UNIFORM ENERGY BOUND AND MORAWETZ ESTIMATE FOR EXTREMECOMPONENTS OF SPIN FIELDS IN THE EXTERIOR OF A SLOWLY

ROTATING KERR BLACK HOLE II: LINEARIZED GRAVITY

SIYUAN MA

Laboratoire Jacques-Louis Lions, Sorbonne Université, Campus Jussieu, 4 place Jussieu 75005Paris, France.

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

Abstract. This second part of the series treats spin ±2 components (or extreme components),that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowlyrotating Kerr black hole. For each of these two components, after performing a first-order differen-tial operator once and twice, the resulting equations together with the Teukolsky master equationitself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin ±2components is proved by treating this system. This is a first step in a joint work [2] in addressingthe linear stability of slowly rotating Kerr metrics.

1. Introduction

The nonlinear stability conjecture of Kerr black holes says that metrics of the subextremal Kerrfamily of spacetimes (M, g = gM,a) (|a| < M) are (expected to be) stable against small perturbationsof initial data as solutions to the vacuum Einstein equations

Ric[g]µν = 0, (1.1)

Ric[g]µν being the Ricci curvature tensor of the metric. An important step toward the resolution ofthis conjecture of nonlinear stability of Kerr metrics is to show the linear stability, i.e., to show theasymptotic decay of linearized gravitational perturbations (also called “linearized gravity”) aroundKerr metrics.

As a beginning step in proving linear stability of Kerr metrics, we consider the gauge invariantextreme components of the Weyl tensor which satisfy the well-known Teukolsky master equation[44] and govern the dynamics of linearized gravity, and prove both a uniform bound of a positivedefinite energy and a Morawetz estimate for these extreme components on a slowly rotating Kerrbackground (where |a|/M � 1 is sufficiently small). In a joint work [2], this basic energy andMorawetz estimate, whereas called “Basic decay condition” or “BEAM condition” in [2], is utilizedto obtain further strong decay estimates for the extreme components and the full linear stability ofslowly rotating Kerr metrics.

1.1. Kerr Metric. For the purpose that this paper can be read independently, we review in thissubsection the setup of the Kerr metric and notation from the first part [30] of this series.

A Kerr spacetime (M, g = gM,a) [26] has a metric given in Boyer–Lindquist (B–L) coordinates[11] (t, r, θ, φ) by

gM,a =−(1− 2MrΣ

)dt2 − 2Mar sin

2 θΣ (dtdφ+ dφdt)

+ Σ∆dr2 + Σdθ2 + sin

2 θΣ

[(r2 + a2)2 − a2∆ sin2 θ

]dφ2 (1.2)

with

∆(r) = r2 − 2Mr + a2 and Σ(r, θ) = r2 + a2 cos2 θ, (1.3)1

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and describes a rotating, stationary (with ∂t Killing), axisymmetric (with ∂φ Killing), asymptoticallyflat solution to vacuum Einstein equations (1.1). The Schwarzschild metric [38] is obtained by settinga = 0.

The region we consider is the domain of outer communication (DOC)

D = {(t, r, θ, φ) ∈ R× (r+,∞)× S2}, (1.4)

where r+ = M+√M2 − a2 is the value of the larger root of ∆(r) = 0 and corresponds to the location

of the event horizon. By symmetry (cf. Section 1.4), we focus only on the future development withboundary the future event horizon H+. In this paper, a slowly rotating Kerr spacetime shouldalways be understood as the DOC of a Kerr spacetime endowed with the Kerr metric g = gM,a withsufficiently small |a|/M � 1.

The tortoise coordinate r∗ is defined by:

dr∗

dr=r2 + a2

∆, r∗(3M) = 0, (1.5)

and we call (t, r∗, θ, φ) the tortoise coordinate system. However, both the B-L and tortoise coordinatesystems fail to extend across the future event horizon H+ due to the singularity in the metriccoefficients. Instead, an ingoing Kerr coordinate system (v, r, θ, φ̃), which is regular on H+, isdefined by: {

dv = dt+ dr∗,

dφ̃ = dφ+ a(r2 + a2)−1dr∗.(1.6)

Moreover, via gluing the coordinate system (ϑ = v− r, r, θ, φ̃) near horizon with the B–L coordinatesystem (t, r, θ, φ) away from horizon smoothly, a global Kerr coordinate system (t∗, r, θ, φ∗) can begiven by {

t∗ = t+ χ1(r) (r∗(r)− r − r∗(r0) + r0) ,

φ∗ = φ+ χ1(r)φ́(r) mod 2π, dφ́/dr = a/∆.(1.7)

The smooth cutoff function χ1(r) here equals 1 in [r+,M + r0/2] and identically vanishes for r ≥ r0with r0 = r0(M) fixed in Section 3.2, and is chosen such that on the initial spacelike hypersurface

Σ0 = {(t∗, r, θ, φ∗)|t∗ = 0} ∩ D, (1.8)

there exist two universal positive constants c(M) and C(M) so that

c(M) ≤ −g(∇t∗,∇t∗)|Σ0 ≤ C(M). (1.9)

Here the initial hypersurface could be taken as {t∗ = D} hypersurface for any real value D, but forconvenience, we take it as in (1.8).

In these coordinate systems, it is manifest that

∂t∗ = ∂t , T and ∂φ∗ = ∂φ̃ = ∂φ. (1.10)

Denote ϕτ as the 1-parameter family of diffeomorphisms generated by T and define constant-timespacelike hypersurfaces satisfying (1.9) as well:

Στ = ϕτ (Σ0) = {(t∗, r, θ, φ∗)|t∗ = τ} ∩ D. (1.11)

We finally adopt the notations for any 0 ≤ τ1 < τ2 that

D(τ1, τ2) =⋃

τ∈[τ1,τ2]

Στ , and H+(τ1, τ2) = ∂D(τ1, τ2) ∩H+.

The reader may refer to the Penrose diagram Fig. 1.We shall make use of a volume element for the hypersurface Στ (τ ≥ 0)

dVolΣτ = Σdr sin θdθdφ∗ in global Kerr coordinates, (1.12)

and the volume form of the manifold is

dVolM ={

Σdtdr sin θdθdφ in B–L coordinates,Σdt∗dr sin θdθdφ∗ in global Kerr coordinates. (1.13)

2

Figure 1. Penrose diagram

Note that dVolΣτ is a convenient reference volume form in calculations and in stating the integralestimates, but it is not the induced volume form on Στ . Unless otherwise specified, we will alwayssuppress these volume forms associated to the integrals in this paper.

1.2. Linearized Gravity and Teukolsky Master Equation. Following Newman–Penrose (N–P)formalim [34, 35], we obtain the complete five N–P components

Φ0 =−Wlmlm, Φ1 = −Wlnlm, Φ2 = −Wlmmn,Φ3 = −Wlnmn, Φ4 = −Wnmnm (1.14)

by projecting the Weyl tensor Wαβγδ onto the Kinnersley null tetrad (l, n,m,m) [27] in B–L coor-dinates:

lµ = 1∆ (r2 + a2,∆, 0, a),

nν = 12Σ (r2 + a2,−∆, 0, a),

mµ = 1√2ρ̄

(ia sin θ, 0, 1, isin θ

), (1.15)

and mµ and ρ̄ being the complex conjugate of mµ and ρ = r − ia cos θ respectively. The full set ofN-P equations, comprising the commutation relations, the Ricci identities, the eliminant relationsand the Bianchi identities in [12, Chapter 1.8], is then a coupled first-order differential system linkingthe tetrad, the spin coefficients and these five N–P components. On a Kerr background,

Φ0 = Φ1 = Φ3 = Φ4 = 0, Φ2 = −Mρ̄−3. (1.16)

We perturb in the N–P equations all the tetrad components, the spin coefficients and the five N-Pcomponents by lT = l + lP , κT = κ + κP ,1 ΦT0 = Φ0 + ΦP0 , etc, and the complete set of equationsfor linearized gravity is then obtained from the N–P equations by keeping the perturbation terms(with superscript P ) only to first order. The perturbed extreme components ΦT0 and ΦT4 (whichare equal to ΦP0 and ΦP4 ) for linearized gravity are the “ingoing and outgoing radiative parts,” andare invariant under gauge transformations and infinitesimal tetrad rotations. From now on, we willdrop the superscript and still denote these perturbed extreme components as Φ0 and Φ4.

Teukolsky [44] derived the decoupled equations on Kerr backgrounds for the spin s = ±2 compo-nents

ψ[+2] = ∆2Φ0 and ψ[−2] = ∆−2ρ4Φ4, (1.17)

1κ is one of the spin coefficients used in [12, Chapter 1.8].3

and showed that these decoupled equations are in fact separable and governed by a single masterequation–the celebrated Teukolsky Master Equation (TME)–given in B–L coordinates by

−[

(r2+a2)2

∆ − a2 sin2 θ

]∂2ψ[s]∂t2 −

4Mar∆

∂2ψ[s]∂t∂φ −

[a2

∆ −1

sin2 θ

]∂2ψ[s]∂φ2

+ ∆s ∂∂r

(∆−s+1

∂ψ[s]∂r

)+ 1sin θ

∂∂θ

(sin θ

∂ψ[s]∂θ

)+ 2s

[a(r−M)

∆ +i cos θsin2 θ

]∂ψ[s]∂φ

+ 2s[M(r2−a2)

∆ − r − ia cos θ]∂ψ[s]∂t − (s

2 cot2 θ + s)ψ[s] = 0.

(1.18)

In fact, TME is valid for general spin s fields with s = n2 , n ∈ Z. In particular, the s = 0 case is thescalar wave equation, and the s = ±1 cases are the governing equations of spin ±1 components ofthe Maxwell field.

The Kinnersley tetrad is, however, singular on H+ in ingoing Kerr coordinates, suggesting thatthe perturbed N–P components are not all regular there. We perform a null rotation by l→ l̃ = ∆/(2Σ) · l,n→ ñ = (2Σ)/∆ · n,

m→ m,(1.19)

and the resulting tetrad (l̃, ñ,m,m), namely the Hawking–Hartle (H–H) tetrad [23], is in fact regularup to and on H+ in global Kerr coordinates. The regular extreme components of linearized gravityin the regular H–H tetrad are then{

Φ̃0(W) = −Wl̃ml̃m =1

4Σ2ψ[+2],

Φ̃4(W) = −Wñmñm = 4Σ2

ρ4 ψ[−2].(1.20)

The results in this paper will be with respect to complex scalars Φ̃0 and Φ̃4.

1.3. Coupled Systems. Denote the future-directed ingoing and outgoing principal null vector fieldsin B–L coordinates

Y , (r2+a2)∂t+a∂φ

∆ − ∂r, V ,(r2+a2)∂t+a∂φ

∆ + ∂r. (1.21)

From TME (1.18), the equations for ψ[+2] and ψ[−2] are(Σ�g + 4i

(cos θsin2 θ

∂φ − a cos θ∂t)− (4 cot2 θ + 2)

)ψ[+2] = −4Zψ[+2], (1.22a)(

Σ�g − 4i(

cos θsin2 θ

∂φ − a cos θ∂t)− (4 cot2 θ − 2)

)ψ[−2] = 4Zψ[−2], (1.22b)

with Z = (r −M)Y − 2r∂t. Construct from ψ[+2] and ψ[−2] the quantities φ0+2 = ψ[+2]/r

4;φ1+2 = (rY r) (φ

0+2);

φ2+2 = (rY r) (rY r) (φ0+2),

(1.23a)

and φ0−2 = ∆

2/r4ψ[−2];φ1−2 = − (rV r) (φ0−2);φ2−2 = (rV r) (rV r) (φ

0−2).

(1.23b)

The upper index here denotes the number of times the differential operator rY r or −(rV r) isperformed. Note that though it is not V but rather ∆r2+a2V which is a regular vector field on H

+,by the second relation in (1.20), the variables

{φi−2

}i=0,1,2

in (1.23b) are indeed smooth up to and

on future horizon if the regular N–P scalar Φ̃4 is. In global Kerr coordinates, the regular vector fieldY equals −∂r + ∂t∗ in [r+,M + r0/2] and is r

2+a2

∆ ∂t∗ +a∆∂φ∗ − ∂r for r ≥ r0.

