· 2020. 3. 23. · MORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD LARS ANDERSSON, PIETER BLUE, AND JINHUA WANG Abstract. The equations governing the perturbations of

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Morawetz estimate for linearized gravity in Schwarzschild

Citation for published version:Andersson, L, Blue, P & Wang, J 2020, 'Morawetz estimate for linearized gravity in Schwarzschild', AnnalesHenri Poincaré, vol. 21, no. 3, pp. 761-813. https://doi.org/10.1007/s00023-020-00886-5

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https://doi.org/10.1007/s00023-020-00886-5https://doi.org/10.1007/s00023-020-00886-5https://www.research.ed.ac.uk/portal/en/publications/morawetz-estimate-for-linearized-gravity-in-schwarzschild(12a66f7e-587d-403d-80b5-6a2b5f063394).html

MORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN

SCHWARZSCHILD

LARS ANDERSSON, PIETER BLUE, AND JINHUA WANG

Abstract. The equations governing the perturbations of the Schwarzschildmetric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the tech-

nique introduced in [3], we prove an integrated local energy decay estimate for

both the Regge-Wheeler and Zerilli equations. In these proofs, we use someconstants that are computed numerically. Furthermore, we make use of the rp

hierarchy estimates [14, 33] to prove that both the Regge-Wheeler and Zerilli

variables decay as t−32 in fixed regions of r.

Contents

1. Introduction 11.1. Regge-Wheeler and Zerilli equations 31.2. Statement of main results 41.3. Comment on the proof 62. Preliminaries 72.1. Notation 72.2. Energy estimate for the wave equation 82.3. Hypergeometric differential equation 83. Integrated local energy decay and uniform bounds for the energy 93.1. Morawetz vector field 93.2. Integrated decay estimate 103.3. Non-degenerate energy 224. Decay estimate 264.1. Energy decay 264.2. Improved decay estimate 37References 43

1. Introduction

The Schwarzschild spacetime is a 1+3−dimensional manifold with the Lorentzianmetric taking the following form in Boyer-Lindquist coordinates (xα) = (t, r, θ, φ),

gµνdxµdxν = −

(1− 2M

r

)dt2 +

(1− 2M

r

)−1dr2 + r2dσS2 ,

in the exterior region which is given by M = R × [2M,∞) × S2. For notationalconvenience, we let

η = 1− µ, µ = 2Mr, ∆ = r2 − 2Mr. (1.1)

We use r∗ to denote the Regge-Wheeler tortoise coordinate

r∗ = r + 2M log(r − 2M)− 3M − 2M logM, (1.2)

Date: Friday 6th December, 2019.

1

2 L. ANDERSSON, P. BLUE, AND J. WANG

and use the retarded and advanced Eddington-Finkelstein coordinates u and vdefined by

u = t− r∗, v = t+ r∗.

In the region near the event horizon H+, located at r = 2M , or inside the blackhole, we are also going to consider the coordinate system (v, r, θ, φ), where v and rare defined as above. In the (v, r, θ, φ) coordinate system, the metric is

gµνdxµdxν = − (1− µ)dv2 + 2drdv + r2dσS2 .

The study of the equations governing the perturbations of the vacuum Schwarzschildmetric was initiated by Regge-Wheeler [31], and their approach was completed byVishveshwara [37] and Zerilli [38]. Such perturbations were classified as being ofodd or even parity, and the different parities were treated separately. The pertur-bations of odd parity are governed by the Regge-Wheeler equation, which is similarto the wave equation for scalar fields on the Schwarzschild spacetime. Later, Zerilliconsidered the even case and showed, by decomposing into spherical harmonics,that the even parity perturbations are governed by the Zerilli equation. A gauge-invariant formulation was also carried out by Moncrief [27, 28] and Clarkson-Barrett[10]. In [10], Clarkson-Barrett extended the 1 + 3 covariant perturbation formal-ism to a ‘1 + 1 + 2 covariant sheet’ formalism by introducing a radial unit vectorin addition to the timelike congruence, and decomposing all covariant quantitieswith respect to this. Bardeen and Press [4] analyzed the perturbation equationsusing the Newman-Penrose formalism. Teukolsky [36] extended this to the Kerrfamily and found that the extreme Newman-Penrose curvature components sat-isfy the Teukolsky equation. The Bardeen-Press equation is the Teukolsky equationrestricted to Schwarzschild case. More relations between the Bardeen-Press, Regge-Wheeler, Zerilli, and Teukolsky equations were established by Chandrasekhar [8, 9].Dafermos-Holzegel-Rodnianski [11] used the double-null gauge to estimate metricperturbations by ascending a hierarchy that goes from variables satisfying a Regge-Wheeler equation all the way to linearised metric components.

We shall prove boundedness, an integrated local energy decay estimate, andpointwise decay for solutions to the Regge-Wheeler and Zerilli equations, both ofwhich take the form

2gψ − Vgψ = 0, (1.3)in the exterior region of the Schwartzchild spacetime. Here, Vg is the Regge-Wheeleror Zerilli potential.

We briefly recall some earlier results about the linear wave equation on blackhole spacetimes. Integrated local energy decay estimates were proved for the waveequation outside Schwarzschild black holes [5, 7, 15]. In the Kerr case, the existenceof a uniformly bounded energy and integrated local energy decay estimates wereproved for |a| �M [3, 12, 34] and more recently for all |a| < M [17]. In addition,there is related work by Finster-Kamran-Smoller-Yau [19] in the |a| < M range.The red-shift effect was first used to control linear waves near the event horizonin [15] (also see [16]). Furthermore, Dafermos and Rodnianski [14] introduced anrp hierarchy of weighted estimates to prove energy decay. Extending this method,Schlue [33] improved the decay rate for linear waves in fixed regions of r outside aSchwarzschild black hole to t−3/2+δ, and for time derivative to t−2+δ. This couldbe compared to an earlier result by Luk [23], where he introduced a commutatorthat is analogous to the scaling and derived similar decay rate. Moschidis [29]extended the rp-weighted energy method to general asymptotically flat spacetimeswith hyperboloidal foliation, and proved the decay rate for wave is at least τ−3/2

(where τ is the hyperboloidal time function) providing that an integrated localenergy decay statement holds. There is much more work using different methods

MORAWETZ ESTIMATE FOR LINEARIZED GRAVITY IN SCHWARZSCHILD 3

to improve the decay rate of linear wave. In fact, the local uniform decay ratefor linear waves can be improved to t−3, see Tataru, Donninger-Schlag-Soffer, etc.[13, 18, 26, 35]. We also mention the results by Pasqualotto [30], where he provedpointwise decay for the Maxwell system in Schwarzschild spacetime, and by Ma[24], where a uniform boundedness of energy and a Morawetz estimate are provedfor each extreme Newman-Penrose component on slowly rotating Kerr background.

There has also been a lot of work on energy bounds for the linearized Ein-stein equation outside a Schwarzschild black hole. Integrated energy decay esti-mates were proved for the Regge-Wheeler equaion [6]. More recently, Dafermos-Holzegel-Rodnianski [11] exhibited t−1 decay for solutions of the Regge-Wheelerequation, and, from this, decay at the same rate for the Teukolsky equation and forcomponents of the linearised metric. Hung-Keller-Wang [20, 21] worked with theRegge-Wheeler-Zerilli-Moncrief system and exhibited t−1 decay for solutions to theRegge-Wheeler and Zerilli equations, from which they also obtained decay for thelinearised metric. Both the approaches of [11] and of [20, 21] rely on a reductionto a scalar equation (or at least to an equation for a section of a bundle that hascomplex dimension 1) and then use the estimate for solutions of the scalar equationto obtain estimates for the full linearised metric. In the recent work of Ma [25],Morawetz estimates for the extreme components are proved in Kerr spacetime.

In this paper, we begin with the Regge-Wheeler-Zerilli-Moncrief system, usingthe techniques in [3] to prove the integrated decay estimate for both Regge-Wheelerand Zerilli variables. Based on this, we apply the rp hierarchy estimate [14, 29, 33]to prove that solutions to Regge-Wheeler or Zerilli equations uniformly decay ast−1, and the time derivatives decay as t−2. Hence, the pointwise decay could beimproved to t−3/2 for finite r. Since both the Regge-Wheeler and Zerilli variableshave angular frequence ` ≥ 2, it should be possible to improve the decay rate inthis paper to at least t−7/2 by vector field methods and to the rate given by thePrice law, t−7, by other methods.

One of the main contributions of this paper is to provide an alternative approachto proving the uniform energy bound and integrated local energy decay for theRegge-Wheeler and Zerilli equations. We do so by extending the method of [3] (seeLemma 3.1), which may prove helpful in treating other problems. This paper alsoproves stronger decay estimates for the Regge-Wheeler and Zerilli variables thanthe t−1 decay appearing in [11, 20, 21]. The observation that stronger decay resultscan be obtained was crucial in the proof of the linear stability of the Kerr spacetimein [2], which appeared since the submission of this work.

1.1. Regge-Wheeler and Zerilli equations. The Regge-Wheeler equation isgiven by

2gψ − V RWg ψ = 0, V RWg = −8M

r3. (1.4)

The Zerilli equation is given, for each spherical harmonic mode with parameter `,by

2gψ − V Zg ψ = 0, V Zg = −8M

r3(2λ̄+ 3)(2λ̄r + 3M)r

4(λ̄r + 3M)2, (1.5)

where 2λ̄ = (` − 1)(` + 2) ≥ 4. These two equations are related by the Chan-drasekhar transformation [8, 9]. We note that it is well known that the sphericalharmonic modes with parameter ` ≥ 2 represent gravitational wave perturbationsthat dynamically evolve and disperse. In contrast, the equations of linearized grav-ity which lead to the Regge-Wheeler and Zerilli equations allow only a finite di-mensional space of solutions for the ` = 0 and ` = 1 spherical harmonic modes.These correspond to perturbations of the mass (corresponding to moving from oneSchwarzschild solution to another) and of the angular momentum (corresponds to


H+

r ≥ rNHr < rNH

ΣiτΣeτ

Figure 1. The hypersurfaces Στ = Σiτ ∪ Σeτ

changing the non-rotating Schwarzschild background to a rotating Kerr solutionand to gauge transformations), see [22, 32].

Remark 1.1 (Modes ` ≥ 2). We only consider solution to the Regge-Wheelerequation (1.4) or Zerilli equation (1.5) with modes ` ≥ 2. For these, the spectrumof the spherical Laplacian −4̊/ S2 acting on functions with ` ≥ 2 is `(` + 1) ≥ 6.Hence upon integrating over S2(t, r),∫

S2(t,r)

|∇/ ψ|2 ≥∫S2(t,r)

6

r2|ψ|2, (1.6)

where ∇/ is the induced covariant derivative on the sphere S2(t, r) of constant r andt.

1.2. Statement of main results. We shall make of the following hypersurfaces,which are illustrated in Figure 1. Near the event horizon H+, we fix 3M > rNH >2M with corresponding tortoise coordinate value r∗NH and let

Σiτ.= {v = τ + r∗NH} ∩ {r < rNH}. (1.7)

While away from the horizon, we use the usual time function t and let

Σeτ.= {t = τ} ∩ {r ≥ rNH}. (1.8)

We define the hypersurfaces Στ by

Στ = Σiτ ∪ Σeτ . (1.9)

For spacelike hypersurfaces Σ, let dµΣ denote the volume form of Σ and nαΣ de-

note the future normal vector of Σ. In this case, for any vector field X,∫

ΣXαn

αΣdµΣ

is the flux of X through Σ. Where a hypersurface becomes null, the quantities nαΣand dµΣ are only defined modulo rescaling, but the flux integral remains uniquellydefined, and we continue to use the notation

∫ΣXαn

αΣdµΣ for this flux. We will

drop the subscript Σ on nαΣ when there is no confusion.We use D to denote the Levi-Civita connection associated with the Schwarzschild

metric gµν . Let {Ωi}, i = 1, 2, 3 be three rotational Killing vector fields aboutorthogonal axes on the 2-sphere S2(t, r) with constant t and r, and ∇/ the in-duced covariant derivative on S2(t, r). We use the short hand notation Ωk for

Ωi11 Ωi22 Ω

i33 , i1 + i2 + i3 ≤ k for all k ∈ N. In section 2, we define a time-like vector

field N in the future development of the initial hypersurface J+(Στ0), such thatN = ∂t for r ≥ rNH > 2M .


