a; ∂t, ∂φ, Q).

This is the form of the d’Alembertian that we shall consider throughout this paper.Let

Gαβ = Σgαβ , µ = sin θ.

Then ψ = 0 is the Euler-Lagrange equation for the action

S =

∫Ld4µ,

with the Lagrangian

L =1

2Gαβ(∂αψ)(∂βψ).

The canonical energy-momentum tensor for S is

T αβ = − δLδ∂αψ

(∂βψ) + δαβL,


and the momentum vector for a vector field X is given by

PαX =T αβXβ

or explicitely

Pα(X,q)[ψ] =

(−Gαγ(∂γψ)(∂βψ) +

1

2δαβ (Gρσ(∂ρψ)(∂σψ))

)Xβ

+ Gαβ(−q(∂βψ)ψ +

1

2(∂βq)ψ

2

).

We have the following relations with the quantities introduced in section 1.1,

∇αPα(X,q)[ψ] =− 1

Σ

1

µ∂α

(µPα(X,q)[ψ]

),

∫

Ω

(∇αPαX,q

)√−|g|d4x =−

∫

Ω

(∂αµPα(X,q)

)d4µ,

E(X,q)[ψ](t) =

∫

Σt

Pt(X,q)[ψ]d3µ.

When it is clear from context what the arguments are, we will frequently writeEX[ψ], EX(t), or EX. Similarly, we will typically write P(X,q) for P(X,q)[ψ]. Towardsthe end of this paper, we also need to evaluate energies on other hypersurfaces. Inthis case, we generate a three form by contracting the vector field P(X,q) against thevolume form generated by the coordinate one forms, and then integrate that 3-formover the corresponding three-dimensional hypersurface. This is explained further insection 5.3.

The advantage of the canonical formalism just introduced is that without com-puting covariant derivatives, one can calculate the divergence of the momentum.

Lemma 2.1. If X is a vector field, q is a function, and if P(X,q) is the associatedmomentum relative to the Lagrangian and weight

L =1

2

(Gαβ(∂αu)(∂βu) + V ψ2

), µ

then the divergence of the corresponding momentum is

1

µ∂α(µP(X,q)[ψ]α

)

=

(−Gαγ(∂γX

β) +1

2(∂γX

γ)Gαβ +1

2

1

µXγ(∂γ µGαβ)

)(∂αψ)(∂βψ)

+1

2(∂γX

γ)V ψ2 +1

2Xγ(∂γ V )ψ2

− q(Gαβ(∂αψ)(∂βψ) + V ψ2) +1

2

(∂βGαβ∂αq

)ψ2

+ Xγ(∂γψ + qψ)

((1

µ∂αµGαβ∂β − V

)ψ

).

If ψ is a solution of(µ−1∂αµGαβ∂β − V

)ψ = 0, then the last term in this formula

vanishes.


Similarly, if Xab and qab are symmetric collections of double indexed vectors andscalars respectively and ψ is a solution of

(µ−1∂αµGαβ∂β − V

)ψ = 0, then

1

µ∂α

(µPα(Xab,qab)

)

=

(−Gαγ(∂γX

abβ) +1

2(∂γX

abγ)Gαβ +1

2Xabγ(∂γ Gαβ)− qabGαβ

)(∂αSaψ)(∂βSbψ)

+1

2(∂γX

abγ)V ψ2 +1

2Xabγ(∂γ V )ψ2

+1

2(∂αGαβ∂βqab)(Saψ)(Sbψ). (2.1)

2.2. The 3 + 1 decomposition. The normal to the surfaces of constant t is

nΣt =

(Π

Σ∆

)1/2

(∂t + ω⊥∂φ) ,

with ω⊥ defined below. The lapse function is defined solely in terms of the foliationand its normal and is

N =(nαΣt∂αt)−1

=

(Π

Σ∆

)−1/2

Unfortunately, the t co-ordinates do not extend beyond the exterior of the blackhole, so the vector field nΣt does not extend continuously beyond the exterior.

We introduce the vector field

T⊥ =∂t + ω⊥∂φ,

ω⊥ =2aMr

Π.

This vector field has the properties that

T⊥ =NnΣt ,

that T⊥ is timelike in the exterior, and that it extends continuously to the eventhorizon and the bifurcation sphere. In fact, it extends smoothly through the eventhorizon and the bifurcation sphere.4 This vector field extends to the null tangentvector on the event horizon and to axial rotation (with coefficient ωH) on the bi-furcation sphere.

2.3. Norms. Given a set of differential operators, X, we use the notation

|ψ|2X =|Xψ|2 =∑

X∈X|Xψ|2.

If no set is specified, simply an index, we mean

|ψ|2n =

n∑

i=0

|Snψ|2,

where Sn is the set of generators of the order-n symmetries given in equation (1.7).

Lemma 2.2 (Spherical Sobolev estimate using symmetries). There is a constant,C, such that ∀(θ, φ) ∈ S2 and all (t, r) ∈ R × (r+,∞), if ψ is sufficiently smooththat the norm on the right is bounded, then

sup(t,r)×S2

|ψ|2 ≤ C∫

(t,r)×S2

|ψ|22d2µ.

if u is sufficiently smooth that the integral on the right is bounded.

4The vector fields ∂t and ∂φ are known to extend smoothly through the bifurcation sphere [26].


Proof. We use ∆/ to denote the spherical Laplacian

∆/ =1

µ∂θµ∂θ +

1

µ2∂2φ.

The absolute value of the spherical Laplacian of u can be estimated by

|∆/ψ| =∣∣∣∣(

1

µ∂θµ∂θ + cot2 θ∂2

φ + ∂2φ

)ψ

∣∣∣∣

≤∣∣∣∣(

1

µ∂θµ∂θ + cot2 θ∂2

φ

)ψ

∣∣∣∣+∣∣∂2φψ∣∣

≤|Qψ|+ a2 sin2 θ|∂2t ψ|+ |∂2

φψ|.|S2ψ|.

By a standard, spherical, Sobolev estimate,

|ψ|2L∞(S2) .∫

S2

(|∆/ψ|2 + |ψ|2

)d2µ.

Since the integrand on the right is bounded by |ψ|2, the desired estimate holds witha uniform constant in (t, r).

In subsection 3.4, we also require the following operator and the associated weakernorms.

Definition 2.3. Let

L =∂2t +Q+ ∂2

φ,

Lε =ε∂2t +Q+ ∂2

φ,

and

|ψ|22,ε =ε|∂2t ψ|2 + |∂t∂θψ|2 +

1

µ2|∂t∂2

φψ|2 + |∆/ψ|2,

|ψ|23,ε =ε2|∂3t ψ|2 + ε|∂2

t∇/ψ|2 + |∂t∆/ψ|2 + |∇/∆/ψ|2.We also introduce the homogeneous norms, generated from the previous norm bytaking ε = 1,

|ψ|n,1.

Lemma 2.4 (The LεL estimate). There is a positive constant C such that, forpositive values of the parameter ε, if |a| ≤ Cε and if ψ smooth, then

(Lεψ)(Lψ) ≥|ψ|22,ε (2.2)

+1

µ∂t(µ(∂tψ)(∆/ψ))(1 + ε)

− 1

µ∇/ · (µ(∂tψ)(∇/∂tψ))

+1

µ2a2 sin2 θ(∂t(µ(∂tψ)(∂2

φψ))− ∂φ(µ(∂tψ)(∂φ∂tψ))).

Proof. This follows by direct computation, but it is important to integrate by partsin t first.

(Lεψ)(Lψ) =((ε∂2

t +Q+ ∂2φ)ψ

) ((∂2t +Q+ ∂2

φ)ψ)

=ε(∂2t ψ)2 + (Qψ)2 + (∂2

φψ)2

+ 2(Qψ)(∂2φψ)

+ (1 + ε)(∂2t ψ)(Qψ) + (1 + ε)(∂2

t ψ)(∂2φψ).


In the last line, the contribution from ∂2t ψ in Qψ gives a strictly positive term, so

it can be dropped, and integration by parts can be applied to the remainder

µ(∂2t ψ)((Q+ ∂2

φ)ψ) ≥µ(∂2t ψ)(∆/ψ)

≥− µ(∂tψ)(∆/∂tψ) + ∂t(µ(∂tψ)(∆/ψ))

≥µ|∇/∂tψ|2 + ∂t(µ(∂tψ)(∆/ψ))−∇/ · (µ(∂tψ)(∇/∂tψ)).

Finally, it remains to show that

ε|dt2ψ|2 + |Qψ|2 + |∂2φψ|2 + 2(Qψ)(∂2

φψ) =ε|∂2t ψ|2 + |(Q+ ∂2

φ)ψ|2

&ε|dt2ψ|2 + |∆/ψ|2.This follows from expanding the left-hand side using Q + ∂2

φ = ∆/ + a2 sin2 θ∂2t ,

estimating the mixed term 2a2 sin2 θ(∂2t ψ)(∆/ψ) by the Cauchy-Schwarz inequality,

and using the fact that |a| . ε.

It is sometimes useful to have an estimate which gives equality, except for somesmall error terms, instead of an inequality. In such cases, we relate Q+ ∂2

φ to ∆/ toobserve that

(Lεψ)(Lψ) =((ε∂2

t + ∆/)ψ) (

(∂2t + ∆/)ψ

)

+ a2 sin2 θ((∂2t ψ)(Lψ) + ((ε∂2

t + ∆/)ψ)(∂2t ψ)).

Thus,∣∣(Lεψ)(Lψ)−

((ε∂2

t + ∆/)ψ) (

(∂2t + ∆/)ψ

)∣∣ ≤a2|ψ|22.Expanding

((ε∂2

t + ∆/)ψ) (

(∂2t + ∆/)ψ

)in the same way as in lemma 2.4, we find

∣∣(Lεψ)(Lψ)− |ψ|22,ε + (time and angular derivatives)∣∣ ≤a2|ψ|22,1, (2.3)

where the time-derivatives terms are time derivatives of terms dominated by |∂tψ||S2ψ|.Similarly,

∣∣(Lεψ)2 − |ψ|23,ε + (time and angular derivatives)∣∣ ≤a2|ψ|23,1, (2.4)

where the time derivative terms are derivatives of quantities bounded by |∂tT1ψ||S2T1ψ|and where the angular derivatives are derivatives of smooth terms.

2.4. Further notation. It is convenient to write the second-order symmetry op-erators with respect to coordinate partial derivatives

Sa =1

µ∂αµS

αβa ∂β .

All other operator built from these, such as L, Lε, andR, can be similarly expanded.For example,

R =1

µ∂αµRαβ∂β .

We use the notation

f = O(rp)

to mean that there is a constant, uniformly in a in some small interval of a valuescontaining 0, such that ∀r > r+, f(r) < Crp. Introduce also the notation

f = O

((∆

r2

)q, rp)

to mean that there is a constant, uniformly in a in some small interval of avaluescontaining 0, such that ∀r > r+,

f(r) < C

(∆

r2

)qrp.


This measures the decay rate at r+ and ∞. If f is continuous, this is all theinformation that is required to bound the function.

We use ∇/ to denote angular partial derivatives and ∆/ for the spherical Laplacianin (θ, φ) coordinates. We use Θi for the rotation vector-fields about the coordinateaxes. With the exception of Θ3 = ∂φ, these are not symmetries in Kerr. We useO1 = Θi to denote the set of these rotations, and we use T1 for ∂t,Θi.

We use 1X to denote the indicator function, which is identically one on X andzero elsewhere.

We define a function to be smooth on a closed interval if it is smooth on theinterior and if all the derivatives are continuous up to the boundary.

3. The bounded-energy argument

In this section, we construct a bounded energy by first constructing an almostconserved energy and then proving a Morawetz estimate to control the growth ofthe energy.

3.1. The blended energy. Recall from (1.8) that for |a| sufficiently small, thevector field

Tχ = ∂t + χωH∂φ

is timelike in the exterior and Killing outside the region [rχ, rχ + M ], since χ isconstant outside this region and decreases from one to zero inside this region. If wechoose rχ sufficiently large so that it corresponds to a larger value of r than anyphoton orbit for our initial choice of small |a|, this property will remain true for anysubsequent decrease in the range of |a| we allow. For specificity, we take rχ = 10M ,which is beyond the range of photon orbits for any Kerr black hole.

The vector field Tχ becomes null on the horizon, so the associated energy degen-

erates there. We compare this with the energy associated with T⊥ = (∆Σ/Π)1/2nΣt

to make clear that the rate of degeneration with respect to the normal is roughtly(∆/(r2+a2))1/2. We also provide a coordinate expression which is useful for makingestimates. The apparently singular contribution to the energy from ∆−1(T⊥ψ)2 isin fact vanishing, since the vector-field T⊥ vanishes on the bifurcation sphere atsuch a rate to exactly compensate for the factor of ∆−1, and then the form dr isdegenerating at a rate of (∆/(r2 + a2))1/2 near the bifurcation sphere.

Lemma 3.1. There is a positive a such that for |a| ≤ a and any smooth functionψ, Tχ is timelike and

PtTχ h(r2 + a2)2

∆(T⊥ψ)2 + ∆(∂rψ)2 +Qαβ(∂αψ)(∂βψ)

h(r2 + a2)2

∆(T⊥ψ) + ∆(∂rψ) + ∆(∂tψ)2 +

∑

i

|Θiψ|2 (3.1)

∫

Σt

PtTχd3µ =

∫

Σt

PαTχdηα

h∫

Σt

PαT⊥dηα. (3.2)

Furthermore, if ψ is a solution of the wave equation ψ = 0, then∣∣∣∣1

µ∂α(µPTχ [ψ]α

)∣∣∣∣ =∆ωH |∂rχ||∂φψ||∂rψ|. (3.3)


Proof. We first expand

Gαβ(∂αψ)(∂βψ) =∆(∂rψ)2 − (r2 + a2)2

∆(∂tψ)2 − 4aMr

∆(∂tψ)(∂φψ)

+Qαβ(∂αψ)(∂βψ) +∆− a2

∆2(∂φψ)2.

We now substitute ∂t = Tχ + χωH∂φ and estimate the terms arising from thedifference. We find

Gαβ(∂αψ)(∂βψ) =∆(∂rψ)2 − (r2 + a2)2

∆(Tχψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2

+

(4aMr

∆− χωH

(r2 + a2)2

∆

)(∂tψ)(∂φψ)

+1

∆

(−a2 + (r2 + a2)2ω2

H

)(∂2φ)

=∆(∂rψ)2 − (r2 + a2)2

∆(Tχψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2

+ aO(1, r−1)(∂tψ)(∂φψ) + a2O(1, r−2)(∂φ)2.

Similarly

Gtβ(∂βψ) =− Π

∆∂tψ −

2aMr

∆ψ

=− (r2 + a2)2Tχψ

+ a2O(1, r−2)∂tψ + aO(1, r−1)∂φψ.

The t component of the momentum associated with Tχ is

PtTχ =− Gtβ(∂βψ)(Tχψ) +1

2Gαβ(∂αψ)(∂βψ)

=1

2

(∆(∂rψ)2 +

(r2 + a2)2

∆(Tχψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2

)

+ a2O(1, r−2)∂tψ + aO(1, r−1)(∂tψ)(∂φψ) + a2O(1, r−2)(∂φψ)2.

Since the asymptotics of the coefficients on the last line grow no faster than thecoefficients of the terms in the first line, we can take a sufficiently small that thefirst line easily dominates the last, and we can conclude

PtTχ &∆(∂rψ)2 +(r2 + a2)2

∆(Tχψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2.

Since the ∂2t term in Q has a bounded factor times a2,

(r2 + a2)2

∆(∂tψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2 &(∂tψ)2 +Qαβ(∂αψ)(∂βψ) + (∂φψ)2

&∑

i

|Θiu|2.

Since

T⊥ =Tχ + aO(∆, r−3)∂φ,

the difference between the two associated energies is

aO(1, r−3)Pt∂φ =aO(∆, r−3)

(2aMr

∆(∂φψ)(∂tψ)

)

=aO(1, r−4)(∂φψ)(∂tψ),

which is easily dominated by PtTχ , and the two momenta are equivalent. Hence,

their integrals are equivalent.


We compute the divergence of the momentum using equation (2.1) and find itbe given by

−Grr(∂rTφχ)(∂φψ)(∂rψ) =−∆(∂rχωH)(∂φψ)(∂rψ),

which we estimate in absolute value.

Recall that we defined higher-order energies by

ETχ,n[ψ] =n∑

i=0

ETχ [Siψ],

where Si is the set of order-i symmetries from (1.7).

Corollary 3.2. If ψ is a solution of the wave equation ψ = 0,

d

dtETχ,n[ψ] ≤C

∫ ∞

r+

∫

S2

1suppχ′ |∂rψ|n|∂φψ|nr2dr sin θdθdφ, (3.4)

where the norms on the right are defined in subsection 2.3.

Proof. This follows from the previous lemma applied to the functions obtainedfrom applying the symmetry operators to the solution ψ and then summing overthe operators.

3.2. Set-up for radial vector fields and their fifth-order analogues. If z andw are smooth functions of r and the parameters M and a, then we can define thefollowing single- and double-indexed quantities

Ra =z

∆Ra,

R′a =∂r

( z∆Ra),

˜R′a =wz1/2

∆1/2R′a,

˜R′′a =∂r

(wz1/2

∆1/2R′a

).

These can be used to define a double-indexed family of vectors and scalars whichwe will use to prove a Morawetz estimate.

Definition 3.3. The Morawetz vector fields and scalar functions are defined to be

Aab =Fab∂r,

qaA =

1

2(∂γA

aγ)− qaA′ , qabA =

1

2(∂γA

abγ)− qabA′ ,

Fa =zwR′a, Fab =zwR′(aLb),

qaA′ =

1

2(∂rz)wR′a, q

abA′ =

1

2(∂rz)wR′(aLb).

For simplicity, we introduce the following notation for the pair consisting of theprevious vector field and function,

A =(Aab, qabA ).

Lemma 3.4. If ψ is a solution to the wave equation ψ = 0, then the divergenceof the momentum associated with these quantities is given by

1

µ∂α (µPA[ψ]α) =Aab(∂rSaψ)(∂rSbψ)

+ Uabαβ(∂αSaψ)(∂βSbψ)

+ Vab(Saψ)(Sbψ), (3.5)


where

Aab =A(aLb),

Uab =1

2wR′aR′b,

Vab =V(aLb),

Aa =z1/2∆3/2(− ˜R′′a),

Va =1

4(∂r∆∂rz(∂rwR′a)),

Uabαβ =Uc(aLb)Sαβc .