The rescalings in the definitions of φ0±2 are such that one can rewrite the first order Z-derivativeterms in terms of φ1±2 plus first order derivative terms with a-dependent coefficients. From thecommutator relations in Appendix A, one can derive the equations satisfied by φ1±2 and φ2±2. Thecoupled systems of equations for these quantities are

L0+2φ0+2 = F

0+2 =

4(r2−3Mr+2a2)r3 φ

1+2 −

8(a2∂t+a∂φ)φ0+2

r , (1.24a)4

L1+2φ1+2 = F

1+2 =

2(r2−3Mr+2a2)r3 φ

2+2 +

6Mr−12a2r φ

0+2

− 4(a2∂t+a∂φ)φ

1+2

r − 6(a2∂t + a∂φ)φ

0+2, (1.24b)

L1+2φ2+2 = F

2+2 =− 8(a2∂t + a∂φ)φ1+2 − 12a2φ0+2, (1.24c)

and

L0−2φ0−2 = F

0−2 =

4(r2−3Mr+2a2)r3 φ

1−2 +

8(a2∂t+a∂φ)φ0−2

r , (1.25a)

L1−2φ1−2 = F

1−2 =

2(r2−3Mr+2a2)r3 φ

2−2 +

6Mr−12a2r φ

0−2

+4(a2∂t+a∂φ)φ

1−2

r + 6(a2∂t + a∂φ)φ

0−2, (1.25b)

L1−2φ2−2 = F

2−2 =8(a

2∂t + a∂φ)φ1−2 − 12a2φ0−2, (1.25c)

respectively.2 The subscript +2 or −2 here indicates the spin weight s = ±2, and the operators L0sand L1s, given by

L0s = Σ�g + 2is(

cos θsin2 θ

∂φ − a cos θ∂t)− s2

(cot2 θ + r

2+2Mr−2a22r2

), (1.26a)

L1s = Σ�g + 2is(

cos θsin2 θ

∂φ − a cos θ∂t)− s2

(cot2 θ + r

2−2Mr+2a2r2

), (1.26b)

are both spin-weighted wave operators, but with different potentials. The equations for φis in (1.24)and (1.25) are in either form of the following equations:

L0sψ = F ; (1.27a)

L1sψ = F, (1.27b)

both of which are called in this paper as inhomogeneous spin-weighted wave equations (ISWWE).When there is no confusion of which spin component we are treating, we may suppress the subscriptof φis and simply write as φi.

Remark 1. After making the substitutions ∂t ↔ −iω, ∂φ ↔ im, and separating the operators Lks(k = 0, 1), the angular parts are the spin-weighted spheroidal harmonic operator of angular Teukolskyequation. The radial operator of L1s is the sum of the radial part of the rescaled scalar wave operatorΣ�g and a potential s2(r2−∆−a2)/r2, and reduces to the radial operator for Regge–Wheeler equation[37] when on Schwarzschild background (a = 0), while the one of L0s is the sum of the radial part ofΣ�g and another potential s2(∆ + a2)/(2r2). See more details in Section 5.2 for Schwarzschild caseand Section 6.2 for Kerr case.

1.4. Main Theorem. The TME admits a symmetry that ∆sψ[−s](−t, r, θ,−φ) and ψ[s](t, r, θ, φ)satisfy the same equation; hence we focus only on the future time development in this paper, andone can obtain the analogous estimates in the past time direction.

For any complex-valued smooth function ψ :M→ C with spin weight s, we define in global Kerrcoordinates that for any τ ≥ 0,

|∂ψ(t∗, r, θ, φ∗)|2 = |∂t∗ψ|2 + |∂rψ|2 + |∇/ψ|2, (1.28)

Eτ (ψ) =

∫Στ

|∂ψ|2, (1.29)

and in ingoing Kerr coordinates that for any τ2 > τ1 ≥ 0,

EH+(τ1,τ2)(ψ) =

∫H+(τ1,τ2)

(|∂vψ|2 + |∇/ψ|2)r2dv sin θdθdφ̃. (1.30)

2This application of the first-order differential operators to the spin ±2 components is closely related to Chan-drasekhar transformation [13].

5

The ∇/ used here are not the standard rotational angular derivatives ∇̌/ on sphere S2(t∗, r), but thespin-weighted version of them, i.e., ∇/ could be any one of ∇/ j (j = 1, 2, 3) defined by

r∇/ 1 = r∇̌/ 1 −is cosφ

sin θ = (− sinφ∂θ −cosφsin θ cos θ∂φ∗)−

is cosφsin θ ,

r∇/ 2 = r∇̌/ 2 −is sinφsin θ = (cosφ∂θ −

sinφsin θ cos θ∂φ∗)−

is sinφsin θ ,

r∇/ 3 = r∇̌/ 3 = ∂φ∗ .(1.31)

In global Kerr coordinates, we can express the squared absolute value of ∇/ψ as

|∇/ψ|2 =∑

i=1,2,3

|∇/ iψ|2

=1

r2

(|∂θψ|2 +

∣∣∣ cos θ∂φ∗ψ+isψsin θ ∣∣∣2 + |∂φ∗ψ|2)=

1

r2

(|∂θψ|2 +

∣∣∣∂φ∗ψ+is cos θψsin θ ∣∣∣2 + s2|ψ|2). (1.32)In particular, note from (1.32) that |∇/ψ|2, and thus |∂ψ|2, already have control over r−2|ψ|2 ifthe spin weight s 6= 0. The same expressions (1.31) and (1.32) hold true in B–L coordinates andingoing Kerr coordinates from (1.10). For convenience of calculations, we may always refer to theseexpressions with ∂φ in place of ∂φ∗ .

For any smooth function ψ with spin weight s, we define for any multi-index k = (k1, k2, k3, k4, k5)with ki ≥ 0 (i = 1, 2, 3, 4, 5) that

∂kψ = ∂k1t∗ ∂k2r ∇/

k31 ∇/

k42 ∇/

k53 ψ. (1.33)

Define the length of such a multi-index k by

|k| = k1 + k2 + k3 + k4 + k5. (1.34)

Denote a few Morawetz densities by3

Mdeg(ψ) = r−1−δ|∂rψ|2 + χtrap(r)(r−1−δ|∂t∗ψ|2 + r−1|∇/ψ|2) + r−3|ψ|2, (1.35a)

M(ψ) = r−1−δ(|∂rψ|2 + |∂t∗ψ|2) + r−1|∇/ψ|2 + r−3|ψ|2, (1.35b)

M̃deg(ψ) = r−1|∂rψ|2 + χtrap(r)r−1(|∂t∗ψ|2 + |∇/ψ|2) + r−3|ψ|2, (1.35c)

M̃(ψ) = r−1|∂ψ|2 + r−3|ψ|2. (1.35d)Here, χtrap(r) = 1 − η[r−trap,r+trap](r), η[r−trap,r+trap](r) is the indicator function in the radius regionbounded by r−trap = 2.9M < 3M < r

+trap = 3.1M , and δ ∈ (0, 1/2) is an arbitrary constant.

Theorem 2. Consider the linearized gravity in the DOC of a slowly rotating Kerr spacetime (M, g =gM,a). Given any smooth4 regular extreme components Φ̃0, Φ̃4 as in Section 1.2 which vanish nearspatial infinity, then for any 0 < δ < 1/2 and nonnegative integer n, there exist universal constantsε0, R = R(M) and C = C(M, δ,Σ0, n) = C(M, δ,Στ , n) such that for all |a|/M ≤ ε0 and any τ ≥ 0,we have the following energy and Morawetz estimate for the regular extreme components:∑

|k|≤n

∫D(0,τ)

(Mdeg(∂kΦ(2)0 ) + M̃(∂

kΦ(1)0 ) + M̃(∂

kΦ(0)0 ))

+∑|k|≤n

2∑i=0

(Eτ (∂

kΦ(i)0 ) + EH+(0,τ)(∂

kΦ(i)0 ))≤ C

∑|k|≤n

2∑i=0

E0(∂kΦ

(i)0 ), (1.36a)

∑|k|≤n

∫D(0,τ)

(Mdeg(∂kΦ(2)4 ) + M(∂

kΦ(1)4 ) + M(∂

kΦ(0)4 ))

+∑|k|≤n

2∑i=0

(Eτ (∂

kΦ(i)4 ) + EH+(0,τ)(∂

kΦ(i)4 ))≤ C

∑|k|≤n

2∑i=0

E0(∂kΦ

(i)4 ). (1.36b)

3We should distinguish among these different notations that a tilde means that there is no extra r−δ power in thecoefficients of ∂r- and ∂t∗ -derivatives term and a subscript deg means there is the trapping degeneracy in the trappedregion, and vice versa.

4In fact, the N–P components should be viewed as sections of a complex line bundle. Therefore, “smooth” meansthat these components and their derivatives to any order with respect to (∂t∗ , ∂r,∇/ 1,∇/ 2,∇/ 3) are continous.

6

Here, the set (Φ(0)j ,Φ(1)j ,Φ

(2)j ) for j = 0, 4 takes

Φ(0)0 = r

4−δΦ̃0, Φ(1)0 = r

4−δY Φ̃0, Φ(2)0 = r

4Y Y Φ̃0; (1.37a)

Φ(0)4 = Φ̃4, Φ

(1)4 =

r∆r2+a2V (rΦ

(0)4 ), Φ

(2)4 =

r∆r2+a2V (rΦ

(1)4 ). (1.37b)

Remark 3. The trapping degeneracy for the Morawetz densities Mdeg(∂kΦ(2)0 ) and Mdeg(∂kΦ(2)4 )

with |k| ≤ n−1 can actually be removed. We shall only focus on proving the n = 0 case until Section6.8. As shown in Section 6.8, the general n ≥ 0 cases follow from an induction in n.

Remark 4. As stated above, consider the n = 0 case. The energy and Morawetz estimate (1.36)is obtained by treating systems (1.24) and (1.25) of φis and is a single estimate at three levels ofregularity for each extreme component, since φ2s involves at most second order derivatives of φ0s.Therefore, in spite of the well-known trapping phenomenon, we prove Morawetz estimates for φ0sand φ1s which are in fact non-degenerate in the trapped region. However, the three levels of regularitymust be treated simultaneously. On one hand, to estimate the inhomogeneous terms on the right-hand side of (1.24) and (1.25), it is necessary to eliminate the trapping degeneracy in the Morawetzestimates for φ0s and φ1s by considering one more order of derivative; on the other hand, it is possibleto close the three estimates simultaneously, because the right-hand side of (1.24) and (1.25) are attwo level of regularity at most, involving no derivatives of φ2s and at most one of φ0s and φ1s.

Note that systems (1.24) and (1.25) are, however, not weakly coupled anymore as in the spin-1case [30], a fact caused by the presence of the φ1s term in (1.24a) and (1.25a) or the φ0s term in(1.24b) and (1.25b). Take system (1.24) for s = +2 for example. Our approach here relies on anestimate bounding φ1+2 from φ2+2 by employing the differential relation (1.23a) between them, whichenables us to treat the system in a rough (but accurate in the Schwarzschild case) sense that the errorterm in the Morawetz estimate for (1.24a) arising from the inhomogeneous term can be controlledby adding a large amount of Morawetz estimate of (1.24c) to it, cf. Section 1.6.

1.5. Previous Results. We briefly review the existed results in the literature on scalar wave equa-tion and Maxwell equations in the exterior of Schwarzschild and Kerr black holes. The uniformboundedness of scalar wave on a Schwarzschild background is first proved in [25], and a robust,powerful tool–Morawetz estimate [33], or integrated local energy decay estimate–was used first in [9]and then in some followup works like [10, 16]. For the scalar wave on the Kerr background, the decayresults are shown in three different approaches [42, 3, 19] where the first two are restricted to slowlyrotating Kerr and the last one is for full subextremal Kerr. Decay behaviours for Maxwell field areproved in [8] on Schwarzschild, and on some spherically symmetric backgrounds or non-stationaryasymptotically flat backgrounds in [31, 41]. These works start with estimating the middle compo-nent from a decoupled, separable Fackerell–Ipser equation [20]. This approach is further generalizedto the slowly rotating Kerr case in [4] to show the uniform boundedness of a positive definite energyand the convergence property of the Maxwell field to a stationary Coulomb field. In contrast, onecan treat first the extreme components satisfying the TME by applying some first-order differentialoperators used here to the extreme components, and the new quantitities satisfy an equation similarto the Fackerell–Ipser equation (or spin-weighted version of Fackerell–Ipser equation). This is carriedout in [36] for the Schwarzschild case and in our first part [30] of this series for the slowly rotatingKerr case.

There are typically two ways of linearizing the vacuum Einstein equations, one via metric pertur-bations and the other via tetrad perturbations. The linear stability of Schwarzschild metric undermetric perturbations was resolved recently in [14, 24]. The former one starts from a Regge–Wheeler[37] type equation satisfied by some scalar constructed by applying a physical-space version offixed-frequency Chandrasekhar transformation [13] to some Riemann curvature components (closelyrelated to extreme components in N–P formalism), and the later one carries out a detailed studyon the Regge–Wheeler–Zerilli–Moncrief [37, 46, 32] system. The energy, Morawetz, and pointwisedecay estimates for this system are obtained in [5] as well.

On a non-static Kerr background, as mentioned already, if one linearizes the vacuum Einsteinequations via tetrad perturbations, the extreme components of the Weyl tensor in Newman–Penrose

7

formalism satisfy a decoupled, separable wave-like equation–the TME (1.18) and govern the dy-namics of linearized gravity. These two extreme components are closely related to each other bydifferential relations: after decomposing spin ±2 components into modes, differential relations be-tween the radial parts of the modes with opposite extreme spin weights, as well as between theangular parts, are derived in [40, 43] and are known as Teukolsky-Starobinsky Identities. See theversion of these identities in the physical space in [1]. In [45], it is shown that the TME admits nomode with frequency having positive imaginary part, or in another way, no exponentially growingmode solution exists, under the assumption of no incoming radiation condition. This mode stabilityresult is recently generalized in [39, 6] to the case of real frequencies. Note that the mode stabil-ity result [39] for scalar field is indispensable in the work [19] treating the scalar wave on the fullsubextremal Kerr backgrounds, and we expect our mode stability result [6] for general spin fieldswill play an essential role in generalizing the results from the slowly rotating Kerr case consideredin this work to the full subextremal Kerr case. Linear stability of slowly rotating Kerr spacetimesis proved in [2, 22], and it is shown in [2] that the linear stability of a subextremal Kerr spacetimecan be proved under an assumption of a basic energy and Morawetz estimate in the same form ofthe main result of this work in the full subextremal Kerr backgrounds. During the submission ofthis work, the authors in [15] obtain a similar result as that in this work. We notice also the works[21] which discusses the stability problem for each azimuthal mode solution to TME and [28] whichproves a first nonlinear stability result for the Schwarzschild metric (though under axially symmetricpolarized perturbations).