The energy-momentum tensor and the corresponding momentum vector for (1.3)is

Tαβ(ϕ) = DαϕDβϕ−1

2gαβ(D

γϕDγϕ+ Vgϕ2),

P ξα(ϕ) = Tαβ(ϕ) · ξβ ,(1.10)

for a vector field ξµ. We take ξ = N or ξ = ∂t to define various energy classes. Thenon-degenerate energy EN (ϕ,Στ ) associated with N is given by

EN (ϕ,Στ ).=

∫Στ

PNα (ϕ)nαΣτdµΣτ

∼∫

Σiτ

(|∂rϕ|2 + |∇/ϕ|2 +

|ϕ|2

r2

)r2drdσS2

+

∫Σeτ

(|∂tψ|2 + |∂rϕ|2 + |∇/ϕ|2 +

|ϕ|2

r2

)r2dr∗dσS2 ,

(1.11)

where the integral in the second line of (1.11) is written in (v, r, θ, φ) coordinatewhile the third is in (t, r, θ, φ) coordinates. For ∂t, the associated energy E

∂t(ϕ, τ)is given by

E∂t(ϕ, τ).=

∫{t=τ}

P ∂tα (ϕ)nαdµ{t=τ}

∼ 12

∫{t=τ}

(|∂tϕ|2 + |∂r∗ψ|2 + (1− µ)

(|∇/ϕ|2 + |ϕ|

2

r2

))r2dr∗dσS2 .

(1.12)

Note that this energy is defined as an integral on the hypersurface {t = τ} not Στ .The main results of this paper are the uniform boundedness of the energy, inte-

grated decay estimates, and pointwise decay for solution to the Regge-Wheeler orZerilli equations.

Theorem 1.2 (Uniform Boundedness of the Energy). Let ψ be a solution to theRegge-Wheeler equation (1.4) or Zerilli equation (1.5), then for τ > τ0,

EN (ψ,Στ ) . EN (ψ,Στ0). (1.13)

The uniform boundedness of higher order energies is stated in Corollary 3.9. Inthe context of this paper, order means the level of regularity.

Theorem 1.3 (Integrated Decay Estimate). Let M > 0 and r̄ > 0. Define thefunction 1r 6h3M to be identically one for |r − 3M | > r̄ and zero otherwise. Forall smooth function ψ solving the Regge-Wheeler equation (1.4) or Zerilli equation(1.5), we have∫ τ

τ0

dt

∫Σ̄t

(∆2

r6|∂rψ|2 +

|ψ|2

r4+ 1r 6h3M

(|∂tψ|2

r2+|∇/ ψ|2

r

))r2drdσS2

. E∂t(ψ, τ0),

(1.14)

where Σ̄τ = {t = τ}.

Remark 1.4. The localising function 1r 6h3M is due to the trapped null geodesic atr = 3M . However, with a loss of regularity, commuting with T , we have,∫ τ

τ0

dt

∫Σ̄t

(∆2

r6|∂rψ|2 +

|ψ|2

r4+|∂tψ|2

r4+|∇/ ψ|2

r4

)r2drdσS2

. E∂t(ψ, τ0) + E∂t(Tψ, τ0).

(1.15)

Alternatively, the sharp cut-off 1r 6h3M can be replaced by a function that vanishesquadratically in (r − 3M).


Combining the red shift effect, the uniform boundedness of the energy (Theorem1.2), and the integrated decay estimate (Theorem 1.3), one can use the globallytime-like vector field N to obtain a non-degenerate local integrated decay estimate(Corollary 3.6). This can be generalized to the high order derivative cases (Corollary3.7 or Remark 3.10).

To state the decay estimate, we introduce additional notation. Let R > 3M bea large constant. We define the interior region

Σ′τ = Στ ∩ {r ≤ R}. (1.16)

Theorem 1.5 (Energy Decay and Pointwise Decay). Let R > 3M be sufficientlylarge. Define the weighted derivatives D = {r∂v, (1 − µ)−1∂u, r∇/ }. Let τ0 > 0,u0 = τ0 −R∗ ,and v0 = τ0 +R∗).

Let ψ be a solution to the Regge-Wheeler equation (1.4) or Zerilli equation (1.5).For n ∈ N, let

In.=∑i≤n

∫ ∞v0

dv

∫S2

∑k+l+j≤7,j≤2,l≤3,k≤5

|Di(r∂v)jΩl∂kt ψ|2r2dσS2∣∣u=u0

+

∫Σ′τ0

∑k+l≤7,l≤3,k≤6

PNµ (Di∂kt Ωlψ)nµdµΣ′τ0


case. However, we are allowed to relax the requirement of smooth lower bound, andfind a continuous lower bound Vjoint for the Morawetz potential. After that, as inthe Regge-Wheeler case, we could work on the second-order ordinary differentialequation with the potential V being replaced by Vjoint, and find a positive C

2

solution. Then the Hardy type estimate follows. The crucial part is finding apositive C2 solution to the hypergeometric differential equation (associated to theZerilli case). This is done by further analysis of the two Frobenius solutions to thehypergeometric differential equation. The asymptotic expansion at the singularityr =∞ is needed to prove the positivity.rp hierarchy. In Section 4, we use a multiplier of the form rp∂v which gives

the rp hierarchy of estimates and this yields the energy decay. This approach, firstintroduced in the context of the wave equation [14], can also be adapted to theRegge-Wheeler and Zerilli equations. Proceeding to the higher-order case, we fur-ther commute the equations with r∂v and derive a first-order r

p weighted inequalityfor all 0 < p ≤ 2. (For the wave equation, the range 0 < p < 2 was treated in [33]and the end point p = 2 was reached in [29].) Based on this, the rp hierarchy ofestimate yields the decay rate of t−3/2 for ψ(t, r) with finite r, which should becompared with [33] (or [23]), where the decay is t−3/2+δ in compact regions. Recallthat in the analysis of the wave equation in [29], the key advance over [33], whichallows the t−3/2+δ decay to be improved to t−3/2 is that the rp hierarchy can beextended to the endpoint p = 2. In particular, although the coefficient in the rp

estimate of |∇/ψ|2 vanishes like 2− p (see (4.9)), the worst error term, which needsto be controlled, can be transformed so that it vanishes at the same 2− p rate (see(4.44)-(4.49)). In this paper, we show that the same holds for the Regge-Wheelerand Zerilli equations.

The paper is organized as follows: We begin in Section 2 with preliminaries,introducing basic notation and background. Section 3 is devoted to the integratedlocal energy decay estimate and uniform boundedness. The energy decay and point-wise decay are proved in Section 4.

Acknowledgements We are grateful to Steffen Aksteiner, Siyuan Ma, and Vin-cent Moncrief for many helpful discussions and suggestions. J.W. was supportedby a Humboldt Foundation post-doctoral fellowship at the Albert Einstein Insti-tute during the period 2014-16, when part of this work was done. She is alsosupported by Fundamental Research Funds for the Central Universities (Grant No.20720170002) and NSFC (Grant No. 11701482).

2. Preliminaries

In this section, we present some notation and basic estimates that we shall usethroughout the paper.

2.1. Notation. Let us introduce the notation T to denote the coordinate vectorfield ∂t with respect to (t, r) coordinates. It is time-like only when r > 2M . Wechoose to work with a globally defined time-like vector field N , defined, in (u, v, θ, φ)coordinates, by

N = (∂u + ∂v) +y1(r)

1− µ∂u + y2(r)∂v,

where y1, y2 > 0 are supported near the event horizon (e.g. where r ≤ rNH),and y1 = 1, y2 = 0 at the event horizon. Notice that we can also write N in the(v, r, θ, φ) coordinates as

N = (1 + 2y2(r))∂v − (y1(r)− y2(r)(1− µ))∂r.


We shall let D denote the covariant derivative associated with the Schwarzschildmetric, ∂ the derivative in terms of coordinates, and ∇/ the induced covariant deriv-ative on the sphere S2(t, r) of constant t and r. Let4/ denote the induced Laplacianon S2(t, r) and −4̊/ S2 to denote the Laplacian on the unit sphere. Recall that forachronal hypersurfaces, nαdµ was already defined.

The notation x . y means x ≤ cy for a universal constant c, and the notationx ∼ y means x . y and y . x. All objects are smooth unless otherwise stated.

2.2. Energy estimate for the wave equation. We would like to study the so-lutions to the wave equation (1.3) on the Schwarzschild spacetime. The energy-momentum tensor for (1.3) is

Tαβ(ϕ) = DαϕDβϕ−1

2gαβ(D

γϕDγϕ+ Vgϕ2). (2.1)

Given a vector field ξµ, the corresponding momentum vector is defined by

P ξα(ϕ) = Tαβ(ϕ) · ξβ . (2.2)

The corresponding energy on a hypersurface Σ is

Eξ(ϕ,Σ) = −∫

Σ

P ξα(ϕ)nαΣdµΣ, (2.3)

where nαΣ is the future normal to Σ. The energy identity takes the form

Eξ(ϕ,Σ′)− Eξ(ϕ,Σ) = −∫∫DDαP ξα(ϕ)dµg (2.4)

where D is the region enclosed between Σ′ and Σ. The associated bulk term Kξ(ϕ)is defined as

Kξ(ϕ) = DαP ξα(ϕ). (2.5)

In applications, ξ will be taken as the vector field ∂t or N .From the dominant energy condition, if ξ is future-directed and causal, and

if Σ is achronal (and the appropriate time-orientation is taken on the normal),

then TαβξαnβΣ ≥ 0. In particular, when considering the hypersurfaces Στ , we willfrequently be able to omit the the nonnegative contribution on the portion of thehorizon or null infinity between Στ1 and Στ2 .

2.3. Hypergeometric differential equation. We refers to [1, §15] for more back-ground on hypergeometric functions. The equation

z(1− z)d2w

d2z+ (c− (a+ b+ 1)z)dw

dz− abw = 0. (2.6)

is the hypergeometric differential equation.

2.3.1. Fundamental solutions. Solutions to the hypergeometric differential equation(2.6) have regular singularities at z = 0, 1,∞ with corresponding exponent pairs{0, 1 − c}, {0, c − a − b}, {a, b} respectively [1, §15]. When none of the exponentpairs differ by an integer, that is, when none of the c, c− a− b, a− b is an integer,we have the following pairs f1(z), f2(z) of fundamental solutions. They are alsonumerically satisfactory in a neighborhood of the corresponding singularity, in thesense that one of the solutions diverges or vanishes at a much more rapid rate thanthe other [1, §2.7(iv)].

• Adapted to Singularity z = 0f1(z) = F (a, b; c; z),

f2(z) = z1−cF (a− c+ 1, b− c+ 1; 2− c; z).

(2.7)


• Adapted to Singularity z = 1f1(z) = F (a, b; a+ b+ 1− c; 1− z),

f2(z) = (1− z)c−a−bF (c− a, c− b; c− a− b+ 1; 1− z).(2.8)

• Adapted to Singularity z =∞

f1(z) = z−aF (a, a− c+ 1; a− b+ 1; 1

z),

f2(z) = z−bF (b, b− c+ 1; b− a+ 1; 1

z).

(2.9)

2.3.2. Integral representations. The hypergeometric function F (a, b; c; z) has thefollowing integral representation [1, §15]: For 0 <


Putting these together, we have for some constant � > 0,

DαPα[ψ,A, q] ≥ �

(A(∂rψ)2 + (V + 6U)|ψ|2

)+ (1− �)U|r∇/ψ|2.

(3.4)

Hence, proving the Morawetz estimate reduces to proving the Hardy inequality∫ ∞2M

(A|∂rϕ|2 + V |ϕ|2

)dr ≥ �Hardy

∫ ∞2M

(∆2

r4|∂rϕ|2 +

ϕ2

r2

)dr (3.5)

with A = A and V = V + 6U . As in the proof of Lemma 3.12 in [3], one has toshow that there is a positive C2 solution to the ordinary differential equation

−∂r(A∂r)φ+ V φ = 0.

3.2. Integrated decay estimate. In this section, we will prove the integrateddecay estimate for both the Regge-Wheeler and Zerilli equations.

First, we introduce a lemma relating the above ODE and the hypergeometricfunctions. Recall that

∆ = r2 − 2Mr.