Proof. In the formula for the divergence of the momentum, (2.1), the terms in-volving Gαβ(∂αSau)(∂βSbu) are refered to as the Lagrangian contributions. The

Lagrangian contribution from (1/2)(∂γAabγ) in q

abA exactly cancels the Lagrangian

contribution from the vector field.The divergence of the momentum is given by

− 1

µ∂α (µPαA) =

(∆(∂rFab)−

1

2Fab(∂r∆)

)(∂rSaψ)(∂rSbψ)

− 1

2Fab

(∂r

(Rαβ∆

))(∂αSaψ)(∂βSbψ)

− qabA′∆(∂rSaψ)(∂rSbψ)− qabA′Rαβ∆

(∂αSaψ)(∂βSbψ).

− 1

2(∂βGαβ∂αqab)(Saψ)(Sbψ).

In the coefficient of the radial derivative terms, the part coming from the vectorfield can be rewritten as

(∆(∂rFab)−

1

2Fab(∂r∆)

)=

(∂r

( Fab∆1/2

))∆3/2.

Expanding using the definitions of z, w, and R′, we have

1

µ∂α (µPαA) =− z1/2∆3/2

(∂r

(z1/2

∆1/2wR′a

))Lb(∂rSaψ)(∂rSbψ)

+1

2wR′aLb

(∂r

(zRαβ

∆

))(∂αSaψ)(∂βSbψ)

+1

2(∂βGαβ∂α(z(∂rwR′(a))Lb)(Saψ)(Sbψ).

The expression R′ was chosen so that it is exactly the derivative in the second term.

Similarly, the quantity ˜R′′ was chosen so that it is the derivative in the first term.Thus, the total bulk term is

1

µ∂α (µPαA) =− z1/2∆3/2 ˜R′′aLb(∂rSaψ)(∂rSbψ)

+1

2(LaR′b)R′αβ(∂αSaψ)(∂βSbψ)

+1

4(∂r∆∂rz(∂rwR′(a))Lb)(Saψ)(Sbψ).

Since ˜R′′aLb is contracted against a quantity which is symmetric in ab, it is not

necessary to distinguish between ˜R′′aLb and ˜R′′(aLb). Substituting the definitionsof Aab, Uabαβ , and Vab gives the desired result.


3.3. Rearrangements. We rearrange the terms related to U to get a strictly pos-itive contribution to the divergence.

Lemma 3.5. If ψ is a solution to the wave equation ψ = 0, then

1

µ∂α(µ(PαA + BαA;I)

)=Aab(∂rSaψ)(∂rSbψ)

+ UabLαβ(∂αSaψ)(∂βSbψ)

+ Vab(Saψ)(Sbψ),

where A, U , and V are defined in lemma 3.4 and

BA;I[ψ]α =(UabLαβ − Uabαβ

)(Saψ)(∂βSbψ).

We will refer to BA;I as the first boundary term.

Proof. Starting from (3.5), it is only the second term on the right side that needsto be manipulated. First, we rearrange the derivative term to get

µUabαβ(∂αSaψ)(∂βSbψ) =µUcaLbSαβc (∂αSaψ)(∂βSbψ)

=− UcaLb(Saψ)(∂αµSαβc ∂βSbψ)

+ ∂α(µUcaLbSαβc (Saψ)(∂βSbψ)).

The first term on the right can be rewritten in terms of Sc, which can be commutedwith Sb, which in turn can be expanded in partial derivatives:

−UcaLb(Saψ)(∂αµSαβc ∂βSbψ) =− µUcaLb(Saψ)(ScSbψ)

=− µUcaLb(Saψ)(SbScψ)

=− UcaLb(Saψ)(∂αµSαβb ∂βScψ).

We can substitute this into the previous calculation, rearrange a derivative in thenew expression, reindex, and use the symmetry of Uab to conclude that

µUabαβ(∂αSaψ)(∂βSbψ) =µUcaLbSαβb (∂αSaψ)(∂βScψ)

− ∂α(µUcaLbSαβb (Saψ)(∂βScψ))

+ ∂α(µUcaLbSαβc (Saψ)(∂βSbψ))

=µUabLαβ(∂αSaψ)(∂βSbψ)

− ∂α(µ(UabLαβ − Uabαβ)(Saψ)(∂βSbψ)

).

Applying the definition BA;I gives the desired result.

3.4. Choosing the weights. In this section, we choose the weights z and w toensure the positivity of the highest order terms in the right-hand side of the estimatein the previous lemma, lemma 3.5.

Definition 3.6. Given a positive value for the parameter ε∂2t, we use the following

weights to define the Morawetz vector field,

z =z1z2, w =w1w2,

z1 =∆

(r2 + a2)2, w1 =

(r2 + a2)4

3r2 − a2,

z2 =1− ε∂2t

(∆

(r2 + a2)2

), w2 =

1

2r.

This choice of weights generate a momentum which has a positive divergence,and for which this positive divergence dominates the square of third derivatives ofψ. The statement and proof of the following lemma make use of the norms givenin subsection 2.3


Remark 3.7. A simpler version of this argument can be run taking z = z1 andw = w1. These weights are chosen so that the coefficient of ∂2

t in R′ and of ∂φ∂t

in ˜R′′ vanish respectively. Eliminating the ∂2t term in R′ is a natural first step,

since, in essence, this is what has been done in all previous vector-field arguments

in the Schwarzschild spacetime. Eliminating the ∂φ∂t term in ˜R′′ is also natural,since this leaves only Q and ∂2

φ terms, which have a natural ellipticity. Takingthese choices gives a collection of vector fields for which the divergence is positive.However, these simpler choices generate a divergence which fails to dominate thirdderivatives which involve two t derivatives, and they generate an energy which is notdominated by the Tχ energy. We have further introduced the weights z2 to obtainbetter control over the ∂2

t derivatives and w2 to “temper” the vector fields, so thatthe associated energy can be controlled by the energies associated with Tχ.

Lemma 3.8. There are positive constants a, ε∂2t, and C such that if |a| ≤ a and

ε∂2t< ε∂2

tand ψ is a solution to the wave equation ψ = 0 then

1

µ∂α(µ(PA[u]α + BαA;I + BαA;II

))(3.6)

≥M ∆2

r2(r2 + a2)|∂rψ|22,ε

∂2t

+1

6

9Mr2 − 46M2r + 54M3

r4|ψ|22,ε

∂2t

+1

4r

(r2 + a2)4

3r2 − a2R′aR′bLαβ(∂αSaψ)(∂βSbψ)

− C ∆2

r2(r2 + a2)(a|∂rψ|22,1 + ε∂2

t|∂rψ|22,a2)

− C 1

r2(a|ψ|22,1 + ε∂2

t|ψ|22,a2).

where

R′ =− 2(r − 3M)r−4Lε∂2t

+ aO(r−4)∂φ∂t + a2O(r−5)Q+ a2O(r−5)∂2φ

+ ε∂2ta2∂2

t + ε∂2tO(r−5)Q+ ε∂2

tO(r−5)∂2

φ

and where the BαA;II satisfy

|BtA;II| .∆2

r2(r2 + a2)|∂r∂tψ|

∑

a

|∂rSaψ|+1

r2|∂tψ|

∑

a

|Saψ|,

BrA;II =0,

and the angular components are smooth functions.

Proof. From the lemma 3.5, there are three terms to control, the U , A, and V terms.Step 1: The U term. The U term can be expanded using the definition in

lemma 3.4 as

1

2UabLαβ(∂αSaψ)(∂βSbψ) =

1

4wR′aR′bLαβ(∂αSaψ)(∂βSbψ),

so it is sufficient to calculate R′. With our choice of z and w, this is

R′ =− ε∂2t(2(r − 3M)r−4 + a2O(r−5))∂2

t

+ aO(r−4)∂φ∂t

− (2(r − 3M)r−4 + a2O(r−5) + ε∂2tO(r−5))Q

− (2(r − 3M)r−4 + a2O(r−5) + ε∂2tO(r−5))∂2

φ.


Step 2: The A term. The A term is

AaLb(∂rSaψ)(∂rSbψ) =∆2

r2 + a2(− ˜R′′a)Lb(∂rSaψ)(∂rSbψ).

With our choices of z and w, we find:

− ˜R′′ =Mε∂2t(r−2 + a2O(r−3))∂2

t

− aMO(r−2)∂φ∂t

+M(r−2 + a2O(r−3) + ε∂2tO(r−3))Q

+M(r−2 + a2O(r−3) + ε∂2tO(r−3))∂2

φ.

We are interested in this because the operator ˜R′′ is very close to Lε∂2t

in the

sense that∣∣∣∣(

(− ˜R′′)− M

r2Lε

∂2t

)∂rψ

∣∣∣∣ =aO(r−3)|T21∂rψ|

+ aO(r−2)|T21∂rψ|

+ aO(r−3)|T21∂rψ|

+ ε∂2tO(r−3)|O2

1∂rψ|+ ε∂2taO(r−3)| sin2 θ∂2

t ∂rψ|=aO(1)|∂rψ|2,1 + ε∂2

tO(1)|∂rψ|2,a2 .

Since L and Lε∂2t

commute with functions of r, we can apply lemma 2.4 to ∂rψ,

to get

M∆2

r2(r2 + a2)(Lε

∂2t

∂rψ)(L∂rψ) ≥M ∆2

r2(r2 + a2)|∂rψ|22,ε

∂2t

+ time and angular derivatives.

The time and angular derivatives are exactly those coming from lemma 2.4. Theterms from the angular derivatives are smooth, and the terms from the time deriv-ative are of the form

M∆2

r2(r2 + a2)∂t((∂t∂rψ)((∆/+ a2 sin2 θ∂2

φ)∂rψ).

Thus, we only need to control contributions from these terms when they appear asboundary terms on hypersurfaces of constant t. They are controlled by

M∆2

r2(r2 + a2)|∂t∂rψ||(∆/+ a2 sin2 θ∂2

φ)∂rψ)| .M ∆2

r2(r2 + a2)|∂t∂rψ|

∑

a

|Sa∂rψ|.

(3.7)

Thus,

Aab(Sa∂rψ)(Sb∂rψ) ≥M ∆2

r2(r2 + a2)|∂rψ|22,ε

∂2t

− C ∆2

r2(r2 + a2)(a|∂rψ|22,1 + ε∂2

t|∂rψ|22,a2)

+ time and angular derivatives

with the time and angular derivatives satisfying the bound given in the statementof this lemma.


Step 3: The V term. By direct computation, the V term is given by

VaSa =1

4∂r∆∂rq

aASa

=

(ε∂2t

1

6(9Mr−2 − 46M2r−3 + 54M3r−4 + aO(r−4))

)∂2t

+ aO(r−4)∂φ∂t

+

(1

6(9Mr−2 − 46M2r−3 + 54M3r−4 + aO(r−4) + ε∂2

tO(r−4))

)Q

+

(1

6(9Mr−2 − 46M2r−3 + 54M3r−4 + aO(r−4) + ε∂2

tO(r−4))

)∂2φ,

=1

6(9Mr−2 − 46M2r−3 + 54M3r−4)Lε

∂2t

+ (a+ ε∂2t)O(r−4)S2,

where we have used O(r−4)S2 to denote terms of the form O(r−4)Sa with Sa ∈ S2.Applying the estimate in (2.3), we find

Vab(Saψ)(Sbψ) =1

6(9Mr−2 − 46M2r−3 + 54M3r−4)(Lε

∂2t

ψ)(Lψ)

+O(r−2)(a|ψ|22,1 + ε∂2t|ψ|22,a)

≥1

6(9Mr−2 − 46M2r−3 + 54M3r−4)|ψ|22,ε

∂2t

+O(r−2)(a|ψ|22,1 + ε∂2t|ψ|22,a)

+ time and angular derivatives.

Again, the time and angular derivatives come from the application of lemma 2.4,so that the terms from the angular derivatives are smooth, and the terms from thetime derivative give a contribution of the form

C

r2|∂tψ||(∆/+ a2 sin2 θ∂2

φ)ψ| ≤Cr2|∂tψ|

∑

a

|Saψ|. (3.8)

The time and angular derivative terms arising in this step and the previous oneare combined into BA;II and are controlled by and (3.7)-(3.8).

Lemma 3.9 (Controlling the boundary terms). If ψ is sufficiently smooth, satisfiesψ = 0, and has initial data which decays sufficiently rapidly at infinity, then

|PA[ψ]t| ≤C(PTχ [S2ψ]t + |S2ψ|2)

|BtA;I| ≤C(PTχ [S2ψ]t + |S2ψ|2)

|BtA;II| ≤C(PTχ [∂tψ]t + PTχ [S2ψ]t + |∂tψ|2 + |S2ψ|2),

limr→r+

PrA =0

limr→r+

BrA;I =0

limr→∞

PrA =0

limr→∞

BrA;I =0.

Here, by “sufficiently rapidly”, we mean that limr→∞ ψ = 0, limr→∞ r∂rψ = 0, andthe same estimates hold for S1ψ and S2ψ.


Proof. By direct computation,

PtA =(−((r2 + a2)2 − a2∆ sin2 θ

)(∂tS2ψ)− 2aMr(∂φS2ψ)

) (O(1)(∂rS2ψ) +O(r−2)(S2ψ)

)

BtA;I =O(r−1)((S2ψ)(T1S2ψ))

PrA =O((∆/r2)2, r2)(∂rS2ψ)(∂rS2ψ)

+O(∆/r2, r)(∂rS2ψ)(S2ψ) +O(∆/r2, 1)(S2ψ)(S2ψ)

BrA;I =0.

Here we have used S2ψ to denote a term which is bounded in absolute value by|S2ψ|. First, the t component of the momentum can be dominated by

1

∆

(((r2 + a2)2 − a2∆ sin2 θ

)(∂tS2ψ) + (2aMr)(∂φS2ψ)

)2

+O((∆/r2), r2)(∂rS2ψ)2 +O(∆/r2, 1)(S2ψ)2

From the expression for the energy, (3.1), the second term is dominated by PtTχ [S2ψ].

The third is clearly dominated by ψ2. This leaves the first. Since the coefficients inthe first term are polynomial, and the ratio of the coefficients of the ∂tψ and ∂φψterm in the first expression is

(r2 + a2)2 − a2∆ sin2 θ

2aMr=

(r2+ + a2)2

2aMr++O(∆/r2, r4)

=(r2

+ + a2)

a+O(∆/r2, r4)

=ω−1H +O(∆/r2, r4),

the first term can be estimated by

Cχ

∆(∂tS2ψ + ωH∂φS2ψ)2 +O(r2)(∂tS2ψ)2 +O(1)(∂φS2ψ)2.

Thus, the expression for the energy in estimate (3.1) shows that this is bounded byPtTχ [S2ψ]. This controls the t component of the momentum.

The t-component of the first boundary term is controlled by

BtA;I =O(r−1)((S2ψ)(∂tS2ψ))

≤O(r2)(∂tS2ψ)2 +O(1)(S2ψ)2

≤PtTχ [S2ψ] + |S2ψ|2.The t component of the second boundary term was partially estimated in the

statement of lemma 3.8, so that

|BtA;II| ≤O((∆/r2)2, 1)|∂r∂tψ||∂rS2ψ|+O(r−2)|∂tψ||S2ψ|

≤O((∆/r2)2, 1)|∂r∂tψ|2 +O((∆/r2)2, 1)|∂rS2ψ|2

+ |∂tψ|2 + |S2ψ|2

≤C(PtTχ [∂tψ] + PtTχ [S2ψ] + |∂tψ|2 + |S2ψ|2).

The limits at r+ and ∞ are easily evaluated. The radial component of themomentum and the first boundary terms are bounded functions times a power of∆, so they vanish at r+. If the initial data decays sufficiently rapidly towardsinfinity, by finite speed of propogation, the same will be true for the solution at anytime, so that in the limit as r → ∞, at fixed t, the momentum and first boundaryterm will go to zero. The radial term of the second boundary term is identicallyzero, so all its limits vanish.


3.5. The Hardy estimate. In (3.6), the coefficient of |ψ|22,ε∂2t

is positive except

in a compact range of r values. The purpose of this section is to prove a Hardyestimate which allows us to get a globally positive coefficient for |ψ|22,ε

∂2t

by using

the positivity of the term involving |∂rψ|22,ε∂2t

. The proof is a bit technical, and the

reader can omit it on a first reading, since the proof is independent of the rest ofthe Morawetz estimate.

Lemma 3.10. There exist positive a and εHardy such that if |a| < a, then for anysmooth function φ on [r+,∞)× S2 which decays sufficiently rapidly at ∞,

∫ ∞

r+

(∆2

r2(r2 + a2)(∂rφ)2 +

1

6

9r2 − 46Mr + 54M2

r4φ2

)dr

≥εHardy

∫ ∞

r+

∆2

r2(r2 + a2)(∂rφ)2 +

1

r2φ2dr. (3.9)

Proof. The proof consists of several parts. The early parts of this proof follow themethod of [5]. First, we will demonstrate that it is sufficient to find a positivesolution to an associated ODE (ordinary differential equation). Second, we rewritethe estimate and ODE in terms of a new function, ϕ. Third, we will construct anexplicit solution for the new ODE when a = 0 and εHardy = 0. Fourth, we will arguethat the construction of the explicit solution can be perturbed to cover nonzero aand εHardy, which will give a perturbed estimate for ϕ. Fifth, we will show that thisgives the estimate for the original function φ. Finally, we will show that boundaryconditions for the ODE do not place restrictions on the function φ.

Step 1: Find a positive solution to the associated ODE. We wish toshow that if the ODE

−∂rA∂ru+ V u = 0,

has a smooth, positive solution u on [r0,∞], then for any smooth function φ on[r0,∞], there is the estimate

∫ ∞

r0

A(∂rφ)2 + V φ2dr ≥ 0,

as long as

φ2A∂ru

u(3.10)

vanishes at r0 and ∞. Recall that we define a function to be smooth on a closedinterval if it is smooth on the interior and all derivatives have a limit at the boundary.

Since u is positive, for any smooth φ, we can define f = φ/u. From integrationby parts,

∫ ∞

r0

A(∂rφ)2 + V φ2dr − [Auf(∂ruf)]∞r0 =

∫ ∞

r0

uf2(−∂rA∂ru+ V u)dr

+

∫ ∞

r0

u2A(∂rf)2dr

− [u2Af(∂rf)]∞r0 .