1.6. Outline of the Proof. It is convenient for the latter discussions to introduce the variableswhich are non-degenerate at H+

φ̃0−2 = ∆−2r4φ0−2, φ̃

1−2 = ∆

−1r2φ1−2, (1.38)

and we may suppress the subindex and simply write as φ̃0 and φ̃1. Let µ0, µ1, µ2, �0, �1, �̂1, A0, andA2 be small positive constants to be fixed. Define two quantities for spin ±2 components respectivelythat

Ξ+2(0, τ) = E0(r4−δφ0+2) + E0(r

2−δφ1+2) + E0(φ2+2)

+ ε1/20

∑ϕ∈{r4−δφ0+2,r2−δφ1+2,φ2+2}

(Eτ (ϕ) + EH+(0,τ)(ϕ)

)+ ε

1/20

∫D(0,τ)

(M̃(r4−δφ0+2) + M̃(r2−δφ1+2) + Mdeg(φ2+2)

), (1.39a)

Ξ−2(0, τ) = E0(φ̃0) + E0(φ̃1) + E0(φ2−2) +

∫Σ0

r(|∇/ φ̃0|2 + |∇/ φ̃1|2

)+ ε

1/20

∑ϕ∈{φ̃0,φ̃1,φ2−2}

(Eτ (ϕ̃) + EH+(0,τ)(ϕ)

)+ ε

1/20

∫D(0,τ)

(M(φ̃0) + M(φ̃1) + Mdeg(φ2−2)

). (1.39b)

We say F1 .ε0 F2 for two functions F1 and F2 in the region D(0, τ) if there exists a universalconstant C and a constant C1 = C1(µ0, µ1, µ2, �0, �1, �̂1, A0, A2) 5 such that

F1 ≤ CF2 + C1Ξ+2(0, τ) (1.40a)

or

F1 ≤ CF2 + C1Ξ−2(0, τ), (1.40b)

depending on which spin component we are considering. We now give the outline of the proof ofestimates (1.36) for spin +2 and −2 components separately.

5The dependence of �̂1 in C1 is not needed for spin −2 component.8

1.6.1. Spin +2 Component. We will first obtain in Section 5 and Section 6.6 the following energyand Morawetz estimates for φ0, φ1 and φ2 defined from the spin +2 component:

Eτ (r4−δφ0) + EH+(0,τ)(r

4−δφ0) +

∫D(0,τ)

M̃deg(r4−δφ0)

.ε0 µ−10

∫D(0,τ)

(�0M̃(r4−δφ0) +

�̂1�0M̃(rφ1) +

1

�0�̂1Mdeg(φ2)

)+ µ0

∫D(0,τ)

(�1M̃(r2−δφ1) + �−11 M̃deg(r

4−δφ0) + �−11 Mdeg(φ2))

+ µ0

∫D(0,τ)

(M̃deg(r4−δφ0) + M̃deg(r2−δφ1) + Mdeg(φ2)

), (1.41a)

Eτ (r2−δφ1) + EH+(0,τ)(r

2−δφ1) +

∫D(0,τ)

M̃deg(r2−δφ1)

.ε0 µ−11

∫D(0,τ)

(�1M̃(r2−δφ1) + �−11 M̃deg(r

4−δφ0) + �−11 Mdeg(φ2))

+ µ1

∫D(0,τ)

(�0M̃(r4−δφ0) +

�̂1�0M̃(rφ1) +

1

�0�̂1Mdeg(φ2)

)+ µ1

∫D(0,τ)

(M̃deg(r4−δφ0) + M̃deg(r2−δφ1) + Mdeg(φ2)

), (1.41b)

Eτ (φ2) + EH+(0,τ)(φ

2) +

∫D(0,τ)

Mdeg(φ2)

.ε0 µ2

∫D(0,τ)

(M̃deg(r4−δφ0) + M̃deg(r2−δφ1) + Mdeg(φ2)

)+ µ2

∫D(0,τ)

(�0M̃(r4−δφ0) +

�̂1�0M̃(rφ1) +

1

�0�̂1Mdeg(φ2)

)+ µ2

∫D(0,τ)

(�1M̃(r2−δφ1) + �−11 M̃deg(r

4−δφ0) + �−11 Mdeg(φ2)). (1.41c)

Here, all the parameters appearing above are small constants to be fixed, and we have assumed thatε0 is much smaller compared to these parameters. We add an A0 multiple of estimate (1.41a) andan A2 multiple of (1.41c) to estimate (1.41b), and find the left-hand side (LHS) of the obtainedestimate is larger than

c

∫D(0,τ)

(A0M̃deg(r4−δφ0) + M̃deg(r2−δφ1) +A2Mdeg(φ2)

)+ c

∫D(0,τ)

(M̃(r4−δφ0) + M̃(r2−δφ1)

). (1.42)

Here, the bound over the second line comes from the following relations

Eτ (r4−δφ0) + E0(r

4−δφ0) +

∫D(0,τ)

(M̃deg(r4−δφ0) + M̃deg(r2−δφ1)

)∼ Eτ (r4−δφ0) + E0(r4−δφ0) +

∫D(0,τ)

(M̃(r4−δφ0) + M̃deg(r2−δφ1)

), (1.43a)

Eτ (r4−δφ0) + Eτ (r

2−δφ1) + E0(r4−δφ0) + E0(r

2−δφ1)

+

∫D(0,τ)

(M̃deg(r4−δφ0) + M̃deg(r2−δφ1) + Mdeg(φ2)

)∼Eτ (r4−δφ0) + Eτ (r2−δφ1) + E0(r4−δφ0) + E0(r2−δφ1)

+

∫D(0,τ)

(M̃(r4−δφ0) + M̃(r2−δφ1) + Mdeg(φ2)

). (1.43b)

9

In the trapped region, M̃deg(r4−δφ0) + M̃deg(r2−δφ1) bounds over |Y φ0|2, |∂r∗φ0|2 and |φ0|2, andhence over |φ0|2 and |Hφ0|2, H = ∂t + a/(r2 + a2)∂φ being a globally timelike vector field in theinterior of D with −g(H,H) = ∆Σ/(r2 + a2)2. Away from the horizon, the wave operator can berewritten as a sum of H2, a second order elliptic operator and up to first order operators. Let χ̃(r) bea smooth cutoff function which equals to 1 in [2.8M, 3.2M ] and vanishes for [r+, 2.7M ]∪[3.3M,+∞).By multiplying the wave equation of r4−δφ0 by χ̃(r)r4−δφ0, integrating over D(0, τ) and applyingintegration by parts, one obtains an integral of |∂r∗φ0|2 + |∇/ φ0|2 over D(0, τ) ∩ [2.8M, 3.2M ] isbounded by a constant times a sum of energy fluxes on Στ and Σ0, an integral of |Hφ0|2 over D(0, τ)∩[2.7M, 3.3M ] and an integral of

Similarly as the spin +2 case, the parameters in the above estimates are small constants to be fixed.We add an A0 multiple of (1.46a) and an A2 multiple of (1.46c) to the estimate (1.46b) and findthe LHS of the gained estimate bounds over∑

i=0,1

(Eτ (φ̃i) + EH+(0,τ)(φ̃i)

)+(Eτ (φ

2) + EH+(0,τ)(φ2))

+ c

∫D(0,τ)

(A0Mdeg(φ̃0) + Mdeg(φ̃1) +A2Mdeg(φ2) + M(φ̃0) + M(φ̃1)

). (1.47)

The reason that the terms M(φ̃0) + M(φ̃1) where trapping degeneracy is removed are present hereis the same as showing the relations (1.43) for the spin +2 component. Fix the parameters in thefollowing order

µ1 � 1, �1 � µ1, A0 � µ1, µ0 � A−10 ,�0 � A−10 µ0, A2 � A0(µ0�

−11 + µ

−10 �−10 ) + µ1�

−10 + µ

−11 �−11 ,

µ2 � min{A−12 �−10 , A

−12 �−11 , �0, �1} (1.48)

such that the RHS of the obtained estimate can be absorbed by (1.47). Hence it holds true forsufficiently small ε0 that∑

i=0,1

(Eτ (φ̃i) + EH+(0,τ)(φ̃i)

)+(Eτ (φ

2) + EH+(0,τ)(φ2))

+

∫D(0,τ)

(M(φ̃0) + M(φ̃1) + Mdeg(φ2)

).ε0 E0(φ̃0) + E0(φ̃1) + E0

(φ2)

. E0(φ̃0) + E0(φ̃1) + E0(φ2), (1.49)

where in the last step, we use Proposition 24 to estimate the integral term∫

Σ0r(|∇/ φ̃0|2 + |∇/ φ̃1|2)

implicit in .ε0 and take ε0 sufficiently small such that the√ε0-dependent terms are absorbed by

the LHS of (1.49). From the estimate (1.49), the desired estimate (1.36b) is proved for the otherregular N-P component Φ̃4 in the case of n = 0.

Overview of the Paper. In Section 2, we give some preliminaries and introduce some furthernotations. Red-shift estimates near horizon and Morawetz estimates in large radius region fordifferent quantities are proved in Section 3. Afterwards, we derive in Section 4 some a prioriestimates on any fixed subextremal Kerr background by considering the definitions (1.23) in thecontext of transport equations. The basic estimates (1.41) and (1.46) are proved in Section 5 onSchwarzschild and in Section 6.6 on a slowly rotating Kerr background. These complete the proofof the estimates (1.36) based on the discussions in Section 1.6 for n = 0 and Section 6.8 for n ≥ 1.

2. Preliminaries and Further Notation

2.1. Well-posedness Theorem. We refer to [30, Section 2.1] for the well-posedness (WP) theoremfor a general system of linear wave equations, which is cited from [7, Chapter 3.2]. Similarly as thereduction in [30, Section 2.1], we can assume that the regular extreme N–P components are smoothand of compact support on the initial hypersurface Σ0.

2.2. Generic constants and general rules. Constants C and c, depending only on ε0, M , δand Σ0, are always understood as large constants and small constants respectively, and may changevalue from term to term throughout this paper based on the algebraic rules: C + C = C, CC = C,Cc ≤ C, etc. When there is no confusion, the dependence on M , ε0, δ, and Σ0 may always besuppressed. Once the constants ε0 and 0 < δ < 1/2 in Theorem 2 are chosen, these constants canbe made to be only dependent on M .

For any two functions F1 and F2, F1 . F2 means that there exists a constant C such thatF1 ≤ CF2 holds everywhere. F1 ∼ F2 indicates that F1 . F2 and F2 . F1, and we say that F1 isequivalent to F2.

11

The standard Laplacian on unit 2-sphere is denoted as 4S2 , and the volume form dσS2 on unit2-sphere is sin θdθdφ∗ or sin θdθdφ depending on which coordinate system is used.

Some cutoff functions will be used in this paper. Denote χR(r) to be a smooth cutoff functionutilized in Section 3.1 which is 1 for r ≥ R and vanishes identically for r ≤ R −M , and χ0(r) asmooth cutoff function which equals 1 for r ≤ r0 and is identically zero for r ≥ r1; see Section 3 forthe choices of r0 and r1. The function χ is a smooth cutoff both to the future time and to the pasttime, which will be applied to the solution in the proof of Theorem 23.

An overline or a bar will always denote the complex conjugate, τ1:

c

∫D(τ1,τ2)∩{r≥R}

{|∂r∗ψ|2

r1+δ+|∂tψ|2

r1+δ+|∇̌/ψ|2

r+|ψ|2

r3+δ

}

≤ C(ĚR−Mτ1 (ψ) + Ě

R−Mτ2 (ψ) +

∫D(τ1,τ2)∩{R−M≤r≤R}

(|∂̌ψ|2 + |ψ|2

))+

∫D(τ1,τ2)

<(G ·Xwψ̄

). (3.5)

12

Here, χR(r) is defined as in Section 2.2, ∇̌/ are the standard rotational angular derivatives on sphereS2(t, r) as in (1.31), and

ĚR−Mτ (ψ) ∼∫

Στ∩{r≥R−M}|∂̌ψ|2 =

∫Στ∩{r≥R−M}

(|∂t∗ψ|2 + |∂rψ|2 + |∇̌/ψ|2). (3.6)

Using this and defining

ER−Mτ (ψ) ∼∫

Στ∩{r≥R−M}|∂ψ|2, (3.7)

we show the following Morawetz estimate in large r region.

Proposition 5. In a subextremal Kerr spacetime, for any fixed 0 < δ < 12 , and for any solution ψto (1.27a) or (1.27b), there exists a constant R0(M) and a universal constant C such that for allR ≥ R0, the following estimate holds for any τ2 > τ1:∫

D(τ1,τ2)∩{r≥R}M(ψ) ≤ CER−Mτ1 (ψ) + CE

R−Mτ2 (ψ)

+ C

∫D(τ1,τ2)∩{R−M≤r≤R}

|∂ψ|2

+ C

∣∣∣∣ ∫D(τ1,τ2)∩{r≥R−M}

<(FXwψ̄

) ∣∣∣∣. (3.8)Remark 6. Recall here the definition of the Morawetz density M(ψ) in (1.35). This estimate willbe applied to ψ = φis defined in (1.23a) and (1.23b) with the corresponding inhomogeneous termF = F is in (1.24) and (1.25).