Lemma 3.1. Let A = M ∆2

r4 and V =Mr4 (V2r

2 + V1Mr1 + V0M

2).Let

α =1

2+

√4V2 + 2V1 + V0 + 1

2,

β =1

2−√

9 + V02

,

a =1 +√

1 + 4V2 + 2V1 + V0 −√

9 + V0 −√

1 + 4V22

,

b =1 +√

1 + 4V2 + 2V1 + V0 −√

9 + V0 +√

1 + 4V22

,

c =1 +√

1 + 4V2 + 2V1 + V0.

Assume none of c, c− a− b, and a− b are integers.For the ODE

−∂r(A∂rφ) + V φ = 0, (3.6)

a pair of fundamental solutions, which we call the Frobenius solutions adapted tor = 2M , is

(r − 2M)αrβF(a, b; c;−r − 2M

2M

),

(r − 2M)αrβ(−r − 2M

2M

)1−cF

(a− c+ 1, b− c+ 1; 2− c;−r − 2M

2M

).

The second can also be expressed as

(r − 2M)αrβ(−r − 2M

2M

)1−c ( r2M

)c−a−bF

(1− a, 1− b; 2− c;−r − 2M

2M

).

Another pair of fundamental solutions, which we call the Frobenius solutions adaptedto r =∞, is

(r − 2M)αrβ(−r − 2M

2M

)−aF

(a, a− c+ 1; a− b+ 1;− 2M

r − 2M

),

(r − 2M)αrβ(−r − 2M

2M

)−bF

(b, b− c+ 1; b− a+ 1;− 2M

r − 2M

).


Proof. We follow the argument from [3]. Let

v = A1/2φ,

x = r − 2M.(3.7)

The ODE (3.6) then becomes

− ∂2xv +Wv = 0, (3.8)with

W =V

A+

1

2

∂2xA

A− (∂xA)

2

A2

=V2r

2 + (V1 − 4)Mr + (V0 + 8)M2

r2(2M − r)2

=V2x

2 + (4V2 + V1 − 4)Mx+ (4V2 + 2V1 + V0)M2

x2(x+ 2M)2.

Now, let ṽ be such that v = xα(x+ d)β ṽ, so that the above ODE becomes1

0 =xα−2(x+ d)β−2P,

P =− x2(x+ d)2∂2xṽ− 2x(x+ d)((α+ β)x+ αd)∂xṽ+(− (α(α− 1)(x+ d)2 + 2αβx(x+ d) + β(β − 1)x2) +Wx2(x+ d)2

)ṽ.

Let d = 2M , so that

Wx2(x+ d)2 =V2x2 + (4V2 + V1 − 4)Mx+ (4V2 + 2V1 + V0)M2,

and the coefficient of ṽ in P above is

(−α(α− 1)− 2αβ − β(β − 1) + V2)x2

+ (−4α(α− 1)− 4αβ + 4V2 + V1 − 4)Mx+ (−4α(α− 1) + 4V2 + 2V1 + V0)M2.

We choose α so that the M2 coefficient vanishes, i.e.

0 =− 4α(α− 1) + 4V2 + 2V1 + V0=− 4α2 + 4α+ (4V2 + 2V1 + V0),

α =1

2±√

4V2 + 2V1 + V0 + 1

2.

From the second line above, we also have the identity −α(α− 1) = −(4V2 + 2V1 +V0)/4. We now choose β so that the ratio of the Mx coefficient to the x

2 coefficientis 2, i.e.

2 =−4α(α− 1)− 4αβ + 4V2 + V1 − 4−α(α− 1)− 2αβ − β(β − 1) + V2

−2α(α− 1)− 4αβ − 2β(β − 1) + 2V2 =− 4α(α− 1)− 4αβ + 4V2 + V1 − 4−2β(β − 1) =− 2α(α− 1) + 2V2 + V1 − 4

=

(−2V2 − V1 −

1

2V0

)+ 2V2 + V1 − 4

=− 12V0 − 4,

0 =4β2 − 4β + (−V0 − 8)

1Observe that equation (3.23) in [3] is missing a minus sign in front of x2(x + d)2∂2xṽ, but therest of the argument there is correct.


β =1

2±√

9 + V02

.

We choose the + sign in α and − sign in β.The substitution x = −2Mz now yields the ODE

0 =z(1− z)∂2z ṽ + (2α− (2α+ 2β)z)∂z ṽ − (2αβ + 1 + V0/2 + V1/2)ṽ.

This is now in the form of a standard hypergeometric differential equations, andthe corresponding parameters thus satisfy

c =2α,

a+ b+ 1 =2α+ 2β,

ab =(2αβ + 1 + V0/2 + V1/2),

which implies the parameters a and b are

a =α+ β − 12−√

4α(α− 1) + 4β(β − 1)− 2V1 − 2V0 − 72

,

b =α+ β − 12

+

√4α(α− 1) + 4β(β − 1)− 2V1 − 2V0 − 7

2.

Thus, the parameters are

a =1 +√

1 + 4V2 + 2V1 + V0 −√

9 + V0 −√

1 + 4V22

,

b =1 +√

1 + 4V2 + 2V1 + V0 −√

9 + V0 +√

1 + 4V22

,

c =1 +√

1 + 4V2 + 2V1 + V0.

Several forms for solutions to the hypergeometric function are given in [1]. Re-versing all the substitutions made so far in the proof, one finds a pair of fundamentalsolutions adapted to z = 0 (i.e. r = 2M) is

(r − 2M)αrβF(a, b; c;−r − 2M

2M

),

(r − 2M)αrβ(−r − 2M

2M

)1−cF

(a− c+ 1, b− c+ 1; 2− c;−r − 2M

2M

).

An alternative way of writing the second solution is

(r − 2M)αrβ(−r − 2M

2M

)1−c ( r2M

)c−a−bF

(1− a, 1− b; 2− c;−r − 2M

2M

).

Another pair of fundamental solutions adapted to z = ∞ (i.e. r approaching thepoint at infinity on the Riemann sphere except along the negative real axis) is

(r − 2M)αrβ(−r − 2M

2M

)−aF

(a, a− c+ 1; a− b+ 1;− 2M

r − 2M

),

(r − 2M)αrβ(−r − 2M

2M

)−bF

(b, b− c+ 1; b− a+ 1;− 2M

r − 2M

).

�


3.2.1. The Regge-Wheeler case.

Proof of Theorem 1.3 for the Regge-Wheeler case. We first prove the statementfor the Regge-Wheeler case, where we have

A = M∆2

r4, (3.9)

V = VRW .= − 52Mr−2 + 15M2r−3 − 23M3r−4. (3.10)

Introducing the notation V = VRW + 6U · 2Mr , we have the lower bound for theMorawetz potential VRW + 6U ≥ V and

V =3

2Mr−2 − 9M2r−3 + 13M3r−4. (3.11)

Recalling A from (3.9), we letA = A. (3.12)

Now we apply lemma 3.1 and find that the resulting ordinary differential equation(3.8) has a solution taking the following form

u = (r − 2M)αrβF (a; b; c; z), (3.13)

where z = − r−2M2M , and F (a; b; c; z) is the associated hypergeometric function. FromLemma 3.1, we find the parameters are

α =1 +√

2

2,

β =1−√

22

2,

a =1 +√

2−√

22−√

7

2,

a =1 +√

2−√

22 +√

7

2,

c = 1 +√

2.

In particular,

a < −2.4 < 0 < 0.1 < b < 0.2 < 2.0 < c.We can use the integral representation (2.10) to show that the hypergeometricfunction F (a; b; c; z) with z < 0 is positive. This further leads to the Hardy estimate(3.5), as in [3]. Combining this estimate with the conservation energy associatedto ∂t, we have∫∫

M

∆2

r4|∂rψ|2 +

|ψ|2

r2+

1

r

(1− 3M

r

)2|r∇/ψ|2dtdrdσS2 . ET (ψ, τ0).

On the other hand, taking the multiplier f∂r∗ , with q = −∂r∗f +(1− 2Mr

)fr and

for instance f =(1− 3Mr

)3, we can obtain, after calculating the associated current,∫∫

MM

(1− 3M

r

)2|∂tψ|2dtdrdσS2

.∫∫M

|ψ|2

r2+

1

r

(1− 3M

r

)2|r∇/ψ|2dtdrdσS2 + ET (ψ, τ0).

Putting these integrated decay estimates together, we achieve∫∫M

∆2

r4|∂rψ|2 +

|ψ|2

r2+ 1r 6h3M

(|∂tψ|2 +

|r∇/ψ|2

r

)dtdrdσS2 . E

T (ψ, τ0).


That is,∫∫M

∆2

r6|∂rψ|2 +

|ψ|2

r4+ 1r 6h3M

(|∂tψ|2

r2+|r∇/ψ|2

r3

)dµg . E

T (ψ, τ0). (3.14)

This completes the proof for the Regge-Wheeler case. �

3.2.2. The Zerilli case. Let V RWg denote the potential in the Regge-Wheeler equa-

tion and V Zg the potential in the Zerilli equation. They are related by VZg =

V RWg (1 + ζ), with

ζ =2λ̄+ 3

4λ̄

(9

(1

2− M

Λ

)2− 1

4

)− 1, (3.15)

whereΛ = λ̄r + 3M, 2λ̄ = `(`+ 1)− 2. (3.16)

Noting that ` ≥ 2, we have λ̄ ≥ 2. We calculate

∂rζ =9M

(2λ̄+ 3

)2

(1

2− M

Λ

)Λ−2. (3.17)

It is easy to check that∂rζ > 0.

Before proceeding to the proof for the Zerilli case, we first state the main ideaand main steps for the proof. In the Zerilli case, it would be difficult to find asmooth lower bound for the Morawetz potential V + 6U (3.4) in the whole region[2M,∞). The key point is: we are allowed to relax the requirement of a smoothlower bound, and find a continuous lower bound Vjoint for the Morawetz potential.Then, much like as in the Regge-Wheeler case, we could work on the second-orderordinary differential equation (3.6) with the potential V being replaced by Vjoint,and find a positive C2 solution. The Hardy inequality then follows and hence theintegrated decay estimate.

We break the proof into 3 steps. In step I, we separate the estimate in the tworegions [2M, 3M ] and (3M,∞) and find a C0 lower bound Vjoint for the Morawetzpotential. Note that the lower bounding potential Vjoint is chosen such that thesecond order ordinary differential equation (3.6) could be transformed to hyperge-ometric differential equations in each of the two intervals. In step II, we analyzethe hypergeometric differential equation associated to the ODE (3.6), and find theFrobenius solutions (adapted to x = 0) in [2M, 3M ] and (3M,∞). The Frobeniussolutions (adapted to x = ∞) follow by making some transformations on the oldFrobenius solutions (adapted to x = 0). In step III, we construct a C1 solution tothe hypergeometric differential equation, which comes from linear combinations ofthe Frobenius solutions (adapted to x = 0). We show that this solution is C2 andshow that it is positive by finding a positive lower bound G(x). The construction ofG(x) is based on an observation that the ratio of two Frobenius solutions (adaptedto x = 0) has a limit at infinity. Remarkably, further analysis for the asymptoticexpansion at infinity shows that G(x) is proportional to one of the Frobenius solu-tions adapted to x = ∞. We can further make use of the integral representation(2.10) to show the positivity of G(x).

Step I: The continuous lower bound for the Morawetz potential.

Lemma 3.2 (Lower bound for the Morawetz potential). In the Zerilli case, wehave the following lower bound for the Morawetz potential V + 6U in (3.4):

For case I, r ≤ 3M ,

V + 6U ≥ Vr≤3M.=

5M

2r2−(

13 +2

3

)M2

r3+

18M3

r4. (3.18)


For case II, r ≥ 3M ,

V + 6U ≥ Vr≥3M.=

M

2r2− 4M

2

r3+

7M3

r4. (3.19)

Furthermore, we have

(Vr≤3M − Vr≥3M )∣∣r=3M

= 0. (3.20)

Proof of Lemma 3.2. As in the Regge-Wheeler case, we still have the bound (3.4)

with A = M∆2

r4 being the same as in (3.9). Now we have to estimate the lowerbound for V + 6U in (3.4). First, we recall the formula for V,

V = VZ .= 14∂r(∆∂r(z∂r(wf))) +

1

2wf∂r(zV

Z), (3.21)

where the first term 14∂r(∆∂r(z∂r(wf))) is the same as in the Regge-Wheeler case.