Since u satisfies the ODE −∂rA∂ru+V u = 0, the first term on the right is positive.Cancelling the boundary terms on the right from those on the left leaves the estimate

∫ ∞

r0

A(∂rφ)2 + V φ2dr =

∫ ∞

r0

u2A(∂rf)2dr + [f2Au(∂ru)]∞r0 .


The boundary term vanishes under condition (3.10), and the integrand on the rightis non-negative, since φ = fu. Therefore,

∫ ∞

r0

A(∂rφ)2 + V φ2dx ≥0.

Step 2: Simplify the estimate to eliminate one of the coefficients. Wewill first introduce the notation

A =∆2

r2(r2 + a2).

We will consider the function

ϕ =A1/2φ.

Since A1/2 is smooth on [r+,∞) and vanishes linearly at r+, the new function ϕ isalso smooth and vanishes at least linearly at r+. Its derivative satisfies

∂rφ =1

A1/2(∂rϕ)− 1

2

∂rA

A3/2ϕ.

Therefore, the right-hand side of (3.9) is given by∫ ∞

r+

(∂rϕ)2 +

(V

A+

1

2

∂2rA

A2− 1

4

(∂rA)2

A2

)ϕ2dr

=

∫ ∞

r+

(∂rϕ)2 − ∂rA

Aϕ(∂rϕ) +

(1

4

(∂rA)2

A2+V

A

)ϕ2dr +

[1

2

∂rA

Aϕ2

]∞

r+

If this satisfies∫ ∞

r+

(∂rϕ)2 +

(V

A+

1

2

∂2rA

A2− 1

4

(∂rA)2

A2

)ϕ2dr ≥εHardy,2

∫ ∞

r+

1

Ar2ϕ2dr

then by multiplying this estimate by 1− εHardy,3

∫ ∞

r+

(∂rϕ)2 +

(V

A+

1

2

∂2rA

A2− 1

4

(∂rA)2

A2

)ϕ2dr

≥∫ ∞

r+

εHardy,3(∂rϕ)2 +

(εHardy,3

(V

A+

1

2

∂2rA

A2− 1

4

(∂rA)2

A2

)+ εHardy,2

1

Ar2ϕ2

)dr.

By taking εHardy,3 sufficiently small, we can conclude inequality (3.9) holds.Step 3: Construction of the explicit solution for a = 0 and εHardy = 0.

Following the arguments in the first section, we could prove the desired estimate(for a = 0 and εHardy = 0) by finding a positive solution to

−∂rA∂ru+ V u = 0 (3.11)

with

A =(r2 − 2Mr)2

r4,

V =1

6

9r2 − 46Mr + 54M2

r4

on the interval [2M,∞). However, by using the argument in the previous section,it is easier to use the transformed function

v =A1/2u =

((r2 − 2r)2

r4

)1/2

u, (3.12)

x =r − 2M, (3.13)


and to solve the ODE (3.11)

−∂2xv +Wv =0, (3.14)

W =V

A+

1

2

∂2xA

A− 1

4

(∂xA)2

A2

=9x2 − 34Mx− 2M2

6x2(x+ 2M)2(3.15)

on the interval x ∈ [0,∞).We first note the following properties of hypergeometric functions [1, 18]. The

hypergeometric function is typically written with parameters F (a, b; c; z). It shouldbe clear in all cases whether a refers to the first parameter of the hypergeomet-ric function or to the angular momentum parameter of the Kerr spacetime. Thehypergeometric function F (a, b; c; z) has the following integral representation fora < 0 < b < c and z 6∈ [1,∞)

F (a, b; c; z) =Γ(c)

Γ(a)Γ(b)

∫ 1

0

tb−1(1− t)c−b−1(1− tz)−adt. (3.16)

It is not obvious from this representation, but it is true, that F is symmetric in itsfirst two arguments, F (a, b; c; z) = F (b, a; c; z). There are a vast number of furtherrelations. The hypergeometric differential equation is

z(1− z)d2w

dz2+ [c− (a+ b+ 1)z]

dw

dz− abw = 0. (3.17)

A pair of solutions to this equation is

F (a, b; c; z),

z1−cF (a− c+ 1, b− c+ 1; 2− c; z).Returning to the ODE arising from the Hardy estimate, we introduce the further

substitution

v =xα(x+ d)β v. (3.18)

The ODE now becomes

v′′ =(α(α− 1)xα−2(x+ 2)β + 2αβxα−1(x+ d)β−1 + β(β − 1)xα(x+ d)β−2

)v

+ 2(αxα−1(x+ d)β + βxα(x+ d)β−1

)v′

+ xα(x+ d)β v′′,

0 =− v′′ +Wv

=xα−2(x+ d)β−2P, (3.19)

P =x2(x+ d)2v′′

− 2x(x+ d)((α+ β)x+ αd)v′

+

(− (α(α− 1)(x+ d)2 + 2αβx(x+ d) + β(β − 1)x2)

+9x2 − 34Mx− 2M2

6x2(x+ 2M)2x2(x+ d)2

)v. (3.20)

The prefactor of xα−2(x+ d)β−2 in the ODE (3.19) can be ignored. If we choose

d =2M, (3.21)

then the rational function in the last term on the right reduces to a polynomial.The coefficient of v′′ is x2(x+ d)2, of v′ is x(x+ d) times a linear function, and

of v a quadratic. If we choose the parameters α and β so that the coefficient ofv is a constant multiple of x(x + d), then an over all factor of x(x + d) can bedropped, leaving the coffecients of v′′, v′, and v as x(x+ d), a linear function, and


a constant respectively. The substitution z = −x/d, then transforms the equationto the hypergeometric differential equation. Our goal is to show such choices of αand β can be made.

It is now merely a matter of checking by direct calculation that this can be done.The coefficient of v is

− α(α− 1)(x2 + 2xd+ d2)− 2αβ(x2 + dx)− β(β − 1)x2

+ (3/2)x2 − (17/3)x− 1/3. (3.22)

In this coefficient, we set the constant order term to zero

−α(α− 1)d2 − 1/3 =0,

α =1

2±√

6

6.

Fortunately, the term αβ(x2 +dx) is already a multiple of x2 +dx, so we may ignoreit when trying to get the coefficient of v to be a multiple of x2 + dx. We set theratios of the remaining coefficients of x2 and of x to be d:

d ((3/2)− α(α− 1)− β(β − 1)) =− 2dα(α− 1)− (17/3).

We can substitute −α(α− 1) = 1/12 to get

2 ((3/2) + (1/12)− β(β − 1)) =(1/3)− (17/3),

β =1

2± 3√

2

2.

The four choices of sign provide four choices of simplified equations to study. Forsimplicity, we will consider only the equation arising from taking the + sign in αand the − sign in β.5

We are left with the differential equation for v

x(x+ 2)v′′ − 2((1 +√

6/6 + 3√

2/2)x+ 1 +√

6/3)v′

+(19/6− 3√

2/2 +√

6/6−√

3)v = 0,

Making the substitutions z = −x/d and ψ(z) = v(x) gives

z(1− z)ψ′′ +(

(1 +√

6/3)− (2− 3√

2 +√

6/3)z)ψ′

+(−19/6 + 3

√2/2 +

√3−√

6/6)ψ = 0. (3.23)

Thus we have a hypergeometric differential equation, with solution v = F (a, b; c,−x/d).We can immediately read off some quantites in terms of the hypergeometric param-eters

c =1 +√

6/3, (3.24)

−a− b− 1 =− 2 + 3√

2−√

6/3,

−ab =− 19/6 + 3√

2/2 +√

3−√

6/6.

We can now solve for the remaining two parameters

a, b =1

2− 3

2

√2 +

√6

6± 1

2

√7. (3.25)

We will make the choice a < b so that

a < −2.5 < 0 < .1 < b < .2 < 1.8 < c.

In particular

a < 0 < b < c.

5This choice simplifies some expressions in the rest of this argument.


Thus, the integral representation (3.16) holds. Multiplying by (−1)−aΓ(c)/(Γ(a)Γ(b)),

we find that ψ(z) is positive when z ≤ 0. This means that v is positive when x ≥ 0,v is also positive when x ≥ 0, and u is positive when r > 2M .

Step 4: The perturbed estimate for v. In this step, we will prove thatthere are 0 < aHardy,4 and 0 < εHardy,4 such that for |a| < aHardy,4 and all suitableϕ,

∫ ∞

0

|∂rϕ|2 + Wϕ2dx ≥ 0,

for

W =9x2 − 34Mx− 2M2

6x2(x+ d)2− εHardy,4

(M + x)2

x2(x+ d)2,

d =r+ − r−r− =M −

√M2 − a2.

This potential is of the form

W =C1x

2 + C2x+ C3

C4x2(x+ d)2, (3.26)

with the coefficients C1, . . . , C4, and d perturbed from their original values in equa-tion (3.15).

From the argument in step 1, it is sufficient to find a positive solution to theassociated ODE (3.14),

−∂2xv + Wv = 0,

with the perturbed potential W . The analysis in step 3 found an explicit, positivesolution for x ∈ [0,∞) for the parameter values dictated by the potential in equation(3.15). This step shows that the previous analysis also applies when the coefficientsare perturbed.

The previous analysis began by making the definition of v in equation (3.18), interms of the parameters α and β. The analysis then proceded by choosing valuesfor α and β by solving quadratic equations coming from the coefficient in formula(3.22), which lead to the new ODE (3.23). This ODE could be solved explicitelyin terms of a hypergeometric function by solving linear and quadratic equations forthe non-zero quantities a, b, and c. Since the coefficients in formula (3.22) dependcontinuously on the parameters C1, C2, C3, C4, and d in the potential; since thecoefficients in the ODE (3.23) depend continuously on α, β, and the coefficientsin the potential; since all the quadratic equations involved had distinct, real roots;and since solutions to linear and quadratic equations depend continuously on thecoefficients; it follows that positive solutions to the ODEs (3.14) and (3.23) can befound explicitely in terms of hypergeometric functions with parameters a, b, andc depending continuously on the parameters in W , at least when those parametervalues are sufficiently close to the values given in equation (3.15). Similarly, whenthe perturbation of the parameter values in the potential W is sufficiently small,then the hypergeometric parameters maintain their order a < 0 < b < c. This givesthe existence of positive aHardy,4 and εHardy,4 which give the desired estimate forthis step.

Step 5: The perturbed estimate for the original function φ. From theargument in step 2, we wish to prove that there exist 0 < a and 0 < εHardy suchthat for 0 ≤ |a| < a and suitable ϕ∫ ∞

r+

(∂rϕ)2 +

(V

A+

1

2

∂2rA

A2− 1

4

(∂rA)2

A2

)ϕ2dr ≥εHardy,2

∫ ∞

r+

1

r2

1

Ar2ϕ2dr, (3.27)


with

A =∆2

r2(r2 + a2),

V =1

6

9r2 − 46Mr + 54M2

r4.

With the benefit of foresight, we introduce a new rotation parameter

a =M −√M2 − a2.

When a is treated as a function of |a| with M fixed, this is a continuous, increasingfunction on the interval [0,M ], which maps the interval [0,M ] to [0,M ]. In addition,since the quantities which appear in our estimate (such as ∆ and r2+a2) only have aquadratic dependence on a, and since a2 can be solved for as a quadratic expressionin a, it follows that the quantities A and V are rational functions in (r,M, a).

The new radial coordinate, analagous to the one defined in (3.13), is now definedto be

x =r − r+ = r − (2M − 2a).

Since r can be solved for linearly in terms of (x,M, a), the quantities A and V arerational functions in (r,M, a).

The quantity

W =V

A+

1

2

∂2rA

A− 1

4

(∂rA)2

A2

is rational in (x,M, a); has degree, with respect to x, two lower in the numeratorthan in the denominator; has singularities in x ∈ [−d,∞) only at x ∈ 0,−d forfixed M and a; these are of order at most two; and, for sufficiently small a, has nosingularities in a for fixed x > 0 and M . Thus, we may expand it as

W =1

∆2

P0 + aP>Q0 + aQ>

,

where the functions P0 and Q0 are polynomials in (x,M), the functions P> and Q>are polynomials in (x,M, a), and Q0 and Q> have no roots in x ∈ [−d,∞). SinceP0/Q0 is determined explicitly by equation (3.15), it follows that

W − 1

∆2

P0

Q0=a

∆2

P>Q0 − P0Q>Q0(Q0 + aQ>)

must decay like r−2 as r →∞ for fixed a and M and has no singularities in [−d,∞).Since this is a rational function, there is a constant C such that

∣∣∣∣W −1

∆2

P0

Q0

∣∣∣∣ ≤aC(M + x)2

∆2.

Thus, there are sufficiently small a and εHardy,2 such that for 0 ≤ |a| < a

W − εHardy,21

Ar2> W ,

with W as in equation (3.26) The smallness of a and εHardy,2 is determined by thesmallness of aHardy,4 and εHardy,4. These then give a and εHardy for which the desiredestimate holds.

Step 6: Controlling the boundary terms. Since the argument from step 1was applied to the function ϕ, the boundary condition which must be imposed forthis argument to hold is that

ϕ2 ∂rv

v


vanishes at r+ and at ∞. Since the positive solution to the ODE is given by

v(r) =xα(x+ d)β v = xα(x+ d)βF (a, b; c;−z/d)

=(r − r+)α(r − r−)βF

(a, b; c;− r − r+

r+ − r−

),

and the hypergeometric function is analytic (in its fourth argument) near zero, theratio ∂rv/v will diverge at most inverse linearly at r = r+. Thus, it is sufficientthat ϕ vanish linearly at r = r+. Since ϕ = ∆/(r(r2 + a2)1/2)φ, it is sufficient thatφ be smooth near r+.

To show the decay near∞, we first note that from the form of the potential W inthe ODE, the solution v(r) will behave like a polynomial as r →∞, so that ∂rv/vwill decay like a constant times 1/r. Thus, it is sufficient that ϕ remains boundedat infinity. Thus, to get decay at both r+ and ∞, it is sufficient that φ be smoothand bounded on [r+,∞).

3.6. Integrating the Morawetz estimate.

Lemma 3.11. For positive parameters a and ε∂2t, there is a positive constant C

such that, for all |a| < a and all smooth ψ solving the wave equation ψ = 0, theestimate

C(ETχ [S2ψ](T2) + ETχ [S1ψ](T2) + ETχ [S2ψ](T1) + ETχ [S1ψ](T1))

≥∫ T2

T1

∫ ∞

r+

∫

S2

(∆2

r4

)|∂rψ|22,1 + r−2|ψ|22,1 + 1r 6h3M

1

r|ψ|23,1d4µ, (3.28)

holds, where 1r 6h3M is identically one, except in an open neighbourhood of the valuesof r for which there are orbiting geodesics. In this neighbourhood, we take 1r 6h3M

to be indentically zero.

Proof. We integrate the result of lemma 3.8 over the coordinate slab (t, r, θ, φ) ∈[T1, T2] × (r+,∞) × S2, from which we get the integral of the right-hand side ofestimate (3.6). From the Hardy estimate (3.9), the integral of the first two termson the right-hand side of (3.6) dominates an absolute constant times

∫ T2

T1

∫

S2

∫ ∞

r+

(∆2

r2(r2 + a2)|∂rψ|22,ε

∂2t

+1

r2|ψ|22,ε

∂2t

)d4µ.

By taking |a| . ε∂2t. 1, these terms will also dominate the fourth and fifth terms,

with a constant factor left over. Since ε∂2t

can be chosen independently of a, the

norms |ψ|2,ε∂2t

can be replaced by |ψ|2,1 at the price of a fixed constant. The same

is true for the norms of ∂rψ.The only term which we still need to estimate is the third,

∫ T2

T1

∫ ∞

r+

∫

S2

(r2 + a2)4

4r(3r2 − a2)Lαβ(∂αR′ψ)(∂βR′ψ)d4µ.

The integrand can be estimated by

(r2 + a2)4

4r(3r2 − a2)Lαβ(∂αR′ψ)(∂βR′ψ) ≥ (r2 + a2)4

4r(3r2 − a2)|T1R′ψ|2

≥1r 6h3M(r2 + a2)4

4r(3r2 − a2)|T1R′ψ|2

≥1r 6h3MO(r−1)|T1Lε∂2t

ψ|2

+ 1r 6h3MO(r−5)ε∂2t|T1La2ψ|2

+ 1r 6h3MO(r−3)a|T1S2ψ|2.


The first term in this expression can be bounded from below by inequality (2.4), sothat

|T1Lε∂2t

ψ|2 &|ψ|23,ε∂2t

+ a2|ψ|23,1 + (time and angular derivatives).

To control the remaining terms, we note that

ε∂2t|T1La2ψ|2 + a|T1S2ψ|2 .(ε∂2

ta2 + a)|∂3

t ψ|2 + (ε∂2ta2 + a)|∂2

t∇/ψ|2

+ (ε∂2t

+ a)|∂t∆/ψ|2 + (ε∂2t

+ a)|∇/∆/ψ|2.

If we impose the conditions that |a| . ε2∂2t. 1, then these terms are controlled by

|ψ|23,ε∂2t

≤|T1Lε∂2t

ψ|2 + time and angular derivatives,

so that

(r2 + a2)4

2r(3r2 − a2)Lαβ(∂αR′ψ)(∂βR′ψ) &1r 6h3Mr

−1|ψ|23,ε∂2t

+ 1r 6h3MO(r−1)(time and angular derivatives).

Thus, if we fix a sufficiently small ε∂2t, and then require |a| < ε2

∂2t, we have control

of |u|3 with the weight 1r 6h3Mr−1.

The time derivative generated in this part of the argument is

∂t(1r 6h3MO(r−1)(∂tT1ψ)(∆/T1ψ)),

where T1 is the set defined in section 2.4 to contain ∂t and the rotations around thecoordinate axes. Thus, the contribution of this time derivative on the boundary ofthe region of integration is bounded by P tTχ [S1ψ] + P tTχ [S2ψ].

We must now control the integral of the momentum and the boundary termsover the boundary of the slab. All the angular derivate terms vanish, since S2 hasno boundary. Similarly, the boundary contributions along r = r+ and r → ∞ arezero by lemma 3.9. (Geometrically, one would expect this, since r = r+ is actu-ally a two-dimensional surface, the bifurcation sphere, and not a three-dimensionalhypersurface, so it should not contribute any boundary terms.)