Proof. Using definition (3.1), the form (1.27b) reads

Σ�̃gψ =(

4ias cos θ∂t + s2 ∆+a2

r2

)ψ + F, (3.9)

and equation (1.27a) can be rewritten as

Σ�̃gψ =(

4ias cos θ∂t + s2 r2+2Mr−2a2

2r2

)ψ + F. (3.10)

Since the difference between the operator (∂φ+is cos θsin θ + a sin θ∂t)2 in Σ�̃g and (

∂φsin θ + a sin θ∂t)

2 inthe expansion of Σ�g has terms with coefficients independent of r, we achieve the same type ofMorawetz estimate in large r region by utilizing the same multiplier Xwψ̄, with |∇/ψ|2 − s2|ψ|2/r2and |∂ψ|2 − s2|ψ|2/r2 in place of |∇̌/ψ|2 and |∂̌ψ|2, ER−M (ψ) replacing ĚR−M (ψ), and G replacedby the the RHS of (3.9) or (3.10) in (3.5).

Consider equation (3.9). The bulk term coming from the source term

G =(

4ias cos θ∂t + s2 ∆+a2

r2

)ψ + F (3.11)

is ∫D(τ1,τ2)

− 1Σ<(((

4ias cos θ∂t + s2 ∆ + a

2

r2

)ψ + F

)(f∂r∗ +

1

4w

)ψ̄

). (3.12)

The term∫D(τ1,τ2)−<

(∆+a2

r2 ψwψ̄)is bounded by

∫D(τ1,τ2)∩[R,∞)−

c|ψ|2r3 , and an integration by parts

applied to the term∫D(τ1,τ2)

−4Σ <

(∆+a2

r2 ψf∂r∗ ψ̄)eventually implies that the term (3.12) is bounded

for R large enough by

−∫D(τ1,τ2)∩[R,∞)

c|ψ|2

r3+ C

∫D(τ1,τ2)∩[R−M,R]

|∂ψ|2

+

∫D(τ1,τ2)∩[R−M,∞)

{Ca2

r2|∂ψ|2 + <

(FXwψ̄

)}. (3.13)

13

We can move this first term to the LHS and find from (1.32) that the resulting LHS bounds over∫D(τ1,τ2)∩{r≥R}

c

{|∂r∗ψ|2

r1+δ+|∂tψ|2

r1+δ+|∇/ψ|2 − s2|ψ|2/r2

r+|ψ|2

r3+δ+|ψ|2

r3

}≥ c

∫D(τ1,τ2)∩{r≥R}

{|∂r∗ψ|2

r1+δ+|∂tψ|2

r1+δ+|∇/ψ|2

r+|ψ|2

r3

}. (3.14)

The spacetime integral of Ca2

r2 |∂ψ|2 over D(τ1, τ2)∩{r ≥ R} is absorbed by this LHS for sufficiently

large R. Therefore, the estimate (3.8) holds true for (3.9), and hence (1.27b).The proof of the Morawetz estimate (3.8) in large radius region for (1.27a) follows in the same

way, and we omit it. � �

In fact, an improved Morawetz estimate in the large radius region for spin +2 component canbe proved. This can be seen by expanding the φ1+2 term on the RHS of (1.24a) into first orderderivatives of φ0+2 and observing that the sign of the coefficient of ∂tφ0+2 provides a damping effectfor sufficiently large r.

Proposition 7. Let 0 < δ < 1/2 be given. In a subextremal Kerr spacetime, there exists a constantR0(M) and a universal constant C such that for all R ≥ R0 and any τ2 > τ1, the following estimateshold true for φ0+2 and φ1+2 respectively:∫

Στ2∩[R,+∞)

∣∣∂ (r4−δφ0+2)∣∣2 + ∫D(τ1,τ2)∩[R,∞)

r−1∣∣∂ (r4−δφ0+2)∣∣2

.∫

Στ2∩[R−M,R)

∣∣∂ (r4−δφ0+2)∣∣2 + ∫Στ1∩[R−M,+∞)

∣∣∂ (r4−δφ0+2)∣∣2+

∫D(τ1,τ2)∩[R−M,R]

∣∣∂ (r4−δφ0+2)∣∣2 , (3.15a)∫Στ2∩[R,+∞)

∣∣∂ (r2−δφ1+2)∣∣2 + ∫D(τ1,τ2)∩[R,∞)

r−1∣∣∂ (r2−δφ1+2)∣∣2

.∫

Στ2∩[R−M,R)

∣∣∂ (r2−δφ1+2)∣∣2 + ∫Στ1∩[R−M,+∞)

∣∣∂ (r2−δφ1+2)∣∣2+

∫D(τ1,τ2)∩[R−M,∞)

|∂(r4−δφ0+2)|2r2 +

∫D(τ1,τ2)∩[R−M,R]

∣∣∂ (r2−δφ1+2)∣∣2 . (3.15b)Proof. We define the variables

φ0,4−δ+2 =(r2+a2√

∆

)4−δ·(ψ[+2]/(r

2 + a2)2), (3.16a)

φ1,2−δ+2 =(r2+a2√

∆

)2−δ·(√

r2 + a2Y(ψ[+2]/(r

2 + a2)3/2))

(3.16b)

and derive the governing equations of them as follows(Σ�g + 4i cos θsin2 θ ∂φ − 4 cot

2 θ + (2 + δ2 − 5δ))φ0,4−δ+2

= (r3−3Mr2+a2r+a2M)

r2+a2

((4−2δ)V (

√r2+a2φ0,4−δ+2 )√r2+a2

+ 2δ(r2+a2

∆ ∂t +a∆∂φ

)φ0,4−δ+2

)+(

4ia cos θ∂t − 8arr2+a2 ∂φ)φ0,4−δ+2 +

P5(r)∆(r2+a2)2φ

0,4−δ+2 , (3.17a)(

Σ�g + 4i cos θsin2 θ ∂φ − 4 cot2 θ + (2 + δ2 − 5δ)

)φ1,2−δ+2

= (r3−3Mr2+a2r+a2M)

r2+a2

((2−2δ)V (

√r2+a2φ1,2−δ+2 )√r2+a2

+ 2δ(r2+a2

∆ ∂t +a∆∂φ

)φ1,2−δ+2

)+

P 5(r)∆(r2+a2)2φ

1,2−δ+2 +

(4ia cos θ∂t − 4arr2+a2 ∂φ

)φ1,2−δ+2

+ 6a∆(a2−r2)

(r2+a2)3 ∂φφ0,4−δ+2 +

6r∆(Mr3−a2r2−3Ma2r−a4)(r2+a2)4 φ

0,4−δ+2 . (3.17b)

14

Here, P5(r) and P 5(r) are both polynomials in r with powers no larger than 5, and the coefficientsof these two polynomials depend only on a,M , and δ and can be calculated explicitly. We shallmake use of the following expansion for any smooth complex scalar ψ of spin weight s(

Σ�g + 2is cos θsin2 θ ∂φ − s2 cot2 θ + |s|+ δ2 − 5δ

)ψ

=(

1sin θ∂θ(sin θ∂θ) +

1sin2 θ

∂2φφ +2is cos θsin2 θ

∂φ − s2 cot2 θ + |s|+ δ2 − 5δ)ψ

−√r2 + a2Y

(∆

r2+a2V(√

r2 + a2ψ))

+ 2arr2+a2 ∂φψ

+(2a∂2tφ + a

2 sin2 θ∂2tt)ψ − 2Mr

3+a2r2−4a2Mr+a4(r2+a2)2 ψ. (3.18)

From Remark 15, the eigenvalues of the operator in the first line on the RHS of (3.18) are not greaterthan δ2 − 5δ which is negative, hence if we choose the multiplier

− Σ−1χRX0φ0,4−δ+2

, − Σ−1χR ∆r2+a2(

(4−2δ)V (√r2+a2φ0,4−δ+2 )√r2+a2

+ 2δ(r2+a2

∆ ∂t +a∆∂φ

)φ0,4−δ+2

)= − Σ−1χR ∆r2+a2

((4−δ)V (

√r2+a2φ0,4−δ+2 )+δY (

√r2+a2φ0,4−δ+2 )√

r2+a2

)(3.19)

for (3.17a), it then follows from integration by parts and Cauchy–Schwarz inequality that∫Στ2∩[R,+∞)

|∂φ0,4−δ+2 |2 +∫D(τ1,τ2)∩[R,∞)

r−1(|X0φ0,4−δ+2 |2 + |∇/ φ

0,4−δ+2 |2

).

(∫Στ2∩[R−M,R)

+

∫Στ1∩{r≥R−M}

)|∂φ0,4−δ+2 |2 +

∫D(τ1,τ2)∩{r≥R−M}

|∂φ0,4−δ+2 |2

r2 . (3.20)

Moreover, by choosing the multiplier −χRr−3(1− 2M/r)φ0,4−δ+2 for (3.17a), we arrive at∫D(τ1,τ2)∩[R,∞)

r−1(|∂rφ0,4−δ+2 |2 + |∇/ φ

0,4−δ+2 |2

).∫

Στ2∩[R−M,+∞)|∂φ0,4−δ+2 |2 +

∫Στ1∩[R−M,+∞)

|∂φ0,4−δ+2 |2

+

∫D(τ1,τ2)∩[R−M,∞)

(r−1|∂t∗φ0,4−δ+2 |2 + r−2|∂φ

0,4−δ+2 |2

). (3.21)

Adding a sufficiently large multiple of (3.20) to (3.21) and taking R sufficiently large, we concludethe inequality (3.15a).

To show the estimate (3.15b), we use the multipliers

− 1ΣχR∆

r2+a2

((2−2δ)V (

√r2+a2φ1,2−δ+2 )√r2+a2

+ 2δ(r2+a2

∆ ∂t +a∆∂φ

)φ1,2−δ+2

)and

−χRr−3(1− 2M/r)φ1,2−δ+2 ,follow the same way as proving the estimate (3.15a) for the equation (3.17b), and use the Cauchy–Schwarz inequality to estimate the bulk integrals arising from the last line of (3.17b). � �

3.2. Red-shift Estimate near H+. The following red-shift estimate near H+ for the inhomoge-neous rescaled scalar wave equation (3.4) is taken from [18, Section 5.2].

Lemma 8. In a slowly rotating Kerr spacetime, there exist constants ε̃0 > 0, r+ ≤ 2M <r0(M, ε̃0) < r1(M, ε̃0) < (1 +

√2)M , c > 0 and C > 0, two smooth real functions y1(r) and

y2(r) on [r+,∞) with y1(r)→ 1 and y2(r)→ 0 as r → r+, and a ϕτ -invariant timelike vector field

N = T + χ0(r) (y1(r)Y + y2(r)T ) (3.22)

such that for all |a|/M ≤ ε̃0, by choosing a multiplier

Nχ0 ψ̄ = −χ0(r)Σ−1Nψ̄, (3.23)15

the following estimate holds for any solution ψ to the inhomogeneous rescaled scalar wave equation(3.4) for any τ2 > τ1:

c

(∫Στ2∩{r≤r0}

∣∣∂̌ψ∣∣2 + ĚH+(τ1,τ2)(ψ) + ∫D(τ1,τ2)∩{r≤r0}

∣∣∂̌ψ∣∣2)≤ C

(∫Στ1∩{r≤r1}

∣∣∂̌ψ∣∣2 + ∫D(τ1,τ2)∩{r0≤r≤r1}

∣∣∂̌ψ∣∣2)+

∫D(τ1,τ2)∩{r≤r1}

<(G ·Nχ0 ψ̄

). (3.24)

Here, the cutoff function χ0(r) is defined as in Section 2.2, and in ingoing E–F coordinates,

ĚH+(τ1,τ2)(ψ) =

∫H+(τ1,τ2)

(|∂vψ|2 + |∇̌/ ψ|2)r2dv sin θdθdφ̃. (3.25)

We generalize it to a spin-weighted wave equation.

Lemma 9. Let ψ be a complex scalar with non-zero spin weight s satisfying a spin-weighted waveequation

Σ�̃gψ = G, (3.26)

with g = gM,a being a slowly rotating Kerr metric. Then there exist constants ε̃0 > 0, r+ ≤ 2M <r0(M, ε̃0) < r1(M, ε̃0) < (1 +

√2)M , c(|s|) > 0 and C(|s|) > 0 such that for all |a|/M ≤ ε̃0 and any

τ2 > τ1,

c(|s|)(EH+(τ1,τ2)(ψ) +

∫Στ2∩{r≤r0}

|∂ψ|2 +∫D(τ1,τ2)∩{r≤r0}

|∂ψ|2)

≤ C(|s|)(∫

Στ1∩{r≤r1}|∂ψ|2 +

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂ψ|2)

+

∫D(τ1,τ2)∩{r≤r1}

<(G ·Nχ0 ψ̄

), (3.27)

where Nχ0 is chosen as in (3.23) in Lemma 8.