Now we consider the second term 12wf∂r(zVZ), which is given by

1

2wf∂r(zV

Z) =1

2wf∂r(zV

RW )(1 + ζ) +1

2wfzV RW∂rζ. (3.22)

Define Z1, Z2 by

Z1 =1

2wf∂r(zV

RW )ζ, Z2 =1

2wfzV RW∂rζ. (3.23)

Then

VZ = VRW + Z1 + Z2, (3.24)where VRW is given in (3.10). In what follows, we will focus on finding lower boundsfor Z1, Z2. Notice that

f = −2(r − 3M)r4

.

In case I: 2M ≤ r ≤ 3M. We have2M(2λ̄+ 3) ≤2Λ ≤ 6M(λ̄+ 1),

(2λ̄+ 2)r ≤2Λ ≤ (2λ̄+ 3)r,(3.25)

and

− 314≤ ζ(2M) = − 3

2(2λ̄+ 3)≤ ζ ≤ ζ(3M) = − 1

4(λ̄+ 1)2< 0. (3.26)

Substituting (3.25) into (3.17), we have

5

14r≤ ∂rζ ≤

3

4r. (3.27)

Since 2M ≤ r ≤ 3M, we have wf ≥ 0. Moreover, in view of (3.27) and the factthat zV RW < 0 (note that V RW = − 8Mr ), we have the estimate for Z2 in (3.23),

− 12wf(r − 2M)6M

r5=

1

2wfzV RW · 3

4r≤ Z2 =

1

2wfzV RW∂rζ < 0. (3.28)

Let us turn to the Z1 in (3.23). Notice that, ∂r(zVRW ) has no sign. Hence we

separate ∂r(zVRW ) into the positive and negative parts:

∂r(zVRW ) = 8Mr−5(3r − 8M) = 24Mr−5(r − 3M) + 8M2r−5.

Noting from (3.26) that ζ < 0 and that wf ≥ 0 for 2M ≤ r ≤ 3M , we know that24Mr−5(r − 3M)wfζ ≥ 0. Thus 12wf∂r(zV

RW )ζ ≥ 12wf · 8M2r−5ζ. We further

use the lower bound of ζ (3.26) to deduce

Z1 =1

2wf∂r(zV

RW )ζ ≥ −12wf

8M2

r5· 3

14= −1

2wf

12M2

7r5. (3.29)


Combining the lower bounds for Z1 and Z2 (3.29), (3.28), we obtain

Z1 + Z2 ≥ Lr≤3M.= −1

2wf

(12M2

7r5+ (r − 2M)6M

r5

). (3.30)

The lower bound of the potential VZ + 6U in the Morawetz estimate is given by

VZ + 6U ≥ VRW + 6U 2Mr

+ Z1 + Z2 ≥ V + Lr≤3M , (3.31)

where V is the lower bound for the Regge-Wheeler potential defined in (3.11). Aftersubstituting V + Lr≤3M , we have for the case 2M ≤ r ≤ 3M ,

VZ + 6U ≥ 5M2r2−(

13 +5

7

)M2

r3+

(18 +

1

7

)M3

r4. (3.32)

Besides, we note that,

5M

2r2−(

13 +5

7

)M2

r3+

(18 +

1

7

)M3

r4

=5M

2r2−(

13 +2

3

)M2

r3+

18M3

r4+M2(3M − r)

21r4.

Therefore, using the fact that r ≤ 3M , we finally get the lower bound for thepotential for r ≤ 3M :

VZ + 6U ≥ Vr≤3M.=

5M

2r2−(

13 +2

3

)M2

r3+

18M3

r4. (3.33)

In case II: r ≥ 3M. We have wf ≤ 0, and ∂r(zV RW ) > 0. Additionally,

ζ ≤ 32λ̄≤ 3

4.

Hence in (3.22) the first term has the lower bound

Z1 =1

2wf∂r(zV

RW )ζ ≥ 38wf∂r(zV

RW ).

In view of the fact that zV RW < 0 and ∂rζ ≥ 0, Z2 is non-negative. Thus,

Z1 + Z2 ≥ Lr≥3M.=

3

8wf∂r(zV

RW ), (3.34)

and, we have the lower bound

VZ ≥ VRW + Lr≥3M , (3.35)

with VRW given by (3.10). Furthermore, we can estimate VZ + 6U for r ≥ 3M by

VZ + 6U ≥ VRW + Lr≥3M + 6U3M

r

= Vr≥3M.=

M

2r2− 4M

2

r3+

7M3

r4.

(3.36)

In summary, for case I, wf ≥ 0, we have found a lower bound Lr≤3M for Z1 +Z2(3.30), such that

VZ + 6U ≥ VRW + Lr≤3M + 6U2M

r= Vr≤3M +

M2(3M − r)21r4

. (3.37)

For case II, wf ≤ 0, we have found a lower bound Lr≥3M for Z1 + Z2 (3.34), suchthat

VZ + 6U ≥ VRW + Lr≥3M + 6U3M

r= Vr≥3M . (3.38)


Both of wf∣∣r=3M

= 0 (and hence Lr≤3M∣∣r=3m

= Lr≥3M∣∣r=3m

= 0) and U∣∣r=3M

=

0 hold, so (Vr≤3M − Vr≥3M )∣∣r=3M

= 0. In particular, we have

Vr≤3M − Vr≥3M = (r − 3M)(

2r − 11M3

)M

r4, (3.39)

which vanishes at r = 3M.�

Step II: The hypergeometric functions associated to the Morawetzpotential. Denote the lower bound for the Morawetz potential V + 6U in eachcase by

V1.= Vr≤3M , and V2

.= Vr≥3M . (3.40)

Notice that (3.39) implies

V1|r=3M = V2|r=3M . (3.41)We therefore define the potential in the whole region r ≥ 2M by

Vjoint.=

{V1, if 2M ≤ r ≤ 3M,V2, if r ≥ 3M.

(3.42)

This is the lower bound for the Morawetz potential in the Zerilli case, and we knowthat Vjoint ∈ C0. In this case, the Hardy type estimate is reduced to finding apositive solution to the ordinary differential equation

− ∂r(A∂rφ) + V φ = 0, (3.43)

on the interval r ∈ [2M,+∞) with

A = A = M∆2

r4, V = Vjoint. (3.44)

If φ is a solution to equation (3.43) with A and V being specified in the Zerilli caseby (3.44), we apply the transformation (see Lemma 3.1)

u = A12φ, x = r − 2M. (3.45)

Then u solves the new ordinary differential equation

− ∂2xu+Wu = 0, (3.46)

on the interval x ∈ [0,+∞) with

W.=

{W1 =

16

15x2−46Mx+4M2x2(x+2M)2 , x ≤M,

W2 =12x2−12Mx+2M2x2(x+2M)2 , x > M.

(3.47)

We will apply the scheme in Lemma 3.1 to this case, and explore the hyperge-ometric functions associated to (3.46)-(3.47) in each of the two region x ≤ M andx > M . To do the calculation explicitly, we set M = 1. In the following lemma,we consider solutions uij . The first index, i, corresponds to the interval on whichthe function solves the hypergoemetric differential equation, with x ≤ 1 and x > 1indexing by i = 1 and i = 2 respectively. The second index j = 1, 2 indexes thetwo Frobenius solutions on that interval.

Lemma 3.3 (The hypergeometric differential equations associated to the Zerilliequation). For x ≤ 1, there are two Frobenius solutions u11, u12 (defined in (3.48))to the ODE (3.46)-(3.47) and u11 is positive (see Figure 2(a)).

For x > 1, there are two Frobenius solutions u21, u22 (defined in (3.51) ) to theODE (3.46)-(3.47) and u21 is positive (see Figure 2(b)).


(a) Positive solution u11 (b) Positive solution u21

Figure 2. Positive hypergeometric functions in Zerilli case

Proof of Lemma 3.3. We find that that there are two linearly independent solutions(adapted to x = 0) to (3.46) in x ≤ 1 taking the form of

u11 = xα1(x+ 2)β1F

(a1, b1; c1;−

x

2

),

u12 = xα1(x+ 2)β1x1−c1F

(b1 − c1 + 1, a1 − c1 + 1; 2− c1;−

x

2

),

(3.48)

where we follow Lemma 3.1 to calculate the parameters

α1 =1

2+

√15

6,

β1 =1

2− 3√

3

2,

a1 =3 +√

15− 9√

3− 3√

11

6,

b1 =3 +√

15− 9√

3 + 3√

11

6,

c1 =

√15

3+ 1.

In particular,

a1 ≈ −3.11 < 0 < b1 ≈ 2.21 < c1 ≈ 2.29. (3.49)and

α1 + β1 =a1 + b1 + 1

2. (3.50)

For F (a1, b1; c1;−x2 ), its second and third parameters satisfy c1 > b1 > 0. We canuse the integral representation (2.10) to show that

F(a1, b1; c1;−

x

2

)> 0, for x > 0,

which says that u11 is positive.Similarly, let u2j , j = 1, 2 denote the Frobenius solutions (adapted to x = 0) to

(3.46) in x > 1,

u21 = xα2(x+ 2)β2F

(a2, b2; c2;−

x

2

),

u22 = xα2(x+ 2)β2x1−c2F

(a2 − c2 + 1, b2 − c2 + 1; 2− c2;−

x

2

),

(3.51)


(a) Positive solution u (b) u near x = 1

Figure 3. The positive C2 solution u

where the parameters are

α2 =1

2+

√2

2, β2 = −

3

2, α2 + β2 =

a2 + b2 + 1

2, (3.52a)

a2 =

√2− 3−

√3

2, (3.52b)

b2 =

√2− 3 +

√3

2, (3.52c)

c2 = 1 +√

2. (3.52d)

In particular,

a2 ≈ −1.66 < 0 < b2 ≈ 0.07 < c2 ≈ 2.41. (3.53)Those value will be useful in proving Theorem 3.4 in Step III. Again we can usethe integral representation (2.10) to show that

F(a2, b2; c2;−

x

2

)> 0, for x > 0.

That is u21 is positive. �

Step III: Constructing the positive C2 solution. We first calculate somequantities which will be useful in constructing the positive C2 solution to the ordi-nary differential equation (3.46)-(3.47). We normalize u11 (3.48) so that u11(1) = 1,and let

w11 =du11dx

∣∣x=1

. (3.54)

We have

w11 ≈ 0.6184539934 · · · (3.55)For j = 1, 2, we normalize u2j (3.51) so that u2j(1) = 1, and let

w2j =du2jdx

∣∣x=1

. (3.56)

We have

w21 = 0.7340312856 · · · , w22 = −0.3321954186 · · · (3.57)Additionally, we observe that

ρ.= − lim

x→+∞

u22u21

= 5.0153723738 · · · (3.58)


Theorem 3.4 (Positive hypergeometric function for the Zerilli equation). We de-fine u, normalized to u(1) = 1, by

u =

{u11, x ≤ 1ωu22 + (1− ω)u21, x > 1

(3.59)

where ω is given by

ω =w11 − w21w22 − w21

= 0.1083984220 · · · (3.60)

Then u is indeed a positive C2 solution (see Figure 3(a), 3(b)) to the ordinarydifferential equation (3.46)-(3.47).

Remark 3.5. In the proof of Theorem 3.4, we use the numerical value of w11 from(3.55), of w2j , j = 1, 2 from (3.57), and of ρ from (3.58) to show the positivity ofu.

Proof for Theorem 3.4. First, we note that, with the choice (3.59), u is actually C1.Furthermore, since u and W are continuous, the original differential equation (3.46)tells that u is also C2. Hence, u defined in (3.59) is a C2 solution to (3.46)-(3.47).

By construction, we have u > 0 for 0 < x ≤ 1, since u11 is positive (see Lemma3.3). It remains to check check that for x > 1, u > 0. Notice that u21 is positive(see Lemma 3.3). We wish to prove that in x > 1, (1 − ω)u21 dominates ωu22, sothat ωu22 + (1− ω)u21 > 0.

From now on, we will focus on the hypergeometric differential equation withW = V2 in x > 1,

− u′′(x) + V2u(x) = 0. (3.61)We set z = −x2 . Then (3.61) has a solution taking the following form

ū = xα2(x+ 2)β2F (z), (3.62)

where F (z) is a solution to the hypergeometric differential equation

z(1− z)d2w

d2z+ (c2 − (a2 + b2 + 1)z)

dw

dz− a2b2w = 0 (3.63)

with the parameters a2, b2, c2 defined in (3.52). Note that, the hypergeometricfunction has possibly regular singularities at z = 0, 1,∞, namely, x = 0,−2,∞.For x > 1, we only focus on the regular singularity x =∞. There are the followingpair f1(z), f2(z) of Frobenius solutions to (3.63), which are adapted to z =∞ (seeLemma 3.1),

f1(z) = z−a2F

(a2, a2 − c2 + 1; a2 − b2 + 1;

1

z

),

f2(z) = z−b2F

(b2, b2 − c2 + 1; b2 − a2 + 1;

1

z

).