We are left to control the integral of the momentum and the boundary termsover the hypersurfaces t = T1 and t = T2. From lemma 3.9, these are controlled, atfixed t, by

∣∣∫PtA + BtA;I + BtA;IId

3µ∣∣

≤C∫ (PtTχ [S2u] + |S2u|2 + PtTχ [∂tu] + |∂tu|2

)d3µ

≤ETχ [S2u] + ETχ [∂tu].

In this, we have used the 1-dimensional Hardy estimate:∫ ∞

0

|ψ|2dx ≤∫ ∞

0

x2|∂xψ|2dx (3.29)

with x = r − r+.

We are unable to make use of the previous lemma since it controls only thirdderivatives, but the boundary terms involve both the second- and third-order ener-gies. (Certain second-derivative terms are controlled, but these are not the impor-tant ones.) In the following lemma, we control the lower-order derivatives. This isthe only place where we use any separability or spectral information.


Lemma 3.12. For the positive parameters a and ε∂2t, there is a positive constant

C such that for all |a| < a and all smooth ψ solving the wave equation ψ = 0, theestimate

C(ETχ,3[ψ](T2) + ETχ,3[ψ](T1)) (3.30)

≥∫ T2

T1

∫ ∞

r+

∫

S2

((∆2

r4

)|∂rψ|22 + r−2|ψ|22 + 1r 6h3M

1

r

(|∂tψ|22 + |∇/ψ|22

))d4µ.

holds, where 1r 6h3M is identically one, except in an open neighbourhood of the valuesof r for which there are orbiting geodesics.

Proof. We must control the weighted integral of the first, second, and third deriva-tives. The Morawetz estimate, (3.28), controls the third derivatives, so we onlyneed to control the lower-order derivatives.

Since ∂φ commutes with the d’Alembertian, it is possible to apply separate outthe zero eigenmode and write ψ in the form

ψ = ψLz=0 + ψLz 6=0.

From the Morawetz estimate, (3.6), we control the integral of the third derivatives.If ψ is real valued, then ψLz=0 and ψLz 6=0 are also real valued.

Since the ∂φ eigenvalue, Lz, satisfies |Lz| ≥ 1, for ψLz 6=0, the integral over thesphere of the third derivatives controls all lower derivatives, i.e. the homogeneousnorm controls the inhomogeneous norm,

∫

S2

|ψLz 6=0|2nd2µ ≤ C∫

S2

|ψLz 6=0|2n,1d2µ.

It remains to control the lower derivatives of ψLz=0. To do this, we prove aMorawetz estimate using a classical, first-order vector-field. Consider the momen-tum associated with

A =zwf∂r,

qA =1

2(∂rA

r)− q′A,

q′A =1

2(∂rz)wf,

f =∂r

(z1/2

∆1/2

∆

(r2 + a2)

).

Using the same sort of calculations as before, we can obtain the analogue of (3.5)

1

µ∂α

(µPα(A,qA)

)=A(∂rψLz=0)2 + Uαβ(∂αψLz=0)(∂βψLz=0) + V|ψLz=0|2,

with

A =1

2z1/2∆3/2∂r

(ww1/2

∆1/2f

),

Uαβ =1

2w(∂rf)R′αβ ,

V =− 1

4∂r∆∂rz(∂rwf).


Taking the same choices of z and w as before, we find

A =1

2

∆2

r2 + a2

(1

r2+ (a+ ε∂2

t)O(r−3)

),

Uαβ(∂αψLz=0)(∂βψLz=0) =1

2(∂rf)2Qαβ(∂αψLz=0)(∂βψLz=0)

+1

2ε∂2t

(∂r

∆

(r2 + a2)2

)(1− 2ε∂2

t

∆

(r2 + a2)2

)(∂tψLz=0)2

V =1

6

9Mr2 − 46M2r + 54M3

r4+ (a+ ε∂2

t)O(r−4).

Note that there are no ∂t∂φ or ∂2φ arising from R′ when acting on ψLz=0. From the

Hardy estimate, (3.9), it follows taht

A(∂rψLz=0)2 + V|ψLz=0|2 &∆2

r2(r2 + a2)|∂rψLz=0|2 +

1

r2|ψLz=0|2.

Thus,

1

µ∂α

(µPα(A,qA)

)& ∆2

r2(r2 + a2)|∂rψLz=0|2 +

1

r2|ψLz=0|2

+ 1r 6h3M (|∂tψLz=0|2 + |∇/ψLz=0|2).

In analogy with the previous results in lemma 3.9, there is a constant and anupper bound on a such that

|E(A,qA)[ψLz=0]| ≤CETχ [ψLz=0].

Thus, we have the Morawetz estimate

C(ETχ(T2) + ETχ(T1)) (3.31)

≥∫ T2

T1

∫ ∞

r+

∫

S2

((∆2

r4

)|∂rψLz=0|2 + r−2|ψLz=0|2

+ 1r 6h3M1

r

(|∂tψLz=0|2 + |∇/ψLz=0|2

))d4µ, (3.32)

which controls the first derivatives.Second derivatives have either two, one, or zero time derivatives. Any term

containing one or more time derivative can be controlled by applying (3.32) to∂tψLz=0. Terms which contain no time derivatives contain at least one angularderivative. Thus, by integrating by parts in the angular derivative and applyingthe Cauchy-Schwarz inequality, we can control such terms by the first and thirdderivatives, which are controlled by (3.32) and (3.6).

By combining the results for ψLz=0 and ψLz 6=0, we have the desired result.

3.7. Closing the argument. We are now able to show that the energy associatedwith Tχ is uniformly bounded by its value on the initial hypersurface. When a = 0,the energy is conserved. When a 6= 0, the energy is no longer conserved, but thefactor by which it can change vanishes linearly in |a|.Theorem 3.13. There are positive constants a and C such that if |a| < a and ψis a solution to the wave equation ψ = 0, then for all t2 ≥ t1 ≥ 0:

ETχ,3[ψ](t2) ≤ (1 + C|a|)ETχ,3[ψ](t1).

Proof. By lemma 3.2

ETχ,3[ψ](t2) + ETχ,3[ψ](t1)

≤|a|C∫

[t1,t2]×(r+,∞)×S2

1suppχ′(|∂rψ|22 + |ψ|23

)d4µ.


By the Morawetz estimate, lemma (3.12), for sufficiently small a, there is a constantC ′ such that the integral of the third derivatives is controlled by the energies. Thus,

ETχ,3[ψ](t2)− ETχ,3[ψ](t1) ≤|a|C ′(ETχ,3[ψ](t2) + ETχ,3[ψ](t1)

).

Thus, for sufficiently small a,

(1− |a|C ′)ETχ,3[ψ](t2) ≤(1 + |a|C ′)ETχ,3[ψ](t1),

ETχ,3[ψ](t2) ≤1 + |a|C ′1− |a|C ′ETχ,3[ψ](t1).

Since, for sufficiently small |a|, the rational function (1+|a|C ′)/(1−|a|C ′) is boundedabove by 1 + C|a| for some C, the desired result holds.

4. Decay estimates for the local energy

In this section, we prove decay using the K vector field. To prove decay, we find itconvenient to work in a different set of coordinates and to work with a transformedfunction. This allows us to write the wave equation as the Euler-Lagrange equationfor a different Lagrangian. A single equation can be the Euler-Lagrange equationfor more than one Lagrangian. The advantage of this Lagrangian is that it allowsus to control terms involving |ψ|2 with the energy.

4.1. Reformulation of the problem. Recall that the r∗ coordinate is defined by

dr

dr∗=

∆

r2 + a2,

and by the choice of initial condition that r∗ = 0 at r = 3M . In the Schwarzschildcase, this choice of initial condition is convenient since it fixes the origin of the r∗coordinates at the photon orbits, and, in the Kerr case, no other choice seems moreconvenient. In this section, we will work with the (t, r∗, θ, φ) coordinates and treatr as a function of r∗. All indices will refer to these coordinates.

Lemma 4.1. Let

ψ =√r2 + a2ψ .

The wave equation ψ = 0 is equivalent to

µ−1∂α

(µGαβ

)∂βψ − V ψ =0, (4.1)

where1

µ∂α

(µGαβ

)∂β =∂2

r∗ +1

(r2 + a2)2R,

V =∆

(r2 + a2)4(2Mr3 + r2a2 − 4Mra2 + a4).

Here, it is understood that R is the operator defined in (1.6).

Proof. Direct computation. The matrix Gαβ is obtained from multiplying the inverse metric gαβ , evaluated in

the (t, r∗, θ, φ) coordinate system by the function Σ∆(r2 + a2)−2. The transformedwave equation in (4.1) can be written as the Euler-Lagrange equation for

S =

∫Ld4µ∗,

L =1

2

(Gαβ(∂αψ)(∂βψ) + V ψ2

), µ = sin θ.

Section 2.1 defines the canonical energy-momentum tensor, the momentum vector,and the energy for this Lagrangian. We denote these

T [ψ], PX[ψ], EX[ψ].


The related bilinear terms are defined similarly.In the r∗ coordinates, we have

Gαβ =− N−2Tα⊥Tβ⊥ + ∂α∗ ∂

β∗ + hαβ , (4.2)

with

VQ =∆

(r2 + a2)2,

N−2 =1− a2 sin2 θVQ, (4.3)

hαβ =VQ∂αθ ∂

βθ

+ VQ

(1− a2 sin2 θ

r2 + 2Mr + a2 cos2 θ

Π

)1

sin2 θ∂αφ∂

βφ . (4.4)

4.2. Energies in the reformulation.

Lemma 4.2 (Dominant energy condition for T ). There are positive constant a andC, such that, if |a| < a, and if X and Y are future-directed, non-spacelike vectors(with respect to ∂t and g), then

−T αβgαγXβYγ ≥ 0 (4.5)

In particular,

CPtTχ ≥ (T⊥ψ)2 + (∂r∗ ψ)2 + VQ|∇/ψ|2 + V |ψ|2. (4.6)

Proof. The energy-momentum tensor can be written as

−T [ψ]αβ =Gαγ(∂γψ)(∂βψ)− 1

2δαβGδε(∂εψ)(∂δψ)− 1

2δαβV ψ

2. (4.7)

=∆Σ

(r2 + a2)2

(gαγ(∂γψ)(∂βψ)− 1

2gαβg

δε(∂εψ)(∂δψ)

)− 1

2gαβV ψ

2

=∆Σ

(r2 + a2)2T [ψ]αβ −

1

2gαβV ψ

2.

Thus, the reformulated energy-momentum tensor can be written as the sum of twoterms. The first satisfies the dominant energy condition because it is a positivemultiple of the standard energy-momentum tensor evaluated on ψ. The secondterm is a negative factor multiplied by the metric, so it also satisfies the dominantenergy condition. Thus, T [ψ]αβgαγ is the sum of two terms which each satisfy thedominant energy condition, so it satisfies the dominant energy condition.

We now turn to the second part of the lemma. From equation (4.7),

T αβTβχ(dt)α =− (Tχψ)Gtγ(∂γψ) +

1

2Gγδ(∂γψ)(∂δψ) +

1

2V ψ2

From the fact that Tαχ∂α = −Gtα∂α, Tχ = T⊥ + aO((∆/r2), r−2)∂φ, and the

expansion of G in (4.2) and (4.4),

T αβTβχ(dt)α =N−2(T⊥ψ)(T⊥ψ) + aO((∆/r2), r−2)(T⊥ψ)(∂φψ)

+1

2

(−N−2(T⊥ψ)2 + (∂∗ψ)2 + hαβ(∂αψ)(∂βψ)

)+

1

2V ψ2.

Since we assume that a is small, and since O((∆/r2)2, r−4) decays more rapidly

than the product of N−2 h 1 and O((∆/r2), r−2), which is the decay rate of the

leading-order piece of h, the term aO((∆/r2), r−2)(T⊥ψ)(∂φψ) is controlled by the

(1/2)(Tχψ)2+hαβ(∂αψ)(∂βψ)) terms with only a small loss. This proves the desiredresult.

Lemma 4.3 (Relation between energy for u and ψ). There are positive constants

a and C, such that, if |a| ≤ a, and if ψ = (r2 + a2)1/2ψ, then


(1) (Global identity)

ETχ [ψ](t) =ETχ [ψ](t). (4.8)

(2) (Local estimate) Given r1 < r2 and r∗1 < 0 < r∗2 such that r∗ = r∗1corresponds to r = r1 and r∗ = r∗2 corresponds to r = r2,

∫ r2

r1

∫

S2

|PtTχ [ψ]|+ |ψ|2d3µ .∫ r∗2

r∗1

∫

S2

PtTχ [ψ]d3µ∗.

Proof. This argument follows by substituting ψ = (r2 + a2)−1/2ψ into the energyETχ , changing from (t, r, θ, φ) to (t, r∗, θ, φ) variables, and then integrating by partsin the r∗ variable.

ETχ [ψ](t)

=

∫T [ψ]tβTβ

χd3µ

=

∫−(Tχψ)Gtγ(∂γψ) +

1

2(1)Gβγ(∂βψ)(∂γψ)d3µ

=

∫−(Tχψ)

1

r2 + a2Gtγ(∂γψ) +

1

2

1

r2 + a2Gβγ(∂βψ)(∂γψ)d3µ (4.9)

+

∫∆√

r2 + a2

(∂r

1√r2 + a2

)ψ(∂rψ) +

1

2∆

(∂r

1√r2 + a2

)2

ψ2d3µ.

We can simplify the final term using integration by parts to obtain

∫∆√

r2 + a2

(∂r

1√r2 + a2

)ψ(∂rψ) +

1

2∆

(∂r

1√r2 + a2

)2

ψ2d3µ

=

∫1

2

(∂r

(∆√

r2 + a2∂r

1√r2 + a2

)+ ∆

(∂r

1√r2 + a2

)2)ψ2d3µ

=

∫1

2

∆

r2 + a2

(∂r

(∆√

r2 + a2∂r

1√r2 + a2

)+ ∆

(∂r

1√r2 + a2

)2)ψ2d3µ

=

∫1

2V ψ2d3µ.

We were able to drop all boundary terms in the integration by parts, since theboundary terms involve bounded factors times ∆, which vanishes at r+, or involve

factors of ψ or its derivative which are assumed to decay rapidly as r → ∞ onsurfaces of fixed t. (Merely decay such that r−1/2ψ → 0 or r1/2ψ → 0 is sufficienthere.) Thus, we find

ETχ [ψ](t) =ETχ [ψ](t).


By a similar argument, using the Cauchy-Schwarz inequality on (4.9)∫ r2

r1

∫

S2

|PTχ [ψ]t|d3µ

=

∫ r2

r1

∫

S2

−(Tχψ)1


1

2

1

r2 + a2Gβγ(∂βψ)(∂γψ)d3µ

+

∫ r2

r1

∫

S2

∆

(r2 + a2)2|ψ||∂rψ|+

1

2

∆

(r2 + a2)3|ψ|2d3µ

.∫ r2

r1

∫

S2

−(Tχψ)1


1

2

1

r2 + a2Gβγ(∂βψ)(∂γψ)d3µ

+

∫ r2

r1

∫

S2

∆

(r2 + a2)3|ψ2|d3µ

.∫ r∗2

r∗1

∫

S2

−(Tχψ)Gtγ(∂γψ) +1

2Gβγ(∂βψ)(∂γψ)d3µ∗

+

∫ r∗2

r∗1

∫

S2

∆2

(r2 + a2)4|ψ2|d3µ∗.

As a consequence of [7, equation (36)], there is the following Hardy estimate for anynon-negative, continuous weight χ which is positive at zero

∫ r∗2

r∗1

1

1 + x2|φ|2dx .

∫ r∗2

r∗1

|∂xφ|2 + χ|φ|2dx, (4.10)

with the implicit constant in . depending on χ, but not on r∗1 and r∗2, as long as

r∗1 < −1 and 1 < r∗2. Since G∗∗ = 1, we can apply this result to the energy for ψ,so that

∫ r∗2

r∗1

∫

S2

PtTχ [ψ]d3µ∗

&∫ r∗2

r∗1

∫

S2

−(Tχψ)Gtγ(∂γψ) +1

2Gβγ(∂βψ)(∂γψ)d3µ∗

+

∫ r∗2

r∗1

∫

S2

1

1 + r2∗|ψ|2d3µ∗.

Since ∆2(r2 + a2)−4 . (1 + r2∗)−1,

∫ r2

r1

∫

S2

|PTχ [ψ]t|d3µ .∫ r∗2

r∗1

∫

S2

PTχ [ψ]d3µ∗.

By the same argument∫ r2

r1

∫

S2

|ψ|2d3µ =

∫ r∗2

r∗1

∫

S2

∆

(r2 + a2)2|ψ|2d3µ∗

.∫ r∗2

r∗1

∫

S2

PtTχ [ψ]d3µ∗.

Corollary 4.4 (Local decay for ψ). If ψ is a solution to the wave equation ψ = 0,then the right-hand side of (3.30) is greater than∫ (

∆

(r2 + a2)

1

r2|∂r∗ ψ|22 +

∆

r2 + a2

1

r4|ψ|22 +

∆

r2 + a2

1r 6h3M

r3(|∂tψ|22 + |∇/ψ|22)

)d4µ∗.

(4.11)


Proof. The right-hand side of (3.30) is∫ T2

T1

∫ ∞

r+

∫

S2

((∆2

r4

)|∂rψ|22 +

1

r2|ψ|22 +

1r 6h3M

r

(|∂tψ|22 + |∇/ψ|22

))d4µ.

Making the substitution ψ = ψ(r2+a2)−1/2 introduces an extra factor of (r2+a2)−1,except in the factor with the radial derivatives, where the situation is a little morecomplicated. The norm |u|2 only introduces derivatives which commute with the

radial derivative, so we may ignore them. We will use v to denote ψ with possiblya derivative operator acting on it.

The sum of the radial derivative and lower-order terms has the form∫

∆2

r4

∣∣∣∣∂rv√

r2 + a2

∣∣∣∣2

+1

r2

1

r2 + a2|v|2dr

=

∫∆2

r4(r2 + a2)|∂rv|2 − 2

r∆2

r4(r2 + a2)2|∂rv||v|

+

(∆2

(r2 + a2)2

r2

(r2 + a2)3+

1

r2

1

r2 + a2

)|v|2dr.