Proof. As is mentioned in the beginning of this section, the operator Σ�̃g is the same as the rescaledscalar wave operator Σ�g except for (

∂φ+is cos θsin θ +a sin θ∂t)

2 in place of the operator ( ∂φsin θ+a sin θ∂t)2

in the expansion of Σ�g. The difference between the operator (∂φ+is cos θ

sin θ + a sin θ∂t)2 in Σ�̃g

and ( ∂φsin θ + a sin θ∂t)2 in the expansion of Σ�g involves only terms with coefficients independent

of t, φ and r, therefore using the multiplier Nχ0ψ chosen in Lemma 8 yields the same red-shiftestimate as in (3.24) but with |∇/ψ|2 − s2|ψ|2/r2 in place of |∂̌ψ|2 in the horizon flux (3.25) and|∂ψ|2 − s2|ψ|2/r2 in place of |∇̌/ψ|2. Since from the lower bound (5.10) of ‖r∇/ψ‖L2(S2(r)), thesphere integrals of |∇/ψ|2− s2|ψ|2/r2 and |∂ψ|2− s2|ψ|2/r2 are bounded below by sphere integrals ofc(|s|)(|∇/ψ|2 + |ψ|2/r2) and c(|s|)(|∂ψ|2 + |ψ|2/r2), respectively, the estimate (3.27) follows. � �

The above red-shift estimate for a general spin-weighted wave equation (3.26) is used here toobtain red-shift estimates for different quantities.

Proposition 10. In a slowly rotating Kerr spacetime , there exist constants ε̃0 > 0, r+ ≤ 2M <r0(M, ε̃0) < r1(M, ε̃0) < (1 +

√2)M and C such that for all |a|/M ≤ ε̃0 and any τ2 > τ1 ≥ 0,

• for ψ ∈ {φ1+2, φ2+2, φ2−2} whose governing equations (1.24b), (1.24c) and (1.25c) can be putinto the form of (1.27b) with the relevant inhomogeneous term F , the following estimateholds:

EH+(τ1,τ2)(ψ) +

∫Στ2∩{r≤r0}

|∂ψ|2 +∫D(τ1,τ2)∩{r≤r0}

|∂ψ|2

≤ C∫

Στ1∩{r≤r1}|∂ψ|2 + C

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂ψ|2

16

+ C

∫D(τ1,τ2)∩[r+,r1]

|F |2; (3.28)

• for the equation (1.24a) of φ0+2, the following estimate near horizon holds:

EH+(τ1,τ2)(φ0+2) +

∫Στ2∩{r≤r0}

|∂φ0+2|2 +∫D(τ1,τ2)∩{r≤r0}

|∂φ0+2|2

≤ C∫

Στ1∩{r≤r1}|∂φ0+2|2 + C

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂φ0+2|2

+ C

∫D(τ1,τ2)∩[r+,r1]

|φ1+2|2. (3.29)

Proof. As in the previous subsection, we refer to the rewritten form (3.9) of the equation (1.27b),which can in turn be put into the form of the equation (3.26). We apply the conclusion in Lemma9 and obtain an estimate as (3.27) with the last line now replaced by∫

D(τ1,τ2)∩{r≤r1}<(((

4ias cos θ∂t +s2(∆ + a2)

r2

)ψ + F

)Nχ0 ψ̄

). (3.30)

After applying integration by parts and Cauchy–Schwarz inequality, this bulk term is bounded by∫D(τ1,τ2)∩{r≤r1}

−N(χ0s2(∆ + a2)

r2|ψ|2

)−∫D(τ1,τ2)∩{r≤r0}

s2(r+ − r−)2r2

|ψ|2

+ C

∫D(τ1,τ2)∩{r≤r1}

(|a||∂ψ|2 + |F |2

)+ C

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂ψ|2. (3.31)

By moving the first two integrals above to the LHS, the first integral produces extra positive energyfluxes and the second one adds additionally positive spacetime integrals of |ψ|2. The spacetimeintegral term |a||∂ψ|2 in the third term is absorbed by the LHS by choosing ε̃0 sufficiently small.Altogether, we conclude the estimate (3.28).

For φ0+2, we write equation (1.24a) into the form of (3.26) as

Σ�̃g(φ0+2) = G(φ

0+2) =

2(r2+2Mr−2a2)r2 φ

0+2 +

4(r2−3Mr+2a2)r3 φ

1+2

+ 8ia cos θ∂tφ0+2 −

8(a2∂t+a∂φ)φ0+2

r . (3.32)

Using the definition of φ1+2, one can eliminate the φ0+2 term and introduce extra Y φ0+2 term with apositive coefficient:

Σ�̃g(φ0+2) = G(φ

0+2) =

2(r2+2Mr−2a2)r Y φ

0+2 +

2r2−16Mr+12a2r3 φ

1+2

− 8(a2∂t+a∂φ)φ

0+2

r + 8ia cos θ∂t(φ0+2). (3.33)

The estimate (3.27) can be applied again, and we need to estimate the bulk term∫D(τ1,τ2)∩{r≤r1}

<(G(φ0+2) ·Nχ0φ0+2

). (3.34)

This integral over r0 ≤ r ≤ r1 is bounded by the RHS of (3.29) from Cauchy–Schwarz inquality,and the left integral over r ≤ r0 is

−∫D(τ1,τ2)∩{r≤r0}

1

Σ<(G(φ0+2)(1 + y2(r))Tφ

0+2

)−∫D(τ1,τ2)∩{r≤r0}

1

Σ<(G(φ0+2)y1(r)Y φ

0+2

). (3.35)

Substituting the expression (3.32) of G(φ0+2) into the first term of (3.35), the integral term fromthe term 2(r

2+2Mr−2a2)r2 φ

0+2 has a good sign after integrating over the spacetime region, and the rest

terms can be either estimated by Cauchy–Schwarz inequality or absorbed by setting ε̃0 sufficientlysmall. For the second term of (3.35), we instead use the expansion of G(φ0+2) on the RHS of (3.33).The first term on the RHS of (3.33) contributes positive bulk terms after moving to the LHS, and

17

we can similarly estimate the remaining integral terms by either requiring ε̃0 sufficiently small or anapplication of the Cauchy–Schwarz inequality. These together prove the estimate (3.29). � �

Notice that the quantities φ0+2 and φ1+2 are, however, degenerate at horizon, hence we shall proveas well the red-shift estimates for φ̃0 and φ̃1 (which are non-degenerate at H+) defined as in (1.38).

Proposition 11. In a slowly rotating Kerr spacetime (M, gM,a), there exist constants ε̃0 > 0,r+ < 2M < r0(M, ε̃0) < r1(M, ε̃0) < (1 +

√2)M and C such that for all |a|/M ≤ ε̃0, the following

red-shift estimates hold for φ̃0−2 and φ̃1−2 for any τ2 > τ1:

EH+(τ1,τ2)(φ̃0) +

∫Στ2∩{r≤r0}

|∂φ̃0|2 +∫D(τ1,τ2)∩{r≤r0}

|∂φ̃0|2

≤ C∫

Στ1∩{r≤r1}|∂φ̃0|2 + C

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂φ̃0|2

+ C

∫D(τ1,τ2)∩[r+,r1]

|φ̃1|2, (3.36)

EH+(τ1,τ2)(φ̃1) +

∫Στ2∩{r≤r0}

|∂φ̃1|2 +∫D(τ1,τ2)∩{r≤r0}

|∂φ̃1|2

≤ C∫

Στ1∩{r≤r1}|∂φ̃1|2 + C

∫D(τ1,τ2)∩{r0≤r≤r1}

|∂φ̃1|2

+ C

∫D(τ1,τ2)∩[r+,r1]

(|φ2−2|2 +

|a|M |∂φ̃0|

2 + |φ̃0|2). (3.37)

Proof. The equation for φ̃0 reads

Σ�̃gφ̃0 =(

4(r−M)r−5∆r Y + 2r∂t

)φ̃0 + 8φ̃0 − 10a

2

r2 φ̃0 − 5∆r2 φ̃0

+ 10r(a2∂t + a∂φ

)φ̃0 + 5r φ̃

1 − 8ia cos θ∂tφ̃0, (3.38)

and the governing equation for r2φ̃1 is

Σ�̃g(r2φ̃1) =

(4(r−M)r−9∆

2r Y + r∂t

)(r2φ̃1) + 72 (r

2φ̃1)− 3a2

2r2 (r2φ̃1) + r2φ

2

+ 6∆r

((Mr − 2a2)φ̃0 + r

(a2∂t + a∂φ

)φ̃0)

+ 5r(a2∂t + a∂φ

)(r2φ̃1)− 8ia cos θ∂t(r2φ̃1). (3.39)

Lemma 9 can be applied to both equations. Near r+, we find for the first term on the RHS of bothequations that (

4(r−M)r−5∆r Y + 2r∂t

)φ̃0 = 2r(Y + ∂t)φ̃0 +O(r − r+)Y φ̃0, (3.40)(

4(r−M)r−9∆2r Y + r∂t

)(r2φ̃1) = r(Y + ∂t)(r

2φ̃1) +O(r − r+)Y (r2φ̃1), (3.41)

and the used operator

ΣNχ0 = − (T + Y )− ((y1(r)− 1)Y + y2(r)T ), (3.42)

where the coefficients y1(r)−1 and y2(r) both converges to 0 as r → r+. Therefore, the bulk integralover D(τ1, τ2) ∩ {r ≤ r0} from the the first term on the RHS of both equations can be absorbed bythe LHS by choosing r0 close to r+ and taking ε̃0 small. The other bulk terms are estimated byeither integration by parts or Cauchy–Schwarz inquality and requiring ε̃0 sufficiently small, and thiscompletes the proof. � �

Remark 12. We fix ε̃0 in this subsection by taking the minimal value among all the ε̃0 in the abovediscussions. Given that ε̃0 is fixed, as have been discussed at the beginning of Section 3, the choicesof r0 = r0(M) and r1 = r1(M) are made by taking the minimal r0 and the maximal r1 in all theabove estimates.

18

4. Estimates for Spacetime Integrals of φ0s and φ1sWe derive in this section some a priori estimates for φ0s and φ1s which are used in Section 1.6.

4.1. Spin +2 Component.

Proposition 13. Let �̂1 > 0 be arbitrary. In a subextremal Kerr spacetime, the following estimateholds for φ1+2 defined as in (1.23a):∫

D(0,τ)

|φ1|2

r2. �̂1

∫D(0,τ)

|rφ1|2

r3+ �̂−11

∫D(0,τ)

|φ2|2

r3+

∫Σ0

|rφ1|2

r2. (4.1)

Proof. We start with a simple identity for any real smooth function f+2(r) and any real value α:

Y(f+2r

α|rφ1|2)

+ f+2αrα−1|rφ1|2 − Y (f+2)rα|rφ1|2 = f+2rα

We prove the inequality (4.6a) below, the proof for (4.6b) being analogous. For a smooth cutofffunction χ2(r) which equals 1 in [6M,∞) and vanishes in [r+, 5M ], any real value β and ∇/ j (j =1, 2, 3) as defined in (1.31), it holds true that

V (f−2χ2rβ |r2∇/ jφ

0|2)− χ2∂rf−2rβ |r2∇/ jφ0|2

− (βχ2f−2 + ∂rχ2f−2r) rβ−1|r2∇/ jφ0|2 = χ2f−2r2+β

Remark 15. This actually shows that the eigenvalues of the operator1

sin θ∂θ(sin θ∂θ) +1

sin2 θ∂2φφ +

2is cos θsin2 θ

∂φ − s2(sin θ)−2 (5.9)

are not greater than −s2 − |s|, and for a scalar ϕ with spin weight s,∫ π0

∫ 2π0

|∇/ ϕ|2 r2 sin θdφdθ ≥ (s2 + |s|)∫ π

0

∫ 2π0

|ϕ|2 sin θdφdθ. (5.10)

Moreover, let ϕm(t, r, θ) = 1√2π∫ 2π

0e−imφϕdφ, then we have∫

S2|r∇/ ϕ|2 dσS2 ≥

∫ π0

∑m∈Z

max{s2 + |s|,m2 + |m|}|ϕm|2 sin θdθ, (5.11)

The equation for ϕ1m` is now

r4∆−1∂2ttϕ1m` − ∂r(∆∂r)ϕ1m` + `(`+ 1)ϕ1m` − 8M/rϕ1m` +G1m` = 0. (5.12)

In the case that the inhomogeneous term G1 = 0, this is exactly the equation one obtains afterdecomposing into spherical harmonics the solution to the classical Regge–Wheeler equation [37] onSchwarzschild:

Σ�gu+ 8Mr u = 0. (5.13)

The equation (5.1a), while, is in a form of an ISWWE with another potential:

L0sϕ0 = Σ�gϕ

0 + 2is cos θsin2 θ

∂φϕ0 − s2

(cot2 θ + r+2M2r

)ϕ0 = G0. (5.14)

After the decomposition into spin-weighted spherical harmonics as above, the equation for ϕ0m` reads

r4∆−1∂2ttϕ0m` − ∂r(∆∂r)ϕ0m` + `(`+ 1)ϕ0m` − (2− 4M/r)ϕ0m` +G0m` = 0. (5.15)

Identity (5.8) holds for ϕ0 as well.We now consider the general form of equations (5.12) and (5.15):

r4∆−1∂2ttϕ− ∂r(∆∂r)ϕ+ `(`+ 1)ϕ+ V (r)ϕ+G = 0, (5.16)

with the potential

V (r) =

{−8M/r, for (5.12),−2 + 4M/r, for (5.15). (5.17)

5.3. Energy Estimate. Multiplying (5.16) by Tϕ = ∂tϕ and taking the real part, we arrive at anidentity:

12∂t

(r4

∆ |∂tϕ|2 + ∆|∂rϕ|2 + `(`+ 1)|ϕ|2 + V |ϕ|2

)− ∂r (

5.4. Morawetz Estimate. In this subsection, following the choices of the multipliers in [3, 5], weprove the Morawetz estimate for the separated equations (5.12) and (5.15) which are both in theform of (5.16), and then derive the Morawetz estimate for the ISWWE (5.3) and (5.14).