(3.64)

Substituting (3.64) into (3.62), we know that there are a pair of Frobenius solutionsū1, ū2 to (3.61), which are adapted to the singularity x =∞,

ū1 = xα2(x+ 2)β2f1, ū2 = x

α2(x+ 2)β2f2.

In view of (3.52a), we can calculate that the characteristic exponents associatedwith the singularity x =∞; these are

b2 − a2 + 12

anda2 − b2 + 1

2,

which can be further specified using the parameters a2, b2 in (3.52). As a result,we have the asymptotic expansion for ū1 and ū2

ū1 ∼ xb2−a2+1

2 →∞, ū2 ∼ xa2−b2+1

2 → 0, as x→∞. (3.65)


We note that the parameters a2, b2 could be chosen in various ways, but the resultingcharacteristic exponents would always be the same. Additionally, we could see that(−1)b2 ū2 is positive. For

(−1)b2 ū2 = 2b2xα2−b2(x+ 2)β2F(b2 − c2 + 1, b2, ; b2 − a2 + 1;−

2

x

), (3.66)

where we had used the fact that the hypergeometric function is symmetric in its firsttwo arguments: F (a, b; c; z) = F (b, a; c; z). For F

(b2 − c2 + 1, b2, ; b2 − a2 + 1;− 2x

),

the second and third arguments satisfy b2 − a2 + 1 > b2 > 0, so we can use theintegral representation (2.10) to conclude that

F

(b2 − c2 + 1, b2, ; b2 − a2 + 1;−

2

x

)> 0 for x > 0.

The general solution to (3.61), which could be written as linear combinations ofthe Frobenius solutions u21, u22 (adapted to x = 0), is either asymptotically decay-ing as ū1 or as ū2. Due to the observation (3.58), we will construct a combinationof the Frobenius solutions u21, u22, which will be denoted by G(x) below, such thatG(x) is asymptotically decaying as ū2, and hence proportional to ū2. Additionally,G(x) serves as a positive lower bound for ωu22 + (1−ω)u21. In this way, we wouldprove the positivity for ωu22 + (1− ω)u21.

Recalling (3.58), we have

ρ = 5.0153723738 · · · (3.67)

We define a new function

G(x) = u22 + ρu21. (3.68)

We wish to prove that G(x) > 0 for x > 0. Now G(x) is a solution to the differentialequation (3.61) with G(1) = 1 + ρ > 0. Moreover, by the construction, we knowthat

limx→∞

G(x)

u21(x)= 0. (3.69)

With respect to the exponents of the two Frobenius solutions (3.65) adapted tosingularity x =∞, we have either

G(x) ∼ xb2−a2+1

2 →∞ or G(x) ∼ xa2−b2+1

2 → 0 as x→∞.

The fact of (3.69) further shows that

G(x) ∼ xa2−b2+1

2 → 0 as x→∞.

As a summary, G(x) is a solution to the ordinary differential equation (3.61) with

G(1) = 1 + ρ, limx→∞

G(x) = 0. (3.70)

On the other hand, G(x) could be expressed in terms of linear combinations of theFrobenius solutions ū1 and ū2: Suppose

G(x) = pū1(x) + qū2(x), (3.71)

where p, q are some constants. Taking limits at at x =∞ yields that

limx→∞

G(x) = p limx→∞

ū1(x) + q limx→∞

ū2(x), (3.72)

which gives 0 = p · ∞+ q · 0. Hence p = 0 and

G(x) = qū2(x). (3.73)

We have known that (−1)b2 ū2 is positive. Hence (3.73) implies that G(x) does notchange sign. Besides, we know that G(1) = 1+ρ > 0, therefore G(x) > 0 for x > 0.


Next, we turn back to ωu22 + (1− ω)u21, which could be written as

ωu22 + (1− ω)u21 = ω(u22 +

1− ωω

u21

). (3.74)

Viewing the value of ω in (3.60) and ρ in (3.67), we know that

1− ωω

> ρ. (3.75)

Additionally, u21 is positive (see Lemma 3.3). Hence,

ωu22 + (1− ω)u21 > ω(u22 + ρu21) = ωG(x) > 0. (3.76)

Therefore, we have proved that u is a positive C2 solution of (3.46). �

The solution u constructed in the previous can be computed and plotted numer-ically, verifying that it is positive (see Figure 3(a)) and has continuous slope (seeFigure 3(b)).

Now we are ready to prove the integrated decay estimate for the Zerilli case.

Proof of Theorem 1.3 for the Zerilli case. We have constructed a C2 function φ(3.45), which is a positive solution to (3.43) with V = Vjoint. Thus the Hardyinequality follows. Combining with the conservation energy associated to ∂t, wecomplete the proof for the Zerilli case. �

3.3. Non-degenerate energy. With the integrated decay estimate, we can usethe vector fieldN to prove the uniform boundedness for the non-degenerated energy.

Proof of Theorem 1.2. For a solution ψ of the equation (1.3), taking the vector fieldξ = T, the associated energy on t = τ slice is

ET (ψ, τ) =1

2

∫t=τ

(|∂tψ|2 + |∂r∗ψ|2 + (1− µ)(|∇/ψ|2 + Vgψ2)

)r2dr∗dσS2 .

It is well-known that, after switching to coordinates that extend smoothly throughr = 2M , the energy ET (ψ, τ) is degenerate, in the sense that it has weights involvingpositive powers of 1 − µ. This is because the vector field T vanishes as r → 2Mon hypersurfaces of constant t. Even on Στ , the vector field T becomes null asr → 2M , which induces a more subtle degeneracy in the T energy. To avoid thesedegeneracies, we use the globally time-like vector fieldN and consider the associatedenergy.

Away from the horizon, define Σeτ = {t = τ} ∩ {r > rNH}. Note that N = T forr > rNH . For solutions of the Regge-Wheeler or Zerilli equations, we only considersolutions supported on angular frequency ` ≥ 2, which implies that (1.6) holds.This indeed gives the positivity of energy on Σeτ . In the case of the Regge-Wheelerequation,

EN (ψ,Σeτ ) ≥1

2

∫Σeτ

(|∂tψ|2 + |∂r∗ψ|2 + (1− µ)

(|∇/ψ|2

4+ψ2

r2

))r2dr∗dσS2 > 0.

In the case of the Zerilli equation,∫Σeτ

|∇/ψ|2 + V Zg ψ2 =∫

Σeτ

(2λ+ 2−

µ(2λ+ 3)(2λ+ 32µ)

(λ+ 32µ)2

)ψ2

r2

≥∫

Σeτ

(2λ− 2 + 6

2λ+ 3

)ψ2

r2> 0.

Note that, (1− µ) = 1− 2Mr ≈ 1 when r > rNH .


Near the horizon, recalling that Σiτ.= {v = τ +r∗NH}∩{r ≤ rNH}, the N energy

is

EN (ψ,Σiτ ) ≈∫

Σiτ

(|∂uψ|2

1− µ+ (1− µ)(|∇/ψ|2 + Vg|ψ|2)

)dudσS2 . (3.77)

The regularity of the integrand in EN (ψ,Σiτ ) is more apparent when written in(v, r, θ, φ) coordinate; in these coordinates, upto a constant, theN energy is

∫Σiτ

(|∂rψ|2+|∇/ψ|2 +Vgψ2)drdσS2 . This is non-degenerate in the sense that N remains timelikeand the integrand controls all tangential derivatives, although, since Σiτ is null,the energy is degenerate in the sense that it fails to control the transverse deriva-tives (∂vψ)

2. Emphasising the importance of the tangential derivatives, we refer toEN (ψ,Σiτ ) as the non-degenerate energy. Due to the fact that ` ≥ 2, we have∫

Σiτ

(|∂rψ|2 + |∇/ψ|2 +

|ψ|2

r2

)drdσS2 . E

N (ψ,Σiτ ). (3.78)

Taking ξ = N , we apply the energy identity with the momentum vector PN (ψ).Noting that N = T is Killing for r > rNH , we have∫

Στ

PNα (ψ)nα +

∫H+

PNα (ψ)nα +

∫∫{r


Corollary 3.6 (Nondegenerate Integrated Decay Estimate). Let ψ be a solution tothe Regge-Wheeler equation (1.4) or Zerilli equation (1.5), we have for all R > 3M∫ τ

τ0

dt

∫Σ′t

PNα (ψ)nαΣτ .

∫Στ0

PNα (ψ)nαΣτ0

+ PTα (Tψ)nαΣτ0

, (3.82)

where Σ′τ = Στ ∩ {r < R}.

To proceed to the higher order case, we use the non-degenerate radial vectorfield

Ŷ =

{(1− µ)−1∂u, if r ≤ rNH ,∂r, if r > rNH ,

(3.83)

where the formula in r ≤ rNH is written in (u, v, θ, φ) coordinates and the one inr > rNH in (t, r, θ, φ) coordinates. Notice that, near horizon we can also write Ŷ

in (v, r, θ, φ) coordinate as Ŷ = ∂r, if r ≤ rNH .

Corollary 3.7 (Nondegenerate High Order Integrated Decay Estimate). Let ψ bea solution to the Regge-Wheeler equation (1.4) or Zerilli equation (1.5), we havefor all R > 3M and all integers n ∈ N,∫ τ

τ0

dt

∫Σ′τ

∑k+l+j≤n

|NkΩlŶ jψ|2dµg

.∫

Στ0

∑k+l+j≤n

PNα (TkΩlŶ jψ)nαΣτ0 +

∑i≤n+1

PTα (Tiψ)nαΣτ0 ,

(3.84)

where Σ′τ = Στ ∩ {r < R}.

Remark 3.8. We remark that, if R < 3M , there is no regularity loss, hence thelast terms in (3.82) and (3.84) should be replaced by PTα (ψ)n

α and∑i≤n

PTα (Tiψ)nα

respectively.

Proof. First, commuting the equation with T, we still have non-degenerate inte-grated decay estimate for Tψ∫ τ

τ0

dt

∫Σ′t

1∑k=0

PNα (Tkψ)nαΣτ .

1∑k=0

∫Στ0

PNα (Tkψ)nαΣτ0 + P

Tα (T

k+1ψ)nαΣτ0 . (3.85)

Elliptic estimate yields the high order integrated decay estimate away from thehorizon. Notice that, to achieve (3.85), we also introduce the ψ̃ as in the proof of

Theorem 1.2 so that ψ̃ is equal to ψ in the future of Στ and satisfies∫Στ

PTα (Tψ)nα =

∫t=τ

PTα (T ψ̃)nα.

Near the horizon, say for r < r0 < rNH , we shall commute the wave operatorwith Y = (1− µ)−1∂u [16]. This commutator has a good sign,

2gY ϕ− Y2gϕ = κY 2ϕ+ f(Y Tϕ, Tϕ, Y ϕ),where f(Y Tϕ, Tϕ, Y ϕ) is linearly dependent on Y Tϕ, Tϕ, Y ϕ, and κ > 0 is anothermanifestation of the red-shift effect. After commuting the equation with Y and T ,we use the energy identity for N to estimate∫

Στ

PNα (Y ψ)nα +

∫H+

PNα (Y ψ)nα +

∫∫{r


where

EN (Y ψ) = −NY ψ(κY 2ψ + f(Y Tψ, Tψ, Y ψ)− ∂rVgψ

)=− κ(Y 2ψ)2 − κ(N − Y )Y ψY 2ψ + (f(Y Tψ, Tψ, Y ψ)− ∂rVgψ)NY ψ,

(3.87)

where Vg is either the Regge-Wheeler VRWg or Zerilli potential V

Zg . The first term

−κ(Y 2ψ)2 has a good sign. For the other terms in the second line of (3.87), weapply the Cauchy-Schwarz inequality. Using the fact that N − Y = T on H+, theycan be bounded in r ≤ r0 by

cKN (Y ψ) + c−1(Y Tψ)2 + c−1KN (ψ), (3.88)

where c is chosen to be a small constant. Furthermore, in view of the local integrateddecay estimate (3.85), we can bound (Y Tψ)2 by∫∫

{r


Σ′τ

Σ′τ0

H+

Nτ

Nτ0Dττ0

I +

r=R

Figure 4. The spacetime foliation⋃τ Σ′τ ∪Nτ

smooth cut-off function supported in {r > R} so that χ ≡ 1 in {r > 2R}. Then,∑k+l+j≤n

∫Στ∪H+

PNα (NkΩlŶ jψ)nα + Iδ[NkΩlŶ jψ](D′ττ0)

.∫

Στ0

∑k+l+j≤n

PNα (TkΩlŶ jψ)nα +

∑i≤n+1

PTα (Tiψ)nα.