Applying Cauchy-Schwarz to the term involving v∂rv and using the extra positivityfrom the original lower-order term, we have that

∫∆2

r4

∣∣∣∣∂rv√

r2 + a2

∣∣∣∣2

+1

r2

1

r2 + a2|v|2dr &

∫∆2

(r2 + a2)3|∂rv|2 +

1

r4|v|2dr

&∫

1

r2 + a2|∂r∗v|2 +

1

r4|v|2dr.

Making the change of variables from r to r∗, introduces an extra factor of ∆/(r2+a2) in the measure. Thus, the right-hand side of (3.30)dominates

∫ (∆

(r2 + a2)2|∂r∗ ψ|22 +

∆

r6|ψ|22 + 1r 6h3M

∆

r5|ψ|23

)d4µ∗.

These coefficients dominate the coefficients given in the statement of this theoremare dominated, so the desired result holds.

Because we will need to work with this later to integrate by parts in the angularvariables, we introduce the following Morawetz density with additional angularderivatives. Note that since the angular derivatives do not commute with S2, thisis not quite the same as the Morawetz density evaluated on angular derivatives ofu.

Definition 4.5. We define the higher-order angular Morawetz densities to

(δPA)[On1 , ψ] =∆

(r2 + a2)

1

r2|On1∂rS2ψ|2

+∆

(r2 + a2)

1

r3(|On1∂tS2ψ|2 + |On1∇/S2ψ|2)

+∆

(r2 + a2)

1

r4|On1S2ψ|2.

(Recall O1 = Θi was defined to be the set of rotations about the coordinate axesin section 2.4.)

4.3. K in Kerr. We now define the K vector field in analogy with the correspond-ing vector field in the Schwarzschild or Minkowski spacestimes. It is well knownthat this type of vector field generates a stronger energy which can be used to provedecay.


Definition 4.6. We define

K =(t2 + r2∗ + 1)Tχ + 2tr∗N

2∂r∗ ,

qK =t(N2 − 1),

where N is defined in (4.3).

We now prove K is timelike. The essence of this argument is the same as inthe Minkowski or Schwarzschild spacetime. The length of Tχ is very close to thatof ∂∗, so (t2 + r2

∗)Tχ + 2tr∂r∗ is timelike when t2 + r2∗ dominates 2tr∗. When

t2 + r2∗ is comparable to 2tr∗, (t2 + r2

∗)Tχ+ 2tr∗∂r∗ can become null or even slightlyspacelike. By adding a little more Tχ, we have built the always timelike vector-field K = (t2 + r2

∗ + 1)Tχ + 2tr∗∂r∗ . This trick of adding a little more of thetimelike vector-field to get a globally timelike K is common in the literature for theMinkowski or Schwarschild spacetime.

Lemma 4.7. There is a positive constant a, such that if |a| < a, then K is timelikewith respect to g.

Proof. Since Tχ is timelike and orthogonal to ∂r∗ , it is sufficient to show that theabsolute value of the norm of (t2 + r2

∗ + 1)Tχ dominates the norm of 2tr∗∂r∗ .For (t, r∗) close to (0, 0), the term Tχ dominates the contribution from all the

other pieces of K, so that K is timelike in this region. We can therefore assumethat at least one of t or |r∗| is bigger than some fixed constant in the rest of thisproof. By taking a sufficiently small, we may assume χ is constant for r∗ biggerthan this constant. (Alternatively, we could take K = (t2 + r2

∗ + C)Tχ + 2trN2∂rwith C sufficiently large, so that this step in the argument would not impose anyfurther restriction on the size of a.)

To prove the lemma, we break the spacetime into three different regions. In thelimit a→ 0, the surfaces t = r∗ describe light cones. For a 6= 0, the surfaces t = r∗are no longer light cones, although give a rough approximation of the location ofthe light cones. In particular, for a small ε > 0, if a is sufficiently small, the regions|r∗| < (1− ε)t remain inside the light cone.

Before considering different regions, we note that

−g(T⊥,T⊥) =∆Σ

(r2 + a2)2N−2 =

∆Σ

(r2 + a2)2

(1− a2 sin2 θ

∆

(r2 + a2)2

)

−g(∂t, ∂t) =1− 2Mr

Σ

g(∂r∗ , ∂r∗) =Σ

∆

(∆

r2 + a2

)2

=∆Σ

(r2 + a2)2.

First, we consider the region r∗ ≤ −(1−ε)t. In this region, the difference in normbetween Tχ and T⊥ is Ca∆2, so the absolute value of the norm of Tχ is boundedbelow by ∆(1 − Ca2∆), and the norm of ∂r∗ is bounded above by ∆(1 + Ca2∆).The coefficient (t2 + r2

∗ + 1)2 dominates (2tr∗)2 + t2. For a sufficiently small

|(t2 + r2∗ + 1)2g(Tχ,Tχ)| ≥(t2 + r2

∗ + 1)2∆(1− Ca2∆)

≥((2tr∗)2 + t2)∆(1− Ca2∆)

≥(2tr∗)2∆ + (1/2)t2∆

≥(2tr∗)2∆(1 + Ca2∆)

≥(2tr∗)2|g(∂r∗ , ∂r∗)|2.

Second, we consider the region −(1 − ε)t ≤ r∗ ≤ (1 − ε)t. From the estimateson the norms of T⊥ and ∂t, it follows that, uniformly in the region currently under


consideration,

−g(Tχ,Tχ) ≥ ∆Σ

(r2 + a2)2

(1− Ca2 ∆

(r2 + a2)2

).

In the region under consideration, we also have

|t± r∗| ≥εtt2 + r2

∗ ≥|2tr∗|+ ε2t2

(t2 + r2∗)

2 ≥(2tr∗)2 + ε4t4

≥(1 + ε4/8)(2tr∗)2 + (ε4/2)t4.

Thus, for sufficiently small a based on ε

|(t2 + r2∗ + 1)2g(Tχ,Tχ)| ≥(1 + ε4/8)(2tr∗)

2g(∂r∗ , ∂r∗)

+ (ε4/2)t4|g(Tχ,Tχ)|.

Thus, K is timelike.Finally, we turn to the region (1−ε)t ≤ r∗. (We may also assume we’re sufficiently

far from (t, r∗) = (0, 0) that Tχ = ∂t.) Here, −g(Tχ,Tχ) = −g(∂t, ∂t) = 1 −2Mr/Σ ≥ (1 − 2Mr−1)(1 − Ca2r−2), and g(∂r∗ , ∂r∗) = ∆Σ(r2 + a2)−2 ≤ (1 −2Mr−1)(1 +Ca2r−2). Following the same type of argument as we used in the firstregion, we have

(t2 + r2∗ + 1)2|g(Tχ,Tχ)|

≥(1− 2Mr−1)((t2 + r2

∗)2(1− Ca2r−2) + (t2 + r2

∗)(1− Ca2r−2))

≥(1− 2Mr−1)((2tr∗)

2 + t2(1− Ca2))

≥(1− 2Mr−1)(2tr∗)2(1 + Ca2r−2)

≥(2tr∗)2g(∂r∗ , ∂r∗).

We now show that the K momentum dominates the Tχ momentum and that, onhypersurfaces of constant t, the energy assciated with qK is small relative to the Kenergy, so that we can freely add or drop the energy associated with qK when wefind it convenient to do so.

Lemma 4.8 (K energy). There are positive constants a, CKE, and CKE,2, suchthat if |a| ≤ a, and if ψ is smooth, then :

(1) (estimate on the energy)

CKEPtK ≥∣∣∣(t+ r∗)(T⊥ + ∂r∗)ψ

∣∣∣2

+∣∣∣(t− r∗)(T⊥ − ∂r∗)ψ

∣∣∣2

+ (t2 + r2∗ + 1)(hαβ(∂αψ)(∂βψ) + V ψ2).

As a consequence, for |r∗|2 ≤ (3/4)t,

CKEPtK ≥ t2PtTχ .

(2) The contribution to the energy from qK is small in the sense that

|EqK(t)| ≤ CKE∫

Σt

|GtβqK(∂βψ)ψ − 1

2Gtβ(∂βqK)ψ2|d3µ∗ ≤a2CKE,2EK(t).


Proof. This follows the same ideas as the previous lemma. We start by computingthe energy,

PtK =− Gtα(∂αψ)Kβ(∂β) +1

2KtGαβ(∂αψ)(∂βψ)

=N−2(T⊥ψ)(t2 + r2∗ + 1)(T⊥ψ) + (T⊥ψ)2tr∗N

2(∂r∗N)

− 1

2(t2 + r2

∗ + 1)(N−2(T⊥ψ)2 − (∂r∗ ψ)2 − hαβ(∂αψ)(∂βψ)− V ψ2)

=1

2(t2 + r2

∗ + 1)(N−2(T⊥ψ)2 + (∂r∗ ψ)2 + hαβ(∂αψ)(∂βψ) + V ψ2) (4.12a)

+ (t2 + r2∗ + 1)N−2(T⊥ψ)(ωHχ− ω⊥)(∂φψ) (4.12b)

+ 2tr∗N2(T⊥ψ)(∂r∗ ψ). (4.12c)

Since ωHχ−ω⊥ . a∆/(r2 +a2)2, the term (4.12b) is dominated by a small multipleof (4.12a). In fact, because of the weights, we are left with

1

2(t2 + r2

∗ +1

2)(N−2(T⊥ψ)2 + (∂r∗ ψ)2 + hαβ(∂αψ)(∂βψ) + V ψ2)

from dominating (4.12a) and (4.12b) together. We now seek to control (4.12c) usingthis remaining term.

First we consider the region ||r∗| − t| < t/2. In this region, |tr∗(N2 − 1)| . a

and |(t2 + r2∗ + 1)(N2 − 1)| . a, so that the term (T⊥ψ)2 + (∂r∗ ψ)2 dominates any

terms arising from replacing all the factors of N by 1. Thus, in this region,

PtK ≥1

2

(t2 + r2

∗ +1

2

)((T⊥ψ)2 + (∂r∗ ψ)2 + hαβ(∂αψ)(∂βψ) + V ψ2

)

+ 2tr∗(T⊥ψ)(∂r∗ ψ)

&∣∣∣(t+ r∗)(T⊥ + ∂r∗)ψ

∣∣∣2

+∣∣∣(t− r∗)(T⊥ − ∂r∗)ψ

∣∣∣2

+ (t2 + r2∗ + 1)(hαβ(∂αψ)(∂βψ) + V ψ2).

In the region ||r∗| − t| ≥ t/2, the dominant term (t2 + r2∗)((T⊥ψ)2 + (∂r∗ ψ)2)

dominates (t2 + r2∗)(N

−2 − 1)((T⊥ψ)2 + (∂r∗ ψ)2) and the terms in (4.12b) and(4.12c) by at least a factor of 2, so that the desired estimate holds. The control

over PtTχ follows from this.

We now turn to the terms arising from qK. These terms are


2Gtβ(∂βqK)ψ2|

.t|N2 − 1||T⊥ψ||ψ|+1

2|N2 − 1|ψ2

.a2t2∆2

(r2 + a2)4ψ2 + a2(T⊥ψ)2 +

a2∆

(r2 + a2)2ψ2.

The first term is controlled by PtK, and the second, by PtTχ . The decay of the final

term is too slow to be directly estimated by the energy, but after integrating alonga hypersurface of constant t and applying the Hardy estimate (4.10), the third term

can be estimated by ETχ . Thus,

∫

Σt


2Gtβ(∂βqK)ψ2|d3µ∗ .a2EtK.


Lemma 4.9 (Bulk term for K). If ψ is a solution to the wave equation (4.1) and

ψ =√r2 + a2ψ, then

1

µ∂α

(µPα(K,qK),3

).

t−2PtTχ,3 for r∗ ≤ −t/2

(t−2 log t)PtK,3 for r∗ ≥ t/2,

and, in the region −t/2 ≤ r∗ ≤ t/2,

1

µ∂α

(µPα(K,qK),3

).t(δPA)[ψ]

+ t(δPA)[O11, ψ]

+ |a|t2(δPA)[ψ]

+ |a|t−1PtK,3.Proof. We begin by considering

K =(t2 + r2∗ + 1)TK + 2tr∗N

2∂r∗ ,

TK =∂t + ωK∂φ,

and derive conditions on ωK. Near the end of this proof we will conclude thatTK = Tχ is permissible. For simiplicity, for most of this proof, we use v to denote

a second-order symmetry operator acting on ψ, and consider the deformation ofP(K,qK)[v]. Similarly, unless otherwise specified, PtTχ and PtK are understood to be

evaluated on v.The bulk term we want to control is

1

µ∂αµPα(K,qK) =

(−Gαγ∂γKβ +

1

2(∂γK

γ)Gαβ +1

2Kγ(∂γ Gαβ)

)(∂αv)(∂β v)

(4.13a)

− qKGαβ(∂αv)(∂β v) (4.13b)

+

(1

2(∂γK

γ)V +1

2Kγ(∂γV )− qKV

)v2, (4.13c)

+1

2(∂βGαβ∂αqK)v2. (4.13d)

With an eye to computing the derivative terms, we first compute

∂γKγ =2t+ 2tN2 + 2tr∗(∂r∗N

2)

=4tN2 + 2t(1− N2 + r∗∂r∗N2).

The terms involving the derivatives and arising from the vector field are

(∂αv)(∂β v)

(−Gαγ∂γKβ +

1

2(∂γK

γ)Gαβ +1

2Kγ(∂γ Gαβ)

)

=N−2(T⊥v)2t(TKv) + (T⊥v)2r∗(∂∗v)

− (∂∗v)2r∗(TKv)− (∂∗v)(t2 + r2∗)(∂r∗ωK)(∂φv)

− (∂∗v)2tN2(∂∗v)− (∂∗v)22tr∗(∂r∗N2)

− hαβ(∂αv)(t2 + r2∗ + 1)(∂βωK)(∂φv)− hαβ2tr∗(∂βN

2)(∂r∗ v)

− (2tN2 + t(1− N2 + r∗∂r∗N2))N−2(T⊥v)2

+ (2tN2 + t(1− N2 + r∗∂r∗N2))(∂∗v)2

+ (2tN2 + t(1− N2 + r∗∂r∗N2))hαβ(∂αv)(∂β v)

− tr∗(∂r∗N−2)(T⊥v)2

+ tr∗(∂r∗ hαβ)(∂αv)(∂β v).


Expanding TK as T⊥ + aO(∆, r−3) and grouping terms, we find

(∂αv)(∂β v)

(−Gαγ∂γKβ +

1

2(∂γK

γ)Gαβ +1

2Kγ(∂γ Gαβ)

)

=− (T⊥v)2t(2(1− N−2) + N−2(1− N2 + r∗∂r∗N2) + r∗∂r∗N

−2)(4.14a)

+ (T⊥v)(∂φv)(2t)(ωK − ω⊥) (4.14b)

− (∂∗v)2t(2r∗(∂r∗N−2)− (1− N2 + r∗∂r∗N

2)) (4.14c)

− (∂∗v)(∂φv)((t2 + r2∗ + 1)(∂r∗ωK) + 2r∗(ωK − ω⊥)) (4.14d)

− (∂∗v)(∂β v)hαβ2tr∗(∂βN2) (4.14e)

− (∂αv)(∂β v)t(−2N2hαβ − r∗(∂r∗ hαβ)− (1− N2 + r∗∂r∗N2)hαβ)

(4.14f)

− (∂αv)(∂φv)hαβ(t2 + r2∗)(∂βωK). (4.14g)

In considering these calculations, it is important to note that the worst terms,involving (T⊥v)(∂∗v) are exactly cancelled. To force this cancellation, we have

included N2 in K = (t2 + r2∗ + 1)Tχ + 2N2tr∗∂r∗ , instead of using (t2 + r2

∗ + 1)∂t +2tr∗∂r∗ , which is a more straight-forward analogue of the vector field used in theMinkowski and Schwarzschild spacetimes. We now analyse the remaining terms.

For the terms (4.14a), (4.14c), and (4.14f), which we consider to be like derivativessquared times t times coefficients, we look at just the coefficients. We also includethe contribution from qKGαβ . The coefficients of (T⊥ψ)2 and (∂r∗ ψ)2 are

Coefficient from (4.14a) =− N−2(N2 − 1 + r∗∂r∗N2)− N2r∗∂r∗N

−2 + qKN−2

=− N−2(N2 − 1 + r∗∂r∗N2) + N−2r∗∂r∗N

2 + qKN−2

=− N−2(N2 − 1− (qK/t))

=0.

Coefficient from (4.14c) =− 2r∗∂r∗N2 + (1− N2 + r∗∂r∗N

2)− (qK/t)

=− N2 + 1− r∗∂r∗N2 − (qK/t)

=− 2(N2 − 1) + N4r∗∂r∗N−2

=a2 sin2 θO(r∗∆, r−3 log r).

Thus, for r∗ < −t/2, with the t(∂r∗ v)2 factor, the term is of the form a2tr∗∆(∂r∗ v)2 .a2t−2(∂r∗ v)2 . t−2PtTχ . For −t/2 ≤ r∗ ≤ t/2, the term is bounded by a2t−1PtK.

For r∗ ≤ t/2, the term is bounded by a2(t−2 log t)PtTχ . a2(t−2 log t)PtK.

The angular derivatives are given by

Coefficient from (4.14f) =2hαβ + r∗(∂r∗ hαβ)

+ (2(N2 − 1) + (1− N2)− r∗(∂r∗N2)− (qK/t))hαβ .

the first two terms of the coefficient of (4.14f) are of a familiar form from theSchwarzschild spacetime. Outside a compact set near the orbiting null-geodesics,they have a good sign, i.e. are negative, and decay like ∆ near the event horizonand not slower than r−3 log r as r →∞. The remaining terms involve only factorsinvolving either (N2−1) or r∗∂r∗N

−2 multiplied by the potential VQ in h. Thus, allthe terms should decay at least like log(∆)∆2 near the horizon and at least as fastas r−4 as r → ∞. In addition, these all have a factor of a on them. Thus, outsidea slightly larger compact set, the negativity of the first two terms is sufficient to


control the rest. Inside that compact set, the sum can be controlled by

(δPA)[O11, ψ].