We multiply (5.16) by

Xϕ̄ = f̂∂rϕ̄+ q̂ϕ̄, (5.22)

take the real part, and arrive at

∂t

(<(r4

∆X(ϕ)∂tϕ̄))

+ 12∂r

(f̂[`(`+ 1)|ϕ|2 − r

4

∆ |∂tϕ|2 −∆|∂rϕ|2 + V |ϕ|2

])+ 12∂r

(

identity (5.8), integrate with respect to the measure dt∗dr over {(t∗, r)|0 ≤ t∗ ≤ τ, 2M ≤ r < ∞},and take (5.30) into account, then we obtain a Morawetz estimate for (5.3) in global Kerr coordinates:∫

D(0,τ)

(∆2

r6 |∂rϕ1|2 + 1r4 |ϕ

1|2 + (r−3M)2

r2

(1r3 |∂t∗ϕ

1|2 + 1r |∇/ϕ1|2))

. ETτ (ϕ1) + ET0 (ϕ

1) +

∫D(0,τ)

r−2(∣∣∣

and is bounded by �0∫D(0,τ) M̃(rφ

0) + 1�0

∫D(0,τ)

|φ1|2r3 by the Cauchy–Schwarz inequality, and hence

by∫D(0,τ)

(�0M̃(r4−δφ0) + �̂1�0 M̃(rφ

1) + 1�0�̂1Mdeg(φ2))

+ E0(r4−δφ0) from the estimate (4.1). For

the other error term Eschw(φ1), it is bounded by∫D(0,τ)

r−2(∣∣∣

Here, the energy densities are

e1(ψ) = dt∗(∂t)e1t (ψ) + dt

∗(∂r∗)e1r∗(ψ), e

1H(ψ) = dr(∂t)e

1t (ψ) + dr(∂r∗)e

1r∗(ψ), (6.2a)

and

e1t (ψ) = Σ−1(|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 − a2∆ |∂φψ|2 + s2(∆+a2)r2 |ψ|2)+ (r

2+a2)2−a2 sin2 θ∆∆Σ |∂tψ|

2 + (r2+a2)2

∆Σ |∂r∗ψ|2, (6.2b)

e1r∗(ψ) = −2(r2+a2)2

Σ∆ <(∂tψ∂r∗ ψ̄

). (6.2c)

Let ψm(t, r, θ) = 1√2π∫ 2π

0e−imφψdφ. It follows from (5.11) and (1.32) that∫

S2

(|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 − a2∆ |∂φψ|2 + s2(∆+a2)r2 |ψ|2)dσS2≥∫ π

0

∑m∈Z

(max{s2 + |s|,m2 + |m|} − a

2m2

∆ + s2 ∆+a2−r2

r2

)|ψm|2 sin θdθ. (6.3)

DenoteA1|m|,|s| = max{s

2 + |s|,m2 + |m|} − a2m2

∆ + s2 ∆+a2−r2

r2 . (6.4)

Recall that |s| = 2. If |m| = 0 or 1, then

A1|m|,2 ≥ 2− a2m2

∆ +4(∆+a2)

r2 , (6.5)

which is positve when r > 2M . When |m| ≥ 2, A1|m|,2 is a monotonically increasing function of |m|in the region r ≥ 2M . Consider an even restricted region r > M +

√M2 + a2. For |m| ≥ 2, by

monotonicity,

A1|m|,2 ≥ A12,2 ≥ 2∆−4a

2

∆ +4(∆+a2)

r2 , (6.6)

and the RHS is positive for r ≥M +√M2 + a2. Therefore, the LHS of (6.3), and hence the energy

density∫S2 e

1(ψ)dσS2 , is positive for r > M +√M2 + a2.

One can similarly choose the multiplier −2Σ−1∂tψ̄ for (1.27a) satisfied by φ0s, and arrive at anenergy identity for any τ2 > τ1:∫

Στ2

e0(ψ) +

∫H+(τ1,τ2)

e0H(ψ) =

∫Στ1

e0(ψ)−∫D(τ1,τ2)

<(2Σ−1F · ∂tψ̄

). (6.7)

Here, the energy densities

e0(ψ) = dt∗(∂t)e0t (ψ) + dt

∗(∂r∗)e0r∗(ψ), e

0H(ψ) = dr(∂t)e

0t (ψ) + dr(∂r∗)e

0r∗(ψ), (6.8a)

e0r∗(ψ) is the same as e1r∗(ψ), and

e0t (ψ) = Σ−1(|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 − a2∆ |∂φψ|2 + s2(r2+2Mr−2a2)2r2 |ψ|2)+ (r

2+a2)2−a2 sin2 θ∆∆Σ |∂tψ|

2 + (r2+a2)2

∆Σ |∂r∗ψ|2. (6.8b)

The estimate (5.11) gives∫S2

(|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 − a2∆ |∂φψ|2 + s2(r2+2Mr−2a2)2r2 |ψ|2)dσS2≥∫ π

0

∑m∈Z

(max{s2 + |s|,m2 + |m|} − a

2m2

∆ − s2 ∆+a2

2r2

)|ψm|2 sin θdθ. (6.9)

DenoteA0|m|,|s| = max{|s|(|s|+ 1), |m|(|m|+ 1)} − a

2m2

∆ − s2 ∆+a2

2r2 . (6.10)25

Recall that |s| = 2. If |m| = 0 or 1, A0|m|,2 ≥ 2−a2m2

∆ +2(r2+2Mr−2a2)

r2 and the RHS is positive whenr > 2M . Note that A0|m|,2 is also a monotonically increasing function of |m| in the region r ≥ 2Mwhen |m| ≥ 2. Hence, for |m| ≥ 2, we have in the region r > M +

√M2 + a2 that

A0|m|,2 ≥ A02,2 ≥ 2∆−4a

2

∆ +2(r2+2Mr−2a2)

r2 > 0.

This implies the energy density∫S2 e

0(ψ)dσS2 is positive for r > M +√M2 + a2. For sufficiently

small |a|/M , M +√M2 + a2 < r0, where r0 > 2M has been fixed in Section 3.2 (see also Remark

12). In conclusion, for r ≥ r0, the energy densities∫S2 e

k(ψ)dσS2 (k = 0, 1) above for both (1.27b)and (1.27a) are strictly positive and satisfy

∫S2 e

k(ψ) ≥ c∫S2 |∂ψ|

2.Since the energy densities

∫S2 e

k(ψ) are both nonnegative in the Schwarzschild case (a = 0), itholds true for sufficiently small |a|/M ≤ ε0 � 1,

−∫S2ek(ψ) ≤ Ca

2

M2

∫S2|∂ψ|2 for r ∈ [r+, r0], (6.11a)∫

H+(τ1,τ2)ekH(ψ) & −

a2

M2EH+(τ1,τ2)(ψ). (6.11b)

These can also be seen in another way by taking k = 1 in (6.11a) as an example. We consider onlyr ≤M + r0/2 since the estimate for M + r0/2 ≤ r ≤ r0 is manifestly valid. One can write e1t (ψ) as

Σ−1(|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 + s2(∆+a2)r2 |ψ|2 + ∆|Y ψ|2)− a

2 sin2 θΣ |∂tψ|

2 − 2aΣ

There exists an ε0 ≥ 0 and a nonnegative differentiable function e0(ε0) ∼ ε20 with e0(0) = 0 suchthat for all |a|/M ≤ ε0 and any ẽ ≥ e0, it holds true that∫

Στ2

|eτ2(ψ̃)|+∫H+(τ1,τ2)

|∂tψ + ∂r∗ψ|2 + ẽEτ2(ψ̃) + ẽEH+(τ1,τ2)(ψ̃)

.∫

Στ1

|eτ1(ψ̃)|+ ẽEτ1(ψ̃) + ẽ∫D(τ1,τ2)∩[r0,r1]

|∂ψ̃|2 + EF,ẽ,τ1,τ2 . (6.16)

Here, for any τ2 > τ1 ≥ 0,

EF,ẽ,τ1,τ2 = ẽ∫D(τ1,τ2)∩[r+,r1]

B(ψ̃, F ) +∣∣∣∣ ∫D(τ1,τ2)

<(Σ−1FTψ̄

) ∣∣∣∣, (6.17)and

B(ψ̃, F ) =

|F |2, for ψ̃ = φ1+2, φ2+2 or φ2−2;|φ1+2|2, for ψ̃ = φ0+2;|φ̃1|2, for ψ̃ = φ̃0−2;|φ2−2|2 + |φ̃0|2 +

|a|M |∂φ̃0|

2, for ψ̃ = φ̃1−2.

(6.18)

We here state a finite in time energy estimate for the ISWWE (1.27a) and (1.27b) based on theabove discussions, which is an analogue of [17, Proposition 5.3.2].

Proposition 17. (Finite in time energy estimate). Given an arbitrary � > 0, there exists anε0 > 0 depending on � and a universal constant C such that for |a|/M ≤ ε0, 1 ≥ ẽ ≥ e0(ε0) ∼ ε20and for any τ1 ≥ 0 and τ1 < τ2 ≤ τ1 + �−1, the following results hold true: For ψ = φis (i = 0, 1, 2),ψ̃ defined as in (6.15) and the corresponding inhomogeneous function F = F is in the linear wavesystems (1.24) and (1.25), we have∫

Στ2

|e(ψ̃)|+ ẽEτ2(ψ̃) + ẽEH+(τ1,τ2)(ψ̃)

≤ (1 + Cẽ)(∫

Στ1

|e(ψ̃)|+ ẽEtotalτ2 (s))

+ CEF,ẽ,τ1,τ2 , (6.19)

and, depending on the spin weight s = ±2 of ψ̃,∫D(τ1,τ2)∩[r0,r1]

|∂ψ̃|2 ≤ CEtotalτ1 (s). (6.20)

Here, EF,ẽ,τ1,τ2 is already defined in (6.17) and, for any τ ≥ 0,

Etotalτ (s) =

{Eτ (r

4−δφ0+2) + Eτ (r2−δφ1+2) + Eτ (φ

2+2), for s = +2;

Eτ (φ̃0−2) + Eτ (φ̃1−2) + Eτ (φ

2−2), for s = −2.

(6.21)

Proof. The first estimate follows from a combination of the previous prop with the second estimate(6.20). The second estimate follows from the fact that it holds true for Schwarzschild case for all �from the discussions in Sections 5 and 1.6 and the well-posedness property in Section 2.1 applied tothe linear wave system of {r4−δφ0+2, r2−δφ1+2, φ2+2} or {φ̃0−2, φ̃1−2, φ2−2}. � �

6.2. Separated Angular and Radial Equations. We consider in this subsection only the op-erators Lis acting on hypothetical integrable

7 functions ψ solving either (1.27a) or (1.27b), and inthe next subsection, a cutoff in time will be applied to the solutions to each subequation in systems(1.24) and (1.25) so as to ensure the integrability condition and allow for the separation introducedbelow.

7A solution to (1.27a) or (1.27b) is integrable if for every integer n ≥ 0, every multi-index 0 ≤ |i| ≤ n and anyr′ > r+, we have ∑

0≤|i|≤n

∫D(−∞,∞)∩{r=r′}

(|∂iψ|2 + |∂iF |2)

If the solution ψ to the equation (1.27b) is integrable, we can write

ψ =1√2π

∫ ∞−∞

e−iωtψω(r, θ, φ)dω, (6.23)

where ψω is defined as the Fourier transform of ψ:

ψω =1√2π

∫ ∞−∞

eiωtψ(t, r, θ, φ)dt. (6.24)

The equality (6.23) can be interpreted in L2(sin θdθdφdt). We further decompose ψω in L2(sin θdθdφ)into

ψω =∑m,`

ψ(aω)m` (r)Y

sm`(aω, cos θ)e

imφ, m ∈ Z. (6.25)

Here, for each m, {Y sm`(aω, cos θ)}`, with min {`} = max{|m|, |s|}, are the eigenfunctions of theself-adjoint operator

Sm = 1sin θ∂θ sin θ∂θ −m2+2ms cos θ+s2

sin2 θ+ a2ω2 cos2 θ − 2aωs cos θ (6.26)

on L2(sin θdθ). These eigenfunctions, called “spin-weighted spheroidal harmonics,” form a completeorthonormal basis on L2(sin θdθ) and have eigenvalues Λ(aω)m` defined by

SmY sm`(aω, cos θ) = −Λ(aω)m` Y

sm`(aω, cos θ). (6.27)

One could similarly define Fω and F(aω)m` .