(3.91)

Here D′ττ0 denotes the domain enclosed between Στ and Στ0 and the improved non-degenerate spacetime integral is defined by (0 < δ < 1)

Iδ[ψ](D).=

∫∫D

(|Nψ|2 + |∂rψ|2

r1+δ+|∇/ ψ|2

r+|ψ|2

r3+δ

)r2dtdrdσS2 . (3.92)

4. Decay estimate

In this section, we will prove quadratic decay for the non-degenerate energy. Weshall employ the rp hierarchy estimate to prove the energy decay estimate [33].

4.1. Energy decay. Let ψ be a solution to the Regge-Wheeler equation (1.4) orZerilli equation (1.5) and define

Ψ.= rψ. (4.1)

Then in the Regge-Wheeler case

LRWΨ.= ∂u∂vΨ− η4/Ψ + V RWΨ = 0; (4.2)

and in the Zerilli case

LZΨ.= ∂u∂vΨ− η4/Ψ + V ZΨ = 0. (4.3)

Here the potentials are defined by

V RW = ηV̂ RW , V̂ RW = −6Mr3

, (4.4)

V Z = ηV̂ Z , V̂ Z = −6Mr3− 8Mζ

r3. (4.5)

We recall that V Zg = VRWg (1 + ζ), with ζ defined in (3.15) and η = 1− µ.

When there is no confusion, we also refer to (4.2) as the Regge-Wheeler equa-tion and (4.3) as the Zerilli equation. We recall some definitions of the space-time foliation

⋃τ Στ (see Section 1.2), where Στ = Σ

iτ ∪ Σeτ with Σiτ = {M|v =

τ+r∗NH}∩{r < rNH} and Σeτ = {M|t = τ}∩{r ≥ rNH}. For our current purposes,we foliate the spacetime region by

⋃τ (Σ

′τ ∪Nτ ) (Figure 4): In the following, we use

R to denote a sufficiently large constant. Technically, in the following arguments,the choice of R depends on the parameter p, but since our estimates are typically


uniform for R sufficiently large and since we only consider a finite number of val-ues of p (in particular p = 1 and p = 2), we do not track the p dependence of Rcarefully. Define the interior region ∪τΣ′τ , where

Σ′τ.= Στ ∩ {r ≤ R}. (4.6)

In the exterior region {r ≥ R}, let Nτ be the outgoing null hypersurface emergingfrom the sphere S2(τ,R) with constant t = τ and constant r = R, that is,

Nτ.= {u = τ −R∗} ∩ {v ≥ τ +R∗}. (4.7)

Let us also define a region bounded by the two null hypersurfaces and the time-likehypersurface (Figure 4):

Dτ2τ1 = {(u, v)∣∣r(u, v) ≥ R and τ2 −R∗ ≥ u ≥ τ1 −R∗}. (4.8)

Where there is no confusion, we use I+ to denote the relevant portion of I+. Forexample, in discussing the boundary of Dτ2τ1 , we use I

+ to denote the portion wheresuch that u ∈ (τ1, τ2). We use H+ similarly.

Lemma 4.1 (Zeroth Order rp Integrated Decay Estimate). Let Ψ be a solutionto the Regge-Wheeler (4.2) or Zerilli equation (4.3). There is the integrated decayestimate, for 0 < p ≤ 2,∫

Nτ2rp|∂vΨ|2dvdσS2 +

∫I+rp(|∇/Ψ|2 + Ψ

2

r2

)dudσS2

+

∫∫Dτ2τ1

rp−1(p

2|∂vΨ|2 +

2− p2

(|∇/Ψ|2 + |Ψ|

2

r2

)+

6pM

r

|Ψ|2

r2

)dudvdσS2

.∫Nτ1

rp|∂vΨ|2dvdσS2 +∫{r=R}

(|∂vΨ|2 + |∇/Ψ|2 + |Ψ|2

)dtdσS2 .

(4.9)

Remark 4.2. Comparing with the zeroth order rp weighted inequality in [33], we

gain an additional bound for the spacetime integralsDτ2τ1

rp−1 6pMr|Ψ|2r2 dudvdσS2 .

This would be crucial in proving the first order rp weighted inequality (for p = 2)in Lemma 4.6.

Proof. Multiplying η−krp∂vΨ with 0 < p ≤ 2, k = 4 on the Regge-Wheeler equation(4.2) or Zerilli equation (4.3), we choose R sufficiently large and integrate withrespect to the measure dudvdσS2 in Dτ2τ1 , to derive the identity∫

Nτ2

rp

ηk|∂vΨ|2 +

∫I+

rp

ηk−1|∇/Ψ|2 + r

pV

ηk|Ψ|2

+

∫∫Dτ2τ1

((2− p)rp−1η−k+2 + 2M(k − 1)rp−2η−k+1

)|∇/Ψ|2

+

∫∫Dτ2τ1−∂u

(rp

ηk

)|∂vΨ|2 − ∂v

(rpV

ηk

)|Ψ|2

=

∫Nτ1

rp

ηk|∂vΨ|2 +

∫{r=R}

rp

ηk|∂vΨ|2 +

rp

ηk−1|∇/Ψ|2 + r

pV

ηk|Ψ|2,

(4.10)

where V could be taken as V RW or V Z in those two different cases. For R suffi-ciently large,

−∂u(rp

ηk

)≥ p

2rp−1 for all 0 < p ≤ 2 and k = 4.

In what follows, we will prove the positivity of the other bulk terms in (4.10) forboth of the Regge-Wheeler and Zerilli cases.


Regge-Wheeler case V = V RW : In the third line of (4.10), the term involving|Ψ|2 reads∫∫

Dτ2τ1−∂v

(rpV RW

ηk

)|Ψ|2 =

∫∫Dτ2τ1

∂v

(6Mrp−3

ηk−1

)|Ψ|2

=

∫∫Dτ2τ1

η−k+1(6M(p− 3)rp−4 + 12M2(4− k − p)rp−5

)|Ψ|2.

(4.11)

Note that (4.11) does not have a good sign. Nevertheless, it is remarkable thatthere is an additional positive term

sDτ2τ1

2M(k − 1)rp−2η−k+1|∇/Ψ|2 in the secondline of (4.10). Using the fact that Ψ has angular frequencies ` ≥ 2 as stated inRemark 1.1, we have ∫∫

Dτ2τ12M(k − 1)rp−2η−k+1|∇/Ψ|2

≥∫∫Dτ2τ1

12M(k − 1)rp−4η−k+1|Ψ|2.(4.12)

Our aim at this stage is to show that this nonnegative term is sufficiently strongthat even only a c1 fraction of it is sufficient to absorb the negative term in (4.11).For any choice of c1 ∈ [0, 1], the bulk terms in (4.10) has the following lower bound∫∫

Dτ2τ1

(6M(p− 3 + 2c1k − 2c1)rp−4 + 12M2(4− k − p)rp−5

)|Ψ|2

+

∫∫Dτ2τ1

p

2rp−1|∂vΨ|2 + (2− p)rp−1|∇/Ψ|2 + 2M(1− c1)(k − 1)rp−2|∇/Ψ|2.

(4.13)

We take c1 =12 . Then for sufficiently large R and 0 < p ≤ 2,

(4.13) ≥∫∫Dτ2τ1

p

2rp−1|∂vΨ|2 +

2− p2

rp−1(|∇/Ψ|2 + |Ψ|

2

r2

)+

6pM

rrp−1

|Ψ|2

r2.

Here we have taken ` ≥ 2 into account. As a result, the identity (4.10) with0 < p ≤ 2 and k = 4 becomes∫

Nτ2

rp

η4|∂vΨ|2dvdσS2 +

∫I+

rp

η3[|∇/Ψ|2 − 6M

r

|Ψ|2

r2]dudσS2

+

∫∫Dτ2τ1

rp−1[p2|∂vΨ|2 +

2− p2

(|∇/Ψ|2 + |Ψ|2

r2) +

6pM

r

|Ψ|2

r2]dudvdσS2

.∫Nτ1

rp


∫r=R

{|∂vΨ|2 + |∇/Ψ|2 + |Ψ|2}dtdσS2 ,

where we should note that due to the fact that ` ≥ 2,∫I+rp(|∇/Ψ|2 + Ψ

2

r2

)dudσS2 .

∫I+

rp

η3

(|∇/Ψ|2 − 6M

r

|Ψ|2

r2

)dudσS2 .

Knowing that η ∼ 1 for R large enough, we achieve the integrated decay estimate(4.9) for the Regge-Wheeler case.

Zerilli case V = V Z : In (4.10), the term involving |Ψ|2 is multiplied by

−∂v(rpV Z

ηk

)= ∂v

(6Mrp−3

ηk−1

)+ ∂v

(8Mrp−3ζ

ηk−1

)> η−k+1

(12M(p− 3)rp−4 + 24M2(4− k − p)rp−5

),

(4.14)

where the facts ζ ≤ 32λ̄≤ 34 , ∂rζ > 0 (see (3.15) and (3.17)) and 0 < p ≤ 2, k ≥ 4

are used. As previously mentioned, those terms in (4.14) do not have a favourable


sign, but the additional positive term 2M(k − 1)rp−2η−k+1|∇/Ψ|2 in (4.10) cancompensate for the negative terms in (4.14). Hence, the bulk terms in (4.10) has alower bound (noting that ` ≥ 2)∫∫

Dτ2τ1

(12M(p− 3 + c2k − c2)rp−4 + 24M2(4− k − p)rp−5

)|Ψ|2

+

∫∫Dτ2τ1

rp−1|∂vΨ|2 + (2− p)rp−1|∇/Ψ|2 + 2M(1− c2)(k − 1)rp−2|∇/Ψ|2,(4.15)

where 0 < c2 ≤ 1 is a universal constant. We here take c2 = 1, k = 4, then forsufficiently large R and 0 < p ≤ 2,

(4.15) ≥∫∫Dτ2τ1

p

2rp−1|∂vΨ|2 +

2− p2

rp−1(|∇/Ψ|2 + |Ψ|

2

r2

)+

6pM

rrp−1

|Ψ|2

r2.

Consequently, the identity (4.10) with 0 < p ≤ 2 and k = 4 turns into∫Nτ2

rp


∫I+

rp

η3

(|∇/Ψ|2 + V̂ Z |Ψ|2

)dudσS2

+

∫∫Dτ2τ1

rp−1(p

2|∂vΨ|2 +

2− p2

(|∇/Ψ|2 + |Ψ|

2

r2

)+

6pM

r

|Ψ|2

r2

)dudvdσS2

.∫Nτ1

rp


∫{r=R}

(|∂vΨ|2 + |∇/Ψ|2 + |Ψ|2

)dtdσS2 .

As before, there is∫I+ r

p(|∇/Ψ|2 + Ψ

2

r2

)dudσS2 .

∫I+

rp

η3 [|∇/Ψ|2 + V̂ Z |Ψ|2], since

` ≥ 2. Thus we conclude the integrated decay estimate (4.9) for the Zerilli case.�

Remark 4.3. We make some remarks about the uniform boundedness and the non-degenerate integrated decay estimate adapted to the new spacetime foliations. Herethe treatment is similar to the one at the horizon (3.80)-(3.81). Since null infinityis also a null hypersurface to which T is tangent, it is again sufficient to take rψ tobe constant along flow lines of the Killing vector field, i.e. to take rψ be independentof u. However, such data fails to vanish at spacelike infinity. Thus, we can firstconsider data such that rψ is constant along the flow lines of the Killing vector fieldand then consider new data constructed from this first set of data on I+ by applyinga smooth cut-off that is 1 for u ≥ τ0, vanishes for u sufficiently negative, and thathas its derivative bounded by K−1 for a large K. Launching data from the union ofΣ′τ0 , Nτ0 , and the portion of I

+ on which u ≤ τ0, we obtain a new solution, ψK , ofthe Regge-Wheeler or Zerilli equation, such that ψK = ψ in the future of Σ

′τ0 ∪Nτ0 .