We now turn to showing the remaining terms are small. The terms we wish tocontrol all involve mixed derivatives. We first estimate these away from photonorbits. We break the remainder of the spacetime into the three regions I, IIa, andIII, where r∗ < −t/2, where |r∗| ≤ t/2 but |r∗| > C to avoid the photon sphere,and where |r∗| > t/2 respectively. Region IIb will refer to the region |r∗| ≤ C.To estimate the remaining terms, we will apply the Cauchy-Schwarz inequality andthen use the following estimates in the regions I, IIa, and III,

PtTχ &(T⊥v)2 + (∂r∗ v) + VQ|∇/v|2 + V v2,

t(δPA)[ψ] &t ∆

(r2 + a2)

1

r2|∂r∗ v|2 + t

∆

(r2 + a2)

1

r3|v|1 + t

∆

(r2 + a2)

1

r4v2,

PtK &(T⊥v)2 + (∂r∗ v) + t2VQ|∇/v|2 + t2V v2

respectively. Since we are applying the Cauchy-Schwarz inequality, these estimateslimit the asymptotic behaviour of the coefficients of the mixed derivatives. Thisimposes restrictions on the behaviour of ωK and its derivatives. Thus, in thesecalculations, we find ourselves having certain coefficients, needing them to satisfycertain asymptotics so that we can apply the relevant estimates, and then requir-ing certain conditions on ωK and its derivatives. We call this “need-have-require”analysis. When there is no condition which we require, we simply say the requiredcondition is “nothing”.

The first term, coming from (4.14b), is

|(T⊥v)(∂φv)(−2tN2(ωK − ω⊥))| .|T⊥v||∂φv|t|ωK − ω⊥|.

In the three regions of interest, the asymptotics we have, those we need, and theconditions we require are

Region Have Need Require

I t(ωK − ω⊥) ∆1/2+ε ωK − ω⊥ . ∆1/2+ε

IIa t(ωK − ω⊥) tO(∆, r−3) ωK − ω⊥ = O(∆, r−3)

III t(ωK − ω⊥) r−2 ωK − ω⊥ . r−3

The second term, from (4.14d), is bounded by the sum of two terms

|(∂∗v)(∂φv)((t2 + r2∗)(∂r∗ωK) + 2r∗(ωK − ω⊥))|.|∂∗v||∂φv|

((t2 + r2

∗)|∂r∗ωK|+ 2|r∗(ωK − ω⊥)|)

We perform a have-need-require analysis on each of these. For the first of theseterms, in region IIa, we will not estimate the term by t(δPA)[ψ] but by at2(δPA)[ψ],which means we “have” an extra factor of t.


I (t2 + r2∗ + 1)∂r∗ωK ∆1/2+ε |∂r∗ωK| . ∆1/2+ε

IIa (t2 + r2∗ + 1)∂r∗ωK at2O(∆, r−5/2) ∂r∗ωK = O(∆, r−5/2)

III (t2 + r2∗ + 1)∂r∗ωK r−2 |∂r∗ωK| . r−4

For the second, we again use the standard estimates.



I r∗(ωK − ω⊥) ∆1/2+ε |ωK − ω⊥| . ∆1/2+ε

IIa r∗(ωK − ω⊥) a2tO(∆, r−5/2) ωK−ω⊥ = O(∆, r−5/2)

III r∗(ωK − ω⊥) r−2 |ωK − ω⊥| . r−3

The remaining derivative terms are estimated similarly. These are

|(∂∗v)(∂β v)hαβ2tr∗(∂βN2)| .|∂∗v||∇/v|V 2

Qt|r∗|a2,

and we require


I V 2Qt|r∗|a2 ∆1/2+ε nothing

IIa V 2Qt|r∗|a2 tO(∆, r−5/2) nothing

III V 2Qt|r∗|a2 r−2 nothing

For the final term, from (4.14g), we again need to use at2(δPA)[ψ] to dominate

|(∂αv)(∂φv)hαβ(t2 + r2∗)(∂βωK)| .|∇/v|2VQ(t2 + r2

∗ + 1)|∇/ωK|.


I VQ(t2 + r2∗ + 1)|∇/ωK| ∆1+ε |∇/ωK| . ∆ε

IIa VQ(t2 + r2∗ + 1)|∇/ωK| at2O(∆, r−3) ∇/ωK = O(1, r−3)

III VQ(t2 + r2∗ + 1)|∇/ωK| r−2 |∇/ωK| . r−2

We turn to controlling these terms in region IIb, near r = 3M . If we assumethat ωK is constant in a neighbourhood of the photon orbits, then the worst termsinvolving t2, coming from (4.14d) and (4.14g), vanish in this region. We will furtherassume that in this region ωK−ω⊥ is bounded by a constant times a, which boundsthe term coming from (4.14b). Applying Cauhcy-Schwartz, all the other terms arebounded by

t1rh3M |∂r∗ v|2 + t1rh3M |∇/v|2 + at1rh3M |∂tv|2

≤(δPA)[ψ]

+ (δPA)[O11, ψ]

+ at−1PK[S1v]t.

By taking the most restrictive conditions, we are left with the following require-ments:

ωK − ω⊥ =O(∆, r−3),

∂r∗ωK =aO(∆, r−4),

∇/ωK =a2O(∆ε, r−3),

and that ωK is constant in a neighbourhood of the photon orbits, and that ωK−ω⊥is bounded by a constant times a in that neighbourhood.

There is a wide range of possibilities for choice of ωK. In particular, our choice

ωK = ωHχ

satisfies all the necessary conditions.


The remaining terms to be estimated, which come from (4.13c)-(4.13d), are givenby v2 times the coefficient

+1

2(∂γK

γ)V +1

2Kγ∂γV − qKV +

1

2(∂αGαβ∂βqK) = + tN2(2V + r∗∂r∗V )

+ t(N2 − 1 + r∗∂r∗N)V

− (N2 − 1)qK −1

2(∂2r∗qK).

As with the angular derivative terms in (4.14f), the first term is familiar from theScwarzschild case and gives a weight which is positive in a compact region, butwhich is negative, and hence trivially bounded, outside this region. The behaviouroutside this region is h −t∆ log ∆ approaching the event horizon and h −tr−3 log rapproaching the horizon. The other terms are all a2O(∆, r−4) at worst, so they areeasily dominated by the first term outside a slightly larger compact region. Thus,the contribution involving v2 without derivatves is easily controlled by t(δPA)[ψ].

4.4. Stronger, light-cone localised Morawetz estimates. In this section, weprove a stronger Morawetz estimate to gain factors of t inside the light cone, |r∗| < t.We do this by using a Morawetz vector field with additional factors of t and localisinginside the light cone.

Definition 4.10. Let χLC : R→ R be a smooth, even function which is identically1 for |x| < 1/2, identically 0 for6 |x| > 3/4, which is weakly decreasing for x > 0,and which is weakly increasing for x < 0. If χLC is written without an argument,it understood to mean

χLC =χLC

(r∗t

).

The stronger, light-cone localised Morawetz vector field (with strength p) is definedto be

Aabp =tpχLCAab,

qabA,p =tpχLCq

abA ,

Ap =(Aabp , q

abA,p).

Note that

Aab0 =AabχLC,

qabA,0 =q

abA χLC.

Lemma 4.11. There are positive constant a and C, such that, if |a| ≤ a, if ψ is a

solution to the wave equation ψ = 0, and if ψ = (r2 +a2)1/2ψ, then for t2 ≥ t1 ≥ 0and p ∈ N+,

C−1

∫ t2

t1

∫tpχLC(δPA)[ψ]d4µ ≤(1 + t2)p−2EK,3[ψ](t2) + (1 + t1)p−2EK,3[ψ](t1)

+

∫ t2

t1

(1 + t)p−3EK,3[ψ](t)dt.

Proof. We begin by relating the divergence of the momentum of the stronger light-cone localised Morawetz vector field to that of the original Morawetz vector-field.

6The parameters 1/2 and 3/4 are not important. The important thing is the ordering 0 <1/2 < 3/4 < 1.


We denote the right-hand side of (3.6) by W[ψ]. The divergence is

1

µ∂α(µPAp) =

1

µ∂α(µtpχLCPA)

=tpχLC1

µ∂α(µPαA) + PαA(∂αχLCt

p),

tpχLCW[ψ] =1

µ∂α(µPAp)

− PαA(∂αtpχLC)− tpχLC

1

µ∂α(µ(BA;I + BA;II)).

We now move the localisation onto the solution itself to find

tpW[χ1/2LCψ]

=1

µ∂α(µPAp)

− PαA(∂αtpχLC)− tpχLC

1

µ∂α(µ(BA;I + BA;II))

− tpχLCM∆2

r2(r2 + a2)

∑

a

(2(∂rSaψ)(∂rχLC)Saψ + (∂rχLC)2(Saψ2)),

=1

µ∂α(µtpχLC(PαA + BαA;I + BαA;II))

− (PαA + BαA;I + BαA;II)(∂αtpχLC)

− tpχLCM∆2

r2(r2 + a2)

∑

a

(2(∂rSaψ)(∂rχLC)Saψ + (∂rχLC)2(Saψ2)).

We now estimate the last term, using χ2 as a smooth localisation which is 1 on thesupport of χLC but still supported well inside the light cone, and using φ to denoteSaψ,

tpχLCM∆2

r2(r2 + a2)(2(∂rφ)(∂rχLC)φ+ (∂rχLC)2φ2)

.tpχLCM∆2

r2(r2 + a2)((∂rφ)2 + (∂rχLC)2φ2)

.tpχLCM∆2

r2(r2 + a2)(∂rφ)2 + tp−2χ2φ

2

.tpχ2

(∆(∂rφ)2 +

1

1 + r2∗φ2

).

Integrating this over a hypersurface of constant t,

∫tpχ2

∆2

(r2 + a2)2((∂rψ)(∂rχLC)ψ + (∂rχLC)2ψ2)d3µ

≤∫tpχ2

(PtTχ +

1

1 + r2∗ψ2

)d3µ

.∫tp−2PtKd3µ∗.

Since PtA, PrA, BtA;I, BrA;I, BtA;II, and BrA;II are all bounded by PtTχ [S2ψ] + |S2ψ|2,

which is bounded by PtTχ [S2ψ], at least in an integrated sense over a portion of the


hypersurface inside the light cone, it follows that

∫tpW[χ

1/2LCψ]d3µ .

∫1

µ∂α(µtpχLC(PαA + BαA;I + BαA;II))d

3µ

+ tp−1

∫ (3/4)t

−(3/4)t

PTχ [ψ]d3µ∗.

Now applying the Hardy estimate and an argument similar to the one earlier in thisproof, it follows that

∫χLCt

p(δPA)[ψ]d3µ .∫

1

µ∂α(µtpχLC(PαA + BαA;I + BαA;II))d

3µ

+ tp−1

∫ (3/4)t

−(3/4)t

PTχ [ψ]d3µ∗.

Finally, integrating in time, we get

∫ t2

t1

∫χLCt

p(δPA)[ψ]d4µ .tp2∫

Σt2

χLC(PtA + BtA;I + BtA;II)d3µ

− tp1∫

Σt1

χLC(PtA + BtA;I + BtA;II)d3µ

+

∫ t2

t1

tp−1

∫ (3/4)t

−(3/4)t

PTχ [ψ]d4µ∗

.tp2∫

Σt2

χLCPtA[S2ψ]d3µ

+ tp1

∫

Σt1

χLCPtA[S2ψ]d3µ

+

∫ t2

t1

tp−1

∫ (3/4)t

−(3/4)t

PtTχ [S2ψ]d4µ∗

.tp−22 EK[S2ψ](t2) + tp−2

1 EK[S2ψ](t1)

+

∫ t2

t1

tp−3EK[S2ψ]dt.

4.5. Closing the K estimate. In this section, we use the stronger, light-conelocalised Morawetz estimate to control the growth of the K energy.

Theorem 4.12. There are positive constants a, C, and C ′ such that, if |a| < a

and if ψ is a solution of the transformed wave equation (4.1), then ∀t ≥ 0

EK,3(t) ≤C(1 + t)C′|a|(EK,3(0) + ETχ,5(0)

).

Proof. Essentially, the stronger Morawetz-estimate, lemma 4.11, controls the growthof the K energy, and the K energy controls the boundary terms in the strongerMorawetz estimate. Thus, a standard Gronwall’s argument closes the estimate. Inthis proof, an integral without limits typically refers to integration over (r∗, θ, φ) ∈R× S2.


First, we can bound the growth of the K energy, using lemma 4.9, by

E(K,qK),3(t2)− E(K,qK),3(t1) ≤∫ t2

t1

ETχ,3(t)(1 + t)−2dt (4.15)

+

∫ t2

t1

∫t(δPA)[ψ]d4µ∗

+

∫ t2

t1

∫t(δPA)[O1

1, ψ]d4µ∗

+ a

∫ t2

t1

∫t2(δPA)[ψ]d4µ∗

+

∫ t2

t1

EK,3(1 + t)−2 log(2 + t)dt.

Since ETχ,3 . EK,3 and (1 + t)−2 . (1 + t)−2 log(2 + t), the first term can beabsorbed into the last.

Before applying the Morawetz estimate, we apply integration by parts in theangular derivatives to the term in (4.15) and apply the Cauchy-Schwarz inequalityto find∫

S2

∫(δPA)[O1

1, ψ]d2µ ≤∫

S2

(δPA)[ψ]1/2(δPA)[O21, ψ]1/2d2µ

≤(∫

S2

(δPA)[ψ]d2µ

)1/2(∫

S2

(δPA)[O21, ψ]d2µ

)1/2

≤(∫

S2

(δPA)[ψ]d2µ

)3/4(∫

S2

(δPA)[O41, ψ]d2µ

)1/4

.

Now applying the integration in the remaining variables and the Cauchy-Schwarzinequality one more time, we find

∫ t2

t1

∫t(δPA)[O1

1, ψ]d4µ∗ .∫ t2

t1

∫t4/3(δPA)[ψ]d4µ∗

+

∫ t2

t1

∫(δPA)[O4

1, ψ]d4µ∗.

From this, we can bound the growth of EK,3 by

EK,3(t2)− EK,3(t1) .∫ t2

t1

∫t4/3(δPA)[ψ]d4µ∗

+

∫ t2

t1

∫(δPA)[O4

1, ψ]d4µ∗

+ a

∫ t2

t1

∫t2(δPA)[ψ]d4µ∗

+

∫ t2

t1

EK,3(1 + t)−2 log(2 + t)dt.

In the first integral, with the aim of applying lemma 4.11, we use the convergenceof∫∞

1t4/3−3dt to get a bounded piece plus what we hope to show is a small piece.

To do this, we choose a time T such that∫∞T

(1 + t)−5/3dt is small relative to theabsolute constants implicit in .. The first integral we break into an integral fromt1 to T and then from T to t2. In the first part, we replace (1 + t)4/3 by (1 +T )4/3,which is also an absolute constant. Thus, the first integral is bounded by

∫ t2

t1

∫t4/3(δPA)[ψ]d4µ∗ .E3,Tχ +

∫ ∞

T

∫t4/3(δPA)[ψ]d4µ∗.


Similarly, in the last integral, we want to use the integrability of∫t−2 log tdt

and the bound EK . (1 + t)2ETχ , so that we can take T sufficiently large that∫∞Tt−2 log tdt is small, and obtain the bound

∫ t2

t1

EK,3(1 + t)−2 log(2 + t)dt .ETχ,3 +

∫ t2

T

EK,3(1 + t)−2 log(2 + t)dt.

We can also apply the original Morawetz estimate, lemma 3.12, with an extrafour derivatives, to control the integral of (δPA)[O4

1, ψ]. This leaves us,

EK,3(t2)− EK,3(t1) .∫ t2

T

t4/3(δPA)[ψ]d4µ∗

+ ETχ,7

+ |a|∫t2(δPA)[ψ]d4µ∗

+

∫ t2

T

EK,3(1 + t)−2 log(2 + t)dt.

Using the stronger Morawetz lemma 4.11, we can bound the integral of theMorawetz terms by,

EK,3(t2)− EK,3(t1) .(1 + t2)−2/3EK,3(t2) + (1 + T )−2/3EK,3(T ) (4.16a)

+

∫ t2

T

(1 + t)−5/3EK,3(t)dt (4.16b)

+ ETχ,7 (4.16c)

+ |a|EK,3(t2) + |a|EK,3(t1) (4.16d)

+ |a|∫ t2

t1

(1 + t)−1EK,3(t)dt (4.16e)

+

∫ t2

T

EK,3(1 + t)−2 log(2 + t)dt., (4.16f)

Let

f(t) = supτ∈[t1,t]

EK,3(τ),

F (t) =

∫ t

t1f(τ)(1 + τ)−1dτ,

K0 =EK,3(t1) + ETχ,7(t1).

We can replace EK,3 on the left of (4.16a) by f . We can make a similar substi-

tution in (4.16a), (4.16b), (4.16d), and (4.16f), and then use the smallness of t−2/32 ,

T−2/3,∫∞Tt−5/3, a, and

∫∞Tt−2 log t to absorb all those terms into the left-hand

side, with only a small loss. Note that by the bounded energy result, one has thatETχ,7(t) . K0 uniformly in t. Thus, we are left with

f(t2) .K0 + a

∫ t2

t1

(1 + t)−1f(t)dt.


We now write this as an integral inequality to apply the standard Gronwall’s in-equality techniques

f(t)− C|a|∫ t

t1(1 + τ)−1f(τ)dτ .K0,

(1 + t)F ′(t)− CaF (t) .K0,

d

dt

((1 + t)−C|a|F (t)

).K0t

−1−C|a|,

F (t) .1

aK0(1 + t)C|a|

f(t) .K0(1 + t)C|a|.

In terms of the original quantities of interest, this gives

EK,3(t) .(1 + t)C|a|(EK,3(0) + ETχ,7(0)

).

We also want to evaluate the K energy on surfaces other than those of constantt. The following lemma allows us to control the K energy on such surfaces. Al-though this lemma is valid for any surface, it is only interesting on time-like or nullhypersurfaces on which the energy should be positive.

Theorem 4.13 (K bounds for other surfaces). Consider the almost-null conesgiven by

Cτ = t− |r∗| = τ.

There are positive constants a, C, and C ′, such that, if |a| ≤ a, and if ψ is a solutionof the transformed wave equation (4.1)2 then for any hypersurface Σ, which neednot be in the futute of Σ0, if Σ can be represented as a graph over (r∗, θ, φ) ∈ R×S2

which lies beneath Cτ and above −Cτ , then

E(K,qK),3(Σ) ≤C max1, τC′|a|(EK,3(0) + ETχ,7(0)

)

Proof. In this proof we will exploit the time symmetry of the previous results. Forsimplicity let K0 = EK,3(0)+ETχ,7(0). Lemma 4.9 allows us to estimate the change

in E(K,qK) from t = 0 to Σ by the four-dimensional integral of µ−1∂α(µPα(K,qK)).