An integration by parts, together with a usage of the Plancherel lemma and the orthonormalityproperty of the basis {Y sm`(aω, cos θ)eimφ}m`, gives∫ +∞

−∞

∑m,`

Λ(aω)m` |ψ

(aω)m` |

2dω

=

∫ ∞−∞

∫S2

{|∂θψ|2 +

∣∣∣∂φψ+is cos θψsin θ ∣∣∣2 − |a cos θ∂tψ + isψ|2 + 2s2|ψ|2}dσS2dt. (6.28)The radial equation for ψ(aω)m` is then{

∂r(∆∂r) + (V )(aω)m`,1(r)

}ψ

(aω)m` = F

(aω)m` , (6.29)

with the potential

(V )(aω)m`,1(r) =

(r2+a2)2ω2+a2m2−4aMrmω∆ −

(λ

(aω)m`,1(r) + a

2ω2). (6.30)

We utilized here a substitution of

λ(aω)m`,1(r) = Λ

(aω)m` −

s2(2Mr−2a2)r2 , (6.31)

by which the above radial equation (6.29) is the same as the radial equation [17, Equation (33)]8 forthe scalar field.

One can obtain for (1.27a) the same angular equation and the following radial equation afterdecomposition: The radial equation for ψ(aω)m` is{

∂r(∆∂r) + (V )(aω)m`,0(r)

}ψ

(aω)m` = F

(aω)m` , (6.32)

with the potential

(V )(aω)m`,0(r) =

(r2+a2)2ω2+a2m2−4aMrmω∆ −

(λ

(aω)m`,0(r) + a

2ω2)

(6.33)

and a substitution ofλ

(aω)m`,0(r) = Λ

(aω)m` −

s2(∆+a2)2r2 . (6.34)

8The authors in [17] missed one term−4aMrmω in the Equation (33), but what is used thereafter is the Schrödingerequation (34) in Section 9 which is correct. Therefore, the validity of the proof will not be influenced by the missingterm.

28

The above two radial equations can now be put into the following form{∂r(∆∂r) + (V )

(aω)m` (r)

}ψ

(aω)m` = F

(aω)m` , (6.35)

with(V )

(aω)m` (r) =

(r2+a2)2ω2+a2m2−4aMrmω∆ −

(λ

(aω)m` (r) + a

2ω2). (6.36)

Here, λ(aω)m` (r) = λ(aω)m`,k(r) and (V )

(aω)m` (r) = (V )

(aω)m`,k(r), and the value of k = 0, 1 depends on which

of the two equations (1.27a) and (1.27b) ψ satisfies.We state here some basic identities for any r > r+ from properties of Fourier transform and

Plancherel lemma:∫ ∞−∞

∫ 2π0

∫ π0

|ψ(t, r, θ, φ)|2 sin θdθdφdt =∫ ∞−∞

∑m,`

∣∣∣ψ(aω)m` (r)∣∣∣2 dω,∫ ∞−∞

∫ 2π0

∫ π0

|∂r∗ψ(t, r, θ, φ)|2 sin θdθdφdt =∫ ∞−∞

∑m,`

∣∣∣∂r∗ψ(aω)m` (r)∣∣∣2 dω,∫ ∞−∞

∫ 2π0

∫ π0

|∂tψ(t, r, θ, φ)|2 sin θdθdφdt =∫ ∞−∞

∑m,`

ω2∣∣∣ψ(aω)m` (r)∣∣∣2 dω.

6.3. Frequency Localised Multiplier Estimates.

6.3.1. Cutoff in Time. To justify the separation procedures in Section 6.2, the assumption that thesolution ψ(t, r, θ, φ) is integrable is sufficient, but this is a priori unknown. Therefore, we applycutoffs to the solution both to the future and to the past, and then do separation for the waveequation which the gained function satisfies.

Let χ2(x) be a smooth cutoff function which equals 0 for x ≤ 0 and is identically 1 when x ≥ 1.Let τ > 2ε−1 be arbitrary with ε > 0 to be chosen. Define

χ = χτ,ε(t∗) = χ2(εt

∗)χ2(ε(τ − t∗)) (6.37)

andψχ = χψ (6.38)

in coordinate system (t∗, r, θ, φ∗). The cutoff function ψχ is now a smooth function supported in0 ≤ t∗ ≤ τ , and ψχ = ψ in ε−1 ≤ t∗ ≤ τ − ε−1. Moreover, it satisfies the following wave equation

Lksψχ = χF + Σ (2∇µχ∇µψ + (�gχ)ψ)− 2isa cos θ∂tχψ

, Fχ. (6.39)

The fact that the afore-defined χ is φ-independent is utilized here in deriving this equation.Note the fact that the functions ψχ and Fχ are compactly supported in 0 ≤ t∗ ≤ τ at each

fixed r > r+, and the assumption that ψ is a compactly supported smooth section solving onesubequation of a linear wave system, hence ψχ is an integrable solution to (6.39) from Section 2.1.In the following discussions, we apply the mode decompositions in Section 6.2 to ψχ and Fχ, andseparate the wave equation (6.39) into the angular equation (6.27) and radial equation (6.35), butwith R(aω)m` = (ψχ)

(aω)m` and (Fχ)

(aω)m` in place of ψ

(aω)m` and F

(aω)m` respectively.

Before introducing the microlocal currents, we give some estimates for the inhomogeneous termFχ here. Due to the fact that ∇χ and �gχ are supported in{

0 ≤ t∗ ≤ ε−1}∪{τ − ε−1 ≤ t∗ ≤ τ

}, (6.40)

one obtains in the coordinate system (t∗, r, θ, φ∗) that

|∂t∗χ| ≤ Cε, |�gχ| ≤ Cε2, (6.41a)

|∇µχ∇µψ|2 +∣∣∣ ias cos θ∂tχψΣ ∣∣∣2 ≤ Cε2 (|∂ψ|2 + a2M−2 ∣∣r−1ψ∣∣2) . (6.41b)

29

6.3.2. Currents in Phase Space. In what follows, we will suppress the dependence on a, ω, m and `of the functions R(aω)m` (r), F

(aω)m` (r), Λ

(aω)m` , λ

(aω)m` (r), (V )

(aω)m` (r) and other functions defined by them.

When there is no confusion, the dependence on r may always be implicit (except for the radial partR(r) to avoid misunderstanding with the radius parameter R). Thus we will write them as R(r), F ,Λ, λ and V .

We transform the radial equation (6.35) into a Schrödinger form, which will be of great use todefine the microlocal currents below, by setting

u(r) =√r2 + a2R(r), H(r) =

∆Fχ(r)

(r2+a2)3/2. (6.42)

The Schrödinger equation for u(r) reads after some calculations

u(r)′′ +(ω2 − V (r)

)u(r) = H(r), (6.43)

where

V =ω2 − ∆(r2+a2)2V +1

(r2+a2)d2

dr∗2 (r2 + a2)1/2

=4Mramω−a2m2+∆(λ+a2ω2)

(r2+a2)2 +∆

(r2+a2)4

(a2∆ + 2Mr(r2 − a2)

), (6.44)

and a prime ′ denotes a partial derivative with respect to r∗ in tortoise coordinates.Given any real, smooth functions y, h and f , define the microlocal currents

Qy = y(|u′|2 +

(ω2 − V

)|u|2), (6.45a)

Qh = h< (u′u)− 12h′|u|2, (6.45b)

Qf = Qh=f′+Qy=f = f ′< (u′u)−

(12f′′ − f

(ω2 − V

))|u|2 + f |u′|2 . (6.45c)

The currents Qy and Qh are constructed via multiplying the equation (6.43) by 2yū′ and hū respec-tively. The derivatives of the above currents are

(Qy)′

= y′(|u′|2 +

(ω2 − V

)|u|2)− yV ′|u|2 + 2y<

(u′H

), (6.46a)(

Qh)′

= h(|u′|2 +

(V − ω2

)|u|2)− 12h

′′|u|2 + h<(uH), (6.46b)(

Qf)′

= 2f ′ |u′|2 − fV ′|u|2 − 12f′′′|u|2 + <

(2fu′H + f ′uH

). (6.46c)

6.3.3. Frequency Regimes. We start to define the separated frequency regimes. Let ωl and λl be(potentially large) parameters and λs be a (potentially small) parameter, all to be determined inthe proof below. The frequency space is divided into

• FT ={

(ω,m, `) : |ω| ≥ ωl, λ < λsω2};

• FTr ={

(ω,m, `) : |ω| ≥ ωl, λ ≥ λsω2};

• FA = {(ω,m, `) : |ω| ≤ ωl,Λ > λl};• FB = {(ω,m, `) : |ω| ≤ ωl,Λ ≤ λl}.

We fix R0 as in Section 3.1 and an arbitrary rc ∈ (2M, r0), with r0 fixed in Section 3.2.

Lemma 18. For all |a| < M and all frequency triplets (ω,m, `),

Λ + 3a2ω2 ≥ 23

max{m2 + |m|, s2 + |s|}, (6.47)

and there exists a sufficiently large R5 ≥ R0+M and two constants c = c(M) > 0 and C = C(M) > 0such that for all r ≥ R5,

V ′(r) < −cr−3(Λ + 4a2ω2 +m2 + 1) + Cr−3a2ω2. (6.48)

Proof. Let

V = Vm + Ve, (6.49a)

Vm =∆(Λ+a2ω2)

(r2+a2)2 . (6.49b)

30

For V = V1 which corresponds to the case (V )(aω)m` (r) = (V )

(aω)m`,1(r), we calculate

Ve =4Mramω−a2m2

(r2+a2)2 −6(Mr−a2)∆r2(r2+a2)2

+ a2∆

r2(r2+a2)4 (3r4 − 8Mr3 + 5a2r2 − 2a2Mr + 2a4), (6.50)

and for V = V0 corresponding to (V )(aω)m` (r) = (V )

(aω)m`,0(r),

Ve =4Mramω−a2m2

(r2+a2)2 −2(r2−3Mr+3a2)∆

r2(r2+a2)2

+ a2∆

r2(r2+a2)4 (3r4 − 8Mr3 + 5a2r2 − 2a2Mr + 2a4). (6.51)

One finds for all r ∈ [r+,∞),V ′m = − 2∆(r2 + a2)−4(Λ + a2ω2)(r3 − 3Mr2 + a2r + a2M), (6.52)V ′e ≤ 4∆(r2 + a2)−1r−3 + C∆r−2

(r−4|amω|+ r−5a2m2 + r−4

). (6.53)

One adds −3a2ω2Y sm`(aω, cos θ) to both sides of (6.27) and obtains

( 1sin θ∂θ sin θ∂θ −m2+2ms cos θ+s2

sin2 θ+ a2ω2 cos2 θ − 2aωs cos θ − 3a2ω2)Y sm`

= − (Λ + 3a2ω2)Y sm`. (6.54)Rewrite the LHS as (

1sin θ∂θ sin θ∂θ −

m2+2ms cos θ+s2

sin2 θ

)Y sm`

−(a2ω2 sin2 θ + 12 (2aω + s cos θ)

2 − 12s2 cos2 θ

)Y sm`. (6.55)

By Remark 15, the operator at the first line has eigenvalues not greater than−max{m2+|m|, s2+|s|},and the eigenvalues of the operator at the second line are less than or equal to 12s

2, hence

Λ + 3a2ω2 ≥ max{m2 + |m|, s2 + |s|} − 12s2 ≥ 2

3max{m2 + |m|, s2 + |s|}. (6.56)

From (6.52), given any � > 0, V ′m ≤ −(2− �)r−3(Λ + a2ω2) for r � 4M large. Together with (6.53),this implies

V ′ − 6r−3a2ω2 ≤ (−(2− �)(Λ + 3a2ω2) + 4)r−3 − 2r−3a2ω2

+ Cr−4(|amω|+ a2m2 + 1). (6.57)For sufficiently small �, there exists a 0 < c0 < 1 such that the coefficient of the first term on theRHS is negative and bounded above by −c0r−3(Λ + 3a2ω2 +m2 + s2). For r ≥ R5 large enough, thesecond line of the RHS is bounded by c02 r

−3(a2ω2 +m2 + 1) using the Cauchy–Schwarz inequality,hence this completes the proof. � �

We shall obtain a phase-space version of Morawetz estimate by choosing different functions y, hand f in each of the four frequency regimes in Sections 6.3.4–6.3.7. The proofs in Sections 6.3.4–6.3.6follow from the discussions in [17, Section 9.4–9.6], nevertheless, we present here the entire proof forcompleteness.

6.3.4. FT Regime (Time-dominated Frequency Regime). In this section, λs will be fixed and ωl willbe left unspecified until Section 6.3.5. For |a|/M ≤ ε0 � 1, by choosing small enough λs and largeenough ωl, there exists a constant c0 < 1 such that we have in FT that

ω2 − V ≥ 1−c02 ω2 in [r+,∞). (6.58)

As to the potential V , apart from the fact in Lemma 18, we have for all r∗ that

|V ′| ≤ C∆/r5((Λ + 3a20ω

2) + 1). (6.59)

Choose a function y to satisfy the following properties:(1) y ≥ 0, y′ ≥ c∆/r4 in (r+, R5 −M ],(2) y ≥ 0, y′ ≥ 0 in [R5 −M,R5],(3) y = 1 in [R5,∞).