From the non-degenerate integrated decay estimate (Corollary 3.6), we obtain∫ ττ0

dt

∫Σ′t

PNµ (ψ)nµ .

∫Στ0

PNµ (ψK)nµ + PTµ (TψK)n

µ. (4.16)

From conservation of energy and taking the limit K →∞, we obtain

limK→∞

∫Στ0

PNµ (ψK)nµ + PTµ (TψK)n

µ =

∫Σ′τ0∪Nτ0

PNµ (ψ)nµ + PTµ (Tψ)n

µ. (4.17)

However, (4.16) is uniform in K, so we find∫ ττ0

dt

∫Σ′t

PNµ (ψ)nµ .

∫Σ′τ0∪Nτ0


µ. (4.18)


Similarly, using the spacetime foliation⋃τ Σ′τ ∪ Nτ , we also have the uniform

boundedness (Theorem 1.2), for any τ2 > τ1,∫Σ′τ2∪Nτ2∪H+∪I+

PNµ (ψ)nµ .

∫Σ′τ1∪Nτ1

PNµ (ψ)nµ + PTµ (ψ)n

µ. (4.19)

Here we still denote the portion of H+ (I+) between Σ′τ2 ∪ Nτ2 and Σ′τ1 ∪ Nτ1 by

H+ (I+). Indeed, there is, analogous to (3.91),∑k+l≤n

∫Σ′τ2∪Nτ2∪H+∪I+

PNα (NkΩlψ)nα + Iδ[NkΩlψ](Dτ2τ1 )

.∫

Σ′τ1∪Nτ1

∑k+l≤n

PNα (TkΩlψ)nα +

∑i≤n+1

PTα (Tiψ)nα.

(4.20)

We will take advantage of the local integrated decay estimate (4.18), the uniformboundedness (4.19) and the rp hierarchy estimate to infer quadratic energy decay[33].

Theorem 4.4 (Energy Decay). Let R > 3M, and let ψ be a solution to the Regge-Wheeler equation (1.4) or Zerilli equation (1.5), with the initial data imposed onΣ′τ0 ∪Nτ0 satisfying∫

Nτ0

∑k≤1

|T k∂v(rψ)|2r2dvdσS2 +∫

Σ′τ0∪Nτ0

∑k≤2

PNµ (Tkψ)nµ τ1,∫ τ2

τ1

dτ

∫Σ′τ∪Nτ

PNµ (ψ)nµ .

∫∫Dτ2τ1

(|∂vΨ|2 + |∇/Ψ|2 +

|Ψ|2

r2

)dudvdσS2

+

∫Σ′τ1∪Nτ1


µ,

(4.23)

where we have utilized the non-degenerate integrated decay estimate (4.18). Takingp = 1 in the rp weighted inequality (Lemma 4.1), we can further estimate the firstterm on the right of (4.23) by∫ τ2

τ1

dτ

∫Σ′τ∪Nτ

PNµ (ψ)nµ .

∫Nτ1

r|∂vΨ|2dvdσS2

+

∫Σ′τ1∪Nτ1


µ.

(4.24)

Next, we take p = 2 in Lemma 4.1, then there exists a sequence {τ ′j}j∈N withτ ′j ∈ (τj , τj+1), τj+1 = 2τj and τ ′0 = τ0, such that∫

Nτ′j

r|∂vΨ|2dvdσS2

.1

τ ′j

(∫Nτ0

r2|∂vΨ|2dvdσS2 +∫

Σ′τ0∪Nτ0PTµ (ψ)n

µ + PTµ (Tψ)nµ

).

(4.25)


In proving (4.25) (and (4.24)), we note that there is an integral on the boundary{r = R} on the right hand side of (4.9), hence we shall use the mean value theoremfor integration in r∗, and the local integrated energy decay (4.18) to bound theboundary term by

∫Σ′τ0∪Nτ0

PTµ (ψ)nµ + PTµ (Tψ)n

µ (Technically, we should shift R

to some R0 ∈ (R,R + 1) [33], but we still denote it by R). Combining (4.24) and(4.25), we arrive at∫ τ ′j+1

τ ′j

dτ

∫Σ′τ∪Nτ

PNµ (ψ)nµ

.1

τ ′j

(∫Nτ0

r2|∂vΨ|2dvdσS2 +∫

Σ′τ0∪Nτ0

PTµ (ψ)nµ + PTµ (Tψ)n

µ

)

+

∫Σ′τ′j∪Nτ′

j


µ.

(4.26)

With the uniform boundedness of energy (4.19), we may estimate the last term in(4.26) as follows: there is a sequence {τ ′′j }j∈N with τ ′′j ∈ (τ ′j−1, τ ′j+1) and τ ′′0 = τ0,such that∫

Σ′τ′j+1∪Nτ′

j+1

(PNµ (ψ)n

µ + PTµ (Tψ)nµ).∫

Σ′τ′′j∪Nτ′′

j

(PNµ (ψ)n

µ + PTµ (Tψ)nµ)

.1

τ ′j

∫ τ ′j+1τ ′j−1

dτ

∫Σ′τ∪Nτ

PNµ (ψ)nµ +

∑i≤1

PNµ (Tiψ)nµ

.(4.27)

We again apply (4.26) to∫ τ ′j+1τ ′j−1

dτ∫

Σ′τ∪NτPNµ (ψ)n

µ +∑i≤1 P

Nµ (T

iψ)nµ in (4.27)

and derive∫Σ′τ′j+1∪Nτ′

j+1

(PNµ (ψ)n

µ + PNµ (Tψ)nµ)

.1

τ ′jτ′j−1

∫Nτ0

∑k≤1

r2|∂vT kΨ|2dvdσS2 +∫

Σ′τ0∪Nτ0

∑i≤2

PTµ (Tiψ)nµ

+

1

τ ′j

∫Σ′τ′j−1∪Nτ′

j−1

PNµ (ψ)nµ +

∑i≤2

PNµ (Tiψ)nµ

,(4.28)

where by the uniform boundness of energy (4.19), the last term could be furtherbounded by

1

τ ′j

∫Σ′τ0∪Nτ0

PNµ (ψ)nµ +

∑i≤2

PNµ (Tiψ)nµ

.Thus, taking (4.26) and (4.28) into account, we have∫ τ ′j+2

τ ′j

dτ

∫Σ′τ∪Nτ

PNµ (ψ)nµ

.1

τ ′j

∫Nτ0

∑k≤1

|∂vT kΨ|2r2dvdσS2 +∫

Σ′τ0∪Nτ0

∑i≤2

PNµ (Tiψ)nµ

. (4.29)


In view of (4.27) and (4.29), we prove quadratic energy decay for∫

Σ′τ′j∪Nτ′

j

PNµ (ψ)nµ,

along the sequence {τ ′j}∞j=0. For any t, we may choose j ∈ N ∪ {0} such thatt ∈ (t′j , t′j+1) and by the uniform boundedness of energy (4.19), so that the qua-dratic decay (4.22) follows for all t not merely the sequence {τ ′j}. �

We further commute the equations with Ω repeatedly to deduce the pointwisedecay estimate.

Theorem 4.5 (Pointwise Decay). Let ψ be a solution to the Regge-Wheeler equa-tion (1.4) or Zerilli equation (1.5), with the initial data prescribed on Σ′τ0 ∪ Nτ0satisfying

∑l≤2

∫Nτ0

∑k≤1

|T kΩl∂v(rψ)|2r2dvdσS2 +∫

Σ′τ0∪Nτ0

∑k≤2

PNµ (TkΩlψ)nµ


On Στ ∩ {2M < r0 < rNH ≤ r ≤ R}, the Sobolev inequality also yields thepointwise decay. �

We next proceed to the high order rp integrated decay estimate. For notationalconvenience, we use Kp−1(Ψ) to denote the spacetime integral in the r

p weightedinequality, i.e.,

Kp−1(Ψ).=

∫∫Dτ2τ1

prp−1(

1

2|∂vΨ|2 +

6M

r

|Ψ|2

r2

)dudvdσS2

+

∫∫Dτ2τ1

2− p2

rp−1(|∇/Ψ|2 + |Ψ|

2

r2

)dudvdσS2 ,

(4.33)

and Sp(Ψ) the energy on I+,

Sp(Ψ).=

∫I+rp(|∇/Ψ|2 + Ψ

2

r2

)dudσS2 . (4.34)

Lemma 4.6 (First Order rp Integrated Decay Estimate). Let Ψ be a solution tothe Regge-Wheeler equation (4.2) or Zerilli equation (4.3), and define

D = {r∂v, (1− µ)−1∂u, r∇/ }. (4.35)

In the region Dτ2τ1 , there is the integrated decay estimate for 0 < p ≤ 2,∑j≤1

∫Nτ2

rp|∂vDjΨ|2dvdσS2 +∫I+rp(|∂vDjΨ|2 +

DjΨ2

r2

)dudσS2

+

∫∫Dτ2τ1

∑j≤1

prp−1(|∂vDjΨ|2 +

M

r

|DjΨ|2

r2

)dudvdσS2

+

∫∫Dτ2τ1

∑j≤1

(2− p)rp−1(|∇/DjΨ|2 + |D

jΨ|2

r2

)dudvdσS2

.∫Nτ1

rp

∑j≤1

|∂vDjΨ|2 +∑l≤1

|∂vΩlΨ|2dvdσS2

+

∫{r=R}

∑i+j+l≤2

|∂iv∇/lDjΨ|2dtdσS2 .

Remark 4.7. In contrast to Lemma 4.6, the end point p = 2 is not achieved inthe first order rp weighted energy inequality of [33, The proof of Proposition 5.6].Since, we cover the p = 2 case, we can further improve the decay estimate for thetime derivative as t−2 (see Section 4.2). This is in contrast to the t−2+δ decay in[33]. This improvement in the rp weighted energy inequality and in the pointwisedecay is achieved by following the argument of [29].

To start with the first order case, we define

Ψ(1).= r∂vΨ, (4.36)

and argue as in the proof leading to Lemma 4.1 to derive the rp integrated decayestimate for Ψ(1). Here we should take into account the error terms arising whencommuting the equations with the operator r∂v. We expect these error terms to becontrolled by the known spacetime integral Kp−1(Ψ) and Kp−1(ΩΨ). However, aswe shall notice below, the commutator (4.39) involves an angular Laplacian 4/Ψ,which corresponds to the error term∫∫

Dτ2τ1−2rp4/Ψ∂vΨ(1), (4.37)


in the rp energy inequality for Ψ(1). If we simply apply the Cauchy-Schwarz inequal-ity to (4.37), then we are faced with

sDτ2τ1

rp−1|∇/ΩΨ|2, which can not be boundedby Kp−1(ΩΨ) when p = 2, since the

sDτ2τ1

2−p2 r

p−1|∇/Ψ|2 in Kp−1(Ψ) vanishes ifp = 2. To get around this difficulty, we perform integration by parts twice on (4.37).Then it follows that the resulted angular derivative term takes the form of∫∫

Dτ2τ1(2− p)2rp−1|∇/Ψ|2 (4.38)

plus boundary terms, as shown in (4.44)-(4.49). Due to the presence of the ad-ditional factor (2 − p)2, (4.38) vanishes when p = 2. Therefore our estimates gothrough even when p = 2. The idea stated above could be found in [29].

Proof. We only give the detailed proof for the Zerilli case, as the Regge-Wheelercase can be carried out in an analogous way. We start with commuting the equationwith r∂v. For any smooth function ϕ ∈ C∞(M), there is the commuting identity,

[ LZ , r∂v]ϕ = η∂u∂vϕ− η∂2vϕ− 2η(

1− 3Mr

)4/ϕ

− η 2Mr2

∂vϕ− r∂vV Zϕ,(4.39)

where LZ is defined in (4.3). In view of the Zerilli equation (4.3) and the commutingidentity (4.39), we obtain the equation for Ψ(1),

LZΨ(1) = −η∂2vΨ + {η2 − 2η(1−

3M

r)}4/Ψ

− η 2Mr2

∂vΨ− (V Z + r∂vV Z)Ψ.(4.40)

It turns out that the leading order term −η∂2vΨ admits a good sign. This would bemore apparent if we rewrite it as

− η∂2vΨ = −η

r∂vΨ

(1) +η2

r∂vΨ, (4.41)

for which the sign of −ηr ∂vΨ(1) is good. In more details, we multiply 2rp(1 −

µ)−k∂vΨ(1) on both sides of (4.40), and integrate on the spacetime region Dτ2τ1 , to

obtain that for 0 < p ≤ 2, k = 4,∫Nτ2

rp|∂vΨ(1)|2dvdσS2 +Kp−1(Ψ(1)) + Sp(Ψ(1))

.∫Nτ1

rp|∂vΨ(1)|2dvdσS2 +∫∫Dτ2τ1−2η−k+1rp−1|∂vΨ(1)|2

+A1 +A2 +A3 + boundary term on{r = R},

(4.42)

where Ai, i = 1, · · · , 3 are defined by

A1.=

∫∫Dτ2τ1

(−4M

rη−k+1 + 2η−k+2

)rp−1∂vΨ∂vΨ

(1)

A2.= +

∫∫Dτ2τ1

2

(η2 − 2η(1− 3M

r)

)η−krp4/Ψ∂vΨ(1)

A3.= −

∫∫Dτ2τ1

(V Z + r∂vVZ)η−krp+1

Ψ

r∂vΨ

(1).