This integral can be broken into three pieces: where r∗ < −|t|/2, where −|t|/2 ≤r∗ ≤ |t|/2, and where |t|/2 < r∗.

We first consider the middle region, where −|t|/2 ≤ r∗ ≤ |t|/2. The four-dimensional volume inside this region and beneath Σ is a subset of the volumeinside this region where |t| ≤ 2τ . Thus, from the proof of lemma 4.5, the integral

of the absolute value of µ−1∂α(µPα(K,qK)) is bounded by C(1 + τ)CaK0.

We now consider the other two regions. First, consider the far region, |t|/2 < r∗.In this region, |µ−1∂α(µPα(K,qK))| is bounded by log(2 + |t|)(1 + |t|)−2Pt(K,qK). If we

now foliate the region beneath Σ by hypersurfaces of constant t, we can bound thefour-dimensional integral of µ−1∂α(µPα(K,qK)) by the integral in time of the integral

over each of the leaves of the foliation. On each leaf, the integral is bounded bylog(2 + |t|)(1 + |t|)−2E(K,qK)(t) . log(2 + |t|)(1 + |t|)Ca−2K0. Thus, the four-

dimensional integral is bounded by K0

∫log(2 + |t|)(1 + |t|)Ca−2dt . K0. The near

region, r∗ < −|t|/2 can be handled similarly.

Thus, the four-dimensional integral is bounded by CτCaK0. Hence, so is E(K,qK)(Σ).


5. Pointwise decay estimates

5.1. Decay in stationary regions. In this subsection, we prove decay in regionsof fixed r bounded away from the event horizon, r+ < r1 < r < r2. Since theseare preserved by the flow of the stationary, Killing field, ∂t, we refer to these asstationary regions.

The essence of the proof of this decay result is that the bound on EK,3 gives abound on t2 times the square of a local H3 norm, which controls the square of thefield, |ψ|2, through a Sobolev estimate. Thus, |ψ| decays like t−1 with a small lossfrom the growth of the K energy.

Theorem 5.1. There are positive constants a and C ′ such that, given r+ < r1 < r2,there is a constant, Cr1,r2 , such that, if |a| ≤ a, and if ψ is a solution to the waveequation ψ = 0, then ∀t ≥ 0, r ∈ [r1, r2], (θ, φ) ∈ S2, there is the estimate

|ψ(t, r, θ, φ)| ≤ Cr1,r2 max1, t−1+C′|a|(EK,3(0)1/2 + ETχ,7(0)1/2

).

Proof. In this proof, a constant of the form C ′r1,r2 refers to a constant which dependsonly on r1 and r2. As is common in analysis, the value of this constant may varyfrom line to line. Since r+ < r1 < r2 < ∞, there are corresponding r∗ values−∞ < r∗1 < r∗2 <∞.

From lemma 2.2, for fixed (t, r∗) and all (θ, φ) ∈ S2

|ψ(t, r∗, θ, φ)|2 .∫

S2

|ψ|2d2µ.

Given any smooth v : R→ R, by a one-dimensional Sobolev estimate (or simplythrough the fundamental theorem of calculus, localisation, and Cauchy-Schwarz),for r∗1 ≤ r∗ ≤ r∗2

|v(r∗)|2 ≤C ′r1,r2∫ r∗2

r∗1

|∂r∗v|2 + |v|2dr∗.

Applying this result with v being a symmetry operator applied to ψ, then summingover the symmetry operators, and then integrating over S2, we find

∫

S2

|ψ|2d2µ ≤ C ′r1,r2∫

S2

∫ r∗2

r∗1

|∂r∗ ψ|22 + |ψ|22d3µ∗.

The left-hand side dominates |ψ|2. Since the range of r is fixed, there is a lowerbound on V , so that

|ψ(t, r∗, θ, φ)|2 ≤C ′r1,r2∫

S2

∫ r∗2

r∗1

|∂r∗ ψ|22 + V |ψ|22d3µ∗.

For sufficiently large t, the region [r∗1, r∗2] × S2 is inside |r∗| ≤ t/2, so that theright-hand side can be bounded by the K energy, and

|ψ(t, r∗, θ, φ)|2 ≤C ′r1,r2t−2EK,3(t)

≤C ′r1,r2t−2−Ca(EK,3(0) + ETχ,7(0)

).

Since (r2 +a2)1/2 is uniformly bounded above and below on (r1, r2) we may replace

ψ(t, r∗, θ, φ) by ψ(t, r∗, θ, φ) to get the desired result.For small t, the result holds from a similar argument using the energy bound.

5.2. Near Decay. To study the wave equation near the event horizon, it is conve-nient to use u+ andu−, which were defined in the introduction, and to use

φ∗ =φ− ωHt.The functions (u+, r, θ, φ∗) are known to form a coordinate system which can beextended to an open set containing the future event horizon.


Theorem 5.2 (Near Decay). There are positive constants a, C and C ′, such that

if |a| < a and if ψ is a solution of the wave equation (4.1), then for all t > r∗,r < 3M , (θ, φ) ∈ S2,

|ψ(t, r, θ, φ)| < C max1, u+−1+C′a(EK,5(0)1/2 + ETχ,9(0)1/2 + EnΣt ,3

(0)1/2).

Proof. Near the horizon, it is convenient to rewrite the wave equation, (4.1), for ψas

0 =− (∂t + ωH∂φ)2ψ + ∂2r∗ ψ

+ 2

(ωH −

2aMr

(r2 + a2)2

)∂t∂φψ

+ ∆(Q+ ∂2φ)ψ

+

(ω2H −

a2

(r2 + a2)2

)ψ

− V ψ=− (∂t + ωH∂φ)2ψ + ∂2

r∗ ψ + Zψ − V ψ,where Z can be expanded as Z = ZaSa and the Za vanish linearly in r − r+.Equivalently, the functions Za/∆ are bounded for r ∈ [r+, 3M ].

To prove decay, it is convenient to use (u+, u−, θ, φ∗) as a coordinate system inthe exterior region. The coordinate derivatives in this coordinate system can berelated to the earlier (t, r∗, θ, φ) coordinate derivatives. The relations that we areinterested in are

∂± = ∂u± =1

2(∂t + ωH∂φ ± ∂r∗) ,

∂θ =∂θ,

∂φ∗ =∂φ.

The relation ∂θ = ∂θ is understood to mean that the θ-coordinate derivatives inthe two systems generate the same vector field. We can, without ambiguity, writeformulas involving coordinate derivatives from both systems. In particular, we canwrite the wave equation, (4.1), as

0 =− 4∂+∂−ψ + Zψ − V ψwith Z defined in terms of ∂t, ∂θ, and ∂φ as above.

To prove decay, we use the one-form

ξ =(∂−ψ

)du−.

For a given (θ, φ∗), let

Ω =(u′+, u′−, θ, φ∗)|u′− ∈ [u+, u−], u′+ ∈ [−u′−, u+]=(t′r′∗, θ′, φ′)|t′ ≥ 0, t+ r∗ ≤ t′ − r′∗ ≤ t− r∗, t′ − r′∗ ≤ t+ r∗.

It is convenient to first consider u+ ≥ 1. We apply Stokes’ theorem to ξ on Ωand in (u+, u− θ, φ∗) coordinates to get

∫

∂Ω

ξ =

∫

Ω

dξ,

ψ(t, r∗, θ, φ− ωHt)− ψ(t+ r∗, 0, θ, φ− ωH(t+ r∗))

+

∫

C(∂−ψ)dr∗ =

∫

Ω

(∂+∂−ψ)du+du−, (5.1)

where C is the curve with t′ = 0, r′∗ ∈ [−u−, u+], and (θ′, φ′) = (θ, φ).


u′+

u′−

u′+ = −u′−

(u+, u−)

(u+, u+)

(−u−, u−) (−u+, u+)

Ω

C

Figure 1. The region Ω in the (u′+, u′−) plane with vertices at

(u+, u−), (−u−, u−), (−u+, u+), and (u+, u+). In the (t, r∗) plane,these vertives correspond to (t, r∗), (0,−t + r∗), (0, t + r∗), and(t+ r∗, 0).

The function ψ evaluated at (u+, u+, θ, φ∗) corresponds to the value at (t′, r′∗, θ′, φ′) =

(u+, 0, θ, φ∗ − ωHu+), so it can be estimated by

|ψ(t+ r∗, 0, θ, φ− ωH(t+ r∗))| . u−1+Ca+

(EK,3(0)1/2 + ETχ,7(0)1/2

).

The integral over C corresponds to an integral in the hypersurface t = 0 withr′∗ ∈ [−u−,−u+] ⊂ (−∞,−u+]. Since ∂− is in the span of T⊥, ∂r∗ , and ∆∂φ withsmooth and uniformally bounded coefficients when r ∈ [r+, 3M ], it follows that

|(∆−1/2∂−)ψ|2 =∆−1|∂−ψ|2 . ∆−1PtT⊥ .Since T⊥ is parallel to the normal, nΣt , it follows that

|(∆−1/2∂−)ψ|2 .∆−1/2PtnΣt.

Therefore,

∫

C(∂−ψ)dr∗ ≤

(∫ −u+

−∞∆1/2dr∗

)1/2(∫ −u+

−∞|(∆−1/2∂−)ψ|2∆1/2dr∗

)1/2

.

The second integral can be bounded by∫ −u+

−∞|(∆−1/2∂−)ψ|2∆1/2dr∗ .

∫

S2

∫ −u+

−∞|(∆−1/2∂−)ψ|22∆1/2d3µ∗

.∫

S2

∫ −u+

−∞PtnΣt ,3

d3µ∗

.EnΣt ,3(0).

Since ∆ decays exponentially in r∗, which is faster than any polynomial

∫

C(∂−ψ)dr∗ ≤

(∫ −u+

−∞∆1/2dr∗

)1/2

EnΣt ,3(0)1/2

≤u−1+Ca+ EnΣt ,3

(0)1/2.


Finally, to estimate the integral over Ω, we break the integral into two regions

A =Ω ∩ r∗ > −t/2,B =Ω ∩ r∗ ≤ −t/2.

We first prove a preliminary estimate in region A. For −t/2 ≤ r∗ ≤ 0, since∆/(r2 + a2) is increasing, it follows that, in (t, r∗, θ, φ) coordinates

∂r∗

((∆

r2 + a2

)1/2

u2+ψ

2

)≥2

(∆

r2 + a2

)1/2

u+ψ2 + 2

(∆

r2 + a2

)1/2

u2+ψ(∂r∗ ψ).

Integrating along a curve of constant (t, θ, φ), we have(

∆

r2 + a2

)1/2

u2+ψ(t, r∗, θ, φ)2

=−∫ 0

r∗

∂r∗

((∆

r2 + a2

)1/2

u2+ψ

2

)dr∗ + Ct2ψ(t, 0, θ, φ)2

.−∫ 0

r∗

(2

(∆

r2 + a2

)1/2

u+ψ2 + 2

(∆

r2 + a2

)1/2

u2+ψ(∂r∗ ψ)

)dr∗

+ t2ψ(t, 0, θ, φ)2

.∫ 0

r∗

1

1 + r2∗u2

+ψ2dr∗ +

∫ 0

r∗

u2+(∂r∗ ψ)2dr∗ + t2ψ(t, 0, θ, φ)2

.EK,3(t)

.tC|a|(EK,3(0) + ETχ,7(0)

).

Therefore,

|ψ(t, r∗, θ, φ)| .(

∆

r2 + a2

)−1/4

t−1+C|a|(EK,3(0)1/2 + ETχ,7(0)1/2

).

Returning to the argument involving Stokes’ theorem, the integral over Ω can beestimated by∫

Ω

(∂+∂−ψ)du+du− =

∫

Ω

(Zψ − V ψ)du+du−

=2

∫

Ω

(Zψ − V ψ)dtdr∗

≤2 supA|∆1/4ψ|2

∫

A

∆3/4dtdr∗

+ 2

(∫

B

∆dtdr∗

)1/2(∫

B

1

∆

((Zψ)2 + (V ψ)2

)dtdr∗

)1/2

.

We now control each of these terms. We start with the first. By the prelimi-nary estimate in region A, there is the estimate ∆1/4|ψ| . t−1+C|a|(EK,3(0)1/2 +

ETχ,7(0)1/2). After applying up to two more derivatives, and noting t′ h u′+ in A,we have that the supremum term decays like

supA|∆1/4ψ|2 . u−1+C|a|

+ (EK,5(0)1/2 + ETχ,9(0)1/2).

Since A ⊂ Ω has |t′ − 2u+| < |r′∗|, the integral in∫A

∆3/4dtdr∗ is bounded by∫A

2∆3/4r∗dr∗, which is bounded, since ∆ decays exponentially.We now turn to the integrals over B. Since Z and V decay linearly in r − r+,

the integral∫

∆−1((Zψ)2 + (V ψ)2)dtdr∗ is bounded by a multiple of ∆ times the

square of up to two derivatives of ψ. If there were no derivatives on ψ, this integral


would be bounded by the Morawetz estimate. Therefore, after applying two moreangular derivatives and a spherical Sobolev estimate, this integral can be controlledby the Morawez estimate

∫

B

1

∆

((Zψ)2 + (V ψ)2

)dtdr∗ . ETχ,5(0).

Finally, the integral∫B

∆dtdr∗ is bounded by an integral of the form∫

∆r∗dr∗, butnow the upper bound on r∗ is h u+, since we are restricted to the region B. Thus,∫B

∆dtdr∗ decays faster than any polynomial in u+. Combining the estimates inregions A and B, we have

∫

Ω

(∂+∂−ψ)du+du− . u−1+C|a|+

(ETχ,5(0)1/2 + ETχ,9(0)1/2

).

Thus, from (5.1), the solution ψ decays like

|ψ(t, r∗, θ, φ− ωHt)| .u−1+C|a|+

(EK,5(0)1/2 + ETχ,9(0)1/2 + EnΣt ,3

(0)1/2).

This gives a bound on ψ at (t, r∗, θ, φ − ωHt). Since the same decay rate holds atall points on the sphere, the desired result holds.

For u+ ∈ [0, 1], the same argument holds to give the boundedness of ψ. Foru+ ≤ 0, a similar argument holds, except that Ω is a triangle in the (u+, u−) plane,with both endpoints on the surface t = 0. The boundedness of the initial data ont = 0 in the region r < 3M , then gives boundedness for u+ < 0.

Since ψ and ψ are uniformly equivalent for r < 3M , this gives the desired result.

5.3. Far decay. We now prove decay in the far region, for r ≥ rχ, with particularattention to the behaviour as r h t and r →∞.

To study the far region, we introduce the radial coordinate, y, and almost nullcoordinates, v±,

y =

∫ r∗

0

hdr′∗,

h =

√1− 2a2

∆

(r2 + a2)2

=

√1− 2a2VQ,

v± =t± y.Since, for large r∗, h h 1 − a2Cr−2, the coordinate y differs from r∗ by at most aconstant. Similarly v± differs from u± = t ± r∗ by at most a fixed constant. Bydirect computation, one finds that the length of the one-form

dv± =dt± hdr∗

is given by

Σ∆

(r2 + a2)2g(dv±,dv±) =Gαβ(dv±)α(dv±)β ,

Gαβ(dv±)α(dv±)β =a2(sin2 θ − 2)VQ ≤ −a2VQ < 0.

Thus, the surfaces of constant v− (or v+) are timelike. Although we have chosen aparticularly convenient form for the factor h, the only property that we use is thath h 1 − Cr−2

∗ , so that the surfaces are timelike, with a fixed rate at which theydegenerate with respect to the T⊥ and ∂r∗ basis, and that the constant in this rateis sufficiently large that a2 . C. We denote the surfaces of constant v− by

Σ+v− .


Unless otherwise stated, we will restrict these to the region where t > max(2y, 0).Outside this region, we extend the surface as a surface of constant t. It is alsoconvenient to introduce the almost null vectors

ξ± =T⊥ ± h−1∂r∗ .

These satisfy ξ+v− = 0 = ξ−v+ and ξ+v+ = 1 = ξ−v−.Before proving decay in the far region, we control the K energy on almost null

hypersurfaces crossing null infinity.

Lemma 5.3. There are positive constants a, C, and C ′ such that, if |a| < a, and

if ψ is sufficiently smooth, then

CEK(Σ+v−) ≥

∫

Σ+v−

(v2+ + 1)

((ξ+ψ)2 |∇/ψ|2

r2+ V ψ2

)dv+d2µ.

Furthermore, if ψ is a solution to the wave equation, (4.1), then

EK(Σ+v−) ≤CvC

′|a|−

(EK,3(Σ0) + ETχ,7(Σ0)

).

Proof. It is convenient to introduce the pair of null vectors

L± =N−1T⊥ ± ∂r∗ .The wave equation can be written as

(1

µ∂αµGαβ∂β − V )ψ =

(−1

2(L+L− + L−L+) +

1

µ∂αµh

αβ∂β − V)ψ = 0.

For any vector in the span of T⊥ and ∂r∗ ,

X =X⊥T⊥ + X∗∂r∗ ,

the vector can be written as

X =X+L+ + X−L−,

X± =1

2

(NX⊥ ±X∗

).

The vectors

Bα± =− Gαβ(dv±)β ,

B± =N−2T⊥ ∓ h∂r∗have null components (ie, components with respect to the L+ and L−)

B±− =1

2

(N ± h

),

B±+ =1

2

(N ∓ h

).

We’re particularly interested in the components of B−, which can be estimated by

B+− h1

B−− ≥a2VQ & a2r−2.

Similarly, there is a decomposition of K as

K =K+L+ + K−L− − (t2 + r2∗ + 1)ω⊥∂φ,

K± =1

2

((t2 + r2

∗ + 1)N ± 2tr∗N2)

=N

2(t± r∗)2 +

N

2± tr∗N(N − 1).


For r∗ ≥ t/2 and a small

|(t2 + r2∗ + 1)ω⊥| .

|a|r,

K± &v2± + 1.