31

In view of the above properties, (Qy)′ ≥ 2y<(u′H

)−Ca2r−3ω2 for r ≥ R5, (Qy)′ ≥ 2y<

(u′H

)for

r ≤ rc, and in the region r ∈ (rc, R5 −M), we havey′(ω2 − V )− yV ′ ≥ 1−c02 y

′ω2 − Cy∆/r5((λ+ a20ω

2) + 1)

≥(c 1−c02 − Cr

−1(λs + a20 + ω

−2l ))

∆/r4ω2. (6.60)

By taking both λs and ε0 sufficiently small and for ωl sufficiently large, the right-hand side is largerthan c∆/r4ω2. Hence, integrating (6.46a) over [r∗−∞, r∗∞] gives

Lemma 19. Let r∗∞ > R∗5 > (R0 +M)∗ and r∗−∞ < r∗c be arbitrary. There exists small enough λsand large enough ωl such that we have in FT frequency regime the following estimate for sufficientlysmall ε0:

c

∫ R∗0r∗c

∆

r4

(|u′|2 +

(ω2 + (Λ + 4a2ω2) + 1

)|u|2)

≤∫ r∗∞r∗−∞

2y<(u′H

)+Qy (r∗∞)−Qy

(r∗−∞

)+ C

∫ r∗∞R∗5

a2r−3ω2|u|2. (6.61)

6.3.5. FTr Regime (Trapped Frequency Regime). We have fixed λs (which is sufficiently small) as inSection 6.3.4, and we will fix ωl here. This is the only frequency regime where trapping phenomenoncould happen. Recall from (6.52) that

V ′m =−2∆

(r2+a2)4 (Λ + a2ω2)(r3 − 3Mr2 + a2r + a2M), (6.62)

and the polynomial P3(r) = r3 − 3Mr2 + a2r + a2M has a unique zero point ra,M in r ∈ [r+,∞)which satisfies moreover |ra,M − 3M | ≤ Ca2. Meanwhile, the estimate (6.47) and the requirementsof |ω| ≥ ω1 and λ ≥ λ2ω2 in this frequency regime imply that for large enough ω1 and sufficientlysmall |a|/M ≤ ε0, Λ + a2ω2 ≥ c(ω2 + max{m2 + |m|, s2 + |s|}). On the other hand,

r3|V ′e | ≤ C∆/r2(a2m2 + |amω|+ 1). (6.63)Therefore, given any small ε4 > 0, we have for sufficiently large ωl (depending on the λs) andsufficiently small ε0 that V ′ has no zeros outside the region [ra,M − ε4, ra,M + ε4]. In fact, V ′ has aunique simple zero in this neighborhood. This can be seen as follows. Note from (6.52), (6.50) and(6.51) that for sufficiently small ε0,(

(r2+a2)4

∆r2 V′m

)′(ra,M ) = − 2∆(ra,M )(Λ+a

2ω2)

r2a,M+a2 (1− a

2

r2a,M− 2a

2Mr3a,M

)

≤ − c(Λ + a2ω2), (6.64a)∣∣∣∣( (r2+a2)4∆r2 V ′e)′∣∣∣∣ . ∆r−2(a2m2 + |amω|+ 1). (6.64b)For ωl sufficiently large and ε0 sufficiently small, Λ + a2ω2 will be much bigger than the RHS of(6.64b). This implies that in the small region [ra,M − ε4, ra,M + ε4], we have(

(r2+a2)4

∆r2 V′)′≤ − c(Λ + a2ω2). (6.65)

Therefore, this proves that for any ε4 > 0, we have for sufficiently large ωl and sufficiently small ε0that there is a unique simple zero, which we denote by r(aω)m` , in [r+,∞) for any (ω,m, `) ∈ FTr, andr

(aω)m` ∈ [ra,M − ε4, ra,M + ε4]. Moreover, for r ≤ r

(aω)m` ,

V ′ ≥ −c(r − r(aω)m` )∆/r6(Λ + a2ω2), (6.66)

and for r ≥ r(aω)m` ,

V ′ ≤ −c(r − r(aω)m` )∆/r6(Λ + a2ω2). (6.67)

Choose a function f associated with Qf current to satisfy the following properties:(1) lim

r∗→−∞f = −1, f = 1 for some large R4 ≥ R0, and f(rc) = −1/2, f(r(aω)m` −M/10) = −1/4,

f(r(aω)m` ) = 0, f(r

(aω)m` +M/10) = 1/4 and f(R0) = 1/2,

32

(2) f ′ ≥ 0 for all r∗, and f ′ ≥ c∆/r4 for rc ≤ r ≤ R0,(3) f ′′′ ≤ −c for r(aω)m` −M/10 ≤ r ≤ r

(aω)m` +M/10, and |f ′′′| ≤ C∆/r5 elsewhere.

Such a function f can be manifestly constructed. Upon making such a choice of function f , it canbe seen from the above properties of V ′ and properties of the chosen function f that for ωl largeenough, we have for all r∗ that

−fV ′ − 12f′′′ ≥

(c+ c(ωl)((Λ + a

2ω2) + ω2)(r − r(aω)m` )2)∆/r7.

By integrating (6.46c) over [r∗−∞, r∗∞], we arrive at the following conclusion.

Lemma 20. Let r∗∞ > R∗4 ≥ R∗0 and r∗−∞ < r∗c be arbitrary. There exists a large ωl such that inFTr frequency regime, it holds true for sufficiently small ε0 that∫ R∗0

r∗c

∆

r4

(c |u′|2 + r−1

[c+ c(ωl)(1− r−1r(aω)m` )

2(ω2 + (Λ + 4a2ω2)

)]|u|2)

≤∫ r∗∞r∗−∞

(2f<

(u′H

)+ f ′<

(uH))

+Qf (r∗∞)−Qf(r∗−∞

). (6.68)

6.3.6. FA Regime (Angular-dominated Frequency Regime). Here, we fix ωl as in Section 6.3.5, andwill choose λl to be sufficiently large. The general idea is as follows. In this regime, for sufficientlysmall |a|/M ≤ ε0, the zero points of V ′(r) in [r+,∞) are located in a small neighborhood ofr = 3M . The Qf current is utilized to achieve the positivity of the bulk term outside this smallneighborhood, while in this small neighborhood, hV |u|2 in (Qh)′ with h(r) being a positive constantis used to dominate the potentially negative bulk term in (Qf )′.

Similarly to the discussions in Section 6.3.5, one calculates

V ′m =−2∆

(r2+a2)4 (Λ + a2ω2)(r3 − 3Mr2 + a2r + a2M), (6.69)

and the polynomial r3 − 3Mr2 + a2r + a2M has a unique zero point ra,M in r ∈ [r+,∞) satisfing|ra,M − 3M | ≤ Ca2. In this frequency regime, it is clear that for λl sufficiently large, Λ + a2ω2 > 0.On the other hand,

r3|V ′e | ≤ C∆/r2(a2m2 + |amω|+ 1). (6.70)Therefore, given any small ε4 > 0, we have for sufficiently small ε0 and sufficiently large λl (dependingon the ωl) that V ′ has no zeros outside the region [ra,M − ε4, ra,M + ε4].

Given ε0 � 1, choose constantsr+ < rc < rleft1 < rleft2 < ra,M − ε4 < ra,M + ε4 < rright1 < rright2 < R0 0 and functions f and h satisfying thefollowing conditions:

(1) f ′ ≥ 0 for r > r+, f ′ ≥ c1 for rc ≤ r ≤ R0,(2) f ∼ −1 + (r − r+) near the horizon, f ≤ − 12 for r ≤ rleft2, f ≥

12 for r ≥ rright1 and f = 1

for r ≥ R0 + 1 such that fV ′ < 0 in (r+, ra,M − ε4) ∪ (ra,M + ε4,∞) and fV ′ ≤ c2V in[ra,M − ε4, ra,M + ε4],

(3) h = 0 in [r+, rleft1] ∪ [rright2,∞) and h = 2c2 in [rleft2, rright1].We then take λl sufficiently large such that for r ∈ [r+, ra,M − ε4] ∪ [ra,M + ε4,∞),

−12fV ′ − 1

2f ′′′ − 1

2h′′ ≥ 0, (6.71)

and for r ∈ [ra,M − ε4, ra,M + ε4],

h(V − ω2)− 12h′′ − fV ′ − 1

2f ′′′ ≥ c2V − 2c2ω2 −

1

2f ′′′ ≥ cc2Λ. (6.72)

Therefore, by integrating over r∗−∞ ≤ r∗ ≤ r∗∞, we conclude

Lemma 21. Let r∗∞ > R∗0 and r∗−∞ < r∗c be arbitrary. Fix ωl as in Section 6.3.5. There exists alarge λl such that in FA frequency regime, we have for sufficiently small ε0 the following estimate

c

∫ R∗0r∗c

(|u′|2 + ∆/r5

(ω2 + (Λ + 4a2ω2) + 1

)|u|2)

33

≤∫ r∗∞r∗−∞

(2f<

(u′H

)+ (f ′ + h)<

(uH))

+Qf (r∗∞)−Qf(r∗−∞

). (6.73)

6.3.7. FB regime (bounded frequency regime). Fix ωl and λl as above. This is a compact frequencyregime. Since the Morawetz estimates in the Schwarzschild case have been proved in Section 5.4,for sufficiently small |a|/M ≤ ε0, the Morawetz estimates in the phase space can be obtained by astability argument. We shall show this below by choosing an appropriate function f in the currentQf .

In this regime, a key fact is that the minimum value of eigenvalues Λ for the separated angularequation (6.27) is close to max{s2 + |s|,m2 + |m|} due to smallness of |aω|. More explicitly, from(6.27),

Λ + a2ω2 ≥ max{s2 + |s|,m2 + |m|} − 2|aωs|≥ max{6,m2 + |m|} − 1− 4a2ω2l . (6.74)

This gives directly that

m2 ≤ C(λl + a2ω2l ). (6.75)We take

f =r3 − 3Mr2 + a2r + a2M

r3(6.76)

in the current Qf and find for sufficiently small |a|/M ,

f ′ =∆

r4(r2 + a2)(3Mr2 − 2a2r − 3a2M) ≥ cM∆

r4. (6.77)

Moreover, f vanishes at a unique point ra,M , which is already defined in Section 6.3.6, with |ra,M −3M | ≤ Ca2 for small enough |a|/M .

To obtain a Morawetz estimate in the phase space, we need to verify −fV ′ − 12f′′′ ≥ 0. Consider

the case that V = V1. The derivative V ′m is as in (6.52), and note from (6.50) that

Ve = −6Mr−1(r2 + a2)−2∆ +O(r−3)Mamω +O(r−4)a2m2 +O(r−4)a2,hence ∣∣V ′e − 6Mr∆(3r−8M)(r2+a2)4 ∣∣ ≤ C∆(Mr−6|amω|+ r−7a2m2 + r−7a2), (6.78a)∣∣∣∣−12f ′′′ + 3M∆(r2 + a2)4 (3r2 − 20Mr + 30M2)

∣∣∣∣ ≤ Ca2∆r−7. (6.78b)This implies

−fV ′ − 12f ′′′ ≥ − f

(V ′0 +

6Mr∆(3r − 8M)(r2 + a2)4

)− 3M∆(3r

2 − 20Mr + 30M2)(r2 + a2)4

− C∆r2 + a2

(Mr−4|amω|+ a2r−5(m2 + 1)). (6.79)

For the RHS of the first line, it equals2∆(r3 − 3Mr2 + a2r + a2M)2(Λ + a2ω2)

r3(r2 + a2)4− 3M∆(3r

2 − 20Mr + 30M2)(r2 + a2)4

− 6Mr∆(3r − 8M)(r3 − 3Mr2 + a2r + a2M)

r3(r2 + a2)4

≥ ∆r3(r2 + a2)4

[2(r − 3M)2r4(6− ε)− 3Mr3(9r2 − 54Mr + 78M2)

]− C∆r2 + a2

r−3ε−1a2ω2l − Ca2r−5. (6.80)

From the above two estimates, to show that there exists a constant c > 0 such that −fV ′ − 12f′′′ ≥

c∆r3

(r2+a2)4 , it is enough to prove there exists a positive constant c3 > 0 such that

4r(r − 3M)2 −M(9r2 − 54Mr + 78M2) ≥ c3r3, (6.81)34

since then the first line on the RHS of (6.80) will dominate over c32∆r3

(r2+a2)4 by taking ε = c3/2,which will in turn dominate over the sum of the second line of (6.79) and the last line of (6.80) bytaking |a|/M ≤ ε0 sufficiently small. The LHS of (6.81) is 4r3 − 33Mr2 + 90M2r − 78M3, and ithas been shown in Section 5.4 that this polynomial has no zero points in [r+,∞), hence the relation(6.81) follows.

In the case that V = V0, one follows the above argument and obtains

− fV ′ − 12f ′′′

≥ − f(V ′m +

r∆(4r2 − 30Mr + 48M2)(r2 + a2)4

)− 3M∆

(r2 + a2)4(3r2 − 20Mr + 30M2)

− C∆r2 + a2

(Mr−4|amω|+ a2r−5(m2 + 1))

≥ r3∆

r3(r2 + a2)4[2(r − 3M)2r(6− ε)− (4r3 − 33Mr2 + 78M2r − 54M3)

]− C∆r2 + a2

r−3ε−1a2ω2l − Ca2r−5. (6.82)

Similarly, we only need to show the polynomial

12r(r − 3M)2 − (4r3 − 33Mr2 + 78M2r − 54M3) = 8r3 − 39Mr2 + 30M2r + 54M3

has no zeros in [r+,+∞), and this has been shown in Section 5.4.In total, by making a choice of function f as in (6.76), it holds true that

(Qf )′ ≥ cM∆r−4|u′|2 + c∆r−5|u|2 + <(2fu′H + f ′uH

). (6.83)

By integrating over