The bulk termsDτ2τ1−2η−k+1rp−1|∂vΨ(1)|2 in (4.42) admits a favourable sign, as it

can be moved to the left hand side and contributes to a positive spacetime integral.


Then (4.42) becomes∫Nτ2

rp|∂vΨ(1)|2dvdσS2 + Sp(Ψ(1)) +Kp−1(Ψ(1)) +∫∫Dτ2τ1

2rp−1|∂vΨ(1)|2

.∫Nτ1

rp|∂vΨ(1)|2dvdσS2 +A1 +A2 +A3 + boundary term on{r = R}.(4.43)

We will estimate the error tems A1, · · · , A3 one by one below.For A1, an application of the Cauchy-Schwarz inequality leads to

|A1| ≤1

c

∫∫Dτ2τ1

rp−1|∂vΨ|2 + c∫∫Dτ2τ1

rp−1|∂vΨ(1)|2,

where c is some universal constant that is chosen to be small enough so thatcsDτ2τ1

rp−1|∂vΨ(1)|2 can be absorbed by Kp−1(Ψ(1)) +sDτ2τ1

2rp−1|∂vΨ(1)|2 on theleft hand side of (4.43). The remaining term

sDτ2τ1

c−1rp−1|∂vΨ|2 can be controlledby Kp−1(Ψ).

For A2, we rewrite it as

A2 =

∫∫Dτ2τ1−2rp4/Ψ∂vΨ(1) + f(r)4/Ψ∂vΨ(1), (4.44)

with |∂jrf | . rp−1−j , j ∈ Z. As mentioned before, for the termsDτ2τ1−2rp4/Ψ∂vΨ(1),

we perform integration by parts twice to obtain∫∫Dτ2τ1−2rp4/Ψ∂vΨ(1) =

∫∫Dτ2τ1−2rp−1|r∇/ ∂vΨ|2 + (2− p)2η2rp−1|∇/Ψ|2

+

∫∫Dτ2τ1

2M(p− 2)ηrp−2∇/Ψ · r∇/ ∂vΨ

+

∫I+rp(2∇/Ψ · r∇/ ∂vΨ− (p− 2)η|∇/Ψ|2

)+ boundary term on{r = R}.

(4.45)

The second termsDτ2τ1

f(r)4/Ψ∂vΨ(1) has lower weights in r. It can be estimatedstraightforwardly by the Cauchy-Schwarz inequality:∫∫

Dτ2τ1|f(r)4/Ψ∂vΨ(1)| ≤

1

c1

∫∫Dτ2τ1

rp−3|∇/ΩΨ|2 + c1∫∫Dτ2τ1

rp−1|∂vΨ(1)|2, (4.46)

where c1 is a small constant. We note that |4/Ψ| . r−1|∇/ΩΨ|. Putting (4.44) and(4.45)-(4.46) together, we arrive at

|A2| ≤∫∫Dτ2τ1−2rp−1|r∇/ ∂vΨ|2 + c1rp−1|∂vΨ(1)|2 +

1

c1rp−3|∇/ΩΨ|2

+

∫∫Dτ2τ1

(2− p)2rp−1|∇/Ψ|2 + 2M(2− p)rp−2(|∂vΩΨ|2 + |∇/Ψ|2)

+

∫I+rp(|∂vΩΨ|2 + |∇/Ψ|2

)+ boundary term on {r = R}.

(4.47)

In (4.47), the first bulk termsDτ2τ1−2rp−1|r∇/ ∂vΨ|2, having a good sign, can be

moved to the left hand side of (4.43), and contributes to the integrated decay es-timate

sDτ2τ1

2rp−1|r∇/ ∂vΨ|2. The next termsDτ2τ1

c1rp−1|∂vΨ(1)|2 can be absorbed

by Kp−1(Ψ(1)) +

sDτ2τ1

2rp−1|∂vΨ(1)|2 on the left hand side of (4.43), for c1 smallenough. The last term in the first line

sDτ2τ1

c−11 rp−3|∇/ΩΨ|2 possesses lower weights


in r. When 0 ≤ p < 2, it is apparent thatsDτ2τ1

c−11 rp−3|∇/ΩΨ|2 . Kp−1(ΩΨ); when

p = 2, we can estimatesDτ2τ1

c−11 rp−3|∇/ΩΨ|2 with p = 2 by∫∫

Dτ2τ1

1

c1rp−3|∇/ΩΨ|2 .

∫∫Dτ2τ1

(2− (p− 1))r(p−1)−1|∇/ΩΨ|2

. K(p−1)−1(ΩΨ), p = 2.

(4.48)

In the second line of (4.47),∫∫Dτ2τ1

(2− p)2rp−1|∇/Ψ|2 ≤ C ·Kp−1(Ψ), for some C > 2(2− p). (4.49)

Note that (4.49) holds even when p = 2. The termsDτ2τ1

2M(2− p)rp−2(|∂vΩΨ|2 +|∇/Ψ|2) can be estimated in the same way. Finally, the integral on I+ in the lastline of (4.47) can be bounded by Sp(Ψ) + Sp(ΩΨ).

As for A3, since |V Z | . Mr3 and |r∂vVZ | . Mr3 , it follows that

|A3| .∫∫Dτ2τ1

rp−1 · Mr

|Ψ|r· |∂vΨ(1)|.

We again use the Cauchy-Schwarz inequality to derive

|A3| .1

c2

∫∫Dτ2τ1

rp−1M2

r2|Ψ|2

r2+ c2

∫∫Dτ2τ1

rp−1|∂vΨ(1)|2,

where we choose c2 to be small enough so that c2sDτ2τ1

rp−1|∂vΨ(1)|2 could be ab-sorbed by Kp−1(Ψ

(1))+sDτ2τ1

2rp−1|∂vΨ(1)|2, while the first term could be boundedby CKp−1(Ψ).

In summary, we conclude that for 0 < p ≤ 2,∫Nτ2

rp|∂vΨ(1)|2dvdσS2 + Sp(Ψ(1)) +Kp−1(Ψ(1))

+

∫∫Dτ2τ1

rp−1(|∂vΨ(1)|2 + |r∇/ ∂vΨ|2)dudvdσS2

.∫Nτ1

rp|∂vΨ(1)|2dvdσS2 +Kp−1(Ψ) + δp2K(p−1)−1(ΩΨ) +Kp−1(ΩΨ)

+ Sp(Ψ) + Sp(ΩΨ) + boundary terms on {r = R}.

Here, the term with the Kronecker delta δp2K(p−1)−1(ΩΨ) exists only when p = 2.Applying Lemma 4.1 with 0 < p ≤ 2 to estimate Kp−1(Ψ) + δp2K(p−1)−1(ΩΨ) +

Kp−1(ΩΨ) + Sp(Ψ) + Sp(ΩΨ), we achieve (for 0 < p ≤ 2)∑j≤1

∫Nτ2

rp|∂v((r∂v)

jΨ)|2dvdσS2 +

∑j,l≤1

∫I+rp(|rj−1∂jvΩlΨ|2 + |∇/Ψ|2

)+∑j≤1

∫∫Dτ2τ1

prp−1(|∂v(r∂v)jΨ|2 +

M

r

|(r∂v)jΨ|2

r2

)dudvdσS2

+∑j≤1

∫∫Dτ2τ1

(2− p)rp−1(|∇/ (r∂v)jΨ|2 +

|(r∂v)jΨ|2

r2

)dudvdσS2

.∫Nτ1

rp

∑j≤1

|∂v(r∂v)jΨ|2 +∑l≤1

|∂vΩlΨ|2 dvdσS2

+∑j+l≤2

∫{r=R}

|∂jv∇/lΨ|2dtdσS2 .

(4.50)


Finally, we commute the equation with T, Ω to derive the rp weighted inequalitiesfor TΨ, ΩΨ. Combining this with (4.50), Lemma 4.6 follows. �

The high order rp integrated decay estimate follows by induction.

Corollary 4.8 (High Order rp Integrated Decay Estimate). Let Ψ be a solutionto the Regge-Wheeler (4.2) or Zerilli equation (4.3) in the region Dτ2τ1 , and definethe weighted derivatives D = {r∂v, (1− µ)−1∂u, r∇/ }. There is the integrated decayestimate for all n ∈ N and 0 < p ≤ 2, ∫

Nτ2

∑j≤n

rp|∂vDjΨ|2dvdσS2

+

∫∫Dτ2τ1

∑j≤n

prp−1(|∂vDjΨ|2 +

6M

r

|DjΨ|2

r2

)dudvdσS2

+

∫∫Dτ2τ1

∑j≤n

(2− p)rp−1(|∇/DjΨ|2 + |D

jΨ|2

r2

)dudvdσS2

.∫Nτ1

∑j≤n

rp|∂vDjΨ|2dvdσS2

+

∫r=R

∑j≤n

{|∂vDjΨ|2 + |∇/DjΨ|2 + |DjΨ|2}dtdσS2 .

4.2. Improved decay estimate. Due to the first order rp integrated decay esti-mate (Lemma 4.6), we can improve the first order energy decay.

Corollary 4.9 (Improved Energy Decay). Let ψ be a solution to the Regge-Wheelerequation (1.4) or Zerilli equation (1.5), with the initial data given on Σ′τ0 ∪ Nτ0satisfying

∫Nτ0

∑k≤1,j≤2

r2|T k(r∂v)jψ|2 +∑

k+l≤4,l≤1

r2|T kΩl∂v(rψ)|2 dvdσS2

+

∫Σ′τ0∪Nτ0

∑k≤3

PNµ (TkΩψ)nµ +

∑k≤5

PNµ (Tkψ)nµ


for 2 < p ≤ 4∫Nτ2

rp|∂vψ(1)|2dvdσS2 +∫∫Dτ2τ1

(p− 2)rp−1(|∂vψ(1)|2 +

M

r

|ψ(1)|2

r2

)dudvdσS2

+

∫∫Dτ2τ1

(4− p)rp−1(|∇/ψ(1)|2 + |ψ

(1)|2

r2

)dudvdσS2

.∫Nτ1

rp|∂vψ(1)|2 +∑j≤1

rp−2|∂vΩjΨ|2dvdσS2 + ∫

{r=R}

∑j+l≤2

|∇/ j∂lvΨ|2dtdσS2 .

(4.54)

We are interested in the quantity ∂vTΨ = ∂2vΨ + ∂v∂uΨ. From the Zerilli equation

(4.3), there is

|∂vTΨ|2 . |∂vψ(1)|2 + |4/Ψ|2 +|M |2

r2|Ψ|2

r4. (4.55)

To control the second term, |4/Ψ|2 . |∇/ΩΨ|2

r2 , we repeat the proof of Theorem 4.4with ΩΨ in the place of Ψ to derive∫

Nτr2|4/Ψ|2dvdσS2 .

∫Nτ

PNµ (Ωψ)nµ .

I

τ2, (4.56)

where I depends only on the initial data (4.51). Hence, we have∫Nτ

r2|∂vTΨ|2dvdσS2 .∫Nτ

r2(|∂vψ(1)|2 +

|M |2

r2|Ψ|2

r4

)dvdσS2 +

I

τ2. (4.57)

Below, we will make use of the rp hierarchy estimate to gain quadratic decay for∫Nτ r

2(|∂vψ(1)|2 + |M |

2

r2|Ψ|2r4

)dvdσS2 . From (4.54) with 2 < p ≤ 4 and Lemma 4.1

with 0 <

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