The functions (v−, r∗, θ, φ) can be used as a coordinate system. (In fact, thecoordinates (v−, r, θ, φ∗) can be used as a coordinate system extending through the(future) event horizon.) Since the new coordinates satisfy

dv− ∧ dr∗ ∧ dµθ ∧ dφ = dt ∧ dr∗ ∧ dµθ ∧ dφ,

the energy associated with any vector-field X is the integral over the 3-surface ofthe contraction of the associated momentum (vector) against the 4-dimensionalmeasure

∫

Σ+v−

PX(dt ∧ dr∗ ∧ dµθ ∧ dφ) =

∫

Σ+v−

PX(dv− ∧ dr∗ ∧ µdθ ∧ dφ)

=

∫

Σ+v−

Pv−X d3µ∗.

In (v−, r∗, θ, φ) coordinates, the energy associated with K on a surface of constantv− is the integral of

Pv−K =PβK(dv−)β

=(B−)α(Kβ)(∂αψ)(∂βψ)

+1

2δβα(dv−)βKα

(−(L+ψ)(L−ψ) + hαβ(∂αψ)(∂βψ)

)

=(B+−)(K+)(L+ψ)2

+

((B+−)(K−) + (B−−)(K+)− 1

2(dv−)αKα

)(L+ψ)(L−ψ)

+ (B−−)(K−)(L−ψ)2

+1

2(dv−)αKα

(hαβ(∂αψ)(∂βψ) + V ψ2

).

By direct computation,

(dv−)αKα =2((B+−)(K−) + (B−−)(K+)

),

so that

Pv−K &(v2+ + 1)(L+ψ)2

+ (v2− + 1)

a2

r2(L−ψ)2

+ (v2+ + v2

− + 1)

(|∇/ψ|2r2

+ V ψ2

).

Since

L± =N−1∂t + Nω⊥∂φ ± ∂r∗ ,

it follows that

|∂tψ|2 + |∂r∗ ψ|2 .|L+ψ|2 + |L−ψ|2 +a2

r6|∇/ψ|2.


The difference between ξ+ and L+ can be estimated by

ξ+ − L+ =(1− N−1)∂t + (1− h−1)∂r∗ + N−1ω⊥∂φ,

|(ξ+ − L+)ψ|2 .a4

r4

(|∂tψ|2 + |∂r∗ ψ|2

)+a2

r6|∂φψ|2

.a4

r4

(|L+ψ|2 + |L−ψ|2

)+a2

r6|∂φψ|2.

A similar estimate holds for ξ− and L−. For r∗ ≥ t/2, the desired derivative can becontrolled by

(v2+ + 1)(ξ+ψ)2 =(v2

+ + 1)(L+ψ)2

+ (v2+ + 1)(−2(L+ψ)((ξ+ − L+)ψ) + (ξ+ − L+)ψ)2

.(v2+ + 1)(L+ψ)2

+ (v2+ + 1)

((ξ+ − Lp)ψ

)2

.(v2+ + 1)(L+ψ)2

+a4

r2(L−ψ)2

+a2

r4|∇/ψ|2

.Pv−K .

Thus, the energy on a surface of constant v−, Σ+v− , associated with the vector K is

EK(Σ+v−) =

∫

Σ+v−

Pv−K d3µ∗

&∫

Σ+v−

(v2+ + 1)

((ξ+ψ)2 +

|∇/ψ|2r2

+ V ψ2

)d3µ∗.

At this point in the argument, we want to show that the energy associated withqK is negligible so that the slow growth of the (K, qK) energy implies that the Kenergy grows (at most) slowly. The energy associated with the scalar qK is

EqK(Σ+v−) =

∫

Σ+v−

Pv−qK d3µ∗

=

∫

Σ+v−

(Gv−β(∂βψ)qKψ −

1

2

(Gv−β∂βqK

)ψ2

)d3µ∗.

If Σ+v− is understood to be restricted to the region r∗ > t/2, then

EqK(Σ+v−) .

∫

Σ+v−

(|Bβ−∂βψ|a2r−1|ψ|+ a2r−2ψ2

)d3µ∗

.∫

Σ+v−

a2|Bβ−∂βψ|2d3µ∗

+

∫

Σ+v−

a2r−2ψ2d3µ∗

.∫

Σ+v−

(a2|L+ψ|2 +

a2

r2|L−ψ|2

)d3µ∗

+

∫

Σ+v−

a2r−2ψ2d3µ∗.


The integrand in the first integral is controlled by Pv−K , so that the integral is

controlled by EK(Σ+v−). By a Hardy estimate, we have

∫ ∞

r∗

a2r−2ψ2dr∗ .∫ ∞

r∗

a2ψ2dr∗ + ψ(v−, r∗, θ, φ)

.∫ ∞

r∗

a2(ξ+ψ)2dr∗ + ψ(v−, r∗, θ, φ).

The energy EK,3 controls the first, integral term on the right.To control the second, we prove a preliminary decay result for 0 ≤ r∗ ≤ t/2,

|ψ(t, r∗, θ, φ)| =∫ r∗

0

|∂r∗ψ|dr∗ + ψ(t, 0, θ, φ)

.(∫ r∗

0

dr∗

)1/2(∫ r∗

0

|∂r∗ ψ|2dr∗

)1/2

+ t−1+C|a|(EK,3(0)1/2 + ETχ,7(0)1/2

)

.r1/2∗ t−1EK,3(t) + t−1+C|a|

(EK,3(0)1/2 + ETχ,7(0)1/2

)

.t−1/2+C|a|(EK,3(0)1/2 + ETχ,7(0)1/2

). (5.2)

In this calculation, we’ve used the spherical Sobolev estimate to relate the integralalong the curve of constant (t, θ, φ) to one over the hypersurface of constant t on

which EK,3(t) is evaluated. This provides uniform control on the end point. Sincewe were interested in the integral over the sphere, it was not necessary to introducethe two, additional derivatives, so that

∫

S2

∫ ∞

r∗

a2r−2|ψ|22d3µ∗

≤∫

S2

∫ ∞

r∗

a2|ξ+ψ|22d3µ∗ +(EK,3(0)1/2 + ETχ,7(0)1/2

).

Thus, for sufficiently small a,

EqK(Σ+v−) .a2(EK(Σ+

v−) +(EK,3(0)1/2 + ETχ,7(0)1/2

)).

Theorem 4.13 gives us a bound on the (K, qK) energy,

E(K,qK)(Σ+v−) .vC|a|− E(K,qK)(Σ0).

Since the qK energy is small relative to the K energy and the initial data, we havethe same estimate on the K energy.

This control of the energy allows us to integrate along almost null surfaces tocontrol the solution ψ near null infinity.

Theorem 5.4. There are constants a, C, and C ′, such that if |a| ≤ a and ψ is asolution to the wave equation ψ = 0, then for all t > 0, r > rχ, (θ, φ) ∈ S2, ifu− ≥ 0,

|ψ(t, r, θ, φ)|

≤ C max1, u−C′|a|(

u+ − u−u+(max1, u−)

)1/2

r−1(EK,3(0)1/2 + ETχ,7(0)1/2

),

and if u− < 0,

|ψ(t, r, θ, φ)|

≤ C max1,−u−−1/2r−1(EK,3(0)1/2 + ETχ,7(0)1/2 + supΣ0

(r3/2ψ

)).


r∗

t

(t, r∗)

(t′, r′∗) ∼ (2v−, v−)

Proof. As usual, we will start by working with the transformed function ψ whichsatisfies (4.1).

With the control from the previous lemma, we can integrate along null curves.Consider the curve, C, from (t, r∗, θ, φ) with tangent ξ+ and parameter λ. Alongthis curve, since the ∂φ component of ξ− vanishes so rapidly, λ will be uniformallyequivalent to t, to r∗, and to v+.

First, we consider when u− ≥ 0. When the curve reaches the surface r′∗ = t′/2,since v− = t − r∗ + O(1), it will reach a point (t′, r′∗, θ

′, φ′) = ((2/3)(t − r∗) +O(1), (1/3)(t−r∗)+O(1), θ, φ+O(1)), at which u′− will be t−r∗+O(1) = v−+O(1).

Integrating along the curve,

ψ(t, r∗, θ, φ) =

∫

Cξ+ψdλ+ ψ

(2

3u− +O(1),

1

3u− +O(1), θ, φ+O(1)

),

|ψ(t, r∗, θ, φ)| ≤(∫

Cv−2

+ dv+

)1/2(∫

Cv2

+(ξ+ψ)2dr∗

)1/2

+

∣∣∣∣ψ(

2

3u− +O(1),

1

3u− +O(1), θ, φ+O(1)

)∣∣∣∣

.(−v−1

+ |u++O(1)u−+O(1)

)1/2(∫

C

v2+(ξ+ψ)2dλ

)1/2

+ |u− +O(1)|−1+C|a|EK,3(0)1/2.

Applying second-order symmetries, integrating in the angular variables, and apply-ing a spherical Sobolev estimate in the usual way allows us to replace the remainingintegral in this expression by the K energy. The end point is controlled by thepreliminary estimate (5.2) in the previous lemma. Combining these we have

|ψ(t, r∗, θ, φ)| .(−(u+ +O(1))−1 + (u− +O(1))−1

)1/2EK,3(Σ+

v−)1/2

+ v−1+C|a|+

(EK,3(0)1/2 + ETχ,7(0)1/2

)

.max1, u−C|a|(

u+ − u−u+(max1, u−)

)−1/2 (EK,3(0)1/2 + ETχ,7(0)1/2

).

To obtain the desired estimate, we note that |ψ| . r−1|ψ|.When u− < 0, a similar argument applies, except that there is no |u−|C|a| loss

arising since the hypersurface of integration lies under the hypersurface u− = 0 andexcept that the endpoint in the integration is at (0, r∗ − t+O(1), θ, φ), so that the


endpoint is estimated by

|ψ(0, t− r∗ +O(1), θ, φ)| ≤(r∗ − t+O(1))−1/2 sup(r−3/2ψ).

This dictates the decay rate, since it is slower than the decay rate from (u+ −u−)/(u+(max1,−u−)). Thus, the second estimate of this theorem holds.

Appendix A. Relevance and global structure of Kerr

In this section, we review the physical relevance and global causal structure ofthe Kerr spacetime. Most of this description can be found in most introductoryrelativity texts (ie [34, 27, 41]).

The Lorentz metric for the Kerr spacetime is most simply given in terms ofthe Boyer-Lindquist coordinates (t, r, θ, φ) by (1.1). For a fixed constant R r+, one might imagine the region (t, r, θ, φ) ∈ R × (R,∞) × S2 to represent theregion of a spacetime which is a vacuum outside some astrophysical object.7 Theset (t, r, θ, φ) ∈ R × (R,∞) × S2 can be endowed both with the Kerr metric inBoyer-Lindquist coordinates (1.1) and with the flat Minkowski metric in sphericalcoordinates η = −dt2 + dr2 + r2(dθ2 + sin2 θdφ2). In this coordinate patch, asr → ∞, all components of g approach those of η, which suggests that the Kerrmetric should be seen as asymptotically flat. 8

At this point, it is useful to review the structure of infinity for the Minkowskispacetime. The Minkowski spacetime can be rewritten in terms of the coordinates(U+, U−, θ, φ) where U± = arctan(t±r). If one then multiplies the Minkowski metricby the conformal factor cos2 U+ cos2 U−, one finds that the Minkowski spacetimecan be conformally embedded into a compact subset of a larger spacetime [27]. Theboundary of the embedded region is called the conformal boundary or points atinfinity. Since conformal transformations preserve null geodesics, it is relatively easyto see that all geodesics terminate (as t → ∞) on the hypersurface U+ = π/2 andoriginate (as t→ −∞) on the hypersurface U− = −π/2. These surfaces are referredto as future and past null infinity, I+ and I−. All timelike geodesics terminate (ast→∞) at future timelike infinity i+, they originiate (as t→ −∞) at past timelikeinfinity i−, and all spacelike geodesics originate and terminate (both as r →∞) ata single point, spacelike infinity, i0. Since the Kerr spacetime is asymptotically flat,it is possible to construct the hypersurfaces I+ and I− on which out-going (witht → ∞ and r → ∞) and in-going (with t → −∞ and r → ∞) geodesics terminateand originate respectively.

The Kerr spacetime, and especially the subcase of the Schwarzschild spacetimewhere a = 0, provides the most important illustrations of the concepts of asymptoticflatness, black holes, and event horizons, which are central to general relativity. Inan asymptotically flat spacetime, a black hole is the complement of the past offuture null infinity, and its boundary is the future part of the event horizon. Theevent horizon must be a null hypersurface. The domain of outer communication, ormore simply the exterior region, is the intersection of the past of future null infinityand the future of past null infinity. Ignoring the physical interpretation given above,one can continue the Kerr spacetime as a vacuum solution of Einstein’s equations bycontinuing it to r < R and even to r < r+. When |a| ≤M , the maximally extendedKerr spacetime contains a black hole, the exterior region is given by r > r+, andthe event horizon is given by r = r+.

7In spherical symmetry, a = 0, it is easy to find explicit, globally smooth solutions of Einstein’s

equation coupled to matter models which can be written in spherical coordinates, in which there

is a spherical material body from r = 0 to r = R, which contain vacuum for r > R, and in whichthe metric exactly coincides with (1.1) for r > R. We are not aware of such solutions for a 6= 0.

8We omit the precise definitions of asymptotically flat[41] and asymptotically simple[27], sincethey are quite technical.


The parameters M and a can be understood by comparing the Kerr spacetimein Boyer-Lindquist coordinates to the Minkowski metric in spherical coordinates.Timelike geodesics, representing the paths of small physical bodies, in the Kerrspacetime do not follow the same trajectories as geodesics in the Minkowski metric.The deviation between these geodesics can be treated as a relative acceleration, and,as r →∞, this acceleration corresponds to that generated by the gravitational forcefrom Newtonian mechanics (also in spherical coordinates) generated by a centralobject with mass M . This is one reason why the parameter M is interpreted as themass of the black hole, even though the Kerr spacetime is a solution of Einstein’sequation in vacuum.

Alternatively, Einstein’s equation can be treated as a Lagrangian theory[36], andthe parameters M and a can be understood as energies associated with a vectorfields. Normally, an energy is computed on a hypersurface, but, under certainconditions, one can apply integration by parts so that the energy can be computedas an integral over the boundary of the hypersurface. For any asymptotically flatspacetime, this allows energies to be defined with respect to vector fields definednear infinity in Minkowski space. In Minkowski space, the energy associated with ∂tis referred to as simply the energy. In relativity, one expects that, with the correctchoice of units, the energy and mass coincide. In Minkowski space, the energyassociated with a rotation about an axis is called the angular momentum aboutthat axis. Computing these quantities on two-surfaces very close to i0, one findsthat the mass of the Kerr black hole is M , that the angular momentum about theaxis of symmetry is Ma, and that the angular momentum about orthogonal axes iszero.

The metric (1.1) is clearly smooth in the exterior region, since neither ∆ nor Σvanish. The roots of ∆ are at

r± = M ±√M2 − a2,

and the roots of Σ occur when both r and a cos θ vanish. Thus, when 0 < |a| <M , the metric is clearly smooth where r > r+, where r− < r < r+, and wherer < r− except at (r, θ) = (0, π/2). We shall refer to these regions as the exterior,intermediate, and deep-interior regions respectively and as block I, block II, andblock III respectively. Using the coordinates (u−, r, θ, φ∗) (with u− defined in theintroduction, r and θ the standard Boyer-Lindquist coordinates, and φ∗ = φ− ωHtgiven in section 5.2), it is possible to construct a smooth coordinate chart whichcovers a copy of both block I and block II.

Through similar choices of changes of coordinates, various copies of the blockscan be be smoothly joined together to form the maximal extension of the Kerrmanifold as illustrated in figure A. The copies of block II and block III meet wherer = r−. The two-dimensional surfaces where two copies of block I and two copiesof block II meet is also a regular point of the maximal extension and is called thebifurcation surface. The diagram in figure A is also a conformal diagram, whichincludes the conformal boundaries of the exterior regions and also of the deep-interior regions. The deep-interior regions contain singularities, in the sense thatthere are affinely parameterised geodesics along which (r, θ) → (0, π/2), on whichthe affine parameter remains bounded, but which cannot be extended, and it is isnot possible to smoothly extend the spacetime so that the geodesics can be extendedto (r, θ) = (0, π/2). The diagram does not reveal other patholgical behaviour in thedeep interior, in particular, the presence of closed timelike geodesics.

The subcritical (|a| < M) Kerr spacetime is expected to have a crucial phys-ical role. From the singularity theorems, it is known that under a wide rangeof circumstances, spacetimes will develop singularities. The weak cosmic censor-ship conjecture asserts that, in asymptotically flat spacetimes, future null infinityis separated from any (future) singularity by an event horizon, as long as some


II

I I

II

III III

II

I I

II

III III

II

I I

II

r=r+ I+

Figure 2. The conformal diagram for the Kerr spacetime: Angu-lar variables have been suppressed. The boundaries of blocks areshown in solid lines. Boundaries at infinity and singularities areindicated by dotted lines. The event horizon at r = r+ and I+

has been indicated for one exterior region. The maximal extensioncontinues with this pattern infinitely forward and backward in time.

as-yet-to-be-determined genericity condition holds. The no hair theorem, or Kerruniqueness theorem, states that the Kerr family of spacetimes (including Minkowskias M = 0) is the unique class of stationary spacetimes satisfying Einstein’s equationin vacuum (see [37] for early work, and [2] for recent progress). It is common inphysics that systems will relax towards stationary configurations, and, in particular,it is widely expected that solutions to Einstein’s equation containing a single blackhole will asymptotically approach a Kerr solution plus gravitational radiation goingto I+, at least in the exterior region. A preliminary step in showing that this istrue would be to show that the Kerr spacetimes are dynamically stable, i.e. that asmall perturbation of the initial data which leads to a Kerr spacetime will generatea solution to Einstein’s equation which asymptotically approaches a Kerr spacetimeplus radiation.

Acknowledgements. A significant part of this work was completed during the“Geometry, Analysis, and General Relativity” programme held at the Mittag-LefflerInstitute, Djursholm, Sweden, in the fall of 2008. Both authors are grateful tothe institute for hospitality and support during this time, and to many of theparticipants in the programme for useful discussions. PB thanks the Albert EinsteinInstute for support during the initial phase of this project. LA was supported inpart by the NSF, under contract no. DMS-0707306.


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E-mail address: [email protected]

Albert Einstein Institute, Am Muhlenberg 1, D-14476 Potsdam, Germany

E-mail address: [email protected]

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Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, Scotland

EH9 3JZ,UK